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Non-Hermitian skin effect and chiral damping in open quantum systems

Fei Song,1 Shunyu Yao,1 and Zhong Wang1, ∗

1Institute for Advanced Study, Tsinghua University, Beijing, 100084, China

One of the unique features of non-Hermitian Hamiltonians is the non-Hermitian skin effect, namelythat the eigenstates are exponentially localized at the boundary of the system. For open quantumsystems, a short-time evolution can often be well described by the effective non-Hermitian Hamil-tonians, while long-time dynamics calls for the Lindblad master equations, in which the Liouvilliansuperoperators generate time evolution. In this Letter, we find that Liouvillian superoperatorscan exhibit the non-Hermitian skin effect, and uncover its unexpected physical consequences. It isshown that the non-Hermitian skin effect dramatically shapes the long-time dynamics, such thatthe damping in a class of open quantum systems is algebraic under periodic boundary condition butexponential under open boundary condition. Moreover, the non-Hermitian skin effect and non-Blochbands cause a chiral damping with a sharp wavefront. These phenomena are beyond the effectivenon-Hermitian Hamiltonians; instead, they belong to the non-Hermitian physics of full-fledged openquantum dynamics.

Non-Hermitian Hamiltonians provide a natural frame-work for a wide range of phenomena such as photonic sys-tems with loss and gain[1–5], open quantum systems[6–16], and quasiparticles with finite lifetimes[17–21]. Re-cently, the interplay of non-Hermiticity and topologicalphases have been attracting growing attentions. Consid-erable attentions have been focused on non-Hermitianbulk-boundary correspondence[22–34], new topologicalinvariants[24, 25, 27, 31, 35–42], generalizations of topo-logical insulators[25, 43–56] and semimetals[57–67], andnovel topological classifications[68–70], among other in-teresting theoretical[71–81] and experimental[82–87] in-vestigations.One of the remarkable phenomena of non-Hermitian

systems is the non-Hermitian skin effect [24, 26](NHSE),namely that the majority of eigenstates of anon-Hermitian operator are localized at bound-aries, which suggests the non-Bloch bulk-boundarycorrespondence[24, 28] and non-Bloch band theory basedon the generalized Brillouin zone[24, 25, 31, 37, 40].Broader implications of NHSE have been underinvestigations[25, 32, 33, 37, 58, 88–99]. Very recently,NHSE has been observed in experiments[100–102].In open quantum systems, non-Hermiticity naturally

arises in the Lindblad master equation that governs thetime evolution of density matrix (see e.g., Refs.[8, 9]):

dρ

dt= −i[H, ρ] +

∑

µ

(

2LµρL†µ − {L†

µLµ, ρ})

≡ Lρ, (1)

where H is the Hamiltonian, Lµ’s are the Lindblad dis-sipators describing quantum jumps due to coupling tothe environment, and L is called the Liouvillian super-operator. Before the occurrence of a jump, the short-time evolution follows the effective non-Hermitian Hamil-tonian Heff = H − i

∑

µ L†µLµ as dρ/dt = −i(Heffρ −

ρH†eff)[11, 12, 103].It is generally believed that when the system size is not

too small, the effect of boundary condition is insignifi-cant. As such, periodic boundary condition is commonly

FIG. 1. (a) SSH model with staggered hopping t1 and t2,with the ovals indicating the unit cells. The Bloch Hamil-tonian is h(k) = (t1 + t2 cos k)σx + t2 sin kσy. The fermionloss and gain are described by the dissipators Ll and Lg

[Eq.(2)] in the master equation framework. (b) A differentrealization of the same model. The hopping Hamiltonianh(k) = (t1 + t2 cos k)σx + t2 sin kσz, and the dissipators are

Llx =

√γlcxA and Lg

x =√γgc

†xA. (b) is equivalent to (a) via

a basis change σy ↔ σz. Because the gain and loss is on-site,(b) is more feasible experimentally.

