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Exact projector Hamiltonian, local integrals of motion,

and many-body localization with symmetry-protected topological order

Takahiro Orito1,2,∗ Yoshihito Kuno3,4,∗ and Ikuo Ichinose11Department of Applied Physics, Nagoya Institute of Technology, Nagoya, 466-8555, Japan

2Department of Physics, Graduate School of Science, Hiroshima University3Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan and

4Department of Physics, Graduate School of Science, Tsukuba University

(Dated: June 16, 2020)

In this work, we construct an exact projector Hamiltonian with interactions, which is given by asum of mutually commuting operators called stabilizers. The model is based on the recently studiedCreutz-ladder of fermions, in which flat-band structure and strong localization are realized. Thesestabilizers are local integrals of motion from which many-body localization (MBL) is realized. Allenergy eigenstates are explicitly obtained even in the presence of local disorders. All states are MBLstates, that is, this system is a full many-body localized (FMBL) system. We show that this systemhas a topological order and stable gapless edge modes exist under the open boundary condition. Bythe numerical study, we investigate stability of the FMBL and topological order.

PACS numbers:

I. INTRODUCTION

Topological order and many-body localization (MBL)are most intensively studied topics in condensed mat-ter physics in the last decade[1–3]. In particular, cer-tain studies suggest that in some systems, all eigenstatesare MBL including those at high energies [4, 5]. Thisphenomenon, which is called full MBL (FMBL), is pro-tected from disturbance due to the existence of localintegrals of motion (LIOMs)[6–14]. Furthermore, someFMBL systems have topologically-ordered states at allenergies [3, 5, 15–17] and under the open boundary con-dition (OBC), all states exhibit degeneracy due to thepresence of gapless edges modes, which exist under theOBC. Actually, the above properties were numericallyverified for some typical topological models [18–20]. Fur-thermore, the above properties, which concern to all en-ergy eigenstates, are closely related to a concept in themodern quantum information called stabilizer [21]. Re-cently, study on LIOMs in topologically-ordered localizedstates was given, in which a notion of topological LIOMs(tLIOMs) was proposed [22].

Stabilizers are mutually commuting local operators,and if Hamiltonian consists of terms including only thestabilizers, this Hamiltonian is called projector Hamilto-nian. Obviously, such Hamiltonian commutes with allthe stabilizers, and the stabilizers play a role of LIOMsif the system is FMBL [4, 5].

The above idea might be applicable for various in-teresting systems, e.g., majorana-fermion system [23–25]. However, LIOMs, stabilizers and local gapless edgemodes can be constructed only perturbatively except forsome toy models [23, 26]. Therefore, discovery of nontriv-ial models, in which LIOMs, stabilizers and edge modes

∗These two authors contributed equally.

can be constructed explicitly, is certainly welcome. Inthis work, we shall present one of them based on a flat-band system.

This work is also motivated by recent theoretical stud-ies on a fermion system in the Creutz-ladder [17, 27–35],which is to be realized by cold atomic gases [36] and canbe constructed in photonic crystals [37]. With specific in-ter and intra-chain hopping amplitudes, the system hasa flat-band dispersion and all eigenstates are localizedwithin a plaquette (FMBL). Moderate interactions be-tween fermions do not change this localization proper-ties, in particular, at low fillings. In Ref. [17], we showedthat the Hamiltonian of the Creutz-ladder fermion can berewritten in terms of linear-combination operators of theoriginal fermions, and the transformed Hamiltonian hasonly local terms in terms of these operators. And very re-cently, some theoretical works have reported the presenceof the MBL in some flat-band systems with interactions[38–41]. In this paper, we show that composites of thenew operators, which appear in the Hamiltonian, are sta-bilizers, and they form LIOMs. This finding presents anew point of view for the localization properties of theCreutz-ladder model.

The original Creutz-ladder model is understood as aprojector system rather straightforwardly. In this work,we extend the model in a nontrivial way by adding cer-tain specific interactions. Even in this case, the systemdoes not lose properties of the projector Hamiltonian. Weexplicitly construct stabilizers and study the structure ofthe energy eigenstates. The Creutz ladder model rep-resented by the stabilizers also have some symmetries,these symmetries seems to be related to the structureof the energy eigenstates and to the gapless edge modeunder the OBC. This study sheds light on perturbativeconstruction of LIOMs proposed in Ref. [4]. In additionto analytical study on the models, numerical study isquite useful to reveal physical properties of models ex-tended by adding extra hopping terms, generic interac-

2

tions between fermions, etc. Stability of the topologicalnature and MBL can be investigated by calculating var-ious quantities.This paper is organized as follows. In Sec. II, we ex-

plain the properties of the original Creutz-ladder modelas the projector Hamiltonian, and explicitly constructgapless edge modes. Then, we introduce interactions,with which non-trivial entanglements between fermionsare generated. We show that stabilizers can be con-structed explicitly. By using these stabilizers, we studyenergy eigenstates in detail. Topological order parameteris given explicitly. Section III gives results of numericalstudy on the models in Sec. II. Existence of the gaplessedge modes is clearly shown for the non-interaction case,and effects of extra hopping terms, which break the topo-logical nature of the system, are also studied. Then, westudy non-ergodic and localized properties of interact-ing models by investigating time evolution of the system.Section IV is devoted for discussion and conclusion.

