Lieb-Robinson Bounds on Entanglement Gaps from Symmetry … · 2019-04-30 · Lieb-Robinson Bounds...

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Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology Zongping Gong, 1 Naoto Kura, 1 Masatoshi Sato, 2 and Masahito Ueda 1, 3, 4 1 Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 2 Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan 3 Institute for Physics of Intelligence, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 4 RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan (Dated: July 23, 2019) A quantum quench is the simplest protocol to investigate nonequilibrium many-body quantum dynamics. Previous studies on the entanglement properties of quenched quantum many-body systems mainly focus on the growth of entanglement entropy. Several rigorous results and phenomenological guiding principles have been established, such as the no-faster-than-linear entanglement growth generated by generic local Hamiltonians and the peculiar logarithmic growth for many-body localized systems. However, little is known about the dynam- ical behavior of the full entanglement spectrum, which is a refined character closely related to the topological nature of the wave function. Here, we establish a rigorous and universal result for the entanglement spec- tra of one-dimensional symmetry-protected topological (SPT) systems evolving out of equilibrium. Our result is derived both for free-fermion SPT systems and interacting ones. For free-fermion systems with Altland- Zirnbauer symmetries, we prove that the single-particle entanglement gap after quenches obeys essentially the same Lieb-Robinson bound as that on the equal-time correlation, provided that there is no dynamical symmetry breaking. As a notable byproduct, we obtain a new type of Lieb-Robinson velocity which is related to the band dispersion with a complex wave number and reaches the minimum as the maximal (relative) group velocity. Within the framework of tensor networks, i.e., for SPT matrix-product states evolved by symmetric and trivial matrix-product unitaries, we also identify a Lieb-Robinson bound on the many-body entanglement gap for gen- eral quenched interacting SPT systems. This result suggests high potential of tensor-network approaches for exploring rigorous results on long-time quantum dynamics. Influence of partial symmetry breaking, effects of disorder, and the relaxation property in the long-time limit are also discussed. Our work establishes a paradigm for exploring rigorous results of SPT systems out of equilibrium. I. INTRODUCTION Recent years have witnessed remarkable experimental de- velopments in atomic, molecular and optical physics, which have enabled us to engineer and control artificial quantum many-body systems at the level of individual atoms, ions and photons [1–3]. Particular attention is focused on nonequilib- rium quantum dynamics [4–7], of which the arguably simplest situation is quantum quenches [8–10] — the system is initial- ized as a wave function |Ψ 0 i which then evolves unitarily by a Hamiltonian H, with respect to which |Ψ 0 i is typically a highly excited superposition state. The wave function at time t is then formally given by |Ψ t i = e -iHt |Ψ 0 i. To model re- alistic quantum simulators, especially ultracold atoms and su- perconducting circuits with short-range interactions, we usu- ally assume H to be local, in the sense that it can be written as a sum of short-range operators. In light of the rapid develop- ment of topological material science [11–14], there is growing interest in topological aspects of quench dynamics [15–36]. A fundamental question in this context is: given |Ψ 0 i as the ground state of a gapped Hamiltonian H 0 , which may be triv- ial or topological, whether the topology of |Ψ t i will change during time evolution, and, if yes, in what way. To make the topology well-defined, we may have to impose certain sym- metries. For the sake of concreteness, we assume that H 0 and H share the same symmetries, if any. The answer to the above question has recently been given and is somewhat negative: for unitary symmetries or/and anti-unitary anti-symmetries [37], |Ψ t i stays in the same symmetry-protected topologi- cal (SPT) phase [16, 17, 33, 34]. For anti-unitary symme- tries or/and unitary anti-symmetries, the topological number of |Ψ t i generally becomes ill-defined (or reduces) due to dy- namical symmetry breaking [33, 34]. To understand this, we only have to note that |Ψ t i is the ground state of [32, 33] H(t) ≡ e -iHt H 0 e iHt , (1) which shares the same spectrum as H 0 . The conservation of topological number follows from the fact that H(t) is gapped and continuously deformed from H 0 in a symmetry- preserving manner. Since topological numbers are rather abstract quantities and take very different forms depending on the specific sys- tems, we need a universal topological indicator to formalize the above qualitative analysis into a general, rigorous and, in principle, experimentally verifiable statement. Entanglement turns out to be an ideal candidate to demonstrate the persis- tence of topology. We can trace the time evolution of the en- tanglement spectrum (ES) for a proper bipartition, which con- tains crucial information of the entanglement pattern and is arguably the most widely used universal topological indicator that is applicable to both noninteracting [38–41] and interact- ing systems [42–47]. The ES is expected to be accessible in near-future ultracold-atom and trapped-ion experiments [48– 50]. The persistence of SPT order thus manifests itself in that the ES stays gapless or degenerate [33, 34]. However, a vital point is missing in the above argument of topology conservation — defining SPT phases requires local- ity in the Hamiltonian [51], while H(t) in Eq. (1) may become highly nonlocal after a long time. That is to say, the dynam- ically generated non-locality could obscure the SPT order in arXiv:1904.12464v2 [quant-ph] 21 Jul 2019

Transcript of Lieb-Robinson Bounds on Entanglement Gaps from Symmetry … · 2019-04-30 · Lieb-Robinson Bounds...

Page 1: Lieb-Robinson Bounds on Entanglement Gaps from Symmetry … · 2019-04-30 · Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology Zongping Gong,1 Naoto Kura,1

Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology

Zongping Gong,1 Naoto Kura,1 Masatoshi Sato,2 and Masahito Ueda1, 3, 4

1Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan2Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

3Institute for Physics of Intelligence, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan4RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan

(Dated: July 23, 2019)

A quantum quench is the simplest protocol to investigate nonequilibrium many-body quantum dynamics.Previous studies on the entanglement properties of quenched quantum many-body systems mainly focus on thegrowth of entanglement entropy. Several rigorous results and phenomenological guiding principles have beenestablished, such as the no-faster-than-linear entanglement growth generated by generic local Hamiltonians andthe peculiar logarithmic growth for many-body localized systems. However, little is known about the dynam-ical behavior of the full entanglement spectrum, which is a refined character closely related to the topologicalnature of the wave function. Here, we establish a rigorous and universal result for the entanglement spec-tra of one-dimensional symmetry-protected topological (SPT) systems evolving out of equilibrium. Our resultis derived both for free-fermion SPT systems and interacting ones. For free-fermion systems with Altland-Zirnbauer symmetries, we prove that the single-particle entanglement gap after quenches obeys essentially thesame Lieb-Robinson bound as that on the equal-time correlation, provided that there is no dynamical symmetrybreaking. As a notable byproduct, we obtain a new type of Lieb-Robinson velocity which is related to the banddispersion with a complex wave number and reaches the minimum as the maximal (relative) group velocity.Within the framework of tensor networks, i.e., for SPT matrix-product states evolved by symmetric and trivialmatrix-product unitaries, we also identify a Lieb-Robinson bound on the many-body entanglement gap for gen-eral quenched interacting SPT systems. This result suggests high potential of tensor-network approaches forexploring rigorous results on long-time quantum dynamics. Influence of partial symmetry breaking, effects ofdisorder, and the relaxation property in the long-time limit are also discussed. Our work establishes a paradigmfor exploring rigorous results of SPT systems out of equilibrium.

I. INTRODUCTION

Recent years have witnessed remarkable experimental de-velopments in atomic, molecular and optical physics, whichhave enabled us to engineer and control artificial quantummany-body systems at the level of individual atoms, ions andphotons [1–3]. Particular attention is focused on nonequilib-rium quantum dynamics [4–7], of which the arguably simplestsituation is quantum quenches [8–10] — the system is initial-ized as a wave function |Ψ0〉 which then evolves unitarily bya Hamiltonian H , with respect to which |Ψ0〉 is typically ahighly excited superposition state. The wave function at timet is then formally given by |Ψt〉 = e−iHt|Ψ0〉. To model re-alistic quantum simulators, especially ultracold atoms and su-perconducting circuits with short-range interactions, we usu-ally assumeH to be local, in the sense that it can be written asa sum of short-range operators. In light of the rapid develop-ment of topological material science [11–14], there is growinginterest in topological aspects of quench dynamics [15–36].A fundamental question in this context is: given |Ψ0〉 as theground state of a gapped Hamiltonian H0, which may be triv-ial or topological, whether the topology of |Ψt〉 will changeduring time evolution, and, if yes, in what way. To make thetopology well-defined, we may have to impose certain sym-metries. For the sake of concreteness, we assume that H0 andH share the same symmetries, if any. The answer to the abovequestion has recently been given and is somewhat negative:for unitary symmetries or/and anti-unitary anti-symmetries[37], |Ψt〉 stays in the same symmetry-protected topologi-cal (SPT) phase [16, 17, 33, 34]. For anti-unitary symme-

tries or/and unitary anti-symmetries, the topological numberof |Ψt〉 generally becomes ill-defined (or reduces) due to dy-namical symmetry breaking [33, 34]. To understand this, weonly have to note that |Ψt〉 is the ground state of [32, 33]

H(t) ≡ e−iHtH0eiHt, (1)

which shares the same spectrum as H0. The conservationof topological number follows from the fact that H(t) isgapped and continuously deformed from H0 in a symmetry-preserving manner.

Since topological numbers are rather abstract quantitiesand take very different forms depending on the specific sys-tems, we need a universal topological indicator to formalizethe above qualitative analysis into a general, rigorous and, inprinciple, experimentally verifiable statement. Entanglementturns out to be an ideal candidate to demonstrate the persis-tence of topology. We can trace the time evolution of the en-tanglement spectrum (ES) for a proper bipartition, which con-tains crucial information of the entanglement pattern and isarguably the most widely used universal topological indicatorthat is applicable to both noninteracting [38–41] and interact-ing systems [42–47]. The ES is expected to be accessible innear-future ultracold-atom and trapped-ion experiments [48–50]. The persistence of SPT order thus manifests itself in thatthe ES stays gapless or degenerate [33, 34].

However, a vital point is missing in the above argument oftopology conservation — defining SPT phases requires local-ity in the Hamiltonian [51], whileH(t) in Eq. (1) may becomehighly nonlocal after a long time. That is to say, the dynam-ically generated non-locality could obscure the SPT order in

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t = 0<latexit sha1_base64="1KC3ezbkRZ2c/AxJti50aXJ6ouo=">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</latexit><latexit sha1_base64="1KC3ezbkRZ2c/AxJti50aXJ6ouo=">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</latexit><latexit sha1_base64="1KC3ezbkRZ2c/AxJti50aXJ6ouo=">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</latexit><latexit sha1_base64="1KC3ezbkRZ2c/AxJti50aXJ6ouo=">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</latexit>

t < t =l

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FIG. 1. Schematic illustration of the main result. (Bottom) Initially,the entanglement gap ∆E of a length-l segment embedded in a 1Dtopological system is of the order of e−κl due to an exponentiallysmall overlap between the tails of the two entanglement edge modes.(Middle) After quench, these edge modes diffuse no faster than lin-early, leading to an exponential increase in ∆E. (Top) After a timeproportional to l, these modes significantly diffuse into the bulk andthe quasi-degeneracy in the ES is clearly lifted.

experimentally relevant length scales, which are often limited.It is thus practically, and of course theoretically important tounderstand at which length scale SPT order survives. Intu-itively, we expect from the bulk-edge correspondence [52] thatthe SPT order should persist up to a time scale t∗ when topo-logical edge modes at different boundaries start to interfereand diffuse into the bulk, though a more refined analysis isneeded to reach a definite conclusion.

A key concept to make the intuitive argument rigorous isthe Lieb-Robinson bound. A natural assumption on localityhas a striking consequence known as the light-cone effect —a local operator evolved by H spreads no faster than an emer-gent “light velocity” vLR. Precisely speaking, there can be anonzero leakage outside the light cone, which nevertheless de-cays exponentially with respect to the distance from the lightcone. This rigorous result was derived by Lieb and Robin-son nearly half a century ago [53], and is known as the Lieb-Robinson bound. An important implication of this bound isthat, for two remote local operators separated by l, the equal-time correlation should stay exponentially small up to a timescale t∗ = l

2vLR, provided that the initial correlation decays

exponentially [54]. Such a light-cone spreading of correla-tion as well as its breakdown for long-range interactions hasrecently been examined in ultracold-atom and trapped-ion ex-periments [55–57].

The Lieb-Robinson bound has been employed to reveal uni-versal behaviors of quantum many-body systems. In addi-tion to correlations between observables, entanglement en-tropy [58–60], which quantifies genuinely quantum correla-tion between a subsystem and its complement, has also beenwidely studied in quench dynamics [61–65]. Again, localityof H sets very fundamental limitations on the growth of en-tanglement entropy. From a quasi-particle viewpoint [66], wecan infer from the Lieb-Robinson bound that the entanglementgrowth cannot be faster than linearly in time [54]. In partic-ular, with the Lieb-Robinson bound applied to quasiadiabaticcontinuation [67], it has been proved that given |Ψ1〉 and |Ψ2〉as the ground states of two gapped local HamiltoniansH1 andH2 that can be adiabatically connected to each other, |Ψ1〉

Equilibrium Nonequilibrium

t (time)

ES

t (time)

EE

l (size)

ES

l (size)

EE

General

Topological

(a) Area Law (b) Growth

(c) Degeneracy (d) Splitting

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FIG. 2. Schematic illustrations of the most fundamental results onentanglement in 1D quantum many-body systems in and out of equi-librium, where the shaded part is rigorously established in this work.(a) The ground state of an arbitrary gapped local Hamiltonian obeysan entanglement-entropy (EE) area law. (b) After a quench, theentanglement-entropy growth exhibits universal features such as lin-ear (red), logarithmic growth (purple) or saturation (blue), depend-ing on whether the postquench Hamiltonian is thermal, many-bodylocalized or Anderson localized, respectively. (c) If the system isin an SPT phase, the ES is degenerate up to an exponentially smallcorrection. (d) Rigorous Lieb-Robinson bound on the lifting of ESdegeneracy for SPT systems after quenches, which constitutes theprimary result of this paper.

obeys an entanglement area law [68] if and only if |Ψ2〉 alsodoes [69]. In one dimension (1D), where all the gapped localHamiltonians are adiabatically connected [70–72], we alwaysobtain an area law by choosing |Ψ1〉 to be a product state. It isworth mentioning that, without referring to any Hamiltonian,all the 1D wave functions with exponentially decaying corre-lations have recently been proved to obey the area law [73–75]. Moreover, the measurement of the entanglement (Renyi)entropy has been achieved in ultracold-atom and trapped-ionexperiments [76–79].

According to the light-cone picture, we can anticipate thatthe time scale of the Leib-Robinson bound roughly estimateshow long it takes for a topological edge mode to completelydiffuse into the bulk, after which the SPT order becomes in-visible. This argument may remain applicable if the physicaledge is replaced by an artificial entanglement cut, which canalways be done even if there are no physical boundaries. Inthis case, reinterpreting l as a subsystem size, we expect thatthe ES should stay nearly gapless or degenerate with precisione−O(l) until t∗ [80]. Such an expectation has been numericallyverified in several free-fermion models [81–83]. It is thus nat-ural to conjecture the existence of a Lieb-Robinson bound onthe entanglement gap (to be exactly defined in a moment):

∆E ≤ Ce−κ(l−vt). (2)

Here v = 2vLR, κ estimates an inverse correlation length andC is a prefactor that is at most polynomial in terms of l and t.See Fig. 1 for a schematic illustration for 1D systems.

In this paper, we prove Eq. (2) for general 1D SPT systemsundergoing nonequilibrium unitary evolution. In fact, a rig-

Page 3: Lieb-Robinson Bounds on Entanglement Gaps from Symmetry … · 2019-04-30 · Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology Zongping Gong,1 Naoto Kura,1

3

orous result has been obtained for intrinsic topological orderfrom the perspective of local indistinguishability between de-generate ground states [54]. However, unlike the derivation inRef. [54] which is a rather straightforward application of theLieb-Robinson bound on operator spreading [53] and is irrel-evant to symmetries, the ES is not a conventional observableand symmetries play a crucial role in protecting 1D topolog-ical phases. Nevertheless, we find that the entanglement gapcan be rigorously bounded by some correlation-related quan-tities, which obey the Lieb-Robinson bound. The central ideais based on the powerful Weyl’s perturbation theorem [84],which guarantees the spectral shift between two Hermitianoperators to be rigorously bounded by the operator norm oftheir difference. In addition to the several well-known resultssuch as the entanglement area law [85–89], the bounds on en-tanglement growth [69, 90] and the entanglement detection oftopological phases [38–47, 91, 92], our work brings about yetanother rigorous and fundamental result on the entanglementin quantum many-body systems [93–95]. As shown Fig. 2,our work lays a corner stone for exploring exact results on theentanglement properties of SPT systems out of equilibrium.Also, we hope that the ideas and methods developed in thiswork could stimulate further exploration of rigorous resultson various spectra, which appear ubiquitously in physics.

This paper is structured as follows. In Sec. II, we review thebasic properties of the ES and the notion of dynamical sym-metry breaking. In Sec. III, we define the entanglement gapand present the main results, i.e., the explicit forms of Lieb-Robinson bounds. In Sec. IV, we focus on the single-particleentanglement gap in quenched free-fermion systems and de-rive the first main result (Theorem 1). In particular, we derivean almost optimal Lieb-Robinson bound for free-fermion sys-tems and justify the quasi-particle picture. In Sec. V, we fo-cus on the many-body entanglement gap in the tensor-networksetting and derive the second main result (Theorem 2). Wediscuss the impact of partial symmetry breaking, the effectsof disorder and long-time dynamics in Sec. VI. Finally, wesummarize the main results of this paper and provide someoutlook in Sec. VII. We relegate some technical details to ap-pendices to avoid digressing from the main subject.

II. PRELIMINARIES

We begin by clarifying the definitions of single-particle andmany-body ES for free-fermion and general systems. We alsobriefly review why 1D SPT order renders the many-body ESto be degenerate. Finally, we review the notion of dynamicalsymmetry breaking and point out the relevant symmetries thatcan protect SPT order in quench dynamics.

A. Definition of the ES

We consider a 1D lattice with the total number of unit cellsdenoted as L. Each unit cell contains d internal degrees offreedoms, including spins, orbitals, sublattices and so on. Forspin systems, d gives the Hilbert-space dimension of a single

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S S

(a) (b) (c)

l l+1

L1

FIG. 3. (a) Entanglement bipartition in a 1D lattice subject to theperiodic boundary condition. The reduced density operator ρS is ob-tained by tracing out the degrees of freedom in S. The total lengthand the length of subsystem S are denoted as L and l, respectively.(b) Typical single-particle ES of a topological particle-hole symmet-ric system. The single-particle entanglement gap ∆sp

E (16) is definedas the splitting between the two modes closest to one half in the ES.(c) Typical many-body ES of an SPT system. The many-body entan-glement gap ∆mb

E (25) is defined to be the width of those modes inthe ground-state manifold of the many-body entanglement Hamilto-nian that become generate in the thermodynamic limit.

unit cell. Given a local basis |j〉dj=1, an arbitrary many-body wave function can be expressed as a superposition ofFock states |j1j2...jL〉 ≡ |j1〉 ⊗ |j2〉 ⊗ ... ⊗ |jL〉 with js =1, 2, ..., d for ∀s = 1, 2, ..., L. For fermionic systems, the localHilbert-space dimension is 2d since each mode can be eitheroccupied or unoccupied. A complete fermionic Fock basisis given by

∏nja(c

†ja)nja |vac〉 with nja = 0, 1 for ∀j =

1, 2, ..., L and a = 1, 2, ..., d, where |vac〉 is the Fock vacuumand c†ja creates a fermion with internal state a in the jth unitcell and satisfies c†j′a′ , cja = δj′jδa′a and cj′a′ , cja = 0.

To study the bipartite entanglement properties, we dividethe entire system into two subsystems S and S (see Fig. 3),which consists of l and L− l adjacent unit cells, respectively.For convenience, we label the unit cells in S as 1, 2, ..., l.Given a many-body wave function |Ψ〉, we can always definethe many-body ES as the eigenvalues of the reduced densityoperator

ρS = TrS |Ψ〉〈Ψ|. (3)

For simplicity, we require the ES to be positive since wecan always truncate the Hilbert space of S into a subspace,within which ρS is positive definite. In particular, for free-fermion systems, ρS is a Gaussian state [96] in the sensethat there exists a quadratic entanglement Hamiltonian HS =∑lj,j′=1

∑da,a′=1[HS ]j′a′,jac

†j′a′cja [97] such that

ρS =e−HS

Tr e−HS. (4)

Diagonalizing the entanglement Hamiltonian as HS =∑ldn=1 εnf

†nfn, where fn’s are related to cja’s via a unitary

transformation, we can identify the single-particle ES as [98]

ξn =1

eεn + 1, n = 1, 2, ..., ld, (5)

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4

in terms of which the many-body ES reads

ζsn =∏

n:0<ξn<1

[1

2+

1

2sn(1− 2ξn)

], (6)

where sn = ±1.

B. SPT-enforced ES degeneracy

For free-fermion systems, it is known that 1 − 2ξnnis exactly the energy spectrum subject to the open bound-ary condition after band flattening [39]. To see this, weconsider a gapped quadratic Hamiltonian in the diagonalizedform H =

∑nEnψ

†nψn, which may not correspond to a

translation-invariant system. Assuming the Fermi energy tobe zero without loss of generality, we can define the single-particle projector onto the Fermi sea as

P< ≡∑

n:En<0|ψn〉〈ψn|, (7)

where |ψn〉 ≡ ψ†n|vac〉 is a single-particle eigenstate. WithPS denoted as the projector onto the single-particle Hilbertspace of S, the single-particle ES ξnn coincides with thespectrum of PSP<PS [98]. The statement made in the begin-ning of this subsection follows from the fact that the involu-tory (i.e., being its own inverse) flattened Hamiltonian is givenby

Hflat = Isp − 2P<, (8)

where Isp is the identity operator in the single-particle sector.From this exact correspondence, we know that there shouldbe ξn = 1

2 modes in the single particle ES for a topologicalinsulator with boundary zero modes. According to Eq. (6), thecorresponding many-body ES is necessarily degenerate.

To analyze noncritical interacting systems in 1D, we em-ploy the matrix-product-state (MPS) formalism [99–102].The validity of this formalism is rooted in the entanglementarea law [68]. For simplicity, we focus on translation-invariantMPSs, which take the form of

|Ψ〉 =∑jsLs=1

Tr[Aj1Aj2 ...AjL ]|j1j2...jL〉, (9)

where js = 1, 2, .., d and Aj’s are D × D matrices with Dbeing the bond dimension. Each MPS is associated with alinear map on CD×D:

E(·) ≡d∑j=1

Aj(·)A†j . (10)

Assuming the MPS to be normal [103], which rules out thepossibility of spontaneous symmetry breaking, we can alwaysperform gauge transformations of Aj’s, i.e., Aj → XAjX

−1

that leaves |Ψ〉 in Eq. (9) invariant, such that E is a unitalchannel:

E(1v) =

d∑j=1

AjA†j = 1v (11)

TABLE I. Dynamical stability of unitary symmetries (a = b = +),anti-unitary symmetries (a = −b = +), unitary anti-symmetries(a = −b = −), and anti-unitary anti-symmetries (a = b = −),where a and b are given in Eq. (14).

a b Dynamical stability Example+ +

√Parity symmetry

+ − × Time-reversal symmetry− + × Chiral symmetry− − √

Particle-hole symmetry

where 1v is the identity in CD×D acting on the virtual Hilbertspace. If we further impose an on-site unitary symmetry, i.e.,ρ⊗Lg |Ψ〉 = |Ψ〉 for ∀g ∈ G with G being a group and ρg ∈U(d) being a unitary representation of G, we can find a pro-jective representation Vg with VgVh = ωg,hVgh, ωg,h ∈ U(1)for ∀g, h ∈ G such that [104]

d∑j′=1

[ρg]jj′Aj′ = V †g AjVg, ∀j = 1, 2, ..., d. (12)

In the thermodynamic limit of subsystem S with length l,i.e., liml→∞ limL→∞, the many-body ES is exactly given byλαλβDα,β=1, where λαDα=1 is the spectrum of Λ (> 0),that is uniquely determined from [100]

E†(Λ) ≡d∑j=1

A†jΛAj = Λ, Tr Λ = 1. (13)

If |Ψ〉 is in an SPT phase, ωg,h must correspond to a nontrivialelement in the second cohomology group H2(G,U(1)) [70–72], leading to degeneracy in λα’s [43]. This is because anondegenerate λα implies that ωg,h must be trivial, as will beproved in Sec. V B.

C. Dynamical symmetry breaking

While there is a huge variety of symmetries, we can classifythem into four groups depending on whether the symmetryoperator S commutes or anti-commutes with the Hamiltonian,and whether it is unitary or anti-unitary:

SHS−1 = aH, SiS−1 = bi, (14)

where a, b = ±. Concretely, S is said to be symmetric (ananti-symmetric) if a = + (a = −), and S is said to be uni-tary (anti-unitary) if b = + (b = −). Note that H in Eq. (14)can be either on the single-particle level (for free-fermion sys-tems) or on the many-body level (for interacting systems).

Now let us impose Eq. (14) to both H0 and H , i.e., theHamiltonians before and after a quench. Regarding the sym-metry action on the parent Hamiltonian (1), we have

SH(t)S−1 = Se−iHtH0eiHtS−1

= e−iabHtaH0eiabHt = aH(abt).

(15)

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5

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FIG. 4. Setting of Theorem 1 — quench in a free-fermion sys-tem with particle-hole symmetry. Initially, only the particle bands(εk < 0) are occupied and the Bloch projector is denoted as P0<(k)(see Eq. (17)). The initial state is, in general, a highly excited stateof the postquench Hamiltonian, whose hole bands (εk > 0) can besignificantly occupied.

Accordingly, when ab = 1, which means that S is either uni-tary and symmetric or anti-unitary and anti-symmetric, wehave [S,H(t)] = 0 for ∀t. Otherwise, H(t) no longer re-spects S in general due to the fact that H(t) generally differsfrom H(−t). This phenomenon is dubbed dynamical symme-try breaking, which reduces the number of symmetries rele-vant to SPT orders in quench dynamics [33]. See Table I for abrief summary.

III. MAIN RESULTS

We are now in a position to present the exact statement ofour main results — Lieb-Robinson bounds on entanglementgaps.

For free-fermion systems, we examine all the Altland-Zirnbauer classes [105]. In 1D, it is known that there are fivenontrivial classes, including one complex class AIII and fourreal classes BDI, D, DIII and CII [106, 107]. According tothe previous analysis on dynamical symmetry breaking (espe-cially Table I), we know that both classes BDI and DIII reduceto class D, class CII reduces to class C and class AIII reducesto class A. Since classes C and A are both trivial, it suffices toconsider class D, which has an involutory particle-hole sym-metry and is characterized by a Z2 topological number in 1D[108]. This is the only Altland-Zirnbauer class in 1D that doesnot suffer from dynamical symmetry breaking so that the SPTorder persists in the thermodynamic limit [33].