adopted, though open-boundary condition is more rele-vant to experiments. In this paper, we show that thelong time Lindblad dynamics of an open-boundary sys-tem dramatically differ from that of a periodic-boundarysystem. Furthermore, this is related to the NHSE of thedamping matrix derived from the Liouvillian. Notableexamples are found that the long time damping is alge-braic (i.e., power law) under periodic boundary conditionwhile exponential under open boundary condition. More-over, NHSE implies that the damping is unidirectional,which is dubbed the “chiral damping”. Crucially, thetheory is based on the full Liouvillian. Although Heff

may be expected to play an important role, it is in factinessential here (e.g., its having NHSE or not does notmatter).

Model.–The system is illustrated in Fig.1(a). Our

Hamiltonian H =∑

ij hijc†i cj , where c†i , ci are fermion

creation and annihilation operators at site i (including

2

additional degrees of freedom such as spin is straight-forward). We will consider single particle loss and gain,with loss dissipators Ll

µ =∑

i Dlµici and gain dissipa-

tors Lgµ =

∑

i Dgµic

†i , respectively. For concreteness, we

take h to be the Su-Schrieffer-Heeger (SSH) Hamiltonian,namely hij = t1 and t2 on adjacent links. A site is alsolabelled as i = xs, where x refers to the unit cell, ands = A,B refers to the sublattice. For simplicity, let eachunit cell contain a single loss and gain dissipator (namely,µ is just x):

Llx =

√

γl/2(cxA − icxB), Lgx =

√

γg/2(c†xA + ic†xB);

(2)

in other words, Dlx,xA = iDl

x,xB =√

γl/2, Dgx,xA =

−iDgx,xB =

√

γg/2. We recognized in Eq.(2) that theσy = +1 states are lost to or gained from the bath. Aseemingly different but essentially equivalent realizationof the same model is shown in Fig.1(b), which can beobtained from the initial model [Fig.1(a)] after a basischange σy ↔ σz . Accordingly, the dissipators in Fig.1(b)are σz = 1 states. As the gain and loss are on-site, itsexperimental implementation is easier. Keeping in mindthat Fig.1(b) shares the same physics, hereafter we focuson the setup in Fig.1(a).To see the evolution of density matrix, it is

convenient to monitor the single-particle correlation∆ij(t) = Tr[c†i cjρ(t)], whose time evolution is d∆ij/dt =

Tr[c†icjdρ/dt]. It follows from Eq.(1) that (see the Sup-plemental Material)

d∆(t)

dt= i[hT ,∆(t)]− {MT

l +Mg,∆(t)}+ 2Mg, (3)

where (Mg)ij =∑

µ Dg∗µiD

gµj and (Ml)ij =

∑

µ Dl∗µiD

lµj ,

and both Ml and Mg are Hermitian matrices. Majoranaversions of Eq.(3) appeared in Refs.[8, 16, 104]. We candefine the damping matrix

X = ihT − (MTl +Mg), (4)

which recasts Eq.(3) as

d∆(t)

dt= X∆(t) + ∆(t)X† + 2Mg. (5)

The steady state correlation ∆s = ∆(∞), to which longtime evolution of any initial state converges, is deter-mined by d∆s/dt = 0, or X∆s + ∆sX

† + 2Mg = 0. Inthis paper, we are concerned mainly about the dynam-ics, especially the speed of converging to the steady state,therefore, we shall focus on the deviation ∆(t) = ∆(t)−∆s, whose evolution is d∆(t)/dt = X∆(t) + ∆(t)X†,which is readily integrated to

∆(t) = eXt∆(0)eX†t. (6)

We can write X in terms of right and lefteigenvectors[105],

X =∑

n

λn|uRn〉〈uLn|, (7)

t1=t2=1,�g=�l=0.4

t1=0.7,t2=1,�g=�l=0.6

t1=1.2,t2=1,�g=�l=0.05

t1=1.2,t2=1,�g=�l=0.1

1 10 100 t 1000

10-2

10-4

10-6

10-8

n(t)