II. MODEL AND LIOMS

A. Creutz-ladder model as projector system

In this section, we introduce a target model and explainits stabilizer structure. Although we can directly intro-duce the stabilizer system, we would like to explain itsphysical background. To this end, we start with the sim-ple Cruetz ladder model, Hamiltonian of which is givenas follows,

HCL =∑

j

[

− iτ1(a†j+1aj − b†j+1bj)

−τ0(a†j+1bj + b†j+1aj) + h.c.]

, (1)

where aj and bj are fermion annihilation operator at sitej, and τ1 and τ0 are intra-chain and inter-chain hop-ping amplitudes, respectively. The shematic figure of theabove model is shown in Fig. 1. In the case of τ1 = τ0,the system has flat-band dispersions. Furthermore, allenergy eigenstates are localized in a plaquette. To seethis, it is useful to introduce the following operators,

wA,j = (aj + ibj)/√2, wB,j = (aj − ibj)/

√2, (2)

where the above w-particles satisfy the fermionic com-

mutation relations, {wα,j , w†β,ℓ} = δαβδjℓ, etc. By using

(wA,j , wB,j) for Hflat ≡ HCL|τ1=τ0 ,

Hflat =∑

j

2[

− iτ0w†A,j+1wBj + iτ0w

†BjwA,j+1

]

, (3)

and therefore,

Hflatw†Aj |0〉 = 2iτ0w

†B,j−1|0〉,

Hflatw†Bj |0〉 = −2iτ0w

†A,j+1|0〉. (4)

FIG. 1: Creutz ladder model: the yellow shade regime is aunit cell, the blue and magenta shade objects are compactlocalized states W+ and W− defined by Eq.(6).

Eq. (4) implies that the w-particle changes its speciesby the nearest-neighbor (NN) hopping, and its locationfluctuates around its original position by successive hop-pings. The Hamiltonian Hflat can be diagonalized by thefollowing operators,

W+j =

1√2(−iwA,j+1 + wB,j),

W+†j =

1√2(iw†

A,j+1 + w†B,j),

W−j =

1√2(−iwA,j+1 − wB,j),

W−†

j =1√2(iw†

A,j+1 − w†B,j), (5)

W+j =

1

2(−iaj+1 + bj+1 + aj − ibj),

W−j =

1

2(−iaj+1 + bj+1 − aj + ibj), (6)

where it is easily to prove that there is one-to-one cor-respondence between (aj , bj) and (W+

j ,W−j ) under the

periodic boundary condition (PBC), and (W+j ,W

−j ) sat-

isfy the fermionic commutation relations,

{Wα†j ,W β

k } = δα,βδjk, (α, β = ±), etc. (7)

Under the PBC, we have∑

j [W+†j W+

j + W−†j W−

j ] =∑

j(a†jaj + b†jbj).

In terms of the W -particle, the Hamiltonian Hflat inEq. (3) is diagonalized

Hflat =∑

j

[−2τ0W+†j W+

j + 2τ0W−†j W−

j ]. (8)

From Eqs. (8) and (7), it is obvious that Hflat is a pro-jector Hamiltonian (stabilizer Hamiltonian), i.e.,

[Hflat,W+†j W+

j ] = [Hflat,W−†j W−

j ] = 0,

[Wα†j Wα

j ,Wβ†k W β

k ] = 0, (α, β = ±). (9)

We define stabilizers, K±0j , as

K+0j =W+†

j W+j , K−

0j =W−†j W−

j , (10)

3

with properties

[Kα0j ,K

β0k] = 0, (α, β = ±), (11)

and then, all energy eigenstates of the Hamiltonian Hflat

are obtained from the eigenstates of K±0j with eigenvalue

0 or 1.It is straightforward to generalize the above system

by introducing random hopping amplitudes by τ0 → tj .Hamiltonian of the random-hopping system is given by

Hrh =∑

j

tj [−2W+†j W+

j + 2W−†j W−

j ]. (12)

Although the flat-band character is destroyed by the ran-dom hopping, the system Hrh in Eq. (12) is still solvable,and all the states are localized.Here, let us examine symmetries of the Hamilto-

nian Hrh [Eq. (12)]. In general, Hamiltonian of non-

interacting fermions, CJ = (cJ , c†J)

t, has the following

form: H =∑

J,K C†JHJKCK , where HJK is a (2 × 2)-

matrix, and indices J,K refer to lattice site and speciesof fermions if there exist. System’s symmetry is defined

by a unitary matrix ΣJK such as∑

J,K Σ†IJHJKΣKN =

±HIN . System of non-interacting ferimons is classifiedby means of the above symmetries. In the following, weexpress ΣJK by showing the transformation of (aj , bj)under ΣJK . There are three non-trivial symmetries. (i)First one is closely related to topological nature of thesystem,

aj → ibj, bj → −iaj, (13)

which induces W±j → W∓

j , and Hrh → −Hrh. As thetime-reversal symmetry is broken by the complex hop-pings in the Hamiltonian, the above symmetry (13) re-veals that Hrh belongs to the AIII class of the topologicalinsulator. In later discussions, we shall consider addi-tional terms. The above symmetry plays an importantrole there. (ii) Second symmetry includes the spatial in-version transformation I, which induces, e.g., under theperiodic boundary condition (PBC), j → −j,

aj → IbjI = b−j bj → IajI = a−j . (14)

The transformation (14) induces W±j → ±W±

−j andHrh → Hrh for the case of tj = t−j . The transformation(14) is nothing but the spatial π-rotation of the Creutzladder system. (iii) Third one is the following particle-hole type transformation,

aj → (−)jib†j, bj → (−)jia†j, (15)

which induces

W+j → −(−)jW−

j

†, W−

j → −(−)jW+j

†,

and Hrh → Hrh. It is rather straightforward to see thatthe spectrum of Hrh in Eq. (12) is doubly degenerate bythe symmetry (15) [under the PBC].