Since the energy spectrum of a free-fermion system in classD is paired as (ε,−ε), the ES should be divided as (ξ, 1− ξ).Hence, we can define the single-particle entanglement gap as

∆spE ≡ 2 min

n

∣∣∣∣ξn − 1

2

∣∣∣∣ . (16)

In the presence of translation invariance, we can simply dealwith the Bloch Hamiltonians and prove the following theo-rem.

Theorem 1 (Free fermions) Consider two 1D translation-invariant lattice systems in class D, whose Bloch Hamiltoni-ans are given by H0(k) and H(k) and the former is gapped

eiHt

(1)

u

v

(2)

(3)

Uv

u=

(4)

U

U=

u†

u(5)

Spec...

...p

p

A

A

l (6)

Spec...

...(7)

... ...

t

A

U

(8)

... ...Oj

U

U

(9)

1

eiHt

(1)

u

v

(2)

(3)

Uv

u=

(4)

U

U=

u†

u(5)

Spec...

...p

p

A

A

l (6)

Spec...

...(7)

... ...

t

A

U

(8)

... ...Oj

U

U

(9)

1

...

...U†OjU

2k0 + 1 (10)

2

(a) (b)

=

FIG. 5. (a) Setting of Theorem 2 — stroboscopic time evolution gov-erned by an MPU (22) generated by Ujj′ starting from an MPS (9)generated by Aj . (b) After evolution by an MPU, an on-site oper-ator Oj stays local but acts nontrivially on (at most) 2k0 + 1 sites.Here k0 is the smallest integer k such that the blocked tensor Uk inEq. (23) is simple.

and topologically nontrivial. We start from the ground state ofH0(k) and make the following two assumptions: (i) the initialBloch projector

P0<(k) ≡∮γ<

dz

2πi

1

z −H0(k)(17)

is analytic on k : |Imk| ≤ κ, |Rek| ≤ π, where κ > 0 andγ< is a loop that encircles all the particle bands; (ii) H(k +iκ) is well-defined and diagonalizable for ∀k ∈ [−π, π], i.e.,

H(k + iκ) =

d∑α=1

εk+iκ,α|uRk+iκ,α〉〈uL

k+iκ,α|, (18)

where |uRk+iκ,α〉 (〈uL

k+iκ,α|) is the right (left) Bloch eigenstateof the α-th band. Then the single-particle entanglement gap(16) of a length-l segment is upper bounded by

∆spE ≤ Ce−κ(l−vt) (19)

during the time evolution governed by H(k). Here

C =

∫ π

−π

dk

2π

(d∑

α=1

‖|uRk+iκ,α〉‖‖|uL

k+iκ,α〉‖)2

‖P0<(k+iκ)‖

(20)with ‖ · ‖ being the operator norm and

v = κ−1 maxk∈[−π,π],α,β

Im(εk+iκ,α − εk+iκ,β) (21)

depends on neither l nor t.

As illustrated in Fig. 4, this theorem depends crucially onthe band picture of translation-invariant free-fermion systems.We can show that a positive κ satisfying (i) and (ii) exists un-der quite general assumptions (see Appendix A).

For interacting SPT systems, we restrict ourselves to thetensor networks formalisms. That is, as illustrated in Fig. 5(a),we start from an MPS and consider the stroboscopic dynamicsgoverned by a matrix-product unitary (MPU) [109–112]

U =∑

js,j′sLs=1

Tr[Uj1j′1Uj2j′2 ...UjLj′L ]|j1j2...jL〉〈j′1j′2...j′L|,

(22)

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6

which is a special class of matrix-product operators [113] sat-isfying U†U = 1⊗L. We assume that the MPU respects thesame symmetries of the initial MPS, i.e., [ρ⊗Lg , U ] = 0 for∀g ∈ G, and that it belongs to the trivial cohomology class sothat the time-evolved MPS stays in the same SPT phase [111].It is known that by putting together k sites into one such thatthe building block U becomes Uk:

Uk ≡ ...U U U

k

,

(23)

the MPU can for sufficiently large k be represented as a bi-layer unitary circuit with each unitary operator acting on twoadjacent blocked sites [114]:

Uk =Uk =Uk =Uk =Uk =u u

v v

.

(24)

Whenever such a representation is possible, we call thebuilding-block tensor simple [110]. In fact, given U and ksuch that Uk is simple, Uk′ is also simple for ∀k′ ≥ k. Thesmallest k that makes Uk simple, which we denote as k0, has aclear physical interpretation as the Lieb-Robinson length — asshown in Fig. 5(b), any on-site operator evolved by the MPUgenerated by U acts nontrivially on at most 2k0 + 1 sites. Fur-ther details on MPUs can be found in Appendix E.

Unlike free-fermion systems in class D, which are charac-terized by a Z2 number so that there is at most one pair ofstable topological entanglement modes near ξ = 1

2 (leadingto 22 = 4-fold degeneracy in the many-body ES), the many-body ES of an SPT MPS can be r2-fold degenerate for ∀r =2, 3, 4, ... in the thermodynamic limit. Here the square r2

arises from the fact that a subsystem has two edges. Minimalsymmetries that realize these SPT MPSs are Zr × Zr, whosesecond-order cohomology groups read H2(Zr ×Zr,U(1)) =Zr. Given r and a many-body ES ζnn with ζn ≥ ζn+1 (notethat larger ζ corresponds to lower eigenvalue of the entangle-ment Hamiltonian), we define the many-body entanglementgap to be

∆mbE ≡ |ζ1 − ζr2 |. (25)

Our main theorem is the following.

Theorem 2 (Interacting systems) Starting from an infiniteSPT MPS with bond dimension D, the many-body entangle-ment gap (25) of a length-l subsystem after t steps of timeevolution by a trivial symmetric MPU generated by U withbond dimension DU is bounded from above by

∆mbE ≤ C(l − 2k0t)

D2−1e−κ(l−vt) (26)

for any l−2k0t ≥ 1+µ1−µ . Here k0 is the smallest integer block-

ing number that makes Uk0 simple and µ is the spectrum ra-dius of E − E∞, where E is defined in Eq. (10) for the initial

MPS and E∞ ≡ limn→∞ En, κ = − lnµ, v = 2k0 − lnDUlnµ ,

and the coefficient

C = 4e2D2(D2 + 1)µ1−D2

(1 +µ)D2+ 1

2 (1−µ)D2− 5

2 (27)

depends only on the initial MPS.

Let us explain why we focus on the stroboscopic dynam-ics generated by an MPU rather than a continuous dynamicsgenerated by a local Hamiltonian. First, we expect this settingto be good enough because we can efficiently approximate afinite-time evolution generated by a local Hamiltonian as abilayer unitary circuit [115], which is equivalent to an MPUor a quantum cellular automaton [110]. The efficiency is en-sured by the conventional Lieb-Robinson bound. By showingthat the spectral shift is rigorously bounded by the approx-imation error, we expect a similar Lieb-Robinson bound onthe many-body entanglement gap for continuous quench dy-namics (see Appendix F). Second, this formalism is of intrin-sic interest for its own sake — it gives exact descriptions forsome Floquet systems [116–118] and quantum circuits [119–127], which have intensively been studied in the context ofnonequilibrium phases of matter and information scrambling.Moreover, this theorem exemplifies the power of tensor net-works as analytical methods for predicting long-time dynam-ical behaviors of interacting quantum many-body systems farfrom equilibrium, which are hardly accessible in numerics.

IV. FREE FERMIONS

As mentioned above, for quench dynamics within the sameAltland-Zirnbauer class [105–107], the only nontrivial classin 1D that does not suffer from (partial) “dynamical symmetrybreaking” is class D [33]. In fact, previous numerical studieson the Su-Schrieffer-Heeger (SSH) model [81] and the Kitaevchain [82] have revealed that the ξ = 1

2 modes split after acharacteristic time scale t∗ ∼ l

2vmax, where l is the length of

the subsystem and vmax is the maximal group velocity of banddispersions. In Fig. 6(b), we reproduce the splitting dynamicsin the SSH model (Fig. 6(a)), which shows that not only thetopological entanglement mode but also the full ES splits. Inthe following, we rigorously establish the underlying Lieb-Robinson bound on the ES splitting stated in Theorem 1.

A. Zero entanglement gap for symmetric bipartition

We first point out a crucial proposition — for half-chainentanglement cut, i.e., l = L

2 , the topological entanglementmodes will be pinned exactly at 1

2 at any time (see Fig. 6(b)).Note that we can always choose the anti-unitary and involu-tory particle-hole-symmetry operator C to be the complex con-jugation K [128]. Under this basis, we have H = iR with Rbeing a skew-symmetric real matrix. Suppose that H is flat-tened so that R2 = −Isp. If Pf R = −1, where Pf denotesthe Pfaffian, there will be a pair of topological entanglementmodes exponentially close to ξ = 1

2 . With the translation

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7

0 10 20 30 40 500.00.20.40.60.81.0

t

ξ

0 10 20 30 40 5010-1610-1310-1010-710-410-1

t

EGCorrLRB

J1 J2(a)

(b) (c)

30 35 400.4950.5000.505

FIG. 6. (a) Quench protocol in the SSH model. (b) Single-particleES dynamics after the quench. The parameters are quenched as(J1, J2) = (0.5, 1) → (1, 0.5) and l = 40. The red solid and bluedashed curves correspond to L = ∞ and L = 2l, respectively. Inthe former case, the single-particle ES splits after t∗ ∼ 33 (indicatedby the red dashed line and determined by threshold ∆sp

E = 10−3

as shown in (c)). Inset: Zoom in on the splitting of the topolog-ical entanglement modes. (c) Dynamics of the single-particle en-tanglement gap for L = ∞ (solid curve). The equal-time correla-tion ‖〈j|P∞< (t)|j+ l〉‖ (purple dotted curve) and the Lieb-Robinsonbound (blue dashed curve) given by Theorem 1 with κ = 0.6 aresuperimposed for the sake of comparison.

invariance assumed, the system at least exhibits a half-chaintranslation symmetry, leading to

R =

[Rd Ro

Ro Rd

]= σ0 ⊗Rd + σx ⊗Ro, (28)

whereRd andRo are both real and skew-symmetric and σ0 =[ 1 00 1 ], σx = [ 0 1

1 0 ]. Moreover, the half-chain ES is given by thespectrum of 1

2 (1− iRd). From R2 = −Isp we obtain

R2d +R2

o = −Ihalf , Rd, Ro = 0half , (29)

where Ihalf and 0half are the half-chain identity and zero oper-ators within the single-particle sector. Provided that R2

o < 0,we can find an anti-unitary operator

A ≡ i(−R2o)−

12RoK, (30)

such that

A2 = −Ihalf , [A, iRd] = 0. (31)

Due to the interplay between the Kramers degeneracy en-forced by A and the nontrivial Z2 topology, two quasi-zeromodes of iRd must be pinned exactly at zero. Since the Z2

index stays unchanged in quench dynamics, the topologicalentanglement mode should always be pinned at 1

2 , leading toa persistent zero single-particle entanglement gap.

Even if Ro is not invertible so that A in Eq. (30) becomesill-defined, we can still show that all the eigenvalues of iRd

falling in the range (−1, 1) are degenerate. For an arbitrarynormalized eigenvector φ with iRdφ = εφ, ε ∈ (−1, 1), wecan construct

φ ≡ 1√1− ε2

Roφ, (32)

such that φ†φ = 1 and iRdφ = εφ. Moreover, we have

φ†φ =1√

1− ε2φ†Roφ =

1√1− ε2

(φ†Roφ)T

= − 1√1− ε2

φ†Roφ = −φ†φ,(33)

which means φ is orthogonal to φ.It is worth mentioning that such an emergent symmetry in

ES has systematically been studied in Ref. [41]. In addition tothe above physical analysis, we also provide a rigorous proofin Appendix B.

B. General idea

To highlight the finite size of a periodic lattice, we willhereafter use P (L)

< instead of P< in Eq. (7) as the single-particle projector onto the Fermi sea, where L denotes thenumber of unit cells. As mentioned in Sec. II B, withthe single-particle projector onto a subsystem S denoted asPS , the single-particle ES coincides with the spectrum ofPSP

(L)< PS [98]. We have shown in the previous subsection

that for L = 2l with l being the length of S, whenever Hflat

in Eq. (8) is characterized by a nontrivial Z2 number protectedby the particle-hole symmetry, the spectrum of PSP

(L)< PS

contains two degenerate eigenstates with eigenvalue 12 and

the entanglement gap exactly vanishes. To prove Theorem 1,it suffices to prove that the spectral shift from PSP

(2l)< PS to

PSP(∞)< PS is exponentially small until a time scale propor-

tional to l. To this end, a natural idea is to utilize the followingWeyl’s perturbation theorem.

Theorem 3 (Weyl’s perturbation theorem) Consider twoHermitian operators O and O′ on a finite Hilbert space.Denoting the jth largest eigenvalue of O and O′ as λj andλ′j , respectively, we have

|λj − λ′j | ≤ ‖O −O′‖. (34)

For self-containedness, we provide a brief proof in Ap-pendix C. According to this theorem, denoting the nth largesteigenvalue of PSP

(L)< PS as ξ(L)

n , we have

|ξ(L)n − ξ(∞)

n | ≤ ‖PSP (L)< PS − PSP (∞)

< PS‖. (35)

According to Eq. (16), we have the following collorary: fora topological class D system, the single-particle entanglementgap is bounded from above as

∆spE ≤ 2‖PSP (2l)

< PS − PSP (∞)< PS‖. (36)

It is thus sufficient to find a Lieb-Robinson bound on the right-hand side (rhs) of Eq. (36), which measures the finite-size cor-rection to the correlation in a finite subsystem S.

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8

Re k

Im k

(1)

1

vmax

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.400.450.500.550.60

κ

v LR

(a) (b)

FIG. 7. (a) Contour deformation used in Eq. (42). For sufficientlysmall κ, the shaded area inside the contour ∂k : |Re k| ≤ π, 0 ≤Im k ≤ κ contains no poles and thus

∫ π−π =

∫ π+iκ−π+iκ. Note that

the two integrals along the vertical edges cancel out due to the 2π-periodicity in Re k. (b) κ dependence of the Lieb-Robinson velocityvLR = 1

2v predicted by Lemma 1 for the SSH model with (J1, J2) =

(1, 0.5). The maximal group velocity vmax = minJ1, J2 natu-rally appears in the limit of κ→ 0. The red dashed lines correspondto the specific choice κ = 0.6 used in Fig. 6(c). The monotonicity ofvLR with respect to κ holds for general analytic Bloch Hamiltonians(see Appendix A 3).

Before going into rigorous proofs, let us first give an intu-itive argument based on the Wannier-function picture [129].Note that the projector onto the Fermi sea can be expressed as

P(L)< =

∑j∈ZL,α∈O

|W (L)jα 〉〈W

(L)jα |, (37)

where |W (L)jα 〉 is a Wannier function of the αth band centered

at the jth site, ZL ≡ 1, 2, ..., L consists of all the sites andO consists of all the occupied bands. Since |W (L)

jα 〉 is ex-ponentially localized in real space [130–132], we expect anexponentially small difference between |W (L)

jα 〉 and |W (∞)jα 〉.

By the same token, although P (∞)< contains infinitely more

Wannier-function projectors than P(L)< , such a difference

should again become exponentially small after being pro-jected by PS , provided that both l and L − l are sufficientlylarge compared with the localization length of a Wannier func-tion. When the system is driven out of equilibrium, we expectthe Wannier function to spread no faster than linearly, leadingto a light-cone behavior of ‖PSP (L)

< PS − PSP (∞)< PS‖. We

will later make this argument rigorous for translation-invariantsystems. Note that the above argument seems to be equallyapplicable to disordered systems with exponentially localizedWannier functions [133].

C. Lieb-Robinson bound on correlations in free-fermionsystems

An important ingredient in our proof is the conventionalLieb-Robinson bound on correlation functions. While theLieb-Robinson bound for general interacting systems is cer-tainly applicable to free-fermion systems, such a bound is usu-ally very loose since it only involves a very limited amount ofinformation about the system such as the hopping range andthe maximal hopping amplitude. Here, we derive a greatly

(a) (b)

(c)-2-1012

ϵ k

-π 0 π

-1

0

1

k

v k

FIG. 8. (a) Dynamics of spatial correlations ‖〈j|P<(t)|0〉‖ (j 6= 0)in an SSH chain involvingL = 101 unit cells. The parameters are thesame as those in Fig. 6. An additional particle-hole symmetric termHPHS =

∑j,a=A,B iJ(c†j+1,acja − H.c.) does not change the dy-

namics of correlations. The dashed lines are given by j = ±vmaxt,where vmax is the largest group velocity of the SSH model with(green) or without (purple) the additional term HPHS. (b) Band dis-persions and (c) the corresponding group velocities of the SSH model(purple curves) and that with the additional term (green curves) withJ = 0.5. The arrows indicate where vkα reaches its maximal abso-lute value. The solid and dotted curves correspond to the particle andhole bands, respectively.

improved Lieb-Robinson bound on the correlation functionsin translation-invariant free-fermion systems. Our result isalmost optimal in the sense that the Lieb-Robinson velocityreaches the maximal (relative) group velocity of band disper-sions in the semiclassical limit.

A nice property of free-fermion systems is that, accordingto Wick’s theorem, all the correlation functions can be decom-posed into two-point correlation functions. Moreover, if thewave function is a particle-number eigenstate, only the corre-lators like 〈c†c〉 are relevant. To compute such a correlator inthe quench dynamics, we can use the formula

〈Ψt|c†j′a′cja|Ψt〉 = 〈j′a′|P<(t)|ja〉, (38)

where |ja〉 = c†ja|vac〉 and P<(t) ≡ e−iHtP0<eiHt is the

time-evolved single-particle projector onto the Fermi sea. Tomeasure the strength of correlation at the length scale |j −j′|, we can consider the norm of 〈j|P<(t)|j′〉, which can beshown to obey a Lieb-Robinson bound given by the followinglemma.

Lemma 1 Consider a time-evolved projector P(L)< (t) ≡

e−iHtP (L)0< eiHt, where P (L)

0< is the projector onto the Fermisea of H0, which is gapped, and denote the Bloch Hamilto-nians of H0 and H as H0(k) and H(k), respectively. Thenfor ∀κ > 0 such that (i) P0<(k) ≡ ∑α∈O |ukα〉〈ukα| (seealso Eq. (17)) is analytic over k : |Re k| ≤ π, |Im k| ≤ κ,where |ukα〉, α ∈ O is the normalized Bloch eigenstate of theαth occupied band, and (ii) H(k + iκ) is well-defined anddiagonalizable for ∀k ∈ [−π, π], we have

‖〈j|P (∞)< (t)|j′〉‖ ≤ Ce−κ(|j−j′|−vt), ∀j, j′ ∈ Z, (39)

where |j〉 is the state localized at the jth site, and C and v aregiven in Eqs. (20) and (21).

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Proof.— After analytic continuation, the Hermiticity con-straints on the Bloch Hamiltonians are generalized toH0(k)† = H0(k) and H(k)† = H(k) (see Appendix A 1).Accordingly, the Bloch projector P0<(k) satisfies

P0<(k)† ≡ −∮γ<

dz

2πi

1

z −H0(k)†

=

∮γ<

dz

2πi

1

z −H0(k)= P0<(k),

(40)

where γ< is a loop that encircles the energies of all the occu-

pied bands. Note that

〈j|P (∞)< (t)|j′〉 =

∫ π

−π

dk

2πeik(j−j′)P<(k, t)

= (〈j′|P (∞)< (t)|j〉)†,

(41)

where P<(k, t) ≡ e−iH(k)tP0<(k)eiH(k)t. This implies‖〈j|P (∞)

< (t)|j′〉‖ = ‖〈j′|P (∞)< (t)|j〉‖, and therefore we can

assume j ≥ j′ without loss of generality. By deforming thecontour of integration (see Fig. 7(a)) and applying spectral de-composition to H(k+ iκ), which is generally non-Hermitian,we can bound the norm of Eq. (41) from above by

‖〈j|P (∞)< (t)|j′〉‖ = e−κ|j−j

′|∥∥∥∥∫ π

−π

dk

2πeik(j−j′)e−iH(k+iκ)tP0<(k + iκ)eiH(k+iκ)t

∥∥∥∥= e−κ|j−j

′|

∥∥∥∥∥∥∑α,β

∫ π

−π

dk

2πeik(j−j′)−i(εk+iκ,α−εk+iκ,β)t|uR

k+iκ,α〉〈uLk+iκ,α|P0<(k + iκ)|uR

k+iκ,β〉〈uLk+iκ,β |

∥∥∥∥∥∥≤∑α,β

e−κ|j−j′|∫ π

−π

dk

2πeIm(εk+iκ,α−εk+iκ,β)t

∥∥|uRk+iκ,α〉〈uL

k+iκ,α|P0<(k + iκ)|uRk+iκ,β〉〈uL

k+iκ,β |∥∥

≤ e−κ|j−j′|+maxk∈[−π,π],α,β Im(εk+iκ,α−εk+iκ,β)t∑α,β

∫ π

−π

dk

2π‖|uR

k+iκ,α〉〈uLk+iκ,α|‖‖P0<(k + iκ)‖‖|uR

k+iκ,β〉〈uLk+iκ,β |‖,

(42)

which completes the proof of Lemma 1. Let us discuss how our result is related to the conventional

group velocity [134]

vkα =dεkαdk

. (43)

In the presence of the sublattice symmetry, as is the caseof the SSH model, the eigenenergies are paired as ±εk for∀k ∈ [−π, π] (see purple curves in Fig. 8(b)), and v in Eq. (21)becomes

v = 2κ−1 maxk∈[−π,π],α

|Im εk+iκ,α|. (44)

The maximal group velocity

vmax ≡ maxk∈[−π,π],α

∣∣∣∣dεkαdk∣∣∣∣ (45)

thus naturally emerges when κ→ 0 (see Fig. 7(b)) due to therelation εk+iκ,α = εkα + idεkαdk κ + O(κ2). Since the boundin Eq. (39) can be made rather small by a sufficiently largel for a given small κ, we expect that the Lieb-Robinson ve-locity vLR = 1

2v [54] is essentially given by vmax at largelength scales. More precisely, suppose that we scale up land t simultaneously while keeping l

t fixed; then, as long aslt < 2vmax, we can always choose κ > 0 such that the boundin Eq. (39) scales like e−O(l) and thus vanishes in the thermo-dynamic limit. This is quite reasonable since the group veloc-ity in Eq. (43) is derived in the semiclassical regime, wherethe length scale is much larger than the lattice constant [134].

In general, however, the energy dispersions are not pairedat each k. This can be the case even if there is a particle-holesymmetry, which only requires εkα = −ε−kα with α being theparticle-hole conjugation of α (see, for example, green curvesin Fig. 8(b)). In the limit of κ → 0, v in Eq. (21) generallyreaches the maximal relative group velocity

vmr ≡ maxk∈[−π,π],α,β

(dεkαdk− dεkβ

dk

), (46)

which is smaller than twice of the maximal group velocity(45) and thus gives a tighter bound on the propagation of cor-relation. For example, as shown in Fig. 8(a), the dynamics ofcorrelation in the SSH model does not change in the presenceof an additional particle-hole symmetric term, which enhancesthe maximal group velocity (see Fig. 8(c)) but leaves the rela-tive group velocity invariant over the entire Brillouin zone.

One may ask whether v in Eq. (21) can be made smallerthan Eq. (46) for some κ > 0 so that the Lieb-Robinson ve-locity can be even tighter. However, by employing a con-tinuous version of the majorization technique [135], we canprove that Eq. (21) increases monotonically with respect to κ(see Fig. 7(b) for example and Appendix A 3 for the generalproof). Therefore, our rigorous bound does not lead to anytighter bound than the maximal relative group velocity. This isphysically reasonable since a completely destructive interfer-ence between the modes with maximal relative group veloci-ties seems impossible due to the difference in wave numbers.Our result thus quantitatively justifies and refines the widelyused quasiparticle picture on the propagation of correlation in

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10

the quench dynamics [9], which, to our knowledge, has ana-lytically been confirmed only in specific situations [136–138].

D. Proof of Theorem 1

We are now in a position to prove the first main result. Tobound the single-particle entanglement gap (19) from the ex-ponential decay in the correlation function, we need the fol-lowing lemma.

Lemma 2 Denoting P (L)< as the projector onto the Fermi sea

of a length-L lattice system, we have

〈j|P (L)< |j′〉−〈j|P (∞)

< |j′〉 =∑

n∈Z\0〈j|P (∞)

< |j′+nL〉 (47)

for ∀j, j′ ∈ ZL.

This result arises from the combination of Eq. (37) and therelation between the Wannier functions on finite and infinitelattices (see Appendix D). Lemma 2 can be used to derive abound on the finite-size correction to the correlation matrix,which determines the ES.

Lemma 3 Consider a length-l segment embedded in a gappedtranslation invariant 1D lattice system with length L (1 ≤ l <L ≤ ∞) under the periodic boundary condition. We denotethe projector onto the Fermi sea as P (L)

< . In the thermody-namic limit (L→∞), we assume that (justified in Lemma 1)

‖〈j|P (∞)< |j′〉‖ ≤ Ce−κ|j−j′|, ∀j, j′ ∈ Z, (48)

where C and κ > 0 do not depend on j and j′. Then, we have

‖PSP (L)< PS − PSP (∞)

< PS‖ ≤2C sinhκl

(eκL − 1) sinhκ, (49)

where PS ≡∑lj=1 |j〉〈j| ⊗ 1I is the projector onto the seg-

ment.

Proof.— According to Eq. (47), the norm of 〈j|P (L)< |j′〉 −

〈j|P (∞)< |j′〉 is bounded from above by

‖〈j|P (L)< |j′〉 − 〈j|P (∞)

< |j′〉‖≤

∑n∈Z\0

‖〈j|P (∞)< |j′ + nL〉‖

≤ C∑

n∈Z\0e−κ|j−j

′−nL|

= C

∞∑n=1

[e−κ(nL+j′−j) + e−κ(nL+j−j′)]

=2C coshκ(j − j′)

eκL − 1,

(50)

where Eq. (48) has been used. Using Eq. (50) and the norminequality [139]

‖O‖2 ≤∑j,j′

‖Ojj′‖2, (51)

with Ojj′ ≡ PjOPj′ , PjPj′ = δjj′Pj and∑j Pj = 1 for an

arbitrary bounded partitioned operator O =∑j,j′ Ojj′ , we

obtain

‖PSP (L)< PS − PSP (∞)

< PS‖2

≤l−1∑j,j′=0

‖〈j|P (L)< |j′〉 − 〈j|P (∞)

< |j′〉‖2

≤ 4C2l∑

j,j′=1

cosh2 κ(j − j′)(eκL − 1)2

=2C2[(

∑lj=1 e

2κj)(∑lj′=1 e

−2κj′) + l2]

(eκL − 1)2

≤[

2C sinhκl

(eκL − 1) sinhκ

]2

,

(52)

where we have used sinhκl ≥ l sinhκ for l ≥ 1. The remaining step to prove Theorem 1 is simply to com-

bine Lemmas 3 and 1 with Eq. 36. By identifying C inEq. (48) with Ceκvt in Eq. (39), we find that Eq. (36) leadsto Theorem 1.