(a)

L=150

A

B

C

D

FIG. 2. (a) The damping of fermion number towards thesteady state of a periodic boundary chain with length L = 150(unit cell). The damping is algebraic for cases A,B with t1 ≤t2, while exponential for C,D with t1 > t2. The initial state isthe completely filled state

∏x,s

c†xs|0〉. (b) The eigenvalues ofthe damping matrix X. Blue: periodic-boundary; Red: open-boundary. The Liouvillian gap of the periodic-boundary chainvanishes for A and B, while it is nonzero for C and D. Forthe open-boundary chain, the Liouvillian gap is nonzero inall four cases. This drastic spectral distinction between openand periodic boundary comes from the NHSE (see text).

and express Eq.(6) as

∆(t) =∑

n,n′

exp[(λn + λ∗n′)t]|uRn〉〈uLn|∆(0)|uLn′〉〈uRn′ |.

(8)

By the dissipative nature, Re(λn) ≤ 0 always holds true.The Liouvillian gap Λ = min[2Re(−λn)] is decisive forthe long-time dynamics. A finite gap implies exponentialconverging rate towards the steady state, while a vanish-ing gap implies algebraic convergence[106].

Periodic chain.–Let us study the periodic boundarychain, for which going to momentum space is more con-venient. It can be readily found that h(k) = (t1 +t2 cos k)σx + t2 sinkσy and

Ml(k) =γl2(1 + σy), Mg(k) =

γg2(1− σy). (9)

3

periodic

open L=30

open L=40

open L=50

open L=60

20 40 60 80t

(a)

n˜(t)

10-4

10-8

10-12

γg=γl=0.2

periodic

open x=1

open x=10

open x=20

open x=30

20 40 60t

(b)

n˜x(t)

10-3

10-7

10-11

γg=γl=0.2L=60

FIG. 3. (a) The particle number damping of a periodic bound-ary chain (solid curve) and open-boundary chains for severalchain length L. The long time damping of a periodic chainfollows a power law, while the open boundary chain followsan exponential law after an initial power law stage. (b) Thesite-resolved damping. The left end (x = 1) enters the expo-nential stage from the very beginning, followed sequentiallyby other sites. For both (a) and (b), the initial state is thecompletely filled state

∏x,s

c†xs|0〉, therefore, ∆(0) the identitymatrix I2L×2L. t1 = t2 = 1, γg = γl = 0.2.

These M(k) matrices are k-independent because the gainand loss dissipators are intra-cell. The Fourier transfor-mation of X is X(k) = ihT (−k) − MT

l (−k) − Mg(k)(the minus sign in −k comes from matrix transposition),therefore, the damping matrix in momentum space reads

X(k) = i[(t1 + t2 cos k)σx + (t2 sin k − iγ

2)σy]−

γ

2I,

(10)

where γ ≡ γl + γg. If we take the realization in Fig.1(b)instead of Fig.1(a), the only modification to X(k) is abasis change σy → σz in Eq.(10), with the physics un-changed. Diagonalizing X(k), we find that the Liouvil-lian gap Λ = 0 for t1 ≤ t2, while the gap opens for t1 > t2[Fig.2(b)]. The damping rate is therefore expected to bealgebraic and exponential in each case, respectively. Toconfirm this, we numerically calculate the site-averagedfermion number deviation from the steady state, definedas n(t) =

√∑

x n2x(t)/L, where nx(t) = nx(t) − nx(∞)

with nx(t) = nxA(t) + nxB(t), nxs ≡ ∆xs,xs being thefermion number at site xs. The results are consistentwith the vanishing (nonzero) gap in the t1 ≤ t2 (t1 > t2)case [Fig.2(a)].Although our focus here is the damping dynamics, we

also give the steady state. In fact, our Ml and Mg

satisfy MTl + Mg = Mgγ/γg, which guarantees that

∆s = (γg/γ)I2L×2L is the steady state solution. It isindependent of boundary conditions.Now we show the direct relation between the algebraic

damping and the vanishing gap of X . The eigenvalues ofX(k) are

λ±(k) = −γ/2± i√

(t21 + t22 + 2t1t2 cos k − γ2/4)− it2γ sin k. (11)