Among the above symmetries, the first one plays animportant role for the topological nature, and thereforewe comment on it rather in detail. For the one-particlestate ψJ with eigenenergy ǫn,

∑

J HIJψJ = ǫnψI , thestate

∑

J ΣIJψJ is also energy eigenstate with energy−ǫn. This means that the quantum Hamiltonian, Hrh,has the ‘symmetry’ defined by the following unitary op-

erator Λ: Λ†ajΛ = ib†j, Λ†bjΛ = −ia†j, Λ†a†jΛ =

−ibj, Λ†b†jΛ = iaj, and

Λ†H∗rhΛ = Hrh. (16)

As Eq. (16) shows, corresponding to the above single-particle symmetry, for many-body state |n〉 with energyEn, Hrh|n〉 = En|n〉, there exists a state with the sameenergy En, given by Λ†|n∗〉, i.e., HrhΛ

†|n∗〉 = EnΛ†|n∗〉.

For the state |n〉 with particle number N , the total par-ticle number of Λ†|n∗〉 is Df − N , where Df is the di-mension of the Hilbert space. Topological invariant canbe constructed from the above properties of the AIII in-sulator. In Ref. [42], it is show that topological nature ofthe AIII insulator is carried over to interacting systemsif the system maintains the above symmetry of the states|n〉 and Λ†|n∗〉. Topological invariant can be constructedby Green function methods for a translationally invariantsystem.We can also add the interactions to Hrh such as

HI =∑

cjk(W+†j W+

j )(W+†k W+

k ) + · · · , (17)

where the model is still solvable as a ‘classical system’

with variables (Wα†j Wα

j ) = ±1. For example, interac-

tions, which respect the symmetries under (13)–(15) (upto irrelevant constants), give

∑

j

[

W+†j W+

j +W−†j W−

j

]2

=1

2

∑

j

[

a†j+1aj+1b†jbj + a†jajb

†j+1bj+1

]

+(scattering terms) (18)

where (scattering terms) =∑

j(a†j+1a

†jbj+1bj), etc.

which induce scattering of a and b-particles.From the above construction of the model,

Htot = Hrh +HI,

there exists a Z2 topological order parameter such as

Z2 = (−1)∑

j(W+†j

W+

j+W

−†j

W−j). (19)

Here, since [Z2, Htot] = 0 holds, both Htot and Z2 canbe diagonalized simultaneously. It should be remarkedthat Z2 operator is invariant under the transformations(13)–(15). As we show shortly, gapless edge modesemerge under the open boundary condition (OBC), andthese modes generate degenerate states with different

4

Z2-quantum number, Z2 = ±1. We can also define astring-order parameter, Ost(i, k), which is a hallmarkof the topological state. It is given by Ost(i, k) =

(−1)∑

kj=i

(W+†j

W+

j+W

−†j

W−j

), and again Ost(i, k) is invari-ant under the transformations (13)–(15). In a specificstate and disorder realization, the string-order parame-ter takes random values Ost(i, k) = ±1.In particular, the above Z2 order parameter plays an

important role for the system with the OBC. In this case,

Z2 → (−1)∑L

j=0(a†

jaj+b

†jbj). It is interesting to see if gap-

less edge modes appear in the system Htot under theOBC as a hall mark of the topological state. We expectthat the edge modes are fermionic, and then, the degen-erate states belong to the different Z2 sectors. To seethe edge modes, we consider the ladder of length L withj = 0, 1, · · · , L, and define

Hop =

L−1∑

j=0

tj [−2W+†j W+

j + 2W−†j W−

j ]

+

L−1∑

j,k=0

cjk(W+†j W+

j )(W+†k W+

k ) + · · · . (20)

From the fact that Wαj is an operator defined on the pla-

quette with j-th and (j + 1)-th rungs, W±−1 → 1

2 (−ia0 +b0) as a−1 = b−1 = 0 due to the present OBC. Then, an

edge operator is expected to be G0 ≡ (a0 + ib0)/√2 =

ωA,0. Here, since {G†0, G0} = 1, (G0)

2 = 0, G0 isfermionic. Furthermore, even for the interacting model ofHop, this operator G0 satisfies four criteria of the gapless(fermionic) edge mode [23, 26]: (I) The G0 commuteswith Hop, [Hop, G0] = 0, (II) The G0 anti-commutes

with Z2, (III) {G†0, G0} = 1, (IV) G0 acts on the edge

portion of the system. Especially, the criterion (II) guar-antees that G0 is a mapping operator between even andodd sectors characterized by the Z2 operator. From theabove, it is obvious that G0 represents the gapless edgemode dictated by the topological order. Similarly for thej = L edge, W±