Finally, let us discuss how to generalize Theorem 1 to afinite lattice with length L > l. The existence of such a Lieb-Robinson bound is already clear from the triangle inequality|ξ(L)n −ξ(2l)

n | ≤ |ξ(2l)n −ξ(∞)

n |+ |ξ(L)n −ξ(∞)

n |, the rhs of whichcan be further bounded by Lemma 3. However, this bound istoo loose since the exact degeneracy is not reproduced whenL = 2l. To obtain a tighter bound, we use the following gen-eralization of Lemma 2:

〈j|P (L1)< |j′〉 − 〈j|P (L2)

< |j′〉=

∑n1∈Z\LCM(L1,L2)

L1Z

〈j|P (∞)< |j′ + n1L1〉

−∑

n2∈Z\LCM(L1,L2)L2

Z

〈j|P (∞)< |j′ + n2L2〉,

(53)

where LCM denotes the least common multiple. Followingthe calculations in Lemma 3, we can use Eq. (53) to derive

∆spE ≤

4C sinhκl

sinhκeκvt

×(

1

eκL − 1+

1

e2κl − 1− 2

eκLCM(L,2l) − 1

),

(54)

which reproduces Eq. (19) for L = ∞ and ∆spE = 0 for L =

2l. Moreover, the bound is O(1) when L is close to l. This isreasonable because the many-body ES of a length-l segmentshould be the same as its complement with lengthL−l. IfL−lis comparable to or even smaller than the localization lengthof the entanglement edge modes, both many-body and single-particle entanglement gaps will be significantly nonzero.

V. INTERACTING SYSTEMS

Let us move on to interacting systems. It suffices to con-sider spin (bosonic) systems since interacting fermions can

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11

be mapped onto spin systems via the Jordan-Wigner transfor-mation, which preserves the locality in the presence of thefermion-parity superselection rule [140]. The only thing weshould be cautious about is that fermionic SPT states with Ma-jorana modes will be transformed into spontaneous symmetrybroken states, as will be discussed in details in Sec. V E. Al-though the very notion of the band is no longer applicable, wecan employ MPSs to efficiently describe the ground state of agapped local Hamiltonian [141]. While the formalism is dif-ferent, we can again upper bound the entanglement gap by aquantity closely related to the correlation of two local observ-ables at the boundaries of the subsystem, as detailed in thefollowing.

A. ES of an MPS

We consider a translation-invariant normal MPS in thecanonical form [142], as given in Eq. (9). For an arbitraryorthonormal basis |α〉Dα=1 on the virtual level, we can de-compose Eq. (9) into

|Ψ〉 =∑α,β

|ψαβ〉|Φβα〉, (55)

where

|ψαβ〉 =∑jsls=1

〈α|Aj1Aj2 ...Ajl |β〉|j1j2...jl〉,

|Φβα〉 =∑

jsLs=l+1

〈β|Ajl+1Ajl+2

...AjL |α〉|jl+1jl+2...jL〉.

(56)

In particular, if |α〉’s are chosen to be the eigenstates of Λ inEq. (13) with eigenvalues λα’s (

∑Dα=1 λα = 1), we have in

the thermodynamic limit L→∞ [100]

〈Φβα|Φβ′α′〉 = limL→∞

〈β′|EL−l(|α′〉〈α|)|β〉

= 〈β′|E∞(|α′〉〈α|)|β〉= 〈β′|Tr[Λ|α′〉〈α|]1v|β〉= λαδβ,β′δα,α′ .

(57)

where E∞(·) ≡ limL→∞ EL(·) = Tr[Λ · ]1v. Therefore,the reduced density matrix of subsystem [1, l] ⊂ Z (i.e., thesegment consisting of sites 1, 2, ..., l) can be written as

ρ[1,l] =∑α,β

λα|ψαβ〉〈ψαβ |. (58)

Unlike |Φβα〉’s, |ψαβ〉’s are not strictly orthogonal to eachother:

〈ψαβ |ψα′β′〉 = λβδα,α′δβ,β′ + εαβ,α′β′ ,

εαβ,α′β′ = 〈α′|(E l − E∞)(|β′〉〈β|)|α〉.(59)

Defining |φα,β〉 ≡√λα|ψα,β〉, we can simplify Eq. (58) as

ρ[1,l] =∑αβ

|φαβ〉〈φα,β | (60)

Spec...

...p

p

A

A

l (1)

Spec...

...(2)

... ...

t

A

U

(3)

... ...Oj

U

U

(4)

...

...U†OjU

2k0 + 1 (5)

1

Spec...

...p

p

A

A

l (1)

Spec...

...(2)

... ...

t

A

U

(3)

... ...Oj

U

U

(4)

...

...U†OjU

2k0 + 1 (5)

1

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FIG. 9. The spectrum of Eq. (60) (top), which is nothing but themany-body ES of a length-l segment embedded in an infinite MPS,coincides with that of Eq. (68) (bottom) according to Lemma 4.

with

〈φαβ |φα′β′〉 = λαλβδα,α′δβ,β′ +√λαλα′εαβ,α′β′ . (61)

To estimate the spectrum of ρ[1,l], we need the followinglemma.

Lemma 4 Given a Hermitian operator ρ =∑Jj=1 |φj〉〈φj | > 0, where |φj〉’s are generally neither

normalized nor orthogonal to each other, the spectrum of ρcoincides with the nonzero part of the spectrum ofM ∈ CJ×J

with Mjj′ = 〈φj |φj′〉.

Proof.— Denoting V = span|φj〉Jj=1 on which ρ acts, wecan expand any |ψ〉 ∈ V as |ψ〉 = |φ〉c =

∑Jj=1 cj |φj〉,

where |φ〉 ≡ [|φ1〉, |φ2〉, ..., |φJ〉] and c = [c1, c2, ..., cJ ]T.Defining 〈φ| = [〈φ1|, 〈φ2|, ..., 〈φJ |]T, we have ρ = |φ〉〈φ|and M = 〈φ|φ〉. Now we prove the following equivalence.

There are r linearly independent eigenstates|ψj〉rj=1 of ρ with the same eigenvalue λ 6= 0.⇔ There are r linearly independent eigenvectorsvjrj=1 of M with eigenvalue λ 6= 0.

This statement implies Lemma 4.⇒: Expanding |ψj〉 as |φ〉vj , by assumption we have

ρ|ψj〉 = |φ〉〈φ|φ〉vj = |φ〉Mvj = λ|ψj〉 = λ|φ〉vj . (62)

Multiplying 〈φ| from the left of Eq. (62), we obtain M2vj =λMvj , implying that Mvj 6= 0 (otherwise ρ|ψj〉 = 0) is aneigenvector of M with eigenvalue λ. Suppose that Mvj’s arenot linearly independent, which means

r∑j=1

kjMvj = 0 (63)

for some kj’s with∑rj=1 |kj |2 6= 0. Operating Eq. (63) on

|φ〉 gives∑rj=1 kjρ|ψj〉 = λ

∑rj=1 kj |ψj〉 = 0, which con-

tradicts the linear independence of |ψj〉rj=1.⇐: By assumption, we have Mvj = λvj for j =

1, 2, ..., r. Defining |ψj〉 = |φ〉vj , we again obtain Eq. (62),

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implying that |ψj〉 is an eigenstate of ρ with eigenvalue λ.Suppose that |ψj〉’s are not linearly independent, which means

r∑j=1

kj |ψj〉 =

r∑j=1

kj |φ〉vj = 0 (64)

for some kj’s with∑rj=1 |kj |2 6= 0. Operating Eq. (64) on

〈φ| gives∑rj=1 kjMvj = λ

∑rj=1 kjvj = 0, which contra-

dicts the linear independence of vjrj=1. According to Lemma 4, with the unimportant zero part ne-glected, the ES is nothing but the spectrum of Mαβ,α′β′ =〈φαβ |φα′β′〉, whose entries are given in Eq. (61).

B. Exact ES degeneracy in the thermodynamic limit

In the limit of l→∞, εαβ,α′β′ in Eq. (59) vanishes. Hence,Mαβ,α′β′ = 〈φαβ |φα′β′〉 becomes diagonalized and the ESis simply given by λαλβDα,β=1, which is the spectrum ofΛ⊗2. In this case, SPT order enforces the ES to be exactlydegenerate, as a result of symmetry fractionalization on thevirtual level.

Lemma 5 The ES of any topologically nontrivial MPS pro-tected by unitary symmetries is at least four-fold degeneratein the thermodynamic limit of a subsystem.

Proof.— Denoting the symmetry group asG and its projectiverepresentation on the virtual level as Vg , we have [104]

[Vg,Λ] = 0, ∀g ∈ G, (65)

where Λ is the unique left fixed point of the unital channel as-sociated with the MPS. Suppose that there is a non-degenerateeigenvalue λ associated with eigenstate |λ〉. From Eq. (65) wehave

Vg|λ〉 = νg|λ〉, νg ∈ U(1), ∀g ∈ G. (66)

Since VgVh = ωg,hVgh for ∀g, h ∈ G, Eq. (66) implies

νgνh = ωg,hνgh ⇔ ωg,h =νgνhνgh

, ∀g, h ∈ G. (67)

Hence, ωg,h belongs to the trivial cohomology class, contra-dicting the assumption that |Ψ〉 is topological. Therefore, theES in the thermodynamic limit of a subsystem, which is noth-ing but the spectrum of Λ⊗2, must be at least four-fold degen-erate.

We mention that anti-unitary symmetries can also supportnontrivial interacting SPT phases with the exactly degenerateES in the thermodynamic limit. A prototypical example is theinvolutary (T 2 = 1) time-reversal symmetry, which may befractionalized into an anti-involutary (T 2

v = −1) anti-unitarysymmetry on the virtual level, leading to the Kramers degener-acy in Λ [43]. However, since anti-unitary symmetries sufferdynamical symmetry breaking [33], the corresponding SPTorder cannot be dynamically stable.

C. Bound on the many-body entanglement gap

In general, Mαβ,α′β′ for a finite l can be decomposed into[Λ⊗2]αβ,α′β′ and a perturbative term:

Mαβ,α′β′ = [Λ⊗2]αβ,α′β′ + Pαβ,α′β′ , (68)

where

Pαβ,α′β′ =√λαλα′〈α′|(E l − E∞)(|β′〉〈β|)|α〉. (69)

It is clear from Eq. (68) that the exact eigenvalue degener-acy in Λ⊗2 is generally lifted by P . However, accordingto Weyl’s perturbation theorem, the many-body entanglementgap should be upper bounded by twice the norm of P :

∆mbE ≤ 2‖P‖. (70)

To proceed further, we upper bound ‖P‖ by ‖P‖2 ≡√Tr[P †P ], which is the Schatten 2-norm and its square takes

a rather simple form:

‖P‖22 =

D∑α,α′,β,β′=1

λαλα′〈α′|(E l − E∞)(|β′〉〈β|)|α〉〈α|(E l − E∞)(|β〉〈β′|)|α′〉

=

D∑α,α′,β,β′=1

λαλα′〈α′α|(E l − E∞)⊗2(|β′β〉〈ββ′|)|αα′〉

=

D∑α,α′,β,β′=1

Tr[Λ⊗2|αα′〉〈α′α|(E l − E∞)⊗2(|β′β〉〈ββ′|)]

= Tr[Λ⊗2S(E l − E∞)⊗2(S)],

(71)

where we have used [E(O)]† = E(O†) and S ≡∑Dα,β=1 |αβ〉〈βα| is the swap operator acting on two copies

of virtual Hilbert spaces. By defining the norm of a superop-

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13

erator as ‖L‖ ≡ max‖O1‖2=‖O2‖2=1 |Tr[O†2L(O1)]|, we canbound ‖P‖22 by

‖P‖22 ≤ ‖Λ⊗2S‖2‖S‖2‖(E l − E∞)⊗2‖

≤ D

2‖E l − E∞‖2,

(72)

where D arises from ‖S‖2, and 12 upper bounds ‖Λ⊗2S‖2 =

Tr[Λ2] since the spectrum of Λ is at least two-fold degenerate.Combining Eq. (72) with Eq. (70), we obtain

∆mbE ≤

√2D‖E l − E∞‖. (73)

Note that E l − E∞ appears routinely in the correlation func-tions of MPSs and leads to an exponential decay [99–101].Quantitatively, ‖E l−E∞‖ can be upper bounded by cε(µ+ε)l

for ∀ε > 0, where µ is the spectral radius of E − E∞ and cεdoes not depend on l [99], implying that ∆mb

E is also expo-nentially small just like the correlation functions mentionedabove. We emphasize that ‖E l−E∞‖may scale like poly(l)µl

[143], in which case a nonzero ε is necessary for giving a rig-orous bound like cε(µ+ ε)l.

D. Proof of Theorem 2

Let us analyze how the bound given in Eq. (73) changeswhen an MPS evolves according to a symmetric and trivialMPU. Denoting the bond dimension of the MPU as DU , thatof the MPS after t steps of evolutions is no more than DDt

U .Moreover, the spectrum of the associated unital channel staysinvariant during the time evolution as a result of unitarity[111] (see also Lemma 10 in Appendix E), and so does µ.Having in mind that ‖E l − E∞‖ is bounded by cε(µ + ε)l,it is natural to expect a Lieb-Robinson bound from Eq. (73).However, this expectation may fail — given ε, cε may growfaster than exponentially in time. To rule out this possiblity,we utilize the function-algebra-based techniques developed inRef. [144] to carefully estimate the growth of ‖E l − E∞‖.

The main idea of Ref. [144] is represented by the followinglemma.

Lemma 6 Given an operatorM on a finite linear space gen-erating a bounded semigroup Mnn∈N, i.e., ‖Mn‖ ≤ CMfor ∀n ∈ N, and an element f in the Wiener algebra W ≡f ∈ Hol(D) : f(z) =

∑p∈N fpz

p, ‖f‖W ≡∑p∈N |fp| <

∞ (Hol(D): set of holomorphic functions within the unit diskD ≡ z ∈ C : |z| < 1), we have

‖f(M)‖ ≤ CM‖f‖W/mMW , (74)

where ‖f‖W/mMW ≡ inf‖g‖W : g = f + mMh, h ∈W and mM ∈ W is the minimal polynomial of M, i.e.,the nonzero polynomial with the lowest degree such that

mM(M) = 0 and the leading coefficient (that of the high-est degree) is equal to 1.

Proof.— Due to ‖Mn‖ ≤ CM for ∀n ∈ N, for ∀f ∈ W , thenorm of f(M) can be upper bounded by

‖f(M)‖ ≤∑p∈N

|fp|‖Mp‖ ≤ CM∑p∈N

|fp| = ‖f‖W . (75)

Since mM is the minimal polynomial ofM, for ∀h ∈W , wehave

f(M) = (f +mMh)(M). (76)

Applying Eq. (75) to the rhs of Eq. (76) gives

‖f(M)‖ ≤ CM‖f +mMh‖W , ∀h ∈W. (77)

Equation (74) then follows from minimization of the rhs ofEq. (77). An arbitrary unital channel E satisfies the condition inLemma 6 due to the fact that En is again a unital channel and

‖En‖ ≤√

D2 , where D is the Hilbert-space dimension [145].

Accordingly, E − E∞ also satisfies the condition.Regarding the minimal polynomial of E − E∞ during time

evolution, we have the following lemma.

Lemma 7 Suppose that a single step of evolution by an MPUgenerated by U changes the associated unital channel of anMPS from E into E ′. Denoting the minimal polynomials ofE − E∞ and E ′ − E ′∞ as m and m′, respectively, we have

m′(z)|z2k0m(z), (78)

namely,m′(z) is a divisor of z2k0m(z), where k0 is the small-est integer such that Uk0 is simple.

Proof.— By assumption, for ∀k ≥ 2k0, the open boundarytensor UkUk can be decomposed into

...U

U

U

U

U

U

k

≡Uk

Uk=

Uk0

Uk0ρ

Uk0

Uk0

1⊗k−2k0

.

(79)We express m explicitly as

∑k ckz

k, which satisfies c0 = 0(due to that the spectrum of E − E∞ contains zero) and

m(E − E∞) =∑k

ck(Ek − E∞) = 0

⇔∑k

ckEk =∑k

ckE∞.(80)

Then, using Eq. (79), we can evaluate z2k0m(z)|z=E′ as

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14

∑k ck ...

A

U

U

A

A

U

U

A

A

U

U

A

k + 2k0

=∑k ck

Ak0

Uk0ρ

Uk0

Ak0

Ak

Ak

Ak0

Uk0

Uk0

Ak0

= liml→∞∑k ck

Ak0

Uk0ρ

Uk0

Ak0

Al

Al

Ak0

Uk0

Uk0

Ak0

(81)

= (∑k ck) liml→∞

Uk0

Uk0

U l

Ul

Uk0

Uk0

Ak0

Ak0

Al

Al

Ak0

Ak0

ρ ρ = (∑k ck) liml→∞

Al

U l

Ul

Al

,

where we have used Eq. (80) (enclosed in dashed rectangles)and Eq. (79). Lemma 7 follows immediately from Eq. (81),which is the tensor-network representation of E ′2k0m(E ′) =∑k ckE ′∞ ⇔ z2k0m(z)|z=E′−E′∞ = 0. A direct corollary of Lemma 7 (applying t times) is

mt(z)|z2k0tm(z), where mt(z) (m(z) ≡ m0(z)) is the mini-mal polynomial of Et−E∞t , with Et determined from the MPSat time t. Using this fact, as long as l > 2k0t, we have

‖zl‖W/mtW≤ inf‖g‖W : g = zl + z2k0tmh, h ∈W=‖zl−2k0t‖W/mW ,

(82)

where we have used ‖g‖W = ‖zng‖W for ∀n ∈ N. Accord-ing to the main result of Ref. [144], which is summarized asTheorem 6 in Appendix G, when l − 2k0t >

µ1−µ , we can

bound ‖E lt − E∞t ‖ from above by

µl−2k0t+1 4Cte2√|mE |(|mE |+ 1)

(l − 2k0t)[1− (1 + 1l−2k0t

)µ]32

× sup|z|=µ(1+ 1

l−2k0t)

1

|B(z)| ,(83)

where Ct ≡ supn∈N ‖Ent ‖ and B(z) is the Blaschke product(G1) with respect to the spectrum of E ≡ E0. If we furtherrequire l− 2k0t ≥ 1+µ

1−µ , even in the worst case, ‖E lt −E∞t ‖ isbounded by an exponentially small quantity up to polynomialcorrections (see Eq. (20) in Ref. [144]):

‖E lt − E∞t ‖ ≤ 4e2Ct√|m|(|m|+ 1)

(1 + µ

1− µ

) 32

×[

1− µ2

µ(l − 2k0t)

]|m|−1

µl−2k0t.

(84)

Denoting the bond dimension of the initial MPS |Ψ0〉 and thatof |Ψt〉 = U t|Ψ0〉 after t steps of time evolutions as D and

Dt, respectively, we have

Dt ≤ DDtU = Det lnDU , (85)

and hence [145]

Ct ≤√Dt

2≤√D

2e

lnDU2 t. (86)

Combing Eq. (73) and Eqs. (84)-(86) with |m| ≤ D2 (D ≥ 2,otherwise |Ψ〉 is a product state), we obtain Theorem 2.

Our theorem rigorously establishes the dynamical stabilityfor general SPT systems in 1D. Yet another important im-plication of Theorem 2 is that the topological discrete time-crystalline oscillation, which is a toggle between differentSPT phases [146] generated by an MPU with nontrivial co-homology [111], persists up to a time scale which increases atleast linearly with respect to the system size. To see this, wehave only to apply the theorem to Up with trivial cohomology,where p is the order of the cohomology group.

E. Applicability to interacting fermions

As mentioned in the beginning of this section, a fermionicsystem in 1D can always be mapped into a spin chain throughthe Jordan-Wigner transformation:

c†ja = σ+(j−1)d+a

∏l<(j−1)d+a

σzl , (87)

where σz =[

1 00 −1

]and σ+ ≡ 1

2 (σx + iσy) = [ 0 10 0 ]. Here

we follow the setup in Sec. II A, i.e., we consider d internalstates in each site of the original fermion system, so that thelength of the corresponding spin chain is Ld. The fermion-parity operator reads

Pf = (−)∑j,a c

†jacja =

∏j,a

σzjd+a, (88)

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15

which sets the superselection rule and is thus a Z2 symmetryof the Hamiltonian. Rigorously speaking, to keep the localityof the obtained spin Hamiltonian, we have to assume the openboundary condition. However, as long as the chain is suffi-ciently long, its subsystem property should not depend on theboundary condition. While a fermionic system never breaksthe Z2 symmetry, the corresponding spin system may sponta-neously break the Z2 symmetry, whenever the fermionic sys-tem exhibits a Majorana mode at the open boundaries [46, 71].If the Z2 symmetry is not broken, the proof for 1D bosonicSPT systems applies directly to the fermionic SPT systems,which have even parities. Remarkably, after a few modifi-cations, essentially the same proof applies even to fermionicSPT systems with odd parities.

While it is possible to directly describe fermionic SPTphases using fermionic MPSs [147, 148], where the anti-commutativity of fermionic operators is encoded in an addi-tional Z2-graded algebraic structure of the tensors, we wouldrather like to follow the approach in Refs. [46, 71] and workin the spin picture after the Jordan-Wigner transformation. Inthe spin picture, denoting |ΨSB〉 as a “physical” symmetry-broken ground state, the corresponding exact (cat) groundstates read [46]

|Ψ〉 =1√2

(|ΨSB〉+ (−)ηPf |ΨSB〉)

=1√2

∑jsLs=1

Tr[ZηAj1 ...AjL ]|j1...jL〉,(89)

where η = 0, 1, |ΨSB〉 is generated by a normal tensor Bjwithout Pf symmetry and

Z = σz⊗1v, Aj = (σz)|j|⊗Bj , j = 1, 2, ..., 2d, (90)

with |j| being the parity of state |j〉, i.e., Pf |j〉 = (−)|j||j〉.Let us consider the spectral property of the associated quan-tum channel: E(ρ) ≡ ∑2d

j=1AjρA†j = λρ, where λ is

an eigenvalue and ρ can generally be expressed as ρ =∑w=0,x,y,z σ

w ⊗ ρw. Defining |w| ∈ Z2 from σwσz =

(−)|w|σzσw, we obtain

E(ρ) =∑

w=0,x,y,z

σw ⊗2d∑j=1

(−)|w||j|BjρwB†j

=∑

w=0,x,y,z

σw ⊗ λρw,(91)

implying that the spectrum of E is the union of those of E±defined by E±(·) =

∑2d

j=1(±)|j|Bj(·)B†j , with each eigen-value doubled. The doubling arises from the fact that, givenρ+ (ρ−) as an eigenstate of E+ (E−) with eigenvalue λ+ (λ−),σ0 ⊗ ρ+ and σz ⊗ ρ+ (σx ⊗ ρ+ and σy ⊗ ρ+) are two de-generate eigenstates of E with the same eigenvalue λ+ (λ−).Recalling that Bj is a normal tensor, we know that, after nor-malization, the spectral radius of E+ is one and that it onlyhas a single eigenvalue equal to one. Let µ− be the spectralradius of E−. Since Bj is not Pf -symmetric, we have µ− < 1

[104]. By gauge transforming Bj’s such that E+(1v) = 1v

and E+(Λ+) = Λ+, we have

E∞(·) ≡ liml→∞

E l(·) =1

2σ0 ⊗ 1v Tr[σ0 ⊗ Λ+(·)]

+1

2σz ⊗ 1v Tr[σz ⊗ Λ+(·)],

(92)

which satisfies ZE∞(Z · Z)Z = E∞(·). According to Fig. 9,the ES in the thermodynamic limit of the subsystem is givenby the spectrum of 1

4 (σ0⊗ σ0 + σz ⊗ σz)⊗Λ⊗2+ , which does

not depend on η in Eq. (89) and is at least two-fold degen-erate. If there are additional symmetries, then following theanalysis for symmetric normal MPSs we know that a nontriv-ial projective representation on the virtual level may enforcean r-fold degeneracy in the spectrum of Λ+, leading to a totalof 2r2-fold degeneracy in the ES.

When the parity symmetry-broken state is evolved by aparity symmetric MPU U , we find that the cat-state struc-ture (89) stays valid since PfU = UPf by assumption. Foran individual |ΨSB〉, we know that the degeneracy in thetime-evolved fixed point Λ+(t) determined from U t|ΨSB〉should stay unchanged. Since the convergence bound given inRef. [144] applies equally to the quantum channels with mul-tiple steady states, the Lieb-Robinson bound on the entangle-ment gap given in Theorem 2 is also applicable to fermionicSPT states with Majorana modes. Moreover, the coeffecientCin Eq. (27) can be tightened by a factor of 2 due to the normal-ization prefactor 1√

2in Eq. (89). Also, denoting the spectral

radius of E+ − E∞+ as µ+, we have µ = minµ+, µ−.

VI. DISCUSSIONS

While the Lieb-Robinson bound places a rigorous upperbound on the maximal splitting between degenerate ES val-ues, it is usually a highly nontrivial problem to determine thedegree of degeneracy, especially when the system undergoespartial symmetry breaking. In addition, the derivation of therigorous Lieb-Robinson bounds on entanglement gaps makesfull use of the translation invariance and the results are mean-ingful only up to t∗ ∼ l

v . It is natural to ask what happenswhen the translation invariance breaks down or/and in longertime scales. Here we give some heuristic arguments to addressthese issues.

A. Partial symmetry breaking

1. Free fermions

For free-fermion systems with Altland-Zirnbauer symme-tries, we have explained the effect of dynamical symmetrybreaking and the reduction of symmetry classes [33]. Theonly two classes whose reduced class is nontrivial are classesBDI and DIII, both of which reduce to class D. In starkcontrast, the former gives a surjective group homomorphismZ→ Z2, while the latter gives a trivial one Z2 → 0 (⊂ Z2). If

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16

… … … …

Spec

......

p

p

¯A A

l

(1)

Spec

......

(2)

...

...

t

A U

(3)

...

...

Oj

U U

(4)

......