Let us consider t1 = t2 ≡ t0 for concreteness (case A inFig.2), then λ−(π) = 0 and the expansion in δk ≡ k − πreads

λ−(π + δk) ≈ −it0δk −t204γ

(δk)4. (12)

Now Eq.(7) becomes X =∑

k,α=± λα(k)|uRkα〉〈uLkα|,and Eq.(8) reads

∆(t) =∑

kk′,αα′

eλα(k)t+λ∗α′ (k

′)t|uRkα〉〈uLkα|∆(0)|uLk′α′〉〈uRk′α′ |.

(13)

For the initial state with translational symmetry,we have 〈uLkα|∆(0)|uLk′α′〉 = δkk′〈uLkα|∆(0)|uLkα′〉.The long-time behavior of ∆(t) is dominated by theα = α′ = − sector, which provides a decay factor∑

δk exp (2Re[λ−(π + δk)]t) ≈∫

d(δk) exp[−t20

2γ (δk)4t] ∼

t−1/4. Similarly, for t1 < t2 we have ∆(t) ∼ t−1/2.Chiral damping.–Now we turn to the open boundary

chain. Although the physical interpretation is quite dif-ferent, our X matrix resembles the non-Hermitian SSHHamiltonian[24, 39], as can be appreciated from Eq.(10).Remarkably, all the eigenstates of X are exponentiallylocalized at the boundary (i.e., NHSE[24]). As such,the eigenvalues of open boundary X cannot be obtainedfrom X(k) with real-valued k; instead, we have to takecomplex-valued wavevectors k + iκ. In other words, theusual Bloch phase factor eik living in the unit circle isreplaced by exp[i(k + iκ)] inhabiting a generalized Bril-louin zone[24], whose shape can be precisely calculatedin the non-Bloch band theory[24, 25, 31, 37, 40].From the non-Bloch band theory[24], we find that κ =

− ln

√

∣

∣

∣

t1+γ/2t1−γ/2

∣

∣

∣, and that the eigenvalues of X of an open

boundary chain are λ±(k + iκ), where λ± are the X(k)eigenvalues given in Eq.(11). We can readily check that,for |γ| < 2|t1|,

λ±(k + iκ) = −γ

2± iE(k), (14)

where E(k) =

√

t21 + t22 −γ2

4 + 2t2

√

t21 −γ2

4 cos k, which

is real. We have also numerically diagonalized X for along open chain [red dots in Fig.2(b)], which confirmsEq.(14). An immediate feature of Eq.(14) is that the real

4

part is a constant, −γ/2, which is consistent with the nu-merical spectrums [Fig.2(b))]. We note that the analyticresults based on generalized Brillouin zone produce thecontinuum bands only, and the isolated topological edgemodes [Fig.2(b), A and B panels] are not contained inEq.(14), though they can be inferred from the non-Blochbulk-boundary correspondence[24, 28]. Here, we focus onbulk dynamics, and these topological edge modes do notplay important roles[107].

It follows from Eq.(14) that the Liouvillian gap Λ = γ,therefore, we expect an exponential long-time dampingof ∆(t). This exponential behavior has been confirmedby numerical simulation [Fig.3(a)]. Before entering theexponential stage, there is an initial period of algebraicdamping, whose duration grows with chain length L[Fig.3(a)]. To better understand this feature, we plotthe damping in each unit cell [Fig.3(b)]. We find thatthe left end (x = 1) enters the exponential damping im-mediately, and other sites enter the exponential stagesequentially, according to their distances to the left end.As such, there is a “damping wavefront” traveling fromthe left (“upper reach”) to right (“lower reach”). Thisis dubbed a “chiral damping”, which can be intuitivelyrelated to the fact that all eigenstates of X are localizedat the right end[24].