L → 12 (aL − ibL) as aL+1 = bL+1 = 0,

and then GL = (aL − ibL)/√2 = ωB,L forms the gapless

edge mode. By the analytical calculation with Hop inEq. (20), we can explicitly verify the above conclusion.However, careful examination is required to see if the

above operators of the gapless modes under the OBCreally produce physically meaningful states. Operation

of G0, G†0 to the energy eigenstates may vanish them. In

fact, the practical calculation shows,

G0W+†0 |0〉 = G0W

−†0 |0〉 = G0W

+†0 W−†

0 |0〉 = 0, (21)

where |0〉 is the empty state. On the other hand, ap-

plication of the creation operator G†0 produces the state

G†0|0〉 = 1√

2(a†0 − ib†0)|0〉 and

G†0W

+†0 |0〉 = 1√

2[(a†0 − ib†0)(ia

†1 + b†1) + 2ia†0b

†0]|0〉, (22)

and similarly for G†0W

−†0 |0〉 and G†

0W+†0 W−†

0 |0〉. All thestates reside on the j = 0 plaquette are doubly degener-ate, and these degenerate states have different Z2-valueswith each other.In the later study by means of the numerical meth-

ods in Sec. III, we consider additional hopping terms[Eq. (38)]. Among them, first one [HIR1] respects thesymmetry under Eq. (13) [and also Eq. (14)], whereassecond [HIR2] does not. In both cases, G0 is not the gen-uine gapless edge mode. However in the first case, wecan construct a gapless mode whose leading contributioncomes from G0. Therefore, the present system can have

a symmetry protected topological state.From the view point of MBL, K±

0j is nothing but LI-OMs, the existence of which indicates that all energyeigenstates are localized. In the present system, inclu-sion of arbitrary hoppings of a and b-particles break theabove LIOM structure, but it is plausible to expect thatthe system maintains its localization properties as in theordinary one-dimensional systems. Our recent study onthe Creutz-ladder revealed that the inclusion of randomon-site potentials induces a crossover from the flat-bandlocalization to the Anderson-type localization as a resultof the breaking of the above LIOM picture.It is interesting to extend the above discussion on the

projector Hamiltonian with LIOMs to interacting sys-tems that have non-trivial LIOMs. Stability of MBLcoming from LIOMs, then, can be investigated. Thiswill be studied in the following subsection.

B. Interacting models

In Sec. IIA, we considered the genuine flat-bandCreutz-ladder model of fermions. There, the stabiliz-ers {Kα

0,j} have only single-plaquette entanglements, andtherefore, each energy eigenstate is a simple productstate. From this aspect, {Kα

0,j} have properties of ℓ-bitoperators (local-bit operators) [3–5].In this subsection, we shall introduce a projector

Hamiltonian with non-trivial interactions and entangle-ments. The interaction, which we consider, is given asfollows,

HN =∑

j

gNj−1[W+†j W−

j +W−†j W+

j ], (23)

where Nj ≡ [W+†j W+

j +W−†j W−

j ] = K+0j +K−

0j, and g isan arbitrary parameter. Total Hamiltonian of the targetmodel is HT = Hrh + HN. The reason to consider theinteraction HN in Eq. (23) will be explained shortly.We shall show that the system HT is a projector

Hamiltonian with topological order. We first seek stabi-lizers by extending Kα

0j of the Creutz-ladder model. Tothis end, we calculate the commutator between Kα

0j andHN to obtain

[K+0j , HN] = gNj−1{W+†

j W−j −W−†

j W+j }. (24)

5

On the other hand, we introduce the following operatorsOj ,

Oj = Nj−1{W+†j W−

j +W−†j W+

j }, (25)

and practical calculation shows

[Oj , Hrh] = 4tjNj−1{W+†j W−

j −W−†j W+

j }. (26)

From Eqs (24) and (26), we can prove

[K+0j −

g

4tjOj , HT ] = 0. (27)

Similar calculation to the above shows

[K−0j +

g

4tjOj , HT ] = 0. (28)

Then we define operators Kαj (α = ±) as,

K+j ≡ K+

0j −g

4tjOj ,

K−j ≡ K−

0j +g

4tjOj . (29)

Practical calculation shows that Kαj (α = ±) commute

with each other, [Kαj ,K

βk ] = 0, and therefore, they are

stabilizers. Finally, we can prove

HT =∑

j

2tj(−K+j +K−

j ). (30)

Here we should stress that the above results are satisfiedexactly for arbitrary tj ’s and g, as a result of the specificchoice of the interaction HN in Eq. (23) [43]. More com-plicated systems with Hamiltonian composed of productsof the stabilizers are also solvable. But they may exhibitdifferent behaviors from HT in Eq. (30) concerning todephasing and temporal developing of entanglement en-tropy, etc., if non-local (long-range) terms of the stabi-lizers exist in the Hamiltonian.