U†O

jU

2k0+1

(5)

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t

ξ

N = 2<latexit sha1_base64="dThvEA/1SZgjSX3BAccwWT1pRF4=">AAAB6nicbZBNS8NAEIYnftb6VfXoZbEInkpSBL0IRS+epKL9gDaUzXbSLt1swu5GKKE/wYsHRbz6i7z5b9y2OWjrCwsP78ywM2+QCK6N6347K6tr6xubha3i9s7u3n7p4LCp41QxbLBYxKodUI2CS2wYbgS2E4U0CgS2gtHNtN56QqV5LB/NOEE/ogPJQ86osdbD3VW1Vyq7FXcmsgxeDmXIVe+Vvrr9mKURSsME1brjuYnxM6oMZwInxW6qMaFsRAfYsShphNrPZqtOyKl1+iSMlX3SkJn7eyKjkdbjKLCdETVDvVibmv/VOqkJL/2MyyQ1KNn8ozAVxMRkejfpc4XMiLEFyhS3uxI2pIoyY9Mp2hC8xZOXoVmteJbvz8u16zyOAhzDCZyBBxdQg1uoQwMYDOAZXuHNEc6L8+58zFtXnHzmCP7I+fwBmYGNVQ==</latexit><latexit sha1_base64="dThvEA/1SZgjSX3BAccwWT1pRF4=">AAAB6nicbZBNS8NAEIYnftb6VfXoZbEInkpSBL0IRS+epKL9gDaUzXbSLt1swu5GKKE/wYsHRbz6i7z5b9y2OWjrCwsP78ywM2+QCK6N6347K6tr6xubha3i9s7u3n7p4LCp41QxbLBYxKodUI2CS2wYbgS2E4U0CgS2gtHNtN56QqV5LB/NOEE/ogPJQ86osdbD3VW1Vyq7FXcmsgxeDmXIVe+Vvrr9mKURSsME1brjuYnxM6oMZwInxW6qMaFsRAfYsShphNrPZqtOyKl1+iSMlX3SkJn7eyKjkdbjKLCdETVDvVibmv/VOqkJL/2MyyQ1KNn8ozAVxMRkejfpc4XMiLEFyhS3uxI2pIoyY9Mp2hC8xZOXoVmteJbvz8u16zyOAhzDCZyBBxdQg1uoQwMYDOAZXuHNEc6L8+58zFtXnHzmCP7I+fwBmYGNVQ==</latexit><latexit sha1_base64="dThvEA/1SZgjSX3BAccwWT1pRF4=">AAAB6nicbZBNS8NAEIYnftb6VfXoZbEInkpSBL0IRS+epKL9gDaUzXbSLt1swu5GKKE/wYsHRbz6i7z5b9y2OWjrCwsP78ywM2+QCK6N6347K6tr6xubha3i9s7u3n7p4LCp41QxbLBYxKodUI2CS2wYbgS2E4U0CgS2gtHNtN56QqV5LB/NOEE/ogPJQ86osdbD3VW1Vyq7FXcmsgxeDmXIVe+Vvrr9mKURSsME1brjuYnxM6oMZwInxW6qMaFsRAfYsShphNrPZqtOyKl1+iSMlX3SkJn7eyKjkdbjKLCdETVDvVibmv/VOqkJL/2MyyQ1KNn8ozAVxMRkejfpc4XMiLEFyhS3uxI2pIoyY9Mp2hC8xZOXoVmteJbvz8u16zyOAhzDCZyBBxdQg1uoQwMYDOAZXuHNEc6L8+58zFtXnHzmCP7I+fwBmYGNVQ==</latexit><latexit sha1_base64="dThvEA/1SZgjSX3BAccwWT1pRF4=">AAAB6nicbZBNS8NAEIYnftb6VfXoZbEInkpSBL0IRS+epKL9gDaUzXbSLt1swu5GKKE/wYsHRbz6i7z5b9y2OWjrCwsP78ywM2+QCK6N6347K6tr6xubha3i9s7u3n7p4LCp41QxbLBYxKodUI2CS2wYbgS2E4U0CgS2gtHNtN56QqV5LB/NOEE/ogPJQ86osdbD3VW1Vyq7FXcmsgxeDmXIVe+Vvrr9mKURSsME1brjuYnxM6oMZwInxW6qMaFsRAfYsShphNrPZqtOyKl1+iSMlX3SkJn7eyKjkdbjKLCdETVDvVibmv/VOqkJL/2MyyQ1KNn8ozAVxMRkejfpc4XMiLEFyhS3uxI2pIoyY9Mp2hC8xZOXoVmteJbvz8u16zyOAhzDCZyBBxdQg1uoQwMYDOAZXuHNEc6L8+58zFtXnHzmCP7I+fwBmYGNVQ==</latexit> N = 3<latexit sha1_base64="0p6e0dzosU1NTEqtDsTbZoTAUYM=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSFfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qDVFxYe3plhZ94gkcKg6345haXlldW14nppY3Nre6e8u9c0caoZb7BYxrodUMOlULyBAiVvJ5rTKJC8FYyup/XWI9dGxOoBxwn3IzpQIhSMorXuby9Pe+WKW3VnIn/By6ECueq98me3H7M04gqZpMZ0PDdBP6MaBZN8UuqmhieUjeiAdywqGnHjZ7NVJ+TIOn0Sxto+hWTm/pzIaGTMOApsZ0RxaBZrU/O/WifF8MLPhEpS5IrNPwpTSTAm07tJX2jOUI4tUKaF3ZWwIdWUoU2nZEPwFk/+C82Tqmf57qxSu8rjKMIBHMIxeHAONbiBOjSAwQCe4AVeHek8O2/O+7y14OQz+/BLzsc3mwWNVg==</latexit><latexit sha1_base64="0p6e0dzosU1NTEqtDsTbZoTAUYM=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSFfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qDVFxYe3plhZ94gkcKg6345haXlldW14nppY3Nre6e8u9c0caoZb7BYxrodUMOlULyBAiVvJ5rTKJC8FYyup/XWI9dGxOoBxwn3IzpQIhSMorXuby9Pe+WKW3VnIn/By6ECueq98me3H7M04gqZpMZ0PDdBP6MaBZN8UuqmhieUjeiAdywqGnHjZ7NVJ+TIOn0Sxto+hWTm/pzIaGTMOApsZ0RxaBZrU/O/WifF8MLPhEpS5IrNPwpTSTAm07tJX2jOUI4tUKaF3ZWwIdWUoU2nZEPwFk/+C82Tqmf57qxSu8rjKMIBHMIxeHAONbiBOjSAwQCe4AVeHek8O2/O+7y14OQz+/BLzsc3mwWNVg==</latexit><latexit sha1_base64="0p6e0dzosU1NTEqtDsTbZoTAUYM=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSFfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qDVFxYe3plhZ94gkcKg6345haXlldW14nppY3Nre6e8u9c0caoZb7BYxrodUMOlULyBAiVvJ5rTKJC8FYyup/XWI9dGxOoBxwn3IzpQIhSMorXuby9Pe+WKW3VnIn/By6ECueq98me3H7M04gqZpMZ0PDdBP6MaBZN8UuqmhieUjeiAdywqGnHjZ7NVJ+TIOn0Sxto+hWTm/pzIaGTMOApsZ0RxaBZrU/O/WifF8MLPhEpS5IrNPwpTSTAm07tJX2jOUI4tUKaF3ZWwIdWUoU2nZEPwFk/+C82Tqmf57qxSu8rjKMIBHMIxeHAONbiBOjSAwQCe4AVeHek8O2/O+7y14OQz+/BLzsc3mwWNVg==</latexit><latexit sha1_base64="0p6e0dzosU1NTEqtDsTbZoTAUYM=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSFfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qDVFxYe3plhZ94gkcKg6345haXlldW14nppY3Nre6e8u9c0caoZb7BYxrodUMOlULyBAiVvJ5rTKJC8FYyup/XWI9dGxOoBxwn3IzpQIhSMorXuby9Pe+WKW3VnIn/By6ECueq98me3H7M04gqZpMZ0PDdBP6MaBZN8UuqmhieUjeiAdywqGnHjZ7NVJ+TIOn0Sxto+hWTm/pzIaGTMOApsZ0RxaBZrU/O/WifF8MLPhEpS5IrNPwpTSTAm07tJX2jOUI4tUKaF3ZWwIdWUoU2nZEPwFk/+C82Tqmf57qxSu8rjKMIBHMIxeHAONbiBOjSAwQCe4AVeHek8O2/O+7y14OQz+/BLzsc3mwWNVg==</latexit>

N = 4<latexit sha1_base64="njBqvjnA9oQyrF71MmXgTJVqVek=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttJu3SzCbsboYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIrg2rvvtFFZW19Y3ipulre2d3b3y/kFTx6li2GCxiFU7oBoFl9gw3AhsJwppFAhsBaObab31hErzWD6acYJ+RAeSh5xRY62Hu6vzXrniVt2ZyDJ4OVQgV71X/ur2Y5ZGKA0TVOuO5ybGz6gynAmclLqpxoSyER1gx6KkEWo/m606ISfW6ZMwVvZJQ2bu74mMRlqPo8B2RtQM9WJtav5X66QmvPQzLpPUoGTzj8JUEBOT6d2kzxUyI8YWKFPc7krYkCrKjE2nZEPwFk9ehuZZ1bN8f16pXedxFOEIjuEUPLiAGtxCHRrAYADP8ApvjnBenHfnY95acPKZQ/gj5/MHnImNVw==</latexit><latexit sha1_base64="njBqvjnA9oQyrF71MmXgTJVqVek=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttJu3SzCbsboYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIrg2rvvtFFZW19Y3ipulre2d3b3y/kFTx6li2GCxiFU7oBoFl9gw3AhsJwppFAhsBaObab31hErzWD6acYJ+RAeSh5xRY62Hu6vzXrniVt2ZyDJ4OVQgV71X/ur2Y5ZGKA0TVOuO5ybGz6gynAmclLqpxoSyER1gx6KkEWo/m606ISfW6ZMwVvZJQ2bu74mMRlqPo8B2RtQM9WJtav5X66QmvPQzLpPUoGTzj8JUEBOT6d2kzxUyI8YWKFPc7krYkCrKjE2nZEPwFk9ehuZZ1bN8f16pXedxFOEIjuEUPLiAGtxCHRrAYADP8ApvjnBenHfnY95acPKZQ/gj5/MHnImNVw==</latexit><latexit sha1_base64="njBqvjnA9oQyrF71MmXgTJVqVek=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttJu3SzCbsboYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIrg2rvvtFFZW19Y3ipulre2d3b3y/kFTx6li2GCxiFU7oBoFl9gw3AhsJwppFAhsBaObab31hErzWD6acYJ+RAeSh5xRY62Hu6vzXrniVt2ZyDJ4OVQgV71X/ur2Y5ZGKA0TVOuO5ybGz6gynAmclLqpxoSyER1gx6KkEWo/m606ISfW6ZMwVvZJQ2bu74mMRlqPo8B2RtQM9WJtav5X66QmvPQzLpPUoGTzj8JUEBOT6d2kzxUyI8YWKFPc7krYkCrKjE2nZEPwFk9ehuZZ1bN8f16pXedxFOEIjuEUPLiAGtxCHRrAYADP8ApvjnBenHfnY95acPKZQ/gj5/MHnImNVw==</latexit><latexit sha1_base64="njBqvjnA9oQyrF71MmXgTJVqVek=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSEfQiFL14kor2A9pQNttJu3SzCbsboYT+BC8eFPHqL/Lmv3Hb5qCtLyw8vDPDzrxBIrg2rvvtFFZW19Y3ipulre2d3b3y/kFTx6li2GCxiFU7oBoFl9gw3AhsJwppFAhsBaObab31hErzWD6acYJ+RAeSh5xRY62Hu6vzXrniVt2ZyDJ4OVQgV71X/ur2Y5ZGKA0TVOuO5ybGz6gynAmclLqpxoSyER1gx6KkEWo/m606ISfW6ZMwVvZJQ2bu74mMRlqPo8B2RtQM9WJtav5X66QmvPQzLpPUoGTzj8JUEBOT6d2kzxUyI8YWKFPc7krYkCrKjE2nZEPwFk9ehuZZ1bN8f16pXedxFOEIjuEUPLiAGtxCHRrAYADP8ApvjnBenHfnY95acPKZQ/gj5/MHnImNVw==</latexit> N = 5<latexit sha1_base64="uLHm+FN3vhaqt31afq1H+jU1c4E=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSUfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qDVFxYe3plhZ94gkcKg6345haXlldW14nppY3Nre6e8u9c0caoZb7BYxrodUMOlULyBAiVvJ5rTKJC8FYyup/XWI9dGxOoBxwn3IzpQIhSMorXuby/PeuWKW3VnIn/By6ECueq98me3H7M04gqZpMZ0PDdBP6MaBZN8UuqmhieUjeiAdywqGnHjZ7NVJ+TIOn0Sxto+hWTm/pzIaGTMOApsZ0RxaBZrU/O/WifF8MLPhEpS5IrNPwpTSTAm07tJX2jOUI4tUKaF3ZWwIdWUoU2nZEPwFk/+C82Tqmf57rRSu8rjKMIBHMIxeHAONbiBOjSAwQCe4AVeHek8O2/O+7y14OQz+/BLzsc3ng2NWA==</latexit><latexit sha1_base64="uLHm+FN3vhaqt31afq1H+jU1c4E=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSUfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qDVFxYe3plhZ94gkcKg6345haXlldW14nppY3Nre6e8u9c0caoZb7BYxrodUMOlULyBAiVvJ5rTKJC8FYyup/XWI9dGxOoBxwn3IzpQIhSMorXuby/PeuWKW3VnIn/By6ECueq98me3H7M04gqZpMZ0PDdBP6MaBZN8UuqmhieUjeiAdywqGnHjZ7NVJ+TIOn0Sxto+hWTm/pzIaGTMOApsZ0RxaBZrU/O/WifF8MLPhEpS5IrNPwpTSTAm07tJX2jOUI4tUKaF3ZWwIdWUoU2nZEPwFk/+C82Tqmf57rRSu8rjKMIBHMIxeHAONbiBOjSAwQCe4AVeHek8O2/O+7y14OQz+/BLzsc3ng2NWA==</latexit><latexit sha1_base64="uLHm+FN3vhaqt31afq1H+jU1c4E=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSUfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qDVFxYe3plhZ94gkcKg6345haXlldW14nppY3Nre6e8u9c0caoZb7BYxrodUMOlULyBAiVvJ5rTKJC8FYyup/XWI9dGxOoBxwn3IzpQIhSMorXuby/PeuWKW3VnIn/By6ECueq98me3H7M04gqZpMZ0PDdBP6MaBZN8UuqmhieUjeiAdywqGnHjZ7NVJ+TIOn0Sxto+hWTm/pzIaGTMOApsZ0RxaBZrU/O/WifF8MLPhEpS5IrNPwpTSTAm07tJX2jOUI4tUKaF3ZWwIdWUoU2nZEPwFk/+C82Tqmf57rRSu8rjKMIBHMIxeHAONbiBOjSAwQCe4AVeHek8O2/O+7y14OQz+/BLzsc3ng2NWA==</latexit><latexit sha1_base64="uLHm+FN3vhaqt31afq1H+jU1c4E=">AAAB6nicbZBNS8NAEIYn9avWr6pHL4tF8FQSUfQiFL14kor2A9pQNttNu3SzCbsToYT+BC8eFPHqL/Lmv3Hb5qDVFxYe3plhZ94gkcKg6345haXlldW14nppY3Nre6e8u9c0caoZb7BYxrodUMOlULyBAiVvJ5rTKJC8FYyup/XWI9dGxOoBxwn3IzpQIhSMorXuby/PeuWKW3VnIn/By6ECueq98me3H7M04gqZpMZ0PDdBP6MaBZN8UuqmhieUjeiAdywqGnHjZ7NVJ+TIOn0Sxto+hWTm/pzIaGTMOApsZ0RxaBZrU/O/WifF8MLPhEpS5IrNPwpTSTAm07tJX2jOUI4tUKaF3ZWwIdWUoU2nZEPwFk/+C82Tqmf57rRSu8rjKMIBHMIxeHAONbiBOjSAwQCe4AVeHek8O2/O+7y14OQz+/BLzsc3ng2NWA==</latexit>

(a)

(b)

FIG. 10. (a) Quench from N decoupled dimer chains described byH0 (93) to another flat-band Hamiltonian H (94) with inter-chaincouplings. Both H0 and H belong to class BDI. (b) Dynamics ofthe single-particle ES for N = 2, 3, 4, 5 demonstrating the nontriv-ial reduction Z → Z2. The red lines correspond to the topologicalentanglement modes at ξ = 1

2, which persist only for odd N . Here

we set J = 1 in all panels.

we look at the ES dynamics for a quenched class BDI systemwith winding number W ∈ Z+, up to the Lieb-Robinson timet∗, we expect two persistent ξ = 1

2 modes if W is odd, whileall the topological modes at ξ = 1

2 immediately multifurcateif W is even. For a nontrivial class DIII system, however, theentanglement gap should immediately open after a quench.

We give an analytically solvable flat-band model to demon-strate the nontrivial Z → Z2 reduction. For simplicity, wefocus on the ES under the open boundary condition, whosedouble gives the exact ES under the periodic boundary con-dition due to the zero correlation length imposed by the bandflatness. Consider N copies of dimer chains described by theHamiltonian

H0 = −J0

∑j

N∑α=1

(c†2j+1,αc2j,α + H.c.), (93)

where cxα denotes the fermion annihilation operator at the xthsite of the αth chain. Starting from the ground state of H0, wesuddenly quench the Hamiltonian into (see Fig. 10(a))

H = −J∑j

N−1∑α=1

(c†2j+1,αc2j,α+1 + H.c.). (94)

As shown in Fig. 10(b), the ES dynamics indeed demonstratesthe expected surjective group homomorphism Z → Z2. Thedetailed calculations are given in Appendix I 1, where weprove the existence/absence of ξ = 1

2 mode for odd/even N .We also note that the Z → Z2 reduction here should be dis-tinguished from that in Ref. [32] which concerns the spatial-temporal topology of quench dynamics starting from a topo-logically trivial state according to a topological Hamiltonian.

TABLE II. Two possible quench protocols that partially break thesymmetry into G = Z2 × Z2 and G = Z3 × Z3 starting from aSPT state protected by G = Z6 × Z6. Here ν and ν label the SPTphases protected by G and G, whose (minimal) degeneracies in theopen-boundary ES are denoted by r and r, respectively. In addition,s ≡ r

ris the number of splitting (s = 1 means no splitting), as

numerically verified in Figs. 16(b) and (c).

G = Z6 × Z6 G = Z2 × Z2 G = Z3 × Z3

ν ∈ Z6 r ν ∈ Z2 r s ν ∈ Z3 r s

0 1 0 1 1 0 1 1

1 6 1 2 3 2 3 2

2 3 0 1 3 1 3 1

3 2 1 2 1 0 1 2

4 3 0 1 3 2 3 1

5 6 1 2 3 1 3 2

2. Interacting systems

The Haldane phase [149], which is a prototypical SPTphase corresponding to the nontrivial element in H2(Z2 ×Z2,U(1)) = Z2, always becomes trivial upon any symmetrybreaking quench since H2(Z2,U(1)) = H2(0,U(1)) = 0.See Ref. [33] for a numerical demonstration for an immediateopening of the entanglement gap. Here, we consider a moregeneral situation where the initial SPT is protected by someunitary symmetries that form a group G, and the postquenchHamiltonian H only respects the symmetries in a subgroupG < G.

It is, in general, very difficult to determine the reduced G-symmetric SPT phase and the corresponding ES degeneracyfrom a given G-symmetric SPT phase. See Appendix I 2 foran exact mathematical formulation of this problem in terms ofcategory-theoretic languages. Here, let us consider a class ofminimal but yet nontrivial examples — G = ZN × ZN →G = Zn×Zn, where N = np with p ∈ Z+. Suppose that theinitial state correspond to ν ∈ ZN = H2(ZN × ZN ,U(1)).By choosing the coboundary gauge properly, the projectiverepresentation V on the virtual level satisfies

V(a,b)V(a′,b′) = ωνa′b

N V(a+a′,b+b′), (95)

where (a, b), (a′, b′) ∈ ZN × ZN . Regarding the elements inthe subgroup Zn × Zn, we find

Va,bVa′,b′ = Vap,bpVa′p,b′p

= ωνa′bp2

N V(a+a′)p,(b+b′)p

= ωpνa′b

n Va+a′,b+b′ ,

(96)

where we have used the fact that (a, b) ∈ Zn × Zn cor-responds to (ap, bp) ∈ ZN × ZN . This implies that thegroup homomorphism from ZN = H2(ZN × ZN ,U(1)) toZn = H2(Zn × Zn,U(1)) is given by

ν → ν = pν mod n. (97)

Page 17: Lieb-Robinson Bounds on Entanglement Gaps from Symmetry … · 2019-04-30 · Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology Zongping Gong,1 Naoto Kura,1

17

Note that such a group homomorphism can be trivial, as is thecase for N = 4 and n = 2 due to ν = 2ν mod 2 = 0.The minimal realizations of nontrivial group homomorphismsturn out to be N = 6 and n = 3, which gives rise toZ6 → Z3 : ν → ν = −ν mod 3, and N = 6 and n = 2,which gives rise to Z6 → Z2 : ν → ν = ν mod 2. Using thefact that the ES of a ZN × ZN -symmetric SPT state charac-terized by ν is (at least) N

GCD(ν,N) -fold degenerate under theopen boundary condition [111], where GCD is the greatestcommon divisor, we can figure out how many branches theoriginal ES should split into. The results are summarized inTable II and are numerically verified by some minimal models(see Appendix I 3).

B. Effects of disorder

As schematically illustrated in the right top panel in Fig. 2,disorder can dramatically alter the universal dynamical behav-ior of the entanglement growth. In this subsection, we quali-tatively address the impact of disorder on the ES dynamics in1D SPT systems.

Since the entanglement-gap opening is ultimately relatedto operator spreading and propagation of correlation, we ex-pect disorder in the postquench HamiltonianH to stabilize theSPT order during quench dynamics, provided that the disor-der does respect the symmetry. For free fermions in 1D, theAnderson localization occurs at an arbitrarily small disorderstrength and introduces a new length scale ξ [150], which isa typical localization length of the eigenfunctions of H andmeasures how far a wave packet can diffuse. As illustrated inFig. 11(a), for l ξ, we expect that ∆sp

E is saturated at e−O(l)

and the SPT order survives even in the limit of t → ∞ (seenumerical signatures in Appendix J 1). On the other hand, forvery weak disorder, l may be comparable to or even smallerthan ξ, and we can still observe the opening of the entangle-ment gap. This occurs even for a half-chain bipartition, unlessthe disorder configuration happens to respect the half-chaintranslation symmetry.

In the presence of interactions, we may expect qualitativelydifferent behavior because the entanglement-entropy growthis unbounded in the many-body localized phase [151], al-though it is logarithmical and hence extremely slow [152–156]. In this case, we conjecture that, after a possible tran-sient similar to the noninteracting case, ∆mb

E grows no fasterthan a power law, which is similar to the behavior of out-of-time-order correlators [157–159]. This is supported by thenumerics on a phenomenological model (see Appendix J 2).

C. Longer time scales

While our rigorous results ensure the persistence of (ap-proximate) ES degeneracy until a time scale that grows lin-early long with the subsystem size, these results are not usefulfor describing the dynamical behavior at longer time scales.In fact, previous numerical studies on some two-band free-fermion models have revealed the possibility for a single-

t

ES

t

ES(a) (b)

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FIG. 11. (a) Expected influence of disorder on the single-particle-ESdynamics starting from a topological state. When the localizationlength ξ is larger or comparable with the subsystem length l, we canstill observe the splitting (purple curve) just like the clean case (redcurve). On the other hand, if l is much larger than ξ, the ES splittingwill become invisible. (b) Power-law relaxation of the single-particleES for t t∗, where t∗ is the time scale set by the Lieb-Robinsonbound. This phenomenon is discovered in Refs. [81–83], where itis also conjectured that the ξ = 1

2modes will revive in the limit of

t→∞ if H∞ given in Eq. (98) is topologically nontrivial.

particle topological entanglement mode to return to ξ = 12

in the long-time limit [81–83]. In these studies, a conjectureis made to the effect that such behavior is determined by thetopology of the time-averaged Hamiltonian

H∞ ≡ limT→∞

1

T

∫ T

0

dte−iHtH0eiHt, (98)

at least for two-band systems in class D and class BDI. Itwould be interesting to examine whether this is indeed trueand, if yes, to what extent (e.g., multiple bands and other sym-metry classes). Moreover, it is found in Ref. [83] that the long-time relaxation dynamics obey some universal power law, i.e.,∆sp

E (t)−∆spE (∞) ∼ t−γ (γ > 0) for sufficiently large t (see

Fig. 11(b)). This behavior can quantitatively be understoodfrom steepest descent calculations, which require the latticesystem to be infinite. Remarkably, the power-law relaxation oflocal observables has been rigorously proved for general finitesingle-band systems in arbitrary dimensions starting from anystate with local correlations (not necessarily Gaussian) andwithin the revival time [160–162]. These rigorous results raiseanother interesting question concerning whether the signatureof free-fermion topology could emerge in quench dynamicsstarting from an interacting quantum many-body state.

As for the long-time behavior of interacting systems, wegenerally do not have an extensive number of conserved quan-tities and the steady state should resemble an excited stateof the posquench Hamiltonian H with energy 〈Ψ0|H|Ψ0〉,provided that the eigenstate thermalization hypothesis holdstrue [151, 163, 164]. Even if the ground state of H is in anSPT phase, the topological features generally disappear in theexcited states. Hence, the retrieval of ES degeneracy in thesteady state seems unlikely. The scenario should be very sim-ple if H is many-body localized — the topological informa-tion in the initial state, i.e., the degeneracy in the many-bodyES, is expected to persist in the long-time limit without a tran-sient loss or revival.

Page 18: Lieb-Robinson Bounds on Entanglement Gaps from Symmetry … · 2019-04-30 · Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology Zongping Gong,1 Naoto Kura,1

18

VII. SUMMARY AND OUTLOOK

The Lieb-Schultz-Mattis-Oshikawa-Hastings theorem[165–167], the Lieb-Robinson bound [53] and the entangle-ment area law [68] are of critical importance in quantummany-body systems. These rigorous results are of greatcurrent interest in light of theoretical insights from topolog-ical material science [168–170] and quantum information[54, 73–75], and of rapidly growing experimental rele-vance [55–57, 76–78]. In addition to these fundamentalachievements, we have established a general principle forthe dynamics of entanglement gaps in SPT systems that aredriven out of equilibrium by a local Hamiltonian or MPU.For free fermions, we have extensively used both the bandand the Wannier-function pictures to derive a Lieb-Robinsonbound on the single-particle entanglement gap (Theorem 1).As a byproduct, we have clarified the relation between theLieb-Robinson velocity and the group velocity of banddispersions. For interacting SPT systems, we have employedthe tensor-network approaches and the techniques of functionalgebra to derive the Lieb-Robinson bound on the many-bodyentanglement gap (Theorem 2). This result illustrates thatthe shortcoming of tensor-network approaches as numericaltools does not hinder its powerfulness as analytical tools fordealing with long-time quantum dynamics. We have alsoconsidered the impact of partial symmetry breaking, in whichcase the ES degeneracy may immediately be lifted partiallyor completely. Possible effects of disorder and relaxationbehaviors at longer time scales have also been discussed.

As future studies, it would be of fundamental importanceto go beyond the tensor-network formalism and prove a Lieb-Robinson bound on the many-body entanglement gap for con-tinuous quench dynamics starting from exact ground statesof local Hamiltonians. As discussed in Appendix F, whileimproving MPUs to continuously generated unitaries aloneseems plausible, it is far from clear how we can improvethe assumption of MPS initial states to exact ground states.It is also natural to consider the generalization to higher di-mensions [47], where we may have to use other indicators tomeasure the sharpness of SPT order. Moreover, some tech-niques used here may break down. For example, we cannotconstruct an exponentially localized Wannier function for atwo-dimensional Chern insulator [132]. Finally, we note thatconsiderable efforts have recently been made for generalizingthe notions of Lieb-Robinson bounds and topological phasesto lattice systems with long-range hoppings and interactions[171–174]. It would also be interesting to relax the localityassumption and consider long-range systems, which can nat-urally be implemented with trapped ions [6, 7, 56, 57], Ryd-berg atoms [5], polar molecules [175] and nitrogen-vacancycenters [4].