More intuitively, the damping of nx(t) = nx(t)−nx(∞)is shown in Fig.4(a). In the periodic boundary chain itfollows a slow power law. In the open boundary chain,a right-moving wavefront is seen. After the wavefrontpasses by x, the algebraically decaying nx(t) enters theexponential decay stage and rapidly diminishes.

The wavefront can be understood as follows. Ac-cording to Eq.(6), the damping of ∆(t) is determinedby the evolution under exp(Xt), which is just theevolution under exp(−γt/2) exp(−iHSSHt), where HSSH

is the non-Hermitian SSH Hamiltonian[24] (with anunimportant sign difference). Now the propagator〈xs| exp(−iHSSHt)|x

′s′〉 can be decomposed as propaga-tion of various momentum modes with velocity vk =∂E/∂k. Due to the presence of an imaginary part κ inthe momentum, propagation from x′ to x acquires anexp[−κ(x − x′)] factor. If this factor can compensateexp(−γt/2), exponential damping can be evaded, givingway to a power law damping. For simplicity we take γsmall, so that κ ≈ −γ/2t1, therefore exp[−κ(x − x′)] ≈exp[vk(γ/2t1)t] and the damping of propagation from x′

to x is exp[(−γ/2 + vkγ/2t1)t] for the k mode. By astraightforward calculation, we have max(vk) = t2 (fort1 > t2) or

√

|t21 − γ2/4| ≈ t1 (for t1 ≤ t2). Let us con-sider t1 ≤ t2 first. When x > max(vk)t, the propagationfrom x′ = x−max(vk)t to x carries a factor exp[(−γ/2+max(vk)γ/2t1)t] = 1; while for x < max(vk)t, we needthe nonexistent x′ = x − max(vk)t < 0, therefore, com-pensation is impossible and we have exponential damp-ing. This indicates a wavefront at x = max(vk)t. Fort1 = t2 = 1, we have max(vk) ≈ 1, which is consistent

FIG. 4. Time evolution of nx(t) = nx(t) − nx(∞), whichshows damping of particle number nx(t) towards the steadystate. (a) nx(t) of the main model with dissipators given byEq.(2) (referred to as “model I”). Left: periodic boundary;Right: open boundary. The chiral damping is clearly seen inthe open boundary case. The dark region corresponds to theexponential damping stage seen in Fig.3. (b) nx(t) of modelII, whose damping matrix X [Eq.(15)] has no NHSE. TheLiouvillian gap is nonzero and the same for periodic and openboundary chains. Common parameters: t1 = t2 = 1; γg =γl = 0.2.

with the wavefront velocity (≈ 1) in Fig.4(a).

As a comparison, we introduce the “model II” (themodel studied so far is referred to as “model I”) thatdiffers from model I only in Lg

x, which is now Lgx =

√

γg

2 (c†xA− ic†xB) [compare it with Eq.(2)]. The damping

matrix is

X(k) = i[(t1 + t2 cos k)σx + (t2 sin k − iγl − γg

2)σy ]−

γ

2I, (15)

which has no NHSE when γl = γg. Accordingly, the openand periodic boundary chains have the same Liouvilliangap, and chiral damping is absent [Fig.4(b)].

In realistic systems, there may be disorders, fluctu-ations of parameters, and other imperfections. Fortu-nately, the main results here are based on the presenceof NHSE, which is a quite robust phenomenon unchangedby modest imperfections. As such, it is expected that ourmain predictions are robust and observable.

Final remarks.–(i) The chiral damping originates fromthe NHSE of the damping matrix X rather than the ef-fective non-Hermitian Hamiltonian. Unlike the damp-ing matrix, the effective non-Hermitian Hamiltonian de-scribes short time evolution. It is found to be Heff =∑

ij c†i (heff)ijcj − iγgL, where heff, written in momentum

5

space, is heff(k) = (t1+t2 cos k)σx+(t2 sin k−iγl−γg

2 )σy−

iγl−γg

2 I. For γg = γl, heff has no NHSE, though X has.Although damping matrices with NHSE can arise quitenaturally (e.g., in Fig.1), none of the previous models(e.g., Ref.[108]) we have checked has NHSE.