C. Eigenstates and topological order

In this section, we study the energy eigenstate of theHamiltonian HT in Eq. (30). This study is useful forlater discussion on the topological order, in particular,existence of gapless edge modes. Energy eigenstates areconstructed from eigenstates of the stabilizers as a prod-uct state.To investigate eigenstates of the stabilizer somewhat

in detail, let us study quantum states residing on site(plaquette) j as a concrete example. From Eqs. (25) and(29), the states at site j have non-trivial entanglementwith the state at site (j−1). We first introduce notationsfor the states;

W+j |0〉+ = 0,W+†

j |0〉+ = |1〉+,W−

j |0〉− = 0,W−†j |0〉− = |1〉−. (31)

For the case of 〈Nj−1〉 = 0, Kαj = Kα

0j = Wα†j Wα

j , and

then, the eigenstates of K+j are |0〉+ and |1〉+ with eigen-

value 0 and 1, respectively. Similarly for K−j . In partic-

ular, the 〈Nj〉 = 1 sector such as Nj |ψ〉 = |ψ〉 is spannedby the states {|1〉+|0〉−, |0〉+|1〉−}.On the other hand, the cases of 〈Nj−1〉 = 1 and 2 are

non-trivial. Let us set the stabilizers as

K+j = K+

0j − λ{W+†j W−

j +W−†j W+

j },K−

j = K−0j + λ{W+†

j W−j +W−†

j W+j }, (32)

where λ = g/(4tj) (g/(2tj)) for 〈Nj−1〉 = 1 (2). Eigen-states of the above stabilizers are obtained as follows forthe 〈Nj〉 = 1 sector,

ψ1 ∝ x1|1〉+|0〉− − λ|0〉+|1〉−,ψ2 ∝ λ|1〉+|0〉− + x1|0〉+|1〉−, (33)

where

K+j ψ1 = x1ψ1, K+

j ψ2 = x2ψ2,

K−j ψ1 = x2ψ1, K−

j ψ2 = x1ψ2, (34)

with x1 = 12 (1 +

√1 + 4λ2) and x2 = 1

2 (1 −√1 + 4λ2). The above result obviously indicates that

unitary rotation in the two-dimensional vector space{|1〉+|0〉−, |0〉+|1〉−} → {ψ1, ψ2} is induced by the inter-action. This unitary rotation is given by operator suchas

U(θ) = eθ(W+†W−−W−†W+), (35)

with a suitable angle θ. Here, [W+†W− −W−†W+)]2 =−1 for the present sector [Nj = 1], and therefore

U(θ) = cos θ + sin θ · (W+†W− −W−†W+). (36)

From Eqs. (33) and (36), value of θ is determined forψ1(2), i.e., tan θ = λ/x1. In Ref. [4], methods of con-structing LIOMs was discussed. There, unitary rotationof subsystem plays a central role. The above discussionshows that the present system is one of examples of theLIOMs construction given in Ref. [4].Finally for the sectors with 〈Nj〉 = 2 and 〈Nj〉 = 0, the

expression of the stabilizers K±j in Eq. (32) shows that

the λ-dependent terms are irrelevant, and the eigenstatesof the stabilizers are simply {|1〉+|1〉−, |0〉+|0〉−}.Let us turn to the system with the OBC, j = 0, · · · , L.

In this case, Hamiltonian of the OBC is derived fromHT ,and for the j = 0 boundary, the corresponding term isgiven by 2t0(−K+

0j=0 +K−0j=0) as 〈N−1〉 = 0. Then, the

discussion on the edge modes given in the previous sec-tion is directly applied to the j = 0 boundary and we findthat all states in the j = 0 plaquette are doubly degener-ate. On the other hand for the j = L− 1 plaquette, thefull stabilizers appear in the Hamiltonian HT . By fixingthe W -particle number at j = L − 2, the above discus-sion on the eigenstates can be applied to the j = L − 1

6

plaquette. There, we show that the existence of the in-teraction term induces the rotation in the {W+,W−}operator plane for the 〈NL−1〉 = 1 case. Then by usingthe rotated W ’s, similar argument to the case of g = 0can be applied to the interacting case. However fromEq. (5), a rotation of {W+,W−} means simply a changeof relative weight of wA,j+1 and wB,j . This indicates that

the edge mode GL = (aL − ibL)/√2 in the g = 0 case

also works in the interacting case. Practical calculationshows,

{G†L,W

+L−1} = {G†

L,W−L−1} = 0,

then, degenerate states are obtained by applying the op-

erator G†L to the states at j = L.

If we introduce interactions expressed in terms of K+j

and K−j such as

∑

cjkK+j K

+k + · · · , there appear corre-

lations between states residing on different sites. Even inthis case, the above gapless edge modes with the OBCsurvive. On the other hand for general interactions,which are not written in terms of K+

j and K−j , energy

eigenstates may not be constructed by simply applyingthe unitary rotation U(θ) in Eq. (35).The topological order for HT is also defined as the

previous case for Htot, and the suitable order parametercharacterizing the above topological states is a ‘general-ization’ of Eq. (19) and Ost(i, k),

Z2 = (−1)∑

j,αKα

j ,

Ost(i, k) = (−1)∑k

j=i(K+

j+K

−j), (37)

where we should note [Kαj ]

2 6= Kαj in general, but from

Eq. (29), K+j +K−

j = K+0j +K−

0j =W+†j W+

j +W−†j W−

j .