ACKNOWLEDGMENTS

We acknowledge Y. Ashida, M. A. Cazalilla, M.-C. Chung,I. Danshita, K. Fujimoto, R. Hamazaki, K. Kawabata, N. Mat-sumoto, and M. McGinley for valuable discussions. In partic-

ular, Z. G. appreciates Z. Wang for providing a simple pictureabout the monotonicity of Eq. (21). This work was supportedby KAKENHI Grant No. JP18H01145, No. JP17H02922and a Grant-in-Aid for Scientific Research on Innovative Ar-eas “Topological Materials Science (KAKENHI Grant No.JP15H05855) from the Japan Society for the Promotion ofScience. Z. G. was supported by MEXT. N. K. was supportedby the Leading Graduate Schools “ALPS. The authors thankthe Yukawa Institute for Theoretical Physics at Kyoto Uni-versity, where this work was initiated during the InternationalMolecule Program YITP-T-18-01 on “Floquet Theory: Fun-damentals and Applications”.

Appendix A: Band theory with a complex wave number

In this appendix, we discuss how to define 1D Bloch Hamil-tonians with complex wave numbers through analytic continu-ation. This is always possible for quasi-local hoppings, whichdecay exponentially with respect to the hopping range. We ar-gue that the analytically continued Hamiltonian should gener-ally be diagonalizable, even if the original one respects certainanti-unitary symmetries. In addition, we prove and explainwhy the Lieb-Robinson velocity in Eq. (21) is monotonic withrespect to the imaginary wave number and thus upper boundsthe maximal relative group velocity. Finally, we provide anexample of the SSH model.

1. Analytic continuation

A Bloch Hamiltonian H(k) can generally be expanded as

H(k) =∑n∈Z

eiknHn, (A1)

where each Fourier component can be obtained as

Hn =

∫ π

−π

dk

2πH(k)e−ikn = H†−n. (A2)

A sufficient condition for the rhs of Eq. (A1) to converge is∑n∈Z

‖Hn‖ <∞, (A3)

which is valid even ifH(k) is non-Hermitian and is obviouslysatisfied by any model with a finite hopping range R, i.e.,

Hn = H−n = 0, ∀n > R. (A4)

For such a class of models, the analytic continuation to thecomplex wave number

H(k + iκ) ≡∑n∈Z

eikn−κnHn (A5)

is well-defined for ∀κ ∈ R and satisfies H(k + iκ + 2π) =H(k + iκ) and H(k + iκ)† = H(k − iκ) for ∀k ∈ [−π, π].

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The analytic continuation can be applied to a wider class ofBloch Hamiltonians whose Fourier components satisfy

‖Hn‖ ≤ C0e−κ0|n|, ∀n ∈ Z, (A6)

where C0, κ0 ∈ R+ do not depend on n. One can easily checkthe validity of Eq. (A3). In this case, for ∀κ ∈ (−κ0, κ0), wehave

‖e−κnHn‖ ≤ C0e−(κ0−|κ|)|n|, ∀n ∈ Z, (A7)

so that H(k + iκ) in Eq. (A5) converges. Now let us showthat ifH(k) is gapped and satisfies Eq. (A6), then the flattenedHamiltonian or the Bloch projector (17) also satisfies Eq. (A6)but with C0 and κ0 modified. To see this, we note that the nthFourier component of the Bloch projector can be written as

Pn =

∫ π

−π

dk

2π

∮γ<

dz

2πi

e−ikn−κn

z −H(k − iκ)(A8)

by deforming the integral contour. With length of γ< denotedas lγ< ≡

∮γ<|dz|, such an integral (A8) can be bounded from

above by

‖Pn‖ ≤∫ π

−π

dk

2π

∮γ<

|dz|2π

∥∥∥∥ 1

z −H(k − iκ)

∥∥∥∥ e−κn≤ lγ<

2πmax

k∈[−π,π],z∈γ<

∥∥∥∥ 1

z −H(k − iκ)

∥∥∥∥ e−κn, (A9)

provided that det[z−H(k− iκ)] 6= 0 for ∀z ∈ γ<, which canalways be satisfied by sufficiently small κ due to a finite gap.This result ensures the existence of κ > 0 such that P<(k)can be analytically continued to k : |Re k| ≤ π, |Im k| ≤κ, implying that condition (i) in Theorem 1 can always besatisfied.

On the other hand, the analytic continuation cannot be ap-plied to long-range hopping, i.e., ‖Hn‖ ∼ O(n−γ) withγ ≥ 0 — while H(k) stays well-defined for γ > 1, Eq. (A5)diverges whenever κ 6= 0. Therefore, Theorem 1 cannot beapplied to long-range systems.

2. Diagonalizability

We argue that condition (ii) in Theorem 1 is also satisfied ingeneral — that is, there always exists a nonzero κ0 such thatH(k+ iκ) is diagonalizable for ∀|κ| < κ0. To show this, it issufficient to show thatH(k+iκ) becomes non-diagonalizableonly at a discrete set of complex wave numbers. This is ex-pected to be true if H(k + iκ) does not respect any sym-metries, since an obvious mechanism for being nondiagonal-izable is the emergence of a second-order exceptional point[176], which requires the fine-tuning of two parameters. Here,these two parameters are k and κ. However, if H(k + iκ) re-spects an anti-unitary symmetry or anti-symmetry S for givenk and κ, i.e.,

SH(k + iκ)S−1 = ±H(k + iκ), (A10)

we only need to fine tune a single parameter to create an ex-ceptional point. This type of Hamiltonians have been activelystudied in the context of non-Hermitian topological systems[177–180], where S is typically the parity-time symmetryand H(k + iκ) may not be analytic and thus κ is no morethan a control parameter. In the following, we will show thatH(k+ iκ), which is an analytic continuation of H(k) with ananti-unitary symmetry S, satisfies

SH(k + iκ)S−1 = ±H(k − iκ) (A11)

instead of Eq. (A10), so we still need to fine-tune both k andκ to make H(k + iκ) non-diagonalizable.

By imposing Eq. (A10) for κ = 0, we can use Eq. (A2) toobtain

SHnS−1 =

∫ π

−π

dk

2πSH(k)S−1eikn

= ±∫ π

−π

dk

2πH(k)eikn = ±H−n.

(A12)

Therefore, the action of S on H(k + iκ) gives

SH(k + iκ)S−1 =∑n∈Z

e−ikn−κnSHnS−1

= ±∑n∈Z

ei(k−iκ)(−n)H−n

= ±H(k − iκ).

(A13)

Similarly, we can show that SH(k)S−1 = ±H(−k), whichis the case of the particle-hole symmetry, gives SH(k +iκ)S−1 = ±H(−k + iκ) by analytic continuation and thusH(k + iκ) is expected to stay diagonalizable for sufficientlysmall κ.

Finally, we note that even if H(k + iκ) happens to be non-diagonalizable, Eq. (19) should still be valid with C replacedby a polynomial of t. This is becauseH(k+iκ) can always betransformed into the Jordan normal form and a nontrivial Jor-dan block with size s ≥ 2 contributes a polynomial prefactorwith degree s− 1 in e−iH(k+iκ)t.

3. Monotonicity of Eq. (21)

We prove that Eq. (21) is monotonic in terms of κ and thusreaches its minimum at κ = 0. To simplify the notation, wefirst introduce

∆αβ(k) ≡ εkα − εkβ , (A14)

which satisfies ∆αβ(k + iκ) = ∆αβ(k−iκ) for k, κ ∈ R. Forfurther simplification, we omit the subscript “αβ” and define

v(k, κ) ≡ Re∆′(k + iκ). (A15)

We then have v(k, κ) = v(k,−κ) and

V (k, κ) ≡ 1

2κ

∫ κ

−κdκ′v(k, κ′) =

Im∆(k + iκ)

κ. (A16)

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To show the monotonicity of maxk∈[−π,π] |V (k, κ)|, it is suf-ficient to show that V (k, κ) as a function of k is majorizedby V (k, κ′) for ∀κ′ > κ. By the statement that an integrablereal function f1 : I ≡ [a, b] → R majorizes f2 : I → R, wemean that there exists a kernel K(x;x′) : I × I → R+

⋃0satisfying∫

I

dxK(x;x′) =

∫I

dx′K(x;x′) = 1 (A17)

and

f2(x) =

∫I

dx′K(x;x′)f1(x′). (A18)

Such a definition is a straightforward generalization of themajorization for real vectors, which can also be regarded asfunctions with I being a discrete set. That is, a real vectora is majorized by b if there exists a doubly stochastic matrixMds, whose entries are all non-negative and the sum of eachrow/column equals to one, such that a = Mdsb [135]. Notethat K(x;x′) is a continuous version of Mds. If f1 majorizesf2, for ∀x ∈ I , we have

f2(x) ≤∫I

dx′K(x;x′) maxx∈I

f1(x) = maxx∈I

f1(x) (A19)

due to K(x;x′) ≥ 0, f1(x′) ≤ maxx∈I f1(x) for ∀x′ ∈ Iand Eq. (A17). Since Eq. (A19) is true for ∀x ∈ I , wehave maxx∈I f2(x) ≤ maxx∈I f1(x). Similarly, we haveminx∈I f2(x) ≤ minx∈I f1(x) and thus maxx∈I |f1(x)| ≥maxx∈I |f2(x)|.

Now let us consider the kernel that transforms V (k, κ1) intoV (k, κ2) with κ1 ≥ κ2. Since v(k, κ) is a harmonic functionwhich is periodic in k and even in κ, it can generally be ex-panded as

v(k, κ) =

∞∑n=1

[an cos(nk) + bn sin(nk)] cosh(nκ), (A20)

where an, bn ∈ R. Accordingly, we can obtain a general formof V (k, κ) in Eq. (A16):

V (k, κ) =

∞∑n=1

[An cos(nk)+Bn sin(nk)]sinh(nκ)

κ, (A21)

where we have redefined the coefficients as An = ann and

Bn = bnn . This general form (A21) implies the following

kernel that transforms V (k, κ1) into V (k, κ2):

K(k, κ2; k′, κ1) =1

2π+

1

π

∞∑n=1

κ1 sinh(nκ2)

κ2 sinh(nκ1)cos[n(k−k′)],

(A22)where the constant term is determined by Eq. (A17). To seewhy the kernel takes this form, we have only to note thatcos[n(k − k′)] = cos(nk) cos(nk′) + sin(nk) sin(nk′) and∫ π−π

dk′

π cos(nk′)V (k′, κ) (∫ π−π

dk′

π cos(nk′)V (k′, κ)) givesthe Fourier coefficient before cos(nk) (sin(nk)), whichshould be modified in an n-dependent manner following the

N=3

N=5

N=7

N=9

-3 -2 -1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

k-k'

Partialsums

N=1

N=3

N=5

Fourier

-3 -2 -1 0 1 2 3k-k'

(a) (b)

Re k

Im kR R

[(2m + 1)1 2]i

[(2m + 1)1 + 2]i

(1)

1

(c)

FIG. 12. Partial sums∑(N−1)/2

n=−(N−1)/2 of the kernel functionK(k, κ2; k′, κ1) with κ1 = 2 and κ2 = 0 expanded as in (a)Eq. (A23) and (b) Eq. (A26). The curve labeled by “Fourier” in (b) isnothing but the curve in (a) for N = 9. Note that the partial sum ofthe Fourier series may be not positive for certain k − k′ (see arrowsin (a)), but K(k, κ2; k′, κ1) is clearly positive from Eq. (A26). (c)Contour integral and poles of Y (k;κ2, κ1). In the limit of R →∞,as in the case in Eq. (A29), we should sum up all the residues withrespect to the poles on the lower-half imaginary axis.

change of κ. We can rewrite Eq. (A22) into a more compactform

K(k, κ2; k′, κ1) =1

2π

∑n∈Z

κ1 sinh(nκ2)

κ2 sinh(nκ1)ein(k−k′). (A23)

We can use Eq. (A23) to check that

K(k, κ; k′, κ) =1

2π

∑n∈Z

ein(k−k′) = δ(k − k′). (A24)

Another special case that transforms V (k, κ) into V (k, 0) =v(k, 0) reads

K(k, 0; k′, κ) =1

2π

∑n∈Z

nκ

sinh(nκ)ein(k−k′). (A25)

In general, whenever κ1 > κ2, the Fourier coefficients inEq. (A23) decays as e−(κ1−κ2)|n| for large |n|, implying theconvergence.

The remaining problem is to confirm whether Eq. (A23) isnon-negative for ∀κ1 > κ2 = 0. This is not clear at firstglance since the partial sum in Eq. (A23) generally containsnegative parts (see Fig. 12(a)), especially when κ2 is close toκ1. Nevertheless, the positivity becomes clear in a differentexpansion:

K(k, κ2; k′, κ1) =∑n∈Z

Y (k − k′ + 2nπ;κ2, κ1), (A26)

where

Y (k;κ2, κ1) =sin(κ2

κ1π)

2κ2[cosh( πκ1k) + cos(κ2

κ1π)]

(A27)

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21

fm j-1 , j- fm j-1 , j-1

fm j , j- fm j-1 , j-1

fm j , j- fm j , j-1

2 4 6 8

2.4

2.5

2.6

2.7

j

-20

-10

0

10

(a)

(b) (c)

(m j , j)

(m j-1, j-1)

(m j+1, j+1)

FIG. 13. (a) 2D array with the maximum of each row marked as(mj , j). The solid arrows denote the differences between the startingpoint to the end point, which are all positive. The dashed arrowshighlight the order relations (descent along the arrow) between thesedifferences. (b) A randomly generated array with L = 500 and (c)the differences in (a) for the jth rows with j = 1, 2, ..., 9.

is positive over R. In perticular, when κ2 = 0, Eq. (A27)becomes

Y (k; 0, κ) =π

2κ[cosh(πκk) + 1]. (A28)

To prove Eq. (A26), we have only to calculate the nth Fouriercoefficient of the rhs:∫ π

−π

dk

2π

∑m∈Z

Y (k + 2mπ;κ2, κ1)e−ink

=

∫ ∞−∞

dk

2πY (k;κ2, κ1)e−ink

=

∫ ∞−∞

dk

4πiκ2

[e−ink

e−π(k+iκ2)

κ1 + 1− e−ink

e−π(k−iκ2)

κ1 + 1

]

=κ1

2πκ2

∞∑m=0

(e−n[(2m+1)κ1−κ2] − e−n[(2m+1)κ1+κ2])

=κ1

2πκ2

sinh(nκ2)

sinh(nκ1),

(A29)

where we have used the residue theorem in deriving the fourthequality (see Fig. 12(c)). The final result in Eq. (A29) indeedcoincides with that in Eq. (A23).

In fact, a simple picture is available from a discrete versionof the problem. We consider a 2D array f : Z×Z→ R, whichsatisfies that for ∀j, j′ ∈ Z, fj,j′ = −fj,−j′ (so that fj,0 = 0),fj+L,j′ = fj,j′ (L ∈ Z+) and the discrete Laplace equation

4fj,j′ = fj+1,j′ + fj−1,j′ + fj,j′+1 + fj,j′−1. (A30)

By imposing the boundary condition

fj,1 = −fj,−1 = h∆′(

2πj

L

), (A31)

we have f(k, κ) ≡ Im∆(k + iκ) = limL→∞,h→0 fb kL2π c,bκh csince f(k, κ) is a harmonic function satisfying

f(k, κ) = f(k + 2π, κ) = −f(k,−κ),

∂κf(k, κ)|κ=0 = ∆′(k).(A32)

The discrete counterpart of the monotonicity ofκ−1 maxk∈[−π,π] f(k, κ) reads

1

j′ + 1maxj∈ZL

fj,j′+1 ≥1

j′maxj∈ZL

fj,j′ (A33)

for ∀j′ ≥ 1. In fact, we can prove a stronger result

maxj∈ZL

fj,j′+1−maxj∈ZL

fj,j′ ≥ maxj∈ZL

fj,j′−maxj∈ZL

fj,j′−1, (A34)

which implies Eq. (A33). Denotingmj′ as the horizontal labelwhere fmj′ ,j′ reaches the maximum for a given j′, we have

maxj∈ZL

fj,j′+1 − maxj∈ZL

fj,j′

≥fmj′ ,j′+1 − fmj′ ,j′=3fmj′ ,j′ − fmj′−1,j′ − fmj′+1,j′ − fmj′ ,j′−1

≥fmj′ ,j′ − fmj′ ,j′−1

≥maxj∈ZL

fj,j′ − maxj∈ZL

fj,j′−1,

(A35)

which completes the proof. We provide a schematic illustra-tion and a numerical verification in Fig. 13. Similarly, wecan prove the monotonicity of 1

j′ minj∈ZL fj,j′ and thus themonotonicity of 1

j′ maxj∈ZL |fj,j′ |.The above idea can also be implemented directly in a con-

tinuous manner, provided that k∗, which makes f(k∗, κ) max-imal or minimal for a given κ, forms a smooth curve of κ. Letus focus on the case of maximum since the minimum counter-part is quite similar. By definition, along this curve we have

∂2kfdk + ∂k∂κfdκ = 0, ∂2

kf = −∂2κf ≤ 0, (A36)

so the second-order differential along this curve satisfies

d2f = ∂2kfdk

2 + ∂2κfdκ

2 + 2∂k∂κfdkdκ

= ∂2κfdκ

2 − ∂2kfdk

2 + 2(∂2kfdk + ∂k∂κfdκ)dk

= ∂2κf

[1 +

(dk

dκ

)2]∣∣∣∣∣k=k∗

dκ2.

(A37)

Introducing F (κ) ≡ f(k∗(κ), κ), we find from Eqs. (A36)and (A37) that

F ′′(κ) ≥ 0. (A38)

This result is sufficient to show

κ−11 F (κ1) ≥ κ−1

2 F (κ2), ∀κ1 > κ2 ≥ 0. (A39)

To see this, we consider the equivalent inequality

F (κ1)− F (κ2)

κ1 − κ2≥ F (κ2)− F (0)

κ2. (A40)

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According to the mean-value theorem, there exists ξ1 ∈[κ2, κ1] and ξ2 ∈ [0, κ2] such that F ′(ξ1) = F (κ1)−F (κ2)

κ1−κ2

and F ′(ξ2) = F (κ2)−F (0)κ2

, so the above inequality becomesF ′(ξ2) ≥ F ′(ξ1), which is ensured by Eq. (A38).

4. Example: SSH model

We consider a general quench in the SSH model:

H0(k) = −(J1 + J2 cos k)σx − J2 sin kσy

→ H(k) = −(J ′1 + J ′2 cos k)σx − J ′2 sin kσy,(A41)

where σx and σy are Pauli matrices. To evaluate vLR, we haveonly to numerically maximize

|Im εk+iκ,±| =√

1

2

[√(J ′21 + J ′22 + 2J ′1J

′2 cos k coshκ)2 + (2J ′1J

′2 sin k sinhκ)2 − (J ′21 + J ′22 + 2J ′1J

′2 cos k coshκ)

](A42)

over k ∈ [−π, π]. Note that κ can be chosen freely, as longas |κ| < | ln J1

J2| (otherwise P<(k + iκ) may become non-

analytic). We plot vLR = κ−1 maxk∈[−π,π] |Im εk+iκ,±|in Fig. 7(b) for the quench protocol used in the main text:J1 = J ′2 = 0.5 and J2 = J ′1 = 1. The κ dependence of vLR

turns out to be rather weak, at least for κ ∈ [0, ln 2).To evaluate C, we first write down the initial Bloch projec-

tor

P<(k) =1

2

1 J1+J2e−ik√

J21+J2

2+2J1J2 cos k

J1+J2eik√

J21+J2

2+2J1J2 cos k1

.(A43)

By introducing

F (k, κ, J1, J2) =J2

1 + J22 e

2κ + 2J1J2eκ cos k√

(J21 + J2

2 )2 + 4J1J2(J21 + J2

2 ) coshκ cos k + 2J21J

22 (cos 2k + cosh 2κ)

, (A44)

we can express the norm of Eq. (A43) with respect to a com-plex wave number as

‖P<(k + iκ)‖ =1

2[F (k, κ, J1, J2) + F (k,−κ, J1, J2)].

(A45)Moreover, the left and right eigenvectors are found to be

〈k = 0,a|uRk+iκ,±〉 =

1√2

[1

∓√

J′1+J′2eik−κ

J′1+J′2e−ik+κ

],

〈k = 0,a|uLk+iκ,±〉 =

1√2

[1

∓√

J′1+J′2eik+κ

J′1+J′2e−ik−κ

],

(A46)

where a labels the sublattice degrees of freedom, leading to

‖|uRk+iκ,±〉‖ =

√1

2[1 + F (k,−κ, J ′1, J ′2)],

‖|uLk+iκ,±〉‖ =

√1

2[1 + F (k, κ, J ′1, J

′2)].

(A47)

Substituting Eqs. (A45) and (A47) into Eq. (20) yields

C =

∫ π

−π

dk

4π[1 + F (k,−κ, J ′1, J ′2)][1 + F (k, κ, J ′1, J

′2)]

× [F (k, κ, J1, J2) + F (k,−κ, J1, J2)].

(A48)

In Fig. 6(c), we choose κ = 0.6, leading to C ' 12.225.

Appendix B: Exact zero modes

In Sec. IV A, we have argued that the emergent Kramersdegeneracy and the nontrivial Z2 topology necessarily supportzero modes in Rd in Eq. (28). Here, we prove this statementby showing that the invertibility of Rd implies a trivial Z2

topological index.

Theorem 4 Given two skew-symmetric real matrices Rd andRo such that R ≡ σ0 ⊗Rd + σx ⊗Ro is unitary and Pf R =−1, we must have detRd = 0.

Proof.— If detRd 6= 0, we can apply the formula

Pf R = Pf Rd Pf(Rd −RoR−1d Ro). (B1)

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This formula can be derived from the identity

Pf(BABT) = detB Pf A, (B2)

which is valid for arbitrary AT = −A and B, and(Rd 0

0 Rd −RoR−1d Ro

)

=

(1 0

−RoR−1d 1

)(Rd Ro

Ro Rd

)(1 −R−1

d Ro

0 1

).

(B3)

Since R is unitary, we have Rd, Ro = 0 and R2d + R2

o =−1, leading to

Rd −RoR−1d Ro = Rd +R−1

d R2o

= Rd +R−1d (−1−R2

d) = −R−1d .

(B4)

Combining Eq. (B4) with Eq. (B1), we have

Pf R = Pf Rd Pf(−R−1d ) =

(Pf Rd)2

detRd= 1, (B5)

where we have used Eq. (B2) with A = B = R−1d and

detA = (Pf A)2 for ∀AT = −A. This result (B5) contra-dicts the assumption Pf R = −1, implying detRd = 0.

Appendix C: Proof of Weyl’s perturbation theorem

In this appendix, we follow Ref. [84] to prove Weyl’s per-turbation theorem. To this end, we first introduce the min-maxprinciple.

Lemma 8 (Min-max principle) For any Hermitian operatorO on an n-dimensional Hilbert space, its jth largest eigen-value λj (j = 1, 2, ..., n) is given by

λj = maxdimV=j

min|ψ〉∈V

〈ψ|O|ψ〉

= mindimW=n+1−j

max|ψ〉∈W

〈ψ|O|ψ〉, (C1)

where V and W are Hilbert subspaces and |ψ〉 is a normal-ized state vector.

Proof.— We denote the eigenstate with eigenvalue λj as |ψj〉(j = 1, 2, ..., n) and define the following two special classesof Hilbert spaces with dimensions j and n+1−j, respectively:

Vj ≡ span|ψ1〉, |ψ2〉, ..., |ψj〉,Wj ≡ span|ψj〉, |ψj+1〉, ..., |ψn〉.

(C2)

With V chosen to be Vj , the rhs of the first line in Eq. (C1)gives λj , implying

λj ≤ maxdimV=j

min|ψ〉∈V

〈ψ|O|ψ〉. (C3)

Similarly, with a choice of W = Wj , the second line inEq. (C1) gives λj , implying

λj ≥ mindimW=n+1−j

max|ψ〉∈W

〈ψ|O|ψ〉. (C4)

On the other hand, for an arbitrary Hilbert subspace V withdimension j, we must have dim(V

⋂Wj) ≥ 1. Otherwise, if

V⋂Wj = ∅, the full Hilbert space containing V

⋃Wj will

be at least n + 1 dimensional, contradicting the assumption.This implies that for ∀V with dimV = j, we have

min|ψ〉∈V

〈ψ|O|ψ〉 ≤ min|ψ〉∈V ⋂

Wj

〈ψ|O|ψ〉

≤ max|ψ〉∈Wj

〈ψ|O|ψ〉 = λj ,(C5)

leading to

λj ≥ maxdimV=j

min|ψ〉∈V

〈ψ|O|ψ〉. (C6)

Combining Eqs. (C3) and (C6), we obtain the first line inEq. (C1). By analogy, from dim(Vj

⋂W ) ≥ 1 for ∀W with

dimW = n− j + 1, we can derive

λj ≤ mindimW=n+1−j

max|ψ〉∈W

〈ψ|O|ψ〉. (C7)

Combining Eqs. (C4) and (C7), we obtain the second line inEq. (C1).

Now let us turn to the proof of Theorem 3. For ∀V withdimV = j, we can decompose O as O′ + (O−O′) to obtain

min|ψ〉∈V

〈ψ|O|ψ〉

= min|ψ〉∈V

(〈ψ|O′|ψ〉+ 〈ψ|O −O′|ψ〉)

≥ min|ψ〉∈V

〈ψ|O′|ψ〉+ min|ψ〉∈V

〈ψ|O −O′|ψ〉

≥ min|ψ〉∈V

〈ψ|O′|ψ〉 − ‖O −O′‖.

(C8)

After maximizing the rhs of Eq. (C8) with respect to V andusing the first line in Eq. (C1), we obtain

λj ≥ λ′j − ‖O −O′‖. (C9)

Following a similar procedure, we can derive

max|ψ〉∈W

〈ψ|O|ψ〉 ≤ max|ψ〉∈W

〈ψ|O′|ψ〉+ ‖O −O′‖, (C10)

which gives rise to (due to the second line in Eq. (C1))

λj ≤ λ′j + ‖O −O′‖. (C11)

Theorem 3 follows from the combination of Eqs. (C9) and(C11).

Appendix D: Proof of Lemma 2

We first show the following lemma which provides the re-lation between |W (L)

jα 〉 and |W (∞)jα 〉.