(ii) The periodic-open contrast between the slow alge-braic and fast exponential damping has important im-plications for experimental preparation of steady states(e.g. in cold atom systems). In the presence of NHSE,approaching the steady states in open-boundary systemscan be much faster than estimations based on periodicboundary condition.

(iii) It is interesting to investigate other rich as-pects of non-Hermitian physics such as PT symmetrybreaking[109] in this platform (Here, we have focused onthe cases that the open-boundary iX is essentially PT-symmetric, meaning that the real parts of X eigenvaluesare constant).

(iv) When fermion-fermion interactions are included,higher-order correlation functions are coupled to the two-point ones, and approximations (such as truncations)are called for. Moreover, the steady states can bemultiple[110, 111], in which case the damping matrix de-pends on the steady state approached, leading to evenricher chiral damping behaviors. These possibilities willbe left for future investigations.

This work is supported by NSFC under grant No.11674189.

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DERIVATION OF THE DIFFERENTIAL

EQUATION OF CORRELATION FUNCTIONS

We shall derive Eq.(3) in the main article, which isreproduced as follows:

d∆(t)

dt= i[hT ,∆(t)]− {MT

l +Mg,∆(t)}+ 2Mg.(16)

In fact, after inserting the Lindblad master equationinto d∆ij/dt = Tr[c†icjdρ/dt] and reorganize the terms,we have

d∆ij(t)

dt= iTr

(

[H, c†i cj ]ρ(t))

+

∑

µ

Tr[(

2L†µ[c

†i cj, Lµ] + [L†

µLµ, c†icj ]

)

ρ(t)]

.(17)

By a straightforward calculation, we have

[H, c†icj ] =∑

mn

hmn[c†mcn, c

†icj ]

=∑

mn

hmn(−δmjc†i cn + δinc

†mcj)

=∑

n

(−hjnc†icn + hnic

†ncj), (18)

therefore, the Hamiltonian commutator term in Eq.(17)is reduced to i[hT ,∆(t)]ij , which is the first term ofEq.(16).

The commutators terms from the loss dissipators Llµ =

9

∑

iDlµici are

2∑

µ

Ll†µ [c

†icj , L

lµ] = 2

∑

µm

Dl∗µmc†m[c†i cj ,

∑

n

Dlµncn]

= 2∑

µmn

Dl∗µmDl

µnc†m[c†i cj , cn]

= 2∑

µmn

Dl∗µmDl

µn(−δinc†mcj)

= −2∑

m

(Ml)mic†mcj , (19)

and∑

µ

[Ll†µL

lµ, c

†icj ] =

∑

µmn

Dl∗µmDl

µn[c†mcn, c

†icj ]

=∑

µmn

Dl∗µmDl

µn(−δmjc†i cn + δinc

†mcj)

=∑

n

(

−(Ml)jnc†i cn + (Ml)nic

†ncj

)

.(20)

The corresponding terms in Eq.(17) sum to−{MT

l ,∆(t)}ij ,

Similarly, for the commutators from the gain dissipa-tors, we have

2∑

µ

Lg†µ [c†i cj , L

gµ] = 2

∑

m

(Mg)mjcmc†i , (21)

and

∑

µ

[Lg†µ Lg

µ, c†icj ] =

∑

n

(

−(Mg)njcnc†i + (Mg)incjc

†n

)

.

(22)

Writing cnc†i = δni − c†icn, we see that the corresponding

terms in Eq.(17) sum to 2(Mg)ijTr(ρ) − {Mg,∆(t)}ij =2(Mg)ij −{Mg,∆(t)}ij . Therefore, all terms at the righthand side of Eq.(17) sum to that of Eq.(16).