Z2 operator also commutes with HT , [Z2, HT ] = 0. Asthere exist zero-energy edge modes, each eigenstate aredoubly degenerate for each Z2-sector. Furthermore, evenif we add interactions expressed in terms of K+

j and K−j

to HT , Ost(i, k) operator acts as Z2-topological orderparameter, and Z2-topological order exists.Finally, let us comment on the symmetries of the in-

teracting model. It is not so difficult to show that thesystem, HT [Eq. (30)], has properties similar to the non-intercation one which is explanied in Sec. II.A. In fact asshown in the above, (−K+

j +K−j )ψj1 = (−x1 + x2)ψj1,

whereas (−K+j + K−

j )ψj2 = (x1 − x2)ψj2. The empty

state |0〉 degenerates with the total full-state, |F〉 =∏

all j ψj1ψj2, and the lowest-energy state (for the case

tj > 0) is the half-filled state |LE〉 =∏

all j ψj1|0〉j−.(We expect that notations are self-evident.) Therefore,the interacting systems, HT , carry over the properties of

the AIII class insulator, and the symmetry is Z2 × Zf2 ,

where Zf2 is the fermion-number parity. More precisely,

there exists a counterpart |n∗〉 for each energy eigenstate|n〉, but their exact degeneracy is broken by the interac-tion. However, the chiral properties of ψJ and

∑

J ΣIJψJ

in the single-particle system are carried over to the in-teracting system, i.e., for each |n〉 with En, there exists

|n〉 with −En. We expect that this Z2 symmetry plays asimilar role as the single-particle Z2 symmetry, and as aresult, there exist the gapless edge modes and the nonlo-cal string order as we saw in the above. According tothe group super-cohomology classification of fermionicsymmetry protected topological (SPT) phases [44, 45],the present interacting models have the Z2-SPT phasesin the spatial one-dimension. Detailed numerical studiesare welcome to verify this observation.

III. NUMERICAL STUDIES

In Sec. II, we introduced models and studied theirphysical properties concerning MBL and topological or-der. In this section, we shall study the target systems indetail by employing numerical exact diagonalization [46].

A. Edge modes in random Creutz-ladder models

We first study the edge modes existing in the randomCreutz-ladder model with the OBC. We calculate the en-ergy eigenvalues of the model Eq. (12) by fixing randomhoppings {tj} and verify that there are two gapless edgemodes. It is interesting to see (in)stability of the edgemodes in the existence of other type of hopping terms. Tostudy this problem, we consider the following two kindsof inter-chain hopping,

HIR1 = v1∑

j

(a†jbj + b†jaj)

= v1i∑

j

(w†A,jwB,j − w†

B,jwA,j),

HIR2 = v2i∑

j

(a†jbj − b†jaj)

= v2∑

j

(w†A,jwA,j − w†

B,jwB,j), (38)

where v1 and v2 are real parameters. We use the wA,B-particle representation as the gapless edge modes in theOBC are nothing but G0 = wA,0, GL = wB,L, and there-fore, the topological nature of the system becomes clearin the description by the wA,B-particle. We call the sys-tem given by the Hamiltonian Hrh + HIR1,2 extendedCreutz-ladder model, where Hrh in Eq. (12) is expressed

as Hrh =∑

j 2tj

[

− iw†A,j+1wBj + iw†

BjwA,j+1

]

. In the

practical calculation, we sometimes employ spin-1/2 op-

erators as w(†)A(B),j → S

−(+)A(B)j , n

A(B)j − 1/2 → Sz

A(B),j

related by a Jordan-Wigner (JW) transformation. Asnumerical bases, it is sufficient to use the product statesof spin-z component base for each sites. Since in the w-particle Hamiltonian under the JW transformation, thetotal spin-z component in the system is conserved. Then,in practical calculation, we fix the total spin-z componentto reduce the numerical cost.

7

2 4 6 8

state

� 12.0

� 11.5

� 11.0

� 10.5

� 10.0

energy

u=0u=3

FIG. 2: Single-shot disordered many-body spectra for differ-ent fillings, the data show the many-body energy from thelowest to the eighth states. Here, 2tj = 1 + δj with the uni-form distribution random values δj ∈ [−0.5, 0.5]. (a) u = 0.There exist four-fold degenerate states with different particlenumbers indicating the existence of zero-energy edge modes.(b) u = 3. The four-fold degeneracy is broken. The systemsize is L = 14.

We display the energy eigenvalues for the case of theuniform and random hoppings in Appendix A. For theuniform-hopping case, addition of HIR1 destroys the flat-band structure, but gapless edge modes survive. Onthe other hand by adding HIR2, the flat-band struc-ture remains but gapless edge modes split into two in-gap edge modes with energy ±∆ ∝ ±v2 [35]. Forthe random-hopping case, similar results are obtained,in particular, for the gapless edge modes. In fact inthe presence of HIR1, the gapless edge modes are ob-tained perturbatively. It is straightforward to verify that

G0 = ωA,0 +vit0ωA,1 +

v21

t0t1ωA,2 + · · · commutes with the

total Hamiltonian, Hrh+HIR1. In the existence of HIR1,the “chiral symmetry” in Eq. (13) is not broken and zero-energy edge modes exist. Therefore in the present sys-tem, the chiral symmetry protects gapless edge modes.We examine the density profile of the wA,B-particles in

the many-body wave function to verify that the gaplessmodes are really localized in the edges. In terms of themany-body wave function, the signal of the presence ofsome localized zero-energy edge modes may be a “fillinganomaly” [47, 48]. For example, for our model with theOBC and in the vicinity of the half filling, if left and rightlocalized edge modes are present in the system, L/2− 1,L/2, and L/2+ 1 particle states are energetically degen-erate. This indicates emergence of four-fold degeneratestates. Actually we demonstrate it numerically for thenon-interacting case with v1 = v2 = 0. In Fig.2 (a), theresults are for the ground-state and low-lying excitationenergies for different particle number near the half filling.From the results of the three different fillings in Fig.2 (a),four-fold degenerate states certainly emerge.Furthermore, we demonstrate how the edge modes

change by adding interactions. As shown in the previ-

ous section, even in some specific interacting case, thelocal stabilizer was found as in Eq. (30). Therefore, thedegeneracy by means of edge modes obviously exists inthat system. Here, we focus not on the specific formof Eq. (30) but on another type of interactions. In thew-particle representation, we consider the following in-teractions,