Lemma 9 Given |W (∞)jα 〉 as a Wannier function of an infinite

1D lattice system, the Wannier function |W (L)jα 〉 of the corre-

sponding finite system with length L reads

|W (L)jα 〉 =

∑n∈Z

PZL |W (∞)j+nL,α〉, (D1)

Page 24: Lieb-Robinson Bounds on Entanglement Gaps from Symmetry … · 2019-04-30 · Lieb-Robinson Bounds on Entanglement Gaps from Symmetry-Protected Topology Zongping Gong,1 Naoto Kura,1

24

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W(1)0

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(a) (b)

W(1)0

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FIG. 14. (a) Wannier functionW (∞)0 centered at the origin of an infi-

nite lattice. (b) Infinite-system Wannier functions W (∞)0 and W (∞)

±Lprojected onto a finite lattice ZL under the periodic boundary con-dition. The finite-system Wannier function W (L)

0 (dashed curve) isrelated to the infinite-system ones by Eq. (D1).

where PZL ≡∑j∈ZL

∑da=1 |ja〉〈ja| is the projector onto

the finite lattice, with d and |j, a〉 being the number of inter-nal states per site and the state localized at the jth site in aninternal state a.

Proof.— Let us first write down the Wannier function definedon an infinite lattice:

|W (∞)jα 〉 =

∫B.Z.

dk

2πeik(x−j)|ukα〉, (D2)

where |ukα〉 =∑da=1 uαa(k)|k = 0, a〉 is the Bloch

wave function with |k = 0, a〉 ≡ ∑j∈Z |ja〉, and x ≡∑

j∈Z j|j〉〈j| ⊗ 1I is the position operator with 1I being theidentity acting on the internal-state Hilbert space. When thelattice is finite, say with length L, the Wannier function is de-fined in the form of a discrete Fourier transformation:

|W (L)jα 〉 =

1

L

∑k∈ 2π

L ZL

eik(xL−j)|ukα〉, (D3)

where |ukα〉 follows the previous definition for an infinitelattice except for |k = 0, a〉 ≡ ∑

j∈ZL|ja〉 and eikxL ≡∑

j∈ZLeikj |j〉〈j| ⊗ 1I.

For ∀j′ ∈ ZL and a = 1, 2, .., d, the coefficient of |j′a〉 onthe rhs in Eq. (D1) is given by

∑n∈Z

〈j′a|W (∞)j+nL,α〉 =

∑n∈Z

∫ 2π

0

dk

2πeik(j′−j−nL)uαa(k).

(D4)

Using the identity∑n∈Z

δ(x− 2nπ) =1

2π

∑n∈Z

einx, (D5)

we can further simplify the rhs of Eq. (D4) as

∑n∈Z

∫ 2π

0

dk

2πeik(j′−j)δ(kL− 2nπ)uαa(k) = 〈j′a|W (L)

jα 〉,

(D6)which completes the proof of Lemma 9.

A schematic illustration of Lemma 9 is shown in Fig. 14.Just like |W (∞)

jα 〉’s, we note that a finite-size Wannier function

in Eq. (D1) satisfies the orthonormal relation:

〈W (L)j′α′ |W

(L)jα 〉

=∑

n,n′∈Z

∑j′′∈ZL,a

〈W (∞)j′+n′L,α′ |j′′a〉〈j′′a|W

(∞)j+nL,α〉

=∑

n,n′∈Z

∑j′′∈ZL,a

〈W (∞)j′α′ |j′′ − n′L, a〉

× 〈j′′ − n′L, a|W (∞)j+(n−n′)L,α〉

=∑m∈Z

〈W (∞)j′α′ |W

(∞)j+mL,α〉 = δj′jδα′α,

(D7)

where we have used 〈j′|W (∞)jα 〉 = 〈j′ + p|W (∞)

j+p,α〉 for ∀p ∈Z, which is a property also inherited by |W (L)

jα 〉 with j′, j ∈ZL.

Now we turn to prove Lemma 2. We first express〈j|P (L)

< |j′〉, which is an operator acting on the internal-stateHilbert space, in terms of Wannier function projectors:

〈j|P (L)< |j′〉 =

∑j′′∈ZL,α∈O

〈j|W (L)j′′α〉〈W

(L)j′′α|j′〉. (D8)

According to Eq. (D1), the above expression can be rewrittenas

〈j|P (L)< |j′〉 =

∑j′′∈ZL,α∈O

∑n,n′∈Z

〈j|W (∞)j′′+nL,α〉〈W

(∞)j′′+n′L,α|j′〉.

(D9)On the other hand, we have

〈j|P (∞)< |j′〉 =

∑j′′∈ZL,α∈O

∑n∈Z

〈j|W (∞)j′′+nL,α〉〈W

(∞)j′′+nL,α|j′〉.

(D10)Combining Eqs. (D9) and (D10), we obtain

〈j|P (L)< |j′〉 − 〈j|P (∞)

< |j′〉=

∑j′′∈ZL,α∈O

∑n 6=n′∈Z

〈j|W (∞)j′′+nL,α〉〈W

(∞)j′′+n′L,α|j′〉

=∑

j′′∈ZL,α∈O

∑n∈Z

∑m∈Z\0

〈j|W (∞)j′′+nL,α〉〈W

(∞)j′′+nL,α|j′ +mL〉

=∑

m∈Z\0〈j|P (∞)

< |j′ +mL〉,

(D11)

where we again use 〈j′|W (∞)jα 〉 = 〈j′+p|W (∞)

j+p,α〉 for ∀p ∈ Z.

Appendix E: Basic properties of matrix-product unitaries

We briefly review Ref. [110] and introduce the several basicproperties of MPUs, which are crucial for proving the mainresult. By definition, for ∀L ∈ Z+, U in Eq. (22) in the maintext obeys

U†U = I ≡ 1⊗L ⇒ d−L Tr[U†U ] = 1, (E1)

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25

implying that the spectrum of the quantum channel EU (·) =

d−1∑dj,j′=1 Ujj′ ·U

†jj′ consists of a single 1 and all the others

being zero. This property enforces d−12Ujj′ to be a normal

tensor [103], in the sense that EU has a unique fixed point,which is Hermitian and positivie-definite. By choosing thegauge properly, i.e., performing a similarity transformationUjj′ → X−1Ujj′X on the virtual level, Ujj′ can always bemade to satisfy

1d

U

Uρ = ρ ,

1d

U

U= ,

(E2)

where ρ† = ρ, ρ > 0 and Tr ρ = 1. It can be proved [110]that for ∀k ∈ Z+

Uk

Ukρ =

1⊗k

,

(E3)

where Uk is related to U by putting together k sites into one,as given in Eq. (23).

An MPU is locality-preserving, in the sense that a local op-erator acting nontrivially on a finite segment stays local afterbeing evolved by an MPU. This property can be understoodfrom the following theorem [110].

Theorem 5 It is always possible to make an MPU generatedby U simple by putting together k ≤ D4

U (DU : bond dimen-sion of the MPU) sites into one (see Eq. (23)). By simple, wemean that the building block U satisfies

U

U

U

U=

U

U

U

Uρ ,

(E4)

where ρ is given in Eq. (E2).

Denoting k0 as the smallest integer such that Uk0 is simple, theabove theorem (5) implies that a local operator spreads by nomore than 2k0 sites upon being evolved by the MPU. To seethis, consider a general local operator OL acting nontriviallyon a segment with length l0. Applying Eqs. (E3) and (E4) toO′L = UOLU

† yields

...

...

... ...

U U U U U U U

U U U U U U U

OL

l0

=

Uk0 Ul0 Uk0

Uk0 Ul0 Uk0

OL ρ

(E5)

=

...

...O′L

l0 + 2k0

,

where local identities are omitted. Note that Fig. 2(c) in themain text is the special case of l0 = 1, which is neverthe-less sufficient for obtaining Eq. (E5) since a local operatorcan generally be expressed asOL =

∑jsl0s=1

cj1j2...jl0Oj1⊗Oj2 ⊗ ...⊗Ojl0 with Ojs being an on-site operator [111].

Besides the locality-preserving property, an MPU is by def-inition unitary, implying the following lemma.

Lemma 10 Given an MPS |Ψ〉 generated by Ajdj=1 and anMPU U generated by Ujj′dj,j′=1, with the associated unitalchannel of the evolved MPS |Ψ′〉 = U |Ψ〉 denoted as E ′(·) ≡∑dj=1A

′j ·A′†j , the spectrum of E ′ is the same as that of E(·) ≡∑d

j=1Aj ·A†j , the unital channel associated with |Ψ〉.

Proof.— The main idea is already mentioned in Ref. [111]— we only have to combine Lemma 9 in Ref. [181] withTr[(

∑j A′j ⊗ A′j)L] = Tr[(

∑j Aj ⊗ Aj)L], ∀L ∈ Z+, which

results from the unitary nature of the time evolution:

...A′

A′

A′

A′

A′

A′

L

= ...

A

U

U

A

A

U

U

A

A

U

U

A

L

= ...A

A

A

A

A

A

L

.

(E6)

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26

A direct corollary of Lemma 10 is that the spectrum of E , i.e.,the transfer matrix of an MPS, stays invariant during the stro-boscopic time evolution governed by an MPU. In particular, µas the spectral radius of E − E∞ is conserved.

Appendix F: Interacting systems undergoing continuousevolution

We argue that the Lieb-Robinson bound on the many-bodyentanglement gap for MPUs implies qualitatively the same re-sult for continuous time evolutions generated by local Hamil-tonians. To this end, we first prove that the difference in time-evolution operators, which can be highly nonlocal, rigorouslybounds the difference in any reduced density operator after thetime evolution.

Lemma 11 Given an arbitrary wave function |Ψ0〉 defined ona bipartite system S

⋃S and two unitaries U and U ′ sat-

isfying ‖U − U ′‖ ≤ ε, denoting ρS ≡ TrS |Ψ〉〈Ψ| andρ′S ≡ TrS |Ψ′〉〈Ψ′| as the density operators of the evolvedwave functions |Ψ〉 ≡ U |Ψ0〉 and |Ψ′〉 ≡ U ′|Ψ0〉, we have‖ρS − ρ′S‖ ≤ ε.Proof.— We first point out a useful norm inequality for thecommutator of a positive-semidefinite Hermitian operator A(A† = A and A ≥ 0) and an arbitrary operator B [182]:

‖[A,B]‖ ≤ ‖A‖‖B‖. (F1)

Note that there is an improvement of factor 12 compared with

‖[A,B]‖ ≤ 2‖A‖‖B‖, which holds for arbitrary A and B.Let us move on to prove Lemma 11. Noting that ρS − ρ′S

is Hermitian, according to the definition of the operator norm,we have

‖ρS − ρ′S‖ = max‖|ψ〉‖=1

〈ψ|ρS − ρ′S |ψ〉

= maxPψ

Tr[Pψ ⊗ 1S(|Ψ〉〈Ψ| − |Ψ′〉〈Ψ′|)]

= maxPψ〈Ψ0|U(Pψ ⊗ 1S)U† − U ′(Pψ ⊗ 1S)U ′†|Ψ0〉

≤maxPψ‖U(Pψ ⊗ 1S)U† − U ′(Pψ ⊗ 1S)U ′†‖

≤maxPψ‖[Pψ ⊗ 1S , U

†U ′]‖,

(F2)

where Pψ ≡ |ψ〉〈ψ| is a rank-one projector, i.e., P 2ψ = Pψ

and TrPψ = 1. Since Pψ ⊗ 1S ≥ 0 and ‖Pψ ⊗ 1S‖ = 1, wefind from Eq. (F1) that for ∀Pψ

‖[Pψ ⊗ 1S , U†U ′]‖

=‖[Pψ ⊗ 1S , U†U ′ − 1]‖

≤‖U†U ′ − 1‖ = ‖U − U ′‖.(F3)

Combining Eq. (F3) with Eq. (F2), we obtain

‖ρS − ρ′S‖ ≤ ‖U − U ′‖ ≤ ε. (F4)

Now let us discuss how to combine Lemma 11 with Theo-rem 2 and the main result of Ref. [115] to bound the many-body entanglement gap in a continuous quench dynamics

eiHt

(1)

u

v

(2)

(3)

Uv

u=

(4)

U

U=

u†

u(5)

Spec...

...p

p

A

A

l (6)

Spec...

...(7)

... ...

t

A

U

(8)

... ...Oj

U

U

(9)

1

eiHt

(1)

u

v

(2)

(3)

Uv

u=

(4)

U

U=

u†

u(5)

Spec...

...p

p

A

A

l (6)

Spec...

...(7)

... ...

t

A

U

(8)

... ...Oj

U

U

(9)

1

eiHt

(1)

u

v

(2)

(3)

Uv

u=

(4)

U

U=

u†

u(5)

Spec...

...p

p

A

A

l (6)

Spec...

...(7)

... ...

t

A

U

(8)

... ...Oj

U

U

(9)

1

eiHt

(1)

u

v

(2)

(3)

Uv

u=

(4)

U

U=

u†

u(5)

Spec...

...p

p

A

A

l (6)

Spec...

...(7)

... ...

t

A

U

(8)

... ...Oj

U

U

(9)

1

eiHt

(1)

u

v

(2)

(3)

Uv

u=

(4)

U

U=

u†

u(5)

Spec...

...p

p

A

A

l (6)

Spec...

...(7)

... ...

t

A

U

(8)

... ...Oj

U

U

(9)

1

(a)

(b)

(c)

(d) (e)

FIG. 15. (a) Continuous time evolution e−iHt generated by a localHamiltonian H and its (b) partial and (c) complete approximationsby local unitaries, which are denoted as U ′c and Uc. If we focuson a subsystem marked in the red rectangle, (b) and (c) make nodifference — starting from the same initial state, the reduced den-sity operators of the states evolved by (b) and (c) coincide with eachother. (d) Building block of (c), which can be regarded as an MPU.(e) Quantum channel associated with (d) takes the form ρTr[...].

starting from an SPT MPS. According to Ref. [115], given alocal Hamiltonian H =

∑j hj , we can approximate it with a

bilayer unitary circuit or quantum cellular automaton Uc suchthat

‖e−iHt − Uc‖ ≤ O(L

|Ω|

)e−κc(|Ω|−vct), (F5)

where L is the total system size, |Ω| is the number of sites thata single unitary acts on, and κc and vc are constants indepen-dent of L. While the bound diverges in the thermodynamiclimit, we can make a cut off of the circuit approximation atthe length scale of the subsystem without changing the re-duced density operator. For such a truncated circuit U ′c, wehave

‖e−iHt − U ′c‖ ≤ O(

l

|Ω|

)e−κc(|Ω|−vct), (F6)

where the rhs is finite in the thermodynamic limit. SeeFigs. (15)(a), (b) and (c) for the relations and distinctions ine−iHt, U ′c and Uc.

On the other hand, to apply Theorem 2, we should still re-gard the approximated reduced density operator as resultingfrom the time evolution byUc. After putting |Ω| sites into one,we can regard Uc as an MPU generated by the block given inFig. 15(d), which is simple (see Fig. 15(e)). Under such arescaling, the parameters in Theorem 2 reads k0 = t = 1,DU = d|Ω| and

µ→ µ|Ω|, l→ l

|Ω| , (F7)

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27

while D stays invariant. Therefore, the many-body entan-glement gap of the approximated density operator ρ′[1,l] isbounded by

∆′mbE ≤ poly

(l

|Ω|

)e−κl+κ

′|Ω|, (F8)

where κ = − lnµ and

κ′ = ln d+ (D2 + 1)κ, (F9)

provided that

l

|Ω| − 2 ≥ coth(κ

2|Ω|). (F10)

As for the many-body entanglement gap of the exact den-sity operator ρ[1,l], we have

∆mbE ≡ |ζr2 − ζ1|≤ |ζr2 − ζ ′r2 |+ |ζr2 − ζ ′1|+ |ζ ′r2 − ζ ′1|≤ 2‖ρ[1,l] − ρ′[1,l]‖+ ∆′mb

E

≤ 2‖e−iHt − U ′c‖+ ∆′mbE ,

(F11)

where we have used Lemma 11 in deriving the last inequality.Combining Eqs. (F6) and (F8) with Eq. (F11) and taking (weuse ' since |Ω| should be an even integer dividing l)

|Ω| ' κcvct+ κ′lκc + κ′

, (F12)

we obtain

∆mbE ≤ O(1)e

− κcκ

κc+κ′(l−κ′κ vct), (F13)

which indeed takes the form of Lieb-Robinson bound. Wenote that, at large length scales such that coth(κ2 |Ω|) = 1 +o(1), Eq. (F10) can be satisfied by

t <κc + κ′

κc

(1

3− κ

κc + κ′− o(1)

)l

vc, (F14)

where the rhs is positive due to κ′ > 5κ, which arises fromEq. (F9) and D ≥ 2 for an arbitrary SPT MPS.

On the other hand, it seems that we cannot derive a Lieb-Robinson bound by simply combining Theorem 2 with theerrors in approximating ground states with MPSs [141]. Evenif we only require the MPS approximation to be locally good[183, 184], the needed bond dimension scales like a polyno-mial of the inverse of error, implying an exponentially largebond dimension and a doubly exponentially large (due to theprefactor) bound predicted by Theorem 2 for an exponentiallysmall error. We leave this problem for future work, whichprobably requires new ideas to directly estimate the entangle-ment gap in the exact ground state.

Appendix G: Convergence bounds for unital channels

We briefly review Ref. [144] and discuss how to bound‖E l − E∞‖ by using function algebra. Applying Lemma 6toM = E − E∞ leads to the following theorem.

Theorem 6 (Main result of Ref. [144] for unital channels)Let E be a unital channel acting on a D × D-dimensionaloperator space such that ‖En‖ ≤ C for ∀n ∈ N. Letµ ≡ limn→∞ ‖En − E∞‖

1n be the spectral radius of

E − E∞. We denote the minimal polynomial of E − E∞as m(z) = mE−E∞(z) =

∑Jj=1(z − µj)

sj , where µj’sare different eigenvalues of E − E∞ and sj is the size ofthe largest Jordan block with eigenvalue µj , and define thecorresponding Blaschke product as

B(z) ≡J∏j=1

(z − µj1− µjz

)sj. (G1)

Then, for l > µ1−µ , we have

‖E l − E∞‖ ≤ 2C‖zl‖W/mW

≤ µl+1 4e2C√|m|(|m|+ 1)

l[1− (1 + l−1)µ]32

× sup|z|=(1+l−1)µ

∣∣∣∣ 1

B(z)

∣∣∣∣ ,(G2)

where |m| =∑Jj=1 sj is the degree of m and C is upper

bounded by√

D2 [145].

Since the detailed proof in Ref. [144] is rather technical, itis worthwhile to sketch the outline here. First, we note that“sup” in Eq. (G2) arises from the following Cauchy-Schwarzinequality:

‖fr‖W =∑p∈N

rp|fp|

≤

√√√√√∑p∈N

r2p

∑p∈N

|fp|2

=

√1

1− r2sup

0≤ρ<1

∫ 2π

0

dφ

2π|f(ρeiφ)|2

≤ ‖f‖H∞√1− r2

, ∀f ∈W,

(G3)

where r ∈ (0, 1) can arbitrarily be chosen, fr(z) ≡ f(rz) and‖f‖H∞ ≡ supz∈D |f(z)|. In the typical case with sj = 1,i.e., ifM = E − E∞ is diagonalizable [185], ‖zl‖W/mW is,by definition, upper bounded by ‖gr‖W as long as gr(µj) =g(rµj) = µlj for ∀j = 1, 2, ..., J . An example is

g(z) =

J∑j=1

µljBj(z)

Bj(rµj), (G4)

where Bj(z) ≡ B(z)z−rµj and B(z) ≡∏J

j=1z−rµj1−rµjz is the mod-

ified Blaschke product. Combining Eqs. (G3) and (G4), we

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28

obtain

‖zl‖W/mW ≤ ‖gr‖W ≤‖g‖H∞√

1− r2=

1√1− r2

sup|z|=1

|g(z)|

=1√

1− r2sup|z|=1

∣∣∣∣∣∣J∑j=1

µlj

(z − rµj)Bj(rµj)

∣∣∣∣∣∣ ,(G5)

where |B(z)| = 1 for ∀|z| = 1 is used. We will eventuallyarrive at Eq. (G2) by further bounding the rightmost expres-sion in Eq. (G5), which can be rewritten in terms of a contourintegral as ∮

|w|=(1+l−1)µ

dw

2πi

wl

Br(w)(z − rw)(G6)

for l > µ1−µ , and set

r =

√1− 1− (1 + l−1)µ

|m| . (G7)

Appendix H: Generalization to finite interacting systems

We generalize Theorem 2 to the case of finite L. While westill have the decomposition given in Eq. (55), |Φαβ〉’s are nolonger orthogonal to each other. To compute the ES, we usethe following generalization of Lemma 4:

Lemma 12 For a bipartite state |Ψ〉 =∑Jj=1 |φj〉|ψj〉,

where |φj〉’s and |ψj〉’s are generally neither normalized nororthogonal to each other, the entanglement spectrum undersuch a bipartition coincides with the spectrum of M

12

ψMφM12

ψ

(Mψ: complex conjugation ofMψ) or M12

φMψM12

φ (except forzeros), where [Mφ]jj′ = 〈φj |φj′〉 and [Mψ]jj′ = 〈ψj |ψj′〉.

Proof.— It is equivalent to consider the spectrum of ρφ =Trψ |Ψ〉〈Ψ| and that of ρψ = Trφ |Ψ〉〈Ψ|. To be specific, wefocus on the former, which can be explicitly written as

ρφ =∑j,j′

|φj〉〈φj′ |Tr[|ψj〉〈ψj′ |]

=∑j,j′

[Mψ]jj′ |φj〉〈φj′ |.(H1)

Since Mψ is Hermitian, it can be expressed as U†ΛU withΛ = diagΛkJk=1 and U being unitary. To be concrete,we have [Mψ]jj′ =

∑j′′ Uj′′j′Λj′′Uj′′j . Introducing |φj〉 =∑

j′

√ΛjUjj′ |φj′〉, we can rewrite ρφ as

ρφ =∑j

|φj〉〈φj | (H2)

to which we can apply Lemma 4 – the spectrum of ρφ coin-cides with that of

〈φj |φj′〉 =∑j′′,j′′′

√ΛjΛj′Uj′j′′′Ujj′′〈φj′′ |φj′′′〉

=∑j′′,j′′′

√ΛjUjj′′ [Mφ]j′′j′′′U

Tj′′′j′

√Λj′

= [√

ΛUMφUT√

Λ]jj′ .

(H3)

From the fact that unitary conjugation preserves the spectrum,we know that the spectrum of ρφ should be given by that of

UT√

ΛUAUT√

ΛU = M12

ψMφM12

ψ . (H4)

Note that the spectrum of M12

ψMφM12

ψ is nothing but the

squared absolute values of the singular values of M12

φ M12

ψ ,

which are the same as those of (M12

φ M12

ψ )T = M12

ψ M12

φ and

thus give the same spectrum as that of M12

φMψM12

φ (this candirectly be obtained by considering ρψ following a similaranalysis as above). In fact, this result has already been obtained in Ref. [47],where it is used to calculate the ES of a projected entangled-pair state.

To bound the many-body entanglement gap, we need thefollowing lemma.

Lemma 13 Let M , M ′, M0, M ′0 be non-negative definiteHermitian matrices and let the jth largest eigenvalue ofM ′

12MM ′

12 and that of M ′

12

0 M0M′ 120 be denoted as λj and

λ0j , respectively. Then for ∀j, we have

|λj − λ0j | ≤ min‖M ′0δ‖+ ‖M12

0 δ′M

12

0 ‖,‖M0δ

′‖+ ‖M ′12

0 δM′ 120 ‖+ ‖δ′δ‖,

(H5)

where δ ≡M −M0 and δ′ ≡M ′ −M ′0.

Proof.— We first note the following useful norm inequality:for any two Hermitian matrices A and B with B ≥ 0 (so thatB

12 is well-defined), we have

‖B 12AB

12 ‖ ≤ ‖AB‖. (H6)

This is because the spectrum of B12AB

12 coincides with that

of AB due to Tr[(B12AB

12 )n] = Tr[(AB)n] for ∀n ∈ N

[181]. Moreover,B12AB

12 is Hermitian so its norm is nothing

but the spectral radius, which is no more than the norm ofAB.This result is a special case of Proposition IX.1.1 in Ref. [84].

Let us turn to the proof of the lemma. Denoting the jthlargest eigenvalue of M ′

12M0M

′ 12 as λ′j , which is also the jth

largest eigenvalue of M12

0 M′M

12

0 , we use Weyl’s perturbation

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29

theorem to obtain

|λj − λ0j | ≤ |λj − λ′j |+ |λ′j − λ0j |≤ ‖M ′ 12MM ′

12 − M ′ 12M0M

′ 12 ‖+ ‖M

12

0 M′M

12

0 − M12

0 M′0M

12

0 ‖= ‖M ′ 12 δM ′ 12 ‖+ ‖M

12

0 δ′M

12

0 ‖≤ ‖M ′δ‖+ ‖M

12

0 δ′M

12

0 ‖≤ ‖M ′0δ‖+ ‖M

12

0 δ′M

12

0 ‖+ ‖δ′δ‖,

(H7)

where we have used ‖A+ B‖ ≤ ‖A‖+ ‖B‖ in the last step.Replacing M and M0 with M ′ and M ′0, respectively, and fol-lowing a similar procedure, we obtain

|λj − λ0j | ≤ ‖M0δ′‖+ ‖M ′

12

0 δM′ 120 ‖+ ‖δδ′‖. (H8)

Combining Eqs. (H7) and (H8), we obtain Eq. (H5). To apply Lemma 13 to the many-body entanglement gap,

we have only to choose

Mαβ,α′β′ = aL〈α′|E l(|β′〉〈β|)|α〉,M ′αβ,α′β′ = aL〈β′|EL−l(|α′〉〈α|)|β〉,M0 = aL1v ⊗ Λ, M ′0 = aLΛ⊗ 1v,

(H9)

where aL = (Tr EL)−12 is a finite-size normalization factor.

Following the derivation of Eq. (73) in the main text, we canupper bound each term on the rhs of Eq. (H5) as

‖M ′0δ‖ ≤ a2L

(D

2

) 34

‖E l − E∞‖,

‖M0δ′‖ ≤ a2

L

(D

2

) 34

‖EL−l − E∞‖,

‖M ′12

0 δM′ 120 ‖ ≤ a2

L

√D

2‖E l − E∞‖,

‖M12

0 δ′M

12

0 ‖ ≤ a2L

√D

2‖EL−l − E∞‖,

‖δ′δ‖ ≤ a2LD

2‖E l − E∞‖‖EL−l − E∞‖.

(H10)

Using the techniques in Sec. V D to bound ‖E l − E∞‖ as-sociated with a time evolved MPS, we obtain the followingtheorem.

Theorem 7 (Finite interacting systems) Starting from anSPT MPS with length L and bond dimension D subject to theperiodic boundary condition, the many-body entanglementgap of a length-l subsystem after t time-evolution steps by a

trivial symmetric MPU with bond dimension DU is boundedfrom above by

∆mbE ≤ (Tr EL)−1[minb1/2(l, t) + b3/4(L− l, t),b1/2(L− l, t) + b3/4(l, t)+ 4b1(l, t)b1(L− l, t)]

(H11)

for any minl, L− l − 2k0t ≥ 1+µ1−µ , where

bα(l, t) = Cα(l − 2k0t)D2−1e−

l−vαtξ , (H12)

with k0, µ and ξ being the same as those in Theorem 2, vα =2k0 − (α+ 1

2 ) lnDUlnµ , and the coefficient

Cα = e2252−αD

32 +α(D2+1)µ1−D2

(1+µ)D2+ 1

2 (1−µ)D2− 5

2

(H13)depends only on the initial state.

Two remarks are in order here. First, we note that in thethermodynamic limit of the entire system, we have Tr EL = 1and bα(∞, t) = 0 so that Theorem 2 is reproduced. Even if Lis finite, we still find that ∆mb

E is exponentially small up to

t ∼ min

minl, L− l

v1/2,

maxl, L− lv3/4

,L

v1

, (H14)

provided that minl, L−l > v1/2(1+µ)

(v1/2−2k0)(1−µ) . Second, in thespecial case of the zero correlation length, i.e., µ = 0, whichcorresponds to fixed points of entanglement renormalization[186], we can infer from Theorem 7 that the degeneracy isexact up to t ∼ l

2k0. This is intuitively rather clear since the

entanglement edge modes are absolutely localized (without anexponential tail) for fixed-point states and it takes a finite timefor a nonzero overlap to develop between the edge modes bya locality-preserving MPU.

Appendix I: Details on ES dynamics upon partial symmetrybreaking

1. Flat-band model for class BDI→ class D

Thanks to the flat-band nature of the Hamiltonians given inEqs. (93) and (94), it suffices to consider the 2N sites clos-est to the entanglement cut (purple dashed line in Fig. 10(a)).Restricted to the single-particle Hilbert subspace of thesesites, the projector onto the Fermi sea of H0 reads P0 =12

⊕Nj=1(σ0 + σx). The ES dynamics is then determined

by the spectrum of ES(t) = PSP (t)PS , where PS =12

⊕Nj=1(σ0 + σz) is the projector onto the left half N sites

and

P (t) = e−iHsptP0e

iHspt, Hsp = 1⊕N−1⊕j=1

eitσx ⊕ 1, (I1)

where we have already set J = 1. After straightforward cal-culations, we obtain the following matrix form of ES(t):

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30

ES(t) =1

2

1 −i sin t 0 0 · · · 0 0

i sin t 1 −i sin t cos t 0 · · · 0 0

0 i sin t cos t 1 −i sin t cos t · · · 0 0

0 0 i sin t cos t 1 · · · 0 0...

......

.... . .

......

0 0 0 0 · · · 1 −i sin t cos t

0 0 0 0 · · · i sin t cos t 1

N×N

, (I2)

whose characteristic polynomial reads

fN (ξ; t) ≡ det[ξIN×N − ES(t)]

=

(ξ − 1

2

)FN−1

(ξ − 1

2;

sin 2t

4

)− sin2 t

4FN−2

(ξ − 1

2;

sin 2t

4

).

(I3)

Here, FN (x; a) is defined recursively as

FN (x; a) = xFN−1(x; a)− a2FN−2(x; a), (I4)

with initial conditions F1(x; a) = x and F2(x; a) = x2 − a2

(or F−1(x; a) = 0 and F0(x; a) = 1). In fact, we have an an-alytic expression FN (x; a) ≡∏N

j=1(x− 2a cos jπN+1 ), which

enjoys the properties FN (bx; ab) = bNF (x; a), FN (x; a) =FN (x;−a) and, in particular, FN (−x; a) = (−)NFN (x; a).Using the recursive relation (I4), we can rewrite fN (ξ; t) into

fN (ξ; t) = FN

(ξ − 1

2;

sin 2t

4

)− sin4 t

4FN−2

(ξ − 1

2;

sin 2t

4

).

(I5)

The roots of fN (ξ; t) give the ES dynamics. Since FN (x; a) isan odd function of x for odd N , we have F2n+1(0; a) = 0 sothat ξ = 1

2 is always a root of fN (ξ; t) if N is odd. Moreover,defining g2n+1(ξ; t) ≡ f2n+1(ξ; t)/(ξ − 1

2 ), we find

g2n+1

(1

2; t

)=

(− sin2 2t

16

)n (1 +

n

cos2 t

),

f2n

(1

2; t

)=

(− sin2 2t

16

)n1

cos2 t,

(I6)

which are generally nonzero except for some special timepoints. Therefore, for an even N , all the initial topologicalentanglement modes at ξ = 1

2 split, while one and only oneξ = 1

2 mode survives for an odd N .Let us calculate the full ES dynamics for N = 1, 2, 3, 4, 5.

When N = 1, we have f1(ξ; t) = ξ − 12 , implying the persis-

tence of the topological entanglement mode at ξ = 12 . This is

a trivial result since H = 0 and there is no dynamics. WhenN = 2, we find f2(ξ; t) =

(ξ − 1

2

)2 − 14 sin2 t, so the two

entanglement modes oscillate as

ξ =1

2(1± sin t). (I7)

WhenN = 3, we find f3(ξ; t) = ξ− 12 )[(ξ− 1

2 )2− 14 sin2 t(1+

cos2 t)], implying a persistent topological mode at ξ = 12 and

two oscillating modes

ξ =1

2(1± sin t

√1 + cos2 t). (I8)

When N = 4, we find f4(ξ; t) = (ξ − 12 )4 − 1

4 sin2 t(1 +

2 cos2 t)(ξ − 12 )2 + 1

16 sin4 t cos2 t, so all of the four modesoscillate in time as

ξ =1

2± sin t

4

√2 + 4 cos2 t± 2

√1 + 4 cos4 t. (I9)

Finally, we find the characteristic polynomial for N = 5 to bef5(ξ; t) = (ξ− 1

2 )[(ξ− 12 )4− 1

4 sin2 t(1+3 cos2 t)(ξ− 12 )2 +

116 sin4 t cos2 t(2 + cos2 t)]. Except for a constant solutionξ = 1

2 , the other four modes oscillate as

ξ =1

2± sin t

4

√2 + 6 cos2 t± 2

√5 cos4 t− 2 cos2 t+ 1.

(I10)These exact ES dynamics are plotted in Fig. 10(b).

2. Mathematical formulation

Having the above simple example in mind, we are ready tointroduce a general mathematical formalism for dealing withpartial symmetry breaking. We first recall that the classifica-tion of gapped free-fermion systems at equilibrium is given bythe homotopy group πds(S), where ds is the spatial dimensionand S is the classifying space satisfying symmetry constraints.If we are interested in the stable topology, we can take thelimit of infinite bands and obtain the well-known K-theoryclassification [107]. For example, S = R1 ≡ limn→∞O(n)for class BDI and S = R2 ≡ limn→∞O(2n)/U(n) forclass D. However, we emphasize that the general formalismapplies equally to the finite-band case, although the practi-cal calculations could be intractable. Denoting S and S asthe classifying spaces subject to symmetries G and G withG < G, we have S ⊂ S since a G-symmetric system isalways G-symmetric but the converse is generally not true.Therefore, we have a natural inclusion ι : S → S, which isa morphism (continuous map) in Top∗ (category of pointedtopological spaces). Such an inclusion induces a group ho-momorphism ι∗ : πds(S) → πds(S) through [f ] → [ι f ],where f : Sds → S is a continuous map from dsD sphere

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31

to S and [f ] is its homotopy equivalence class. In fact, πdscan be regarded as a functor from Top∗ to Grp (categoryof groups) or Ab (category of Abelian groups) if ds ≥ 2,which maps not only pointed topological spaces into homo-topy groups but also continuous maps between topologicalspaces into group homomorphisms between the correspond-ing homotopy groups [187]. In particular, ι∗ is the image of ιthat makes the following diagram commute:

S S

πds(S) πds(S)

ι

ι∗

πds πds

(I11)

For class BDI→ class D in 1D, we have ds = 1, π1(R1) =Z, π1(R2) = Z2 and ι∗(N) = N mod 2 for ∀N ∈ Z, asillustrated in the previous subsection.

Let us move on to discuss interacting SPT systems clas-sified by group cohomology [188]. We note that the group-cohomology classification is actually complete in 1D [70–72], although not in higher dimensions [189]. Instead ofthe classifying spaces, we focus directly on the symmetrygroups. The inclusion ι of G into G, which is a natu-ral group homomorphism, induces another group homomor-phism from Hdst(G,U(1)) to Hdst(G,U(1)), where dst ≡ds + 1 ∈ Z+ is the spacetime dimension. To see this, weonly have to note that an dst-cocycle ω : G×dst → U(1)

can naturally be restricted to ω : G×dst → U(1) throughω(g1, g2, ..., gdst) = ω(g1, g2, ..., gdst), which obviously sat-isfies the cocycle property dω = 1. In fact, any group co-homology h : G → Hdst(G,U(1)) (U(1) can actually bereplaced by other Abelian groups) is a contravariant functorfrom Grp to Ab, which map not only groups into Abelian co-homology groups but also group homomorphisms into thosebetween cohomology groups [190]. By contravariant, wemean that the directions of morphisms are reversed by thefunctor. In particular, the reduction of SPT phases is deter-mined by the induced map ι∗ that makes the following dia-gram commute:

G G

Hdst(G,U(1)) Hdst(G,U(1))

ι

ι∗

h h

(I12)

In the main text, we have given the simplest nontrivial exam-ple with dst = 2, H2(G,U(1)) = ZN , H2(G,U(1)) = Znand ι∗(ν) = N

n ν mod n.Finally, let us comment on the impact of SPT-order reduc-

tion on the ES dynamics. For free-fermion systems, the bulk-edge correspondence usually has a very simple form — thenumber of edge states, or the degeneracy in ES, is simplygiven by the bulk topological number [52, 191, 192]. As aprototypical example (e.g., class BDI→ class D in 1D), a sin-gle (no) ξ = 1

2 mode survives a surjective Z→ Z2 reduction ifthe original topological number is odd (even). For interacting

SPT systems in 1D, the open boundary ES degeneracy r is de-termined by the minimal dimension of the irreducible projec-tive representations of the symmetry group. Such a minimaldimension is 1 if the projective representation can be lifted to alinear representation, but is otherwise no less than 2 accordingto Lemma 5. In fact, there is a character theory for projectiverepresentations [193], which shares many similarities with theconventional character theory for linear representations. Forexample, denoting the dimension of the αth irreducible pro-jective representation with respect to a 2-cocycle ω as dα, wehave dα||G| and

∑Rωα=1 d

2α = |G|, where Rω is the number

of ω-regular conjugacy classes, i.e., those conjugacy classeswith a representative element g satisfying ω(g, h) = ω(h, g)for ∀h ∈ Ng ≡ h ∈ G : gh = hg < G. An immediatecollorary is that, for G = ZN ×ZN with N being a prime, wehave r = N for any G-symmetric SPT state. By simple anal-ysis, we can also infer that possible r > 1 for G = Z6×Z6 is2, 3 and 6. These conclusions are consistent with the resultsin Table II.

3. Minimal models for quenched interacting SPT systems

Let us introduce the interacting counterparts of flat-bandfree-fermion models, in the sense that these minimal mod-els have zero correlation lengths. In (1 + 1)D spacetime,a G-symmetric SPT state with zero correlation length canbe built from a 2-cocyle (which satisfies ω(gh, k)ω(g, h) =ω(g, hk)ω(h, k) for ∀g, h, k ∈ G) as [188]

|Ψ〉 =1

|G|L2∑gjLj=1

L∏j=1

ω(g−1j gj+1, g

−1j+1)−1|g1g2...gL〉.

(I13)Here the local Hilbert space C|G| is spanned by |g〉 : g ∈ Gand the on-site symmetry representation is regular: ρg =∑h∈G |gh〉〈h| for ∀g ∈ G. Note that such a construction

(I13) applies equally to continuous symmetries if we replace∑g by

∫dg [188]. To demonstrate the impact of partial sym-

metry breaking on the ES dynamics, we consider the simplestcase in which the Hamiltonian is a sum of commutative two-site operators:

H =

L∑j=1

hj , hj =∑

gj ,gj+1

h(gj , gj+1)|gjgj+1〉〈gjgj+1|,

(I14)whose eigenstates are simply Fock states. To partially breakthe symmetry from G to G, we require

h(g, g′) = h(gg, gg′) ∈ R, ∀g, g′ ∈ G and g ∈ G, (I15)

and otherwise

h(g, g′) 6= h(g′′g, g′′g′), ∀g′′ ∈ G\G. (I16)

As illustrated in Fig. 16(a), due to the fact that hj’s commutewith each other, the open boundary ES at time t are simply thesquared singular values of

[M(t)]gg′ = e−ih(g,g′)tω(g−1g′, g′−1)−1. (I17)

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32

0 2 4 6 8 1010-410-310-210-11

t

ζ

0 2 4 6 8 1010-410-310-210-11

t

ζ

0 2 4 6 8 1010-410-310-210-11

t

ζ

0 2 4 6 8 1010-410-310-210-11

t

ζ

0 2 4 6 8 1010-410-310-210-11

t

ζ

0 2 4 6 8 1010-410-310-210-11

t

ζ

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1 ! 2<latexit sha1_base64="7pCcxS5oAwJ2+tTAzmccJIQjD9c=">AAACC3icbVA9T8MwEHXKVwlfAUYWq1UkpiopRTBWsDAWibZIbVQ5rttadezIdpCqqDsLf4WFAYRY+QNs/BucNAO0PMm6d+/udL4Xxowq7XnfVmltfWNzq7xt7+zu7R84h0cdJRKJSRsLJuR9iBRhlJO2ppqR+1gSFIWMdMPpdVbvPhCpqOB3ehaTIEJjTkcUI22kgVPx+1rUbbdugme7Zyb4tttYZOd5NnCqXs3LAVeJX5AqKNAaOF/9ocBJRLjGDCnV871YBymSmmJG5nY/USRGeIrGpGcoRxFRQZrfMoeuUYZwJKR5XMNc/T2RokipWRSazgjpiVquZeJ/tV6iR5dBSnmcaMLxYtEoYVALmBkDh1QSrNnMEIQlNX+FeIIkwtrYl5ngL5+8Sjr1mm/4baPavCrsKIMTUAGnwAcXoAluQAu0AQaP4Bm8gjfryXqx3q2PRWvJKmaOwR9Ynz+ZoZbz</latexit><latexit sha1_base64="7pCcxS5oAwJ2+tTAzmccJIQjD9c=">AAACC3icbVA9T8MwEHXKVwlfAUYWq1UkpiopRTBWsDAWibZIbVQ5rttadezIdpCqqDsLf4WFAYRY+QNs/BucNAO0PMm6d+/udL4Xxowq7XnfVmltfWNzq7xt7+zu7R84h0cdJRKJSRsLJuR9iBRhlJO2ppqR+1gSFIWMdMPpdVbvPhCpqOB3ehaTIEJjTkcUI22kgVPx+1rUbbdugme7Zyb4tttYZOd5NnCqXs3LAVeJX5AqKNAaOF/9ocBJRLjGDCnV871YBymSmmJG5nY/USRGeIrGpGcoRxFRQZrfMoeuUYZwJKR5XMNc/T2RokipWRSazgjpiVquZeJ/tV6iR5dBSnmcaMLxYtEoYVALmBkDh1QSrNnMEIQlNX+FeIIkwtrYl5ngL5+8Sjr1mm/4baPavCrsKIMTUAGnwAcXoAluQAu0AQaP4Bm8gjfryXqx3q2PRWvJKmaOwR9Ynz+ZoZbz</latexit><latexit sha1_base64="7pCcxS5oAwJ2+tTAzmccJIQjD9c=">AAACC3icbVA9T8MwEHXKVwlfAUYWq1UkpiopRTBWsDAWibZIbVQ5rttadezIdpCqqDsLf4WFAYRY+QNs/BucNAO0PMm6d+/udL4Xxowq7XnfVmltfWNzq7xt7+zu7R84h0cdJRKJSRsLJuR9iBRhlJO2ppqR+1gSFIWMdMPpdVbvPhCpqOB3ehaTIEJjTkcUI22kgVPx+1rUbbdugme7Zyb4tttYZOd5NnCqXs3LAVeJX5AqKNAaOF/9ocBJRLjGDCnV871YBymSmmJG5nY/USRGeIrGpGcoRxFRQZrfMoeuUYZwJKR5XMNc/T2RokipWRSazgjpiVquZeJ/tV6iR5dBSnmcaMLxYtEoYVALmBkDh1QSrNnMEIQlNX+FeIIkwtrYl5ngL5+8Sjr1mm/4baPavCrsKIMTUAGnwAcXoAluQAu0AQaP4Bm8gjfryXqx3q2PRWvJKmaOwR9Ynz+ZoZbz</latexit><latexit sha1_base64="7pCcxS5oAwJ2+tTAzmccJIQjD9c=">AAACC3icbVA9T8MwEHXKVwlfAUYWq1UkpiopRTBWsDAWibZIbVQ5rttadezIdpCqqDsLf4WFAYRY+QNs/BucNAO0PMm6d+/udL4Xxowq7XnfVmltfWNzq7xt7+zu7R84h0cdJRKJSRsLJuR9iBRhlJO2ppqR+1gSFIWMdMPpdVbvPhCpqOB3ehaTIEJjTkcUI22kgVPx+1rUbbdugme7Zyb4tttYZOd5NnCqXs3LAVeJX5AqKNAaOF/9ocBJRLjGDCnV871YBymSmmJG5nY/USRGeIrGpGcoRxFRQZrfMoeuUYZwJKR5XMNc/T2RokipWRSazgjpiVquZeJ/tV6iR5dBSnmcaMLxYtEoYVALmBkDh1QSrNnMEIQlNX+FeIIkwtrYl5ngL5+8Sjr1mm/4baPavCrsKIMTUAGnwAcXoAluQAu0AQaP4Bm8gjfryXqx3q2PRWvJKmaOwR9Ynz+ZoZbz</latexit>

2 ! 1<latexit sha1_base64="IjcJJxTflNmZ6yyC4xnbIJRZT88=">AAACBHicbVC7TsMwFHXKq4RXgLGLRVWJqUpKEYwVLIxFoi1SG1WO67RWHTuyHaQq6sDCr7AwgBArH8HG3+CkGaDlSNY9Pude2fcEMaNKu+63VVpb39jcKm/bO7t7+wfO4VFXiURi0sGCCXkfIEUY5aSjqWbkPpYERQEjvWB6nfm9ByIVFfxOz2LiR2jMaUgx0kYaOpXGQAvPrp0tStMU166d57ehU3Xrbg64SryCVEGB9tD5GowETiLCNWZIqb7nxtpPkdQUMzK3B4kiMcJTNCZ9QzmKiPLTfIk5rBllBEMhzeEa5urviRRFSs2iwHRGSE/UspeJ/3n9RIeXfkp5nGjC8eKhMGFQC5glAkdUEqzZzBCEJTV/hXiCJMLa5JaF4C2vvEq6jbpn+G2z2roq4iiDCjgBp8ADF6AFbkAbdAAGj+AZvII368l6sd6tj0VrySpmjsEfWJ8/ngiU3Q==</latexit><latexit sha1_base64="IjcJJxTflNmZ6yyC4xnbIJRZT88=">AAACBHicbVC7TsMwFHXKq4RXgLGLRVWJqUpKEYwVLIxFoi1SG1WO67RWHTuyHaQq6sDCr7AwgBArH8HG3+CkGaDlSNY9Pude2fcEMaNKu+63VVpb39jcKm/bO7t7+wfO4VFXiURi0sGCCXkfIEUY5aSjqWbkPpYERQEjvWB6nfm9ByIVFfxOz2LiR2jMaUgx0kYaOpXGQAvPrp0tStMU166d57ehU3Xrbg64SryCVEGB9tD5GowETiLCNWZIqb7nxtpPkdQUMzK3B4kiMcJTNCZ9QzmKiPLTfIk5rBllBEMhzeEa5urviRRFSs2iwHRGSE/UspeJ/3n9RIeXfkp5nGjC8eKhMGFQC5glAkdUEqzZzBCEJTV/hXiCJMLa5JaF4C2vvEq6jbpn+G2z2roq4iiDCjgBp8ADF6AFbkAbdAAGj+AZvII368l6sd6tj0VrySpmjsEfWJ8/ngiU3Q==</latexit><latexit sha1_base64="IjcJJxTflNmZ6yyC4xnbIJRZT88=">AAACBHicbVC7TsMwFHXKq4RXgLGLRVWJqUpKEYwVLIxFoi1SG1WO67RWHTuyHaQq6sDCr7AwgBArH8HG3+CkGaDlSNY9Pude2fcEMaNKu+63VVpb39jcKm/bO7t7+wfO4VFXiURi0sGCCXkfIEUY5aSjqWbkPpYERQEjvWB6nfm9ByIVFfxOz2LiR2jMaUgx0kYaOpXGQAvPrp0tStMU166d57ehU3Xrbg64SryCVEGB9tD5GowETiLCNWZIqb7nxtpPkdQUMzK3B4kiMcJTNCZ9QzmKiPLTfIk5rBllBEMhzeEa5urviRRFSs2iwHRGSE/UspeJ/3n9RIeXfkp5nGjC8eKhMGFQC5glAkdUEqzZzBCEJTV/hXiCJMLa5JaF4C2vvEq6jbpn+G2z2roq4iiDCjgBp8ADF6AFbkAbdAAGj+AZvII368l6sd6tj0VrySpmjsEfWJ8/ngiU3Q==</latexit><latexit sha1_base64="IjcJJxTflNmZ6yyC4xnbIJRZT88=">AAACBHicbVC7TsMwFHXKq4RXgLGLRVWJqUpKEYwVLIxFoi1SG1WO67RWHTuyHaQq6sDCr7AwgBArH8HG3+CkGaDlSNY9Pude2fcEMaNKu+63VVpb39jcKm/bO7t7+wfO4VFXiURi0sGCCXkfIEUY5aSjqWbkPpYERQEjvWB6nfm9ByIVFfxOz2LiR2jMaUgx0kYaOpXGQAvPrp0tStMU166d57ehU3Xrbg64SryCVEGB9tD5GowETiLCNWZIqb7nxtpPkdQUMzK3B4kiMcJTNCZ9QzmKiPLTfIk5rBllBEMhzeEa5urviRRFSs2iwHRGSE/UspeJ/3n9RIeXfkp5nGjC8eKhMGFQC5glAkdUEqzZzBCEJTV/hXiCJMLa5JaF4C2vvEq6jbpn+G2z2roq4iiDCjgBp8ADF6AFbkAbdAAGj+AZvII368l6sd6tj0VrySpmjsEfWJ8/ngiU3Q==</latexit>

3 ! 0<latexit sha1_base64="ukq6E7zt0zIG9JUa/zQs6NKTRdQ=">AAAB/XicbVDLSsNAFJ3UV42v+Ni5GSwFVyXRii6LblxWsA9oQ5lMJ+3QySTM3Ag1FH/FjQtF3Pof7vwbp20W2npguIdz7uXeOUEiuAbX/bYKK6tr6xvFTXtre2d3z9k/aOo4VZQ1aCxi1Q6IZoJL1gAOgrUTxUgUCNYKRjdTv/XAlOaxvIdxwvyIDCQPOSVgpJ5zdN6F2LXL1Xm5MMWze07Jrbgz4GXi5aSEctR7zle3H9M0YhKoIFp3PDcBPyMKOBVsYndTzRJCR2TAOoZKEjHtZ7PrJ7hslD4OY2WeBDxTf09kJNJ6HAWmMyIw1IveVPzP66QQXvkZl0kKTNL5ojAVGGI8jQL3uWIUxNgQQhU3t2I6JIpQMIFNQ/AWv7xMmmcVz/C7aql2ncdRRMfoBJ0iD12iGrpFddRAFD2iZ/SK3qwn68V6tz7mrQUrnzlEf2B9/gCsV5LF</latexit><latexit sha1_base64="ukq6E7zt0zIG9JUa/zQs6NKTRdQ=">AAAB/XicbVDLSsNAFJ3UV42v+Ni5GSwFVyXRii6LblxWsA9oQ5lMJ+3QySTM3Ag1FH/FjQtF3Pof7vwbp20W2npguIdz7uXeOUEiuAbX/bYKK6tr6xvFTXtre2d3z9k/aOo4VZQ1aCxi1Q6IZoJL1gAOgrUTxUgUCNYKRjdTv/XAlOaxvIdxwvyIDCQPOSVgpJ5zdN6F2LXL1Xm5MMWze07Jrbgz4GXi5aSEctR7zle3H9M0YhKoIFp3PDcBPyMKOBVsYndTzRJCR2TAOoZKEjHtZ7PrJ7hslD4OY2WeBDxTf09kJNJ6HAWmMyIw1IveVPzP66QQXvkZl0kKTNL5ojAVGGI8jQL3uWIUxNgQQhU3t2I6JIpQMIFNQ/AWv7xMmmcVz/C7aql2ncdRRMfoBJ0iD12iGrpFddRAFD2iZ/SK3qwn68V6tz7mrQUrnzlEf2B9/gCsV5LF</latexit><latexit sha1_base64="ukq6E7zt0zIG9JUa/zQs6NKTRdQ=">AAAB/XicbVDLSsNAFJ3UV42v+Ni5GSwFVyXRii6LblxWsA9oQ5lMJ+3QySTM3Ag1FH/FjQtF3Pof7vwbp20W2npguIdz7uXeOUEiuAbX/bYKK6tr6xvFTXtre2d3z9k/aOo4VZQ1aCxi1Q6IZoJL1gAOgrUTxUgUCNYKRjdTv/XAlOaxvIdxwvyIDCQPOSVgpJ5zdN6F2LXL1Xm5MMWze07Jrbgz4GXi5aSEctR7zle3H9M0YhKoIFp3PDcBPyMKOBVsYndTzRJCR2TAOoZKEjHtZ7PrJ7hslD4OY2WeBDxTf09kJNJ6HAWmMyIw1IveVPzP66QQXvkZl0kKTNL5ojAVGGI8jQL3uWIUxNgQQhU3t2I6JIpQMIFNQ/AWv7xMmmcVz/C7aql2ncdRRMfoBJ0iD12iGrpFddRAFD2iZ/SK3qwn68V6tz7mrQUrnzlEf2B9/gCsV5LF</latexit><latexit sha1_base64="ukq6E7zt0zIG9JUa/zQs6NKTRdQ=">AAAB/XicbVDLSsNAFJ3UV42v+Ni5GSwFVyXRii6LblxWsA9oQ5lMJ+3QySTM3Ag1FH/FjQtF3Pof7vwbp20W2npguIdz7uXeOUEiuAbX/bYKK6tr6xvFTXtre2d3z9k/aOo4VZQ1aCxi1Q6IZoJL1gAOgrUTxUgUCNYKRjdTv/XAlOaxvIdxwvyIDCQPOSVgpJ5zdN6F2LXL1Xm5MMWze07Jrbgz4GXi5aSEctR7zle3H9M0YhKoIFp3PDcBPyMKOBVsYndTzRJCR2TAOoZKEjHtZ7PrJ7hslD4OY2WeBDxTf09kJNJ6HAWmMyIw1IveVPzP66QQXvkZl0kKTNL5ojAVGGI8jQL3uWIUxNgQQhU3t2I6JIpQMIFNQ/AWv7xMmmcVz/C7aql2ncdRRMfoBJ0iD12iGrpFddRAFD2iZ/SK3qwn68V6tz7mrQUrnzlEf2B9/gCsV5LF</latexit>