HS =∑

j

u

4

[

(nAj − 1

2)(nB

j − 1

2) + (nB

j − 1

2)(nA

j+1 −1

2)

]

,

(39)

where nA(B)j = w†

A(B),jwA(B),j . It should be remarked

here that the interactions in Eq.(39) as well as the hop-ping amplitudes in Eqs. (38) cannot be expressed in termsof the stabilizers K+

0j and K−0j , and therefore, the system

described by the total Hamiltonian loses the projector na-ture and the edge modes tend to unstable for sufficientlylarge u. Figure 2 (b) displays the typical result for it.The four-fold degenerate states existing for u = 0 splitinto three states with different particle filling for u = 3.0.Therefore, a “phase transition” to a non-topological statetakes place there, We numerically estimate the criticalcoupling, uc, as uc ∼ 2. We note that similar dynamicshave been reported in a diamond flat band system withinteractions, where fate of the localization dynamics wasstudied [49].Next, we shall examine the stability of edge modes by

studying time evolution of the states that include theedge modes. In particular, we focus on effects of theinter-chain hopping, HIR1 in Eq (38), on the stability.That is, we focus on Hflat +HIR1.As the initial state for the time evolution, we consider

the following states,

|Ψ0〉 = G†0|Φq〉, |ΨL〉 = G†

L|Φq〉, (40)

with

|Φq〉 =2qL∏

i=1

Aq|0〉, (41)

where Aq = w†A,(2q)−1(i−1)+1, and q is given by q = 1/2

for the density-wave like bulk state, which we consider inthe following. By numerical methods, we obtain the wavefunction at t, Ψ(t), and calculate the following quantityto observe the (in)stability of the edge modes,

P0(t) = 〈Ψ0(t)|G†0G0|Ψ0(t)〉 (42)

If P0(t) decreases from unity, the edge modes lose theircoherence [coherence of |a〉 and |b〉]. On the other hand, ifit keeps a value close to unity for large t, the edge modesand topological nature of the system are stable againstthe decoherence.Numerical results are shown in Fig. 3 for various val-

ues of v1. There, we calculate P0(t) and also the particledensity at typical sites in the bulk. We consider the case

8

FIG. 3: Short-time exact dynamics for the Hflat +HIR1 sys-tem. (a) P0(t) in Eq. (42). (b) Dynamics of the w-particledensity at the center of the system. System size L = 8. (c)Short-time exact dynamics of P0(t) in the presence of inter-action of HS and weak HIR1. Here, v1 = 0.1, L = 8. For alldata, the unit of time is h/(2τ0), and unit of energy 2τ0.

of the uniform hopping for the flat-band localization aswe are first interested in the stability of edge modes, be-havior of the bulk states, and effects of the interactionHS

in the genuine system. Study on the random systems is afuture work. Figure 3(a) shows that as v1 increases, thevalue of P0(t) is getting small for the initial stage of thetime evolution, but it has a fairly large value for t→ largefor all v1. This result strongly indicates the existence ofstable zero modes in the vicinity of the edges. On theother hand, Fig. 3(b) shows that the w-particle densityat the central regime of the system oscillates around 0.5for any value of v1. Hence, the dynamics of the local-ized edge states is clearly different from that of the bulkdynamics.In addition to the effects of HIR1, we investigate the

effects of the interaction of Eq. (39) with a weak but finitev1 = 0.1, i.e., the model is Hflat+HIR1+HS . The resultsare displayed in Fig. 3(c). The stability of the localizededge states is gradually lost as increasing the interactionu. On the other hand, for the u/4 = 2.5 case, the decayis somewhat weakened. Possible reason for this behavioris that the entanglement spreading is faster compared to

the other cases, and the spreading saturates in a finitesystem. Anyway, detailed analysis is interesting and thisis a future work.

B. (Non)ergodicity of interacting model

In this subsection, we shall study MBL for the systemHT [Eq. (30)] including the NN interactions. Even inthe presence of the interaction of HN [Eq. (23)], the totalHamiltonian can be described by the local stabilizer form.Accordingly, if we consider the dynamics for a certainquantum state based on this system, the information ofthe quantum state can be conserved. Here, as a typicalexample, we investigate time evolution where the initialstates is given by Eq. (41). To this end, as explained inthe previous subsection, the w-particle representation isemployed and for HT , we use

Nj = (w†A,j+1wA,j+1 + w†

B,jwB,j),

W+†j W−

j +W−†j W+

j = (w†A,j+1wA,j+1 − w†

B,jwB,j).