4 ! 2<latexit sha1_base64="I+eut8i7IAnmXbpQp2p6POX2CJg=">AAAB9HicbZDLSgMxFIYzXut4q7p0EywFV2WmVHRZdOOygr1AO5RMmmlDM8mYnCmU0udw40IRtz6MO9/GTDsLbT0Q8vH/55CTP0wEN+B5387G5tb2zm5hz90/ODw6Lp6ctoxKNWVNqoTSnZAYJrhkTeAgWCfRjMShYO1wfJf57QnThiv5CNOEBTEZSh5xSsBKQa0HquqWr+zlu/1iyat4i8Lr4OdQQnk1+sWv3kDRNGYSqCDGdH0vgWBGNHAq2NztpYYlhI7JkHUtShIzE8wWS89x2SoDHCltjwS8UH9PzEhszDQObWdMYGRWvUz8z+umEN0EMy6TFJiky4eiVGBQOEsAD7hmFMTUAqGa210xHRFNKNicshD81S+vQ6ta8S0/1Er12zyOAjpHF+gS+ega1dE9aqAmougJPaNX9OZMnBfn3flYtm44+cwZ+lPO5w9U75B/</latexit><latexit sha1_base64="I+eut8i7IAnmXbpQp2p6POX2CJg=">AAAB9HicbZDLSgMxFIYzXut4q7p0EywFV2WmVHRZdOOygr1AO5RMmmlDM8mYnCmU0udw40IRtz6MO9/GTDsLbT0Q8vH/55CTP0wEN+B5387G5tb2zm5hz90/ODw6Lp6ctoxKNWVNqoTSnZAYJrhkTeAgWCfRjMShYO1wfJf57QnThiv5CNOEBTEZSh5xSsBKQa0HquqWr+zlu/1iyat4i8Lr4OdQQnk1+sWv3kDRNGYSqCDGdH0vgWBGNHAq2NztpYYlhI7JkHUtShIzE8wWS89x2SoDHCltjwS8UH9PzEhszDQObWdMYGRWvUz8z+umEN0EMy6TFJiky4eiVGBQOEsAD7hmFMTUAqGa210xHRFNKNicshD81S+vQ6ta8S0/1Er12zyOAjpHF+gS+ega1dE9aqAmougJPaNX9OZMnBfn3flYtm44+cwZ+lPO5w9U75B/</latexit><latexit sha1_base64="I+eut8i7IAnmXbpQp2p6POX2CJg=">AAAB9HicbZDLSgMxFIYzXut4q7p0EywFV2WmVHRZdOOygr1AO5RMmmlDM8mYnCmU0udw40IRtz6MO9/GTDsLbT0Q8vH/55CTP0wEN+B5387G5tb2zm5hz90/ODw6Lp6ctoxKNWVNqoTSnZAYJrhkTeAgWCfRjMShYO1wfJf57QnThiv5CNOEBTEZSh5xSsBKQa0HquqWr+zlu/1iyat4i8Lr4OdQQnk1+sWv3kDRNGYSqCDGdH0vgWBGNHAq2NztpYYlhI7JkHUtShIzE8wWS89x2SoDHCltjwS8UH9PzEhszDQObWdMYGRWvUz8z+umEN0EMy6TFJiky4eiVGBQOEsAD7hmFMTUAqGa210xHRFNKNicshD81S+vQ6ta8S0/1Er12zyOAjpHF+gS+ega1dE9aqAmougJPaNX9OZMnBfn3flYtm44+cwZ+lPO5w9U75B/</latexit><latexit sha1_base64="I+eut8i7IAnmXbpQp2p6POX2CJg=">AAAB9HicbZDLSgMxFIYzXut4q7p0EywFV2WmVHRZdOOygr1AO5RMmmlDM8mYnCmU0udw40IRtz6MO9/GTDsLbT0Q8vH/55CTP0wEN+B5387G5tb2zm5hz90/ODw6Lp6ctoxKNWVNqoTSnZAYJrhkTeAgWCfRjMShYO1wfJf57QnThiv5CNOEBTEZSh5xSsBKQa0HquqWr+zlu/1iyat4i8Lr4OdQQnk1+sWv3kDRNGYSqCDGdH0vgWBGNHAq2NztpYYlhI7JkHUtShIzE8wWS89x2SoDHCltjwS8UH9PzEhszDQObWdMYGRWvUz8z+umEN0EMy6TFJiky4eiVGBQOEsAD7hmFMTUAqGa210xHRFNKNicshD81S+vQ6ta8S0/1Er12zyOAjpHF+gS+ega1dE9aqAmougJPaNX9OZMnBfn3flYtm44+cwZ+lPO5w9U75B/</latexit>

5 ! 1<latexit sha1_base64="TBHysJewffktg0GTBw86qlcyjqc=">AAAB7XicbZBNS8NAEIYnftb6VfXoJVgETyURRY9FLx4r2A9oQ9lsN+3azW7YnQgl9D948aCIV/+PN/+NmzYHbX1h4eGdGXbmDRPBDXret7Oyura+sVnaKm/v7O7tVw4OW0almrImVULpTkgME1yyJnIUrJNoRuJQsHY4vs3r7SemDVfyAScJC2IylDzilKC1Wpc9VH65X6l6NW8mdxn8AqpQqNGvfPUGiqYxk0gFMabrewkGGdHIqWDTci81LCF0TIasa1GSmJkgm207dU+tM3Ajpe2T6M7c3xMZiY2ZxKHtjAmOzGItN/+rdVOMroOMyyRFJun8oygVLio3P90dcM0oiokFQjW3u7p0RDShaAPKQ/AXT16G1nnNt3x/Ua3fFHGU4BhO4Ax8uII63EEDmkDhEZ7hFd4c5bw4787HvHXFKWaO4I+czx97MI5l</latexit><latexit sha1_base64="TBHysJewffktg0GTBw86qlcyjqc=">AAAB7XicbZBNS8NAEIYnftb6VfXoJVgETyURRY9FLx4r2A9oQ9lsN+3azW7YnQgl9D948aCIV/+PN/+NmzYHbX1h4eGdGXbmDRPBDXret7Oyura+sVnaKm/v7O7tVw4OW0almrImVULpTkgME1yyJnIUrJNoRuJQsHY4vs3r7SemDVfyAScJC2IylDzilKC1Wpc9VH65X6l6NW8mdxn8AqpQqNGvfPUGiqYxk0gFMabrewkGGdHIqWDTci81LCF0TIasa1GSmJkgm207dU+tM3Ajpe2T6M7c3xMZiY2ZxKHtjAmOzGItN/+rdVOMroOMyyRFJun8oygVLio3P90dcM0oiokFQjW3u7p0RDShaAPKQ/AXT16G1nnNt3x/Ua3fFHGU4BhO4Ax8uII63EEDmkDhEZ7hFd4c5bw4787HvHXFKWaO4I+czx97MI5l</latexit><latexit sha1_base64="TBHysJewffktg0GTBw86qlcyjqc=">AAAB7XicbZBNS8NAEIYnftb6VfXoJVgETyURRY9FLx4r2A9oQ9lsN+3azW7YnQgl9D948aCIV/+PN/+NmzYHbX1h4eGdGXbmDRPBDXret7Oyura+sVnaKm/v7O7tVw4OW0almrImVULpTkgME1yyJnIUrJNoRuJQsHY4vs3r7SemDVfyAScJC2IylDzilKC1Wpc9VH65X6l6NW8mdxn8AqpQqNGvfPUGiqYxk0gFMabrewkGGdHIqWDTci81LCF0TIasa1GSmJkgm207dU+tM3Ajpe2T6M7c3xMZiY2ZxKHtjAmOzGItN/+rdVOMroOMyyRFJun8oygVLio3P90dcM0oiokFQjW3u7p0RDShaAPKQ/AXT16G1nnNt3x/Ua3fFHGU4BhO4Ax8uII63EEDmkDhEZ7hFd4c5bw4787HvHXFKWaO4I+czx97MI5l</latexit><latexit sha1_base64="TBHysJewffktg0GTBw86qlcyjqc=">AAAB7XicbZBNS8NAEIYnftb6VfXoJVgETyURRY9FLx4r2A9oQ9lsN+3azW7YnQgl9D948aCIV/+PN/+NmzYHbX1h4eGdGXbmDRPBDXret7Oyura+sVnaKm/v7O7tVw4OW0almrImVULpTkgME1yyJnIUrJNoRuJQsHY4vs3r7SemDVfyAScJC2IylDzilKC1Wpc9VH65X6l6NW8mdxn8AqpQqNGvfPUGiqYxk0gFMabrewkGGdHIqWDTci81LCF0TIasa1GSmJkgm207dU+tM3Ajpe2T6M7c3xMZiY2ZxKHtjAmOzGItN/+rdVOMroOMyyRFJun8oygVLio3P90dcM0oiokFQjW3u7p0RDShaAPKQ/AXT16G1nnNt3x/Ua3fFHGU4BhO4Ax8uII63EEDmkDhEZ7hFd4c5bw4787HvHXFKWaO4I+czx97MI5l</latexit>

(a)

(b)

(c)

Z6 ! Z2<latexit sha1_base64="GxCr5IDqBFQJeJf6GhR0e5JLAF0=">AAACFnicbVDLSsNAFJ34rPEVdekmWApuWpLia1l002UF+8A2hsl00g6dPJi5EUroV7jxV9y4UMStuPNvnLRZ1NYDA2fOuZd77/FiziRY1o+2srq2vrFZ2NK3d3b39o2Dw5aMEkFok0Q8Eh0PS8pZSJvAgNNOLCgOPE7b3ugm89uPVEgWhXcwjqkT4EHIfEYwKMk1yqXzHkS2XqIPaZnVYaL3AgxDz0vvJ+6Fsua+Vd01ilbFmsJcJnZOiihHwzW+e/2IJAENgXAsZde2YnBSLIARTtWwRNIYkxEe0K6iIQ6odNLpWROzpJS+6UdCvRDMqTrfkeJAynHgqcpsSbnoZeJ/XjcB/8pJWRgnQEMyG+Qn3ITIzDIy+0xQAnysCCaCqV1NMsQCE1BJZiHYiycvk1a1Yit+e1asXedxFNAxOkGnyEaXqIbqqIGaiKAn9ILe0Lv2rL1qH9rnrHRFy3uO0B9oX78oU54k</latexit><latexit sha1_base64="GxCr5IDqBFQJeJf6GhR0e5JLAF0=">AAACFnicbVDLSsNAFJ34rPEVdekmWApuWpLia1l002UF+8A2hsl00g6dPJi5EUroV7jxV9y4UMStuPNvnLRZ1NYDA2fOuZd77/FiziRY1o+2srq2vrFZ2NK3d3b39o2Dw5aMEkFok0Q8Eh0PS8pZSJvAgNNOLCgOPE7b3ugm89uPVEgWhXcwjqkT4EHIfEYwKMk1yqXzHkS2XqIPaZnVYaL3AgxDz0vvJ+6Fsua+Vd01ilbFmsJcJnZOiihHwzW+e/2IJAENgXAsZde2YnBSLIARTtWwRNIYkxEe0K6iIQ6odNLpWROzpJS+6UdCvRDMqTrfkeJAynHgqcpsSbnoZeJ/XjcB/8pJWRgnQEMyG+Qn3ITIzDIy+0xQAnysCCaCqV1NMsQCE1BJZiHYiycvk1a1Yit+e1asXedxFNAxOkGnyEaXqIbqqIGaiKAn9ILe0Lv2rL1qH9rnrHRFy3uO0B9oX78oU54k</latexit><latexit sha1_base64="GxCr5IDqBFQJeJf6GhR0e5JLAF0=">AAACFnicbVDLSsNAFJ34rPEVdekmWApuWpLia1l002UF+8A2hsl00g6dPJi5EUroV7jxV9y4UMStuPNvnLRZ1NYDA2fOuZd77/FiziRY1o+2srq2vrFZ2NK3d3b39o2Dw5aMEkFok0Q8Eh0PS8pZSJvAgNNOLCgOPE7b3ugm89uPVEgWhXcwjqkT4EHIfEYwKMk1yqXzHkS2XqIPaZnVYaL3AgxDz0vvJ+6Fsua+Vd01ilbFmsJcJnZOiihHwzW+e/2IJAENgXAsZde2YnBSLIARTtWwRNIYkxEe0K6iIQ6odNLpWROzpJS+6UdCvRDMqTrfkeJAynHgqcpsSbnoZeJ/XjcB/8pJWRgnQEMyG+Qn3ITIzDIy+0xQAnysCCaCqV1NMsQCE1BJZiHYiycvk1a1Yit+e1asXedxFNAxOkGnyEaXqIbqqIGaiKAn9ILe0Lv2rL1qH9rnrHRFy3uO0B9oX78oU54k</latexit><latexit sha1_base64="GxCr5IDqBFQJeJf6GhR0e5JLAF0=">AAACFnicbVDLSsNAFJ34rPEVdekmWApuWpLia1l002UF+8A2hsl00g6dPJi5EUroV7jxV9y4UMStuPNvnLRZ1NYDA2fOuZd77/FiziRY1o+2srq2vrFZ2NK3d3b39o2Dw5aMEkFok0Q8Eh0PS8pZSJvAgNNOLCgOPE7b3ugm89uPVEgWhXcwjqkT4EHIfEYwKMk1yqXzHkS2XqIPaZnVYaL3AgxDz0vvJ+6Fsua+Vd01ilbFmsJcJnZOiihHwzW+e/2IJAENgXAsZde2YnBSLIARTtWwRNIYkxEe0K6iIQ6odNLpWROzpJS+6UdCvRDMqTrfkeJAynHgqcpsSbnoZeJ/XjcB/8pJWRgnQEMyG+Qn3ITIzDIy+0xQAnysCCaCqV1NMsQCE1BJZiHYiycvk1a1Yit+e1asXedxFNAxOkGnyEaXqIbqqIGaiKAn9ILe0Lv2rL1qH9rnrHRFy3uO0B9oX78oU54k</latexit>

Z6 ! Z3<latexit sha1_base64="+JDd5IBS1cOL+nL5xl+4j8SboSE=">AAACFnicbVDLSsNAFJ34rPEVdekmWApuWhLfy6KbLivYB7YxTKaTdujkwcyNUEK/wo2/4saFIm7FnX/jpM2ith4YOHPOvdx7jxdzJsGyfrSl5ZXVtfXChr65tb2za+ztN2WUCEIbJOKRaHtYUs5C2gAGnLZjQXHgcdryhjeZ33qkQrIovINRTJ0A90PmM4JBSa5RLp13IbL1En1Iy6wGY70bYBh4Xno/di+UNfM91V2jaFWsCcxFYuekiHLUXeO724tIEtAQCMdSdmwrBifFAhjhVA1LJI0xGeI+7Sga4oBKJ52cNTZLSumZfiTUC8GcqLMdKQ6kHAWeqsyWlPNeJv7ndRLwr5yUhXECNCTTQX7CTYjMLCOzxwQlwEeKYCKY2tUkAywwAZVkFoI9f/IiaZ5UbMVvz4rV6zyOAjpER+gY2egSVVEN1VEDEfSEXtAbeteetVftQ/ucli5pec8B+gPt6xcp2J4l</latexit><latexit sha1_base64="+JDd5IBS1cOL+nL5xl+4j8SboSE=">AAACFnicbVDLSsNAFJ34rPEVdekmWApuWhLfy6KbLivYB7YxTKaTdujkwcyNUEK/wo2/4saFIm7FnX/jpM2ith4YOHPOvdx7jxdzJsGyfrSl5ZXVtfXChr65tb2za+ztN2WUCEIbJOKRaHtYUs5C2gAGnLZjQXHgcdryhjeZ33qkQrIovINRTJ0A90PmM4JBSa5RLp13IbL1En1Iy6wGY70bYBh4Xno/di+UNfM91V2jaFWsCcxFYuekiHLUXeO724tIEtAQCMdSdmwrBifFAhjhVA1LJI0xGeI+7Sga4oBKJ52cNTZLSumZfiTUC8GcqLMdKQ6kHAWeqsyWlPNeJv7ndRLwr5yUhXECNCTTQX7CTYjMLCOzxwQlwEeKYCKY2tUkAywwAZVkFoI9f/IiaZ5UbMVvz4rV6zyOAjpER+gY2egSVVEN1VEDEfSEXtAbeteetVftQ/ucli5pec8B+gPt6xcp2J4l</latexit><latexit sha1_base64="+JDd5IBS1cOL+nL5xl+4j8SboSE=">AAACFnicbVDLSsNAFJ34rPEVdekmWApuWhLfy6KbLivYB7YxTKaTdujkwcyNUEK/wo2/4saFIm7FnX/jpM2ith4YOHPOvdx7jxdzJsGyfrSl5ZXVtfXChr65tb2za+ztN2WUCEIbJOKRaHtYUs5C2gAGnLZjQXHgcdryhjeZ33qkQrIovINRTJ0A90PmM4JBSa5RLp13IbL1En1Iy6wGY70bYBh4Xno/di+UNfM91V2jaFWsCcxFYuekiHLUXeO724tIEtAQCMdSdmwrBifFAhjhVA1LJI0xGeI+7Sga4oBKJ52cNTZLSumZfiTUC8GcqLMdKQ6kHAWeqsyWlPNeJv7ndRLwr5yUhXECNCTTQX7CTYjMLCOzxwQlwEeKYCKY2tUkAywwAZVkFoI9f/IiaZ5UbMVvz4rV6zyOAjpER+gY2egSVVEN1VEDEfSEXtAbeteetVftQ/ucli5pec8B+gPt6xcp2J4l</latexit><latexit sha1_base64="+JDd5IBS1cOL+nL5xl+4j8SboSE=">AAACFnicbVDLSsNAFJ34rPEVdekmWApuWhLfy6KbLivYB7YxTKaTdujkwcyNUEK/wo2/4saFIm7FnX/jpM2ith4YOHPOvdx7jxdzJsGyfrSl5ZXVtfXChr65tb2za+ztN2WUCEIbJOKRaHtYUs5C2gAGnLZjQXHgcdryhjeZ33qkQrIovINRTJ0A90PmM4JBSa5RLp13IbL1En1Iy6wGY70bYBh4Xno/di+UNfM91V2jaFWsCcxFYuekiHLUXeO724tIEtAQCMdSdmwrBifFAhjhVA1LJI0xGeI+7Sga4oBKJ52cNTZLSumZfiTUC8GcqLMdKQ6kHAWeqsyWlPNeJv7ndRLwr5yUhXECNCTTQX7CTYjMLCOzxwQlwEeKYCKY2tUkAywwAZVkFoI9f/IiaZ5UbMVvz4rV6zyOAjpER+gY2egSVVEN1VEDEfSEXtAbeteetVftQ/ucli5pec8B+gPt6xcp2J4l</latexit>

FIG. 16. (a) Reduction of the many-body ES into a two-site ES from the commutativity of all the two-site uni-taries. The array of circles is the product state |G〉⊗Lwith |G〉 ≡ |G|− 1

2∑g |g〉. The red and blue two-site

unitaries are given by∑g,g′ ω(g−1g′, g′−1)−1|gg′〉〈gg′| and∑

g,g′ e−ih(g,g′)t|gg′〉〈gg′|, respectively. The two-site state marked

in the red rectangle reads∑g,g′ [M(t)]gg′ |gg′〉, where [M(t)]gg′

is given in Eq. (I17). (b) Many-body ES dynamics for a topologi-cal number reduction Z6 → Z2 from a partial symmetry breakingquench G = Z6 × Z6 → G = Z2 × Z2. (c) Same as (b) but forZ6 → Z3 from G = Z6×Z6 → G = Z3×Z3. In (b) and (c), blue,purple and red curves are of degeneracy 1, 2 and 3, respectively. Theexpressions j → j′ (j ∈ Z6 and j′ ∈ Z2,3) specify the group homo-morphism from H2(G,U(1)) to H2(G,U(1)). See also Table II.

We can then numerically determine the degeneracies r and rin the initial ES and that after the quench.

In Figs. 16(b) and (c), we plot the ES dynamics for twodifferent partial symmetry breaking quenchesG = Z6×Z6 →G = Z2 × Z2 and G = Z6 × Z6 → G = Z3 × Z3. Werandomly sample h(g, g′) among [−1, 1] while keeping theG-symmetry requirement (I15). The behaviors of ES splittingagree perfectly with the results in Table II. While not shownhere, we have also checked that the ES always becomes non-degenerate upon the partial symmetry breaking quench G =Z4 × Z4 → G = Z2 × Z2. This is fully consistent with thefact that the induced group homomorphism from Z4 to Z2 istrivial, as mentioned in Sec. VI A 2.

0 10 20 30 40 5010-1510-1210-910-610-3

0.1 10 1000 10510-1510-1210-910-610-3

t

ΔE

l=10

l=20

l=30

l=40

l=50

0.1 10 1000 1050

1

2

3

4

t

S E

(a)

(b)

FIG. 17. Dynamics of (a) the entanglement entropy and (b) thesingle-particle entanglement gap after a quench to the disorderedSSH model (J1) from a clean state. Inset in (b): the same as the mainpanel but in the log-linear scale. We chooseL = 2l+1 so that the en-tanglement bipartition is asymmetric and the initial entanglement gapis finite. The numbers of disorder realizations for l = 10, 20, 30, 40and 50 are 104, 5×103, 2×103, 103 and 5×102, respectively. Theparameters are quenched as (J , J ′, f) = (0.5, 1, 0)→ (1, 0.5, 0.6).Note that the early-time data of ∆sp

E for l = 50 (only a part of whichis visible) are not reliable due to a finite numerical resolution.

Appendix J: Numerical simulations for disordered systems

In this appendix, we provide some numerical pieces of evi-dence to support the qualitative discussions in Sec. VI B.

1. Disordered SSH model

We consider the entanglement dynamics in a disorderedSSH model described by the Hamiltonian

H = −∑j

(Jjb†jaj + J ′ja

†j+1bj + H.c.), (J1)

where aj and bj denote the sublattice fermionic modes in thejth unit cell, and the hopping amplitudes

Jj ∈ [(1−f)J , (1+f)J ], J ′j ∈ [(1−f)J ′, (1+f)J ′] (J2)

are uniformly and independently sampled. We start from atopological state with (J , J ′, f) = (0.5, 1, 0) (no disorder)and quench the parameters to (J , J ′, f) = (1, 0.5, 0.6). Thelength of the subsystem is chosen to be l = 1

2 (L − 1) so thatthe entanglement gap in the initial state becomes nonzero dueto the asymmetric entanglement bipartition.

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L=12

L=10

L=8

L=6

L=4

0.1 100 105 10801234567

t

S E

0.1 100 105 10810-10

10-7

10-4

10-1

t

ΔE

(a)

(b)

FIG. 18. Dynamics of (a) the half-chain entanglement entropy and(b) the many-body entanglement gap starting from the Z2 ×Z2 SPTMPS (J3) and evolving in time according to the disordered Hamil-tonian given in Eq. (J7). The numbers of disorder realizations forL = 4, 6, 8, 10 and 12 are 5 × 104, 104, 5 × 103, 5 × 102 and3× 102, respectively. The parameters in Eqs. (J4) and (J7) are set tobe p = q = 0.49, J0 = 3 and κ = 3.

As shown in Fig. 17(a), we find that the entanglement en-tropy grows logarithmically — a feature usually associatedwith many-body localized systems. Since there is no inter-action in our model, one may naıvely expect that the entan-glement grows extremely slowly like ln ln t [194], as mightbe inferred from a dynamical version of the strong-disorderrenormalization group [154]. However, there is a crucial dif-ference in our setup — the intial state |Ψ0〉 is not a productstate (in real space) and has a nonzero correlation length. Wedenote U0 as a local unitary such that U0|Ψ0〉 becomes a prod-uct state; then the quench dynamics in this new frame is gov-erned by U0HU

†0 (H is given by Eq. (J1) with the postquench

parameters), which now involves small but finite long-rangehoppings. We expect this effect to dramatically change thecommon paradigm of entanglement growth in Anderson insu-lators. On the other hand, the entanglement entropy eventuallybecomes saturated at a constant independent of (sufficientlylarge) l, as a manifestation of a finite localization length.

In stark contrast to the slow growth of entanglement en-tropy, the numerical results (see Fig. 17(b)) suggests an expo-nentially fast growth of the single-particle entanglement gapbefore saturation, which is similar to the clean case. Never-theless, due again to the finiteness of the localization length,the saturation value decreases exponentially with respect tol, implying that the topological entanglement edge modes arestable even in the long-time limit.

2. Phenomenological model for many-body localization

The minimal group that supports an interacting SPT phaseprotected by unitary symmetries is G = Z2×Z2 = (m,n) :m,n ∈ Z2, whose second cohomology group readsH2(Z2×Z2,U(1)) = Z2. By specifying the on-site symmetry as theregular representation ρ(m,n) = Zm ⊗Zn with Z = |0〉〈0| −|1〉〈1| acting on a local qubit, we can construct a nontrivialMPS as

|Ψ0〉 =∑

js=00,01,10,11Tr[Aj1

Aj2...AjL ]|j1j2...jL〉, (J3)

where a local state |j〉 consists of two qubits and the injectivetensor Aj is given by

A00 =√

(1− p)(1− q)σ0, A01 =√q(1− p)σx,

A10 = i√p(1− q)σy, A11 =

√pqσz,

(J4)

with p, q ∈ (0, 1). We can check that the projective represen-tation on the virtual level is nontrivial:

V(0,0) = σ0, V(1,0) = σx,

V(0,1) = σy, V(1,1) = σz.(J5)

To slightly lift the exact four-fold degeneracy in the ES of afinite segment, we choose p = q = 0.49 in Eq. (J4).

To simulate the effect of many-body localization, we recallthat a fully many-body localized Hamiltonian can generallybe written as

HMBL =∑j

hjτzj +

∑j,j′

Jjj′τzj τ

zj′ + · · · , (J6)

where τzj = UlocZjU†loc is the spin operator of the logic qubit

which is related to the physical one by a local unitary trans-formation [151]. Inspired by this phenomenology, we expectthat an Ising Hamiltonian with random and exponentially de-cay long-range interations,

H ′ =∑j<j′

Jjj′ZjZj′ , Jjj′ ∈ [−J0e−κ(j′−j), J0e

−κ(j′−j)],

(J7)where Jjj′’s are sampled uniformly and independently, wouldbe sufficient to capture the essential physics of the dynami-cal interplay between many-body localization and SPT order.Note that this Ising Hamiltonian (J7) respects the Z2 × Z2

symmetry.Figure 18(a) shows that the half-chain entanglement en-

tropy of the state |Ψt〉 = e−iH′t|Ψ0〉 essentially follows a

logarithmic growth. The separation between two successivelocal peaks is very close to eκ (as a multiple, since we takethe logarithmic scale), consistent with the argument that thelogarithmic growth of entanglement entropy results basicallyfrom the decoherence of remote spins [155]. In stark con-trast with the previous noninteracting case, the entanglemententropy eventually becomes saturated at an extensive quantity(i.e., proportional to L). Regarding the growth of the many-body entanglement gap, the numerical results in Fig. 18(b)

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seem to suggest a power law. This result is to some extentexpected once we accept that the entanglement gap in an SPT

state is bounded essentially by the correlation at the subsystemlength scale, which we have proved for clean systems.

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