To study the dynamics, we calculate the expectationvalue of the particle density in the central regime of thesystem as a function of time, which is set to unity in theinitial state.Figure 4 is the result of the time evolution of the par-

ticle density at the center of the system. For Fig. 4(a),the clear and perfect oscillating behavior is observed forany value of g, which indicates the particle-density re-vivals. This means that the information of the densitydistribution at the initial state is preserved, and there-fore, the ergodicity is broken. Furthermore, we add HIR1

and investigate the effect of HN. Here, we set v1 = 0.1and employ the same initial state and calculate the sameobservable in Fig. 4 (a). The result of the dynamics isshown in Fig. 4 (b). Interestingly enough, even for thepresence of HIR1 term, the revival oscillation gets recov-ered as increasing the strength of HN. This shows thatas increasing the interaction, the nature as stabilizer ofthe projector Hamiltonian HT gets dominant in the dy-namics of the system. Strong correlations between NNplaquettes generated by HN enhances the non-ergodicity.Very recently, interesting observation concerning to oscil-lating behavior in MBL states, a quantum time crystal,was given by using notion of extensive dynamical sym-metries [50]. The above numerical results suggest thatthey exist in the present system, HT [Eq. (30)]. This isan interesting future problem.

IV. CONCLUSION

In this work, we studied the Creutz-ladder model offermions and its extensions from the view point of theprojector system. It is known that the Creutz-laddermodel has a flat-band dispersion and exhibits the strong

9

0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0

t ime

0.0

0.2

0.4

0.6

0.8

1.0

density

FIG. 4: Short-time exact dynamics of the particle density atthe center of the system. (a) Dynamics for the system HT .Data show perfect oscillating behavior of all values of g. Pe-riod of the oscillation is longer for smaller g. (b) Dynamicsfor the system HT + HIR1. Here, v1 = 0.1. Particle densityexhibits stable oscillation for larger value of g, whereas it de-creases to 1/2 for small g. For all data, L = 8. The hoppingamplitude in HT is set to be uniform: tj = τ0. The unit oftime is h/(2τ0), and unit of energy 2τ0.

localization properties for specific choice of the intra-chain and inter-chain couplings. We revealed that themodel with the above hopping amplitudes has a projec-tor Hamiltonian. The stabilizers are explicitly obtained.We showed that they play a role of LIOMs, which dic-tate localization of all energy eigenstates. Furthermore,we showed that the systems have topological nature andthere exist gapless edge modes under the OBC. Numer-ical study indicates that the above single-particle prop-erties are preserved even in the existence of interactionsup to moderate coupling constants.

Next, we generalized the Creutz-ladder model withpreserving its projector nature. The new model hasstrong correlations between particles in NN plaquettes.The stabilizers are explicitly obtained and they work onparticles in NN plaquettes. This generalized model re-tains the topological nature, and there exist gapless edgemodes. Typical energy eigenstates are analytically ob-tained, and they give an important insight on construc-tion of LIOMs in generic systems.

Finally, by numerical methods, we investigated time-evolution of the states with and without edge modes inorder to examine the (non)ergodicity of the system. Theinter-chain hopping amplitude, which breaks the chiralsymmetry, plays an important role for the time evolution.

Our findings in the present work are quite interesting

and useful for the investigation of the full localization andtopological order from the view point of the projectorHamiltonian. The present models are feasible by recentexperiments on cold atomic gases, and we hope that ourfindings will be verified by experiments. In particular,the time evolution of the systems is very interesting andimportant for the quantum information physics.

Acknowledgments

This work is supported by the Grant-in-Aid for JSPSFellows (No.17J00486).

Appendix A: Single-particle spectrum of variousextended models of Creutz-ladder fermion system

0 25 50

state

− 2.0

− 1.5

− 1.0

− 0.5

0.0

0.5

1.0

1.5

2.0

energy

0 25 50

state

0 25 50

state

0 25 50

state

(a) (b) (c) (d)

FIG. A.1: Energy spectra in the w-particle representationunder the OBC. (a) Uniform coupling 2tj = 1. Flat-banddispersion and zero-energy modes exist in the band cen-ter. (b) Single-shot disordered spectra for random coupling:2tj = 1 + δj with the uniformly-distributed random valuesδj ∈ [−0.5, 0.5]. (c) v1 = 0.3. Flat bands disappear, butzero-energy modes exist in the band center. (d) v2 = 0.3.Flat bands exist, but zero-energy modes split into inter-bandstates with finite energy.

As we explain in the main text, we study a single-particle spectrum for various extended models of theCreutz-ladder fermions. Through this investigation, wecan observe relevant terms in the Hamiltonian for pre-serving/ breaking topological character of the Creutz-ladder model with the specific hopping amplitude, τ0 =τ1.We consider the OBC, and results are shown in

Fig. A.1. Fig. A.1 (a) is the energy spectrum for thenon-random uniform hopping amplitude with τ0 = τ1.There exist two flat bands and two energy-zero modesbetween the flat bands. Figure A.1 (b) shows the resultof the random hopping amplitude, and there still existtwo gapless modes. The above two cases are analyticallystudied in the main text and the numerical results arewhat we expected. Figure A.1 (c) and (d) are results

10

for the models with uniform coupling τ1 = τ0 and in-cluding inter-chain hopping such as (v1 = 0.3, v2 = 0)and (v2 = 0.3, v1 = 0), respectively. These results areinteresting. In the case (c), there still exist two gaplessmodes in the band center. On the other hand in the case(d), two flat bands are preserved, whereas the inter-band

modes split into two gapful modes.From the above observation, we conclude that the chi-

ral symmetry given by the transformation of Eq. (13) isessential ingredient for the existence of the gapless modes.In the main text, we show that these gapless modes areconfined near the boundaries of the system.

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