Anomalously large relaxation times in dissipative lattice models beyond the non-Hermitian skin effect Gideon Lee1Alexander McDonald1 2 3and Aashish Clerk1

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Anomalously large relaxation times in dissipative lattice models beyond the

non-Hermitian skin eﬀect

Gideon Lee,1Alexander McDonald,1, 2, 3 and Aashish Clerk1

1Pritzker School of Molecular Engineering, The University of Chicago, Chicago, Illinois 60637, USA

2Institut Quantique & D´epartement de Physique,

Universit´e de Sherbrooke, Sherbrooke, Qu´ebec, J1K 2R1, Canada

3Department of Physics, University of Chicago, Chicago, IL 60637, USA

(Dated: August 29, 2023)

We show for generic quantum non-Hermitian tight-binding models that relaxation timescales of

local observables are not controlled by the localization length ξloc associated with the non-Hermitian

skin eﬀect, contrary to popular belief. Instead, interference between eigenvectors eﬀectively makes

the extreme localization of modes largely irrelevant to relaxation; this is ultimately a consequence

of causality and locality. Focusing on the paradigmatic Hatano-Nelson model, we demonstrate that

there exists instead a much larger length scale ξprop which controls the rate of decay towards the

steady state. Further, varying ξprop can lead to anomalously large relaxation times that scale with

system size, or to the expected behavior where the dissipative gap correctly predicts the rate of

decay. Our work highlights an important aspect of the non-Hermitian skin eﬀect: the exceptional

sensitivity to boundary conditions here necessarily takes a ﬁnite amount of time to manifest itself.

I. INTRODUCTION

The exotic eﬀects of non-Hermiticity in classical and

quantum systems alike has generated widespread inter-

est in recent years [1–3]. One such unusual phenomenon

is the existence of anomalously long relaxation times:

generic local observables reach their steady-state value

much more slowly than the characteristic decay rates,

the smallest of which is known as the dissipative gap,

would suggest [4,5]. This behavior has been observed

in a wide range of models such as random quantum cir-

cuits [6–9] and non-Hermitian tight-binding models [10–

12]. The latter also exhibits counter-intuitive behavior

known as the non-Hermitian skin eﬀect (NHSE) [13–16],

which occurs when the Hamiltonian under open bound-

ary conditions has a macroscopic number of localized

edge modes and a drastically diﬀerent spectrum than its

periodic boundary condition counterpart. The NHSE has

been probed in experiments [17–19].

Recent work has suggested that these two counter-

intuitive eﬀects are in fact intimately related. By in-

terpreting a Lindblad superoperator as a non-Hermitian

Hamiltonian, Ref. [10] argues that exponentially-large-in-

system-size expansion coeﬃcients due the NHSE leads to

a unexpectedly-large relaxation time

τEVec∼L

ξloc∆,(1)

where τis a reasonably-deﬁned relaxation timescale for

a local observable, Lis the size of the system, ∆ is the

dissipative gap [20], and ξloc is the localization length of

the corresponding least-damped Liouvillian eigenmode.

Since the argument leading to Eq. (1) is based solely on

the localized Lindblad eigenvectors, we refer to it as the

eigenvector (EVec) prediction. A similar argument was

put forth in Refs. [11,21].

In this work, we argue that this analysis is not valid

for generic 1D non-Hermitian tight-binding models; not

only is the relaxation time insensitive to ξloc, but τ∆

need not scale with Leven when the model exhibits

the NHSE. The mechanism leading to the breakdown of

Eq. (1) is simple but ubiquitous in tight-binding mod-

els exhibiting the NHSE: the dynamics of local observ-

ables involve a large number of eigenvectors, and inter-

ference between these modes can eﬀectively make their

exponentially-localized nature largely irrelevant. This

conclusion, unlike the prediction of Eq. (1), cannot be

reached by considering the eigenvectors or eigenvalues

separately. Rather, the combination of both is essen-

tial to understand the physics at play, as is done natu-

rally when one considers the system’s Green’s functions.

While we focus on a few well-studied models (the quan-

tum realization of the Hatano-Nelson model [22,23] and

two additional models in App. Ffor concreteness, we ar-

gue that this cancellation is generic and must occur due

to locality. Note that one can also study anomalous re-

laxation in non-Hermitian models using pseudo-spectral

methods [7,24,25]; however, these do not generally allow

one to make precise statements, nor do they provide the

intuition we present here.

II. QUANTUM HATANO-NELSON MODEL

The Hatano-Nelson model is a paradigmatic model ex-

hibiting the NHSE. It can be realized in an uncondi-

tional, fully quantum setting using engineered dissipation

[26]. The most natural realization is through the use of

nearest-neighbour correlated loss [18,27]. The equation

of motion for the density matrix using this setup reads

arXiv:2210.14212v2 [quant-ph] 28 Aug 2023

FIG. 1. (a) Schematic of the quantum Hatano-Nelson (QHN) model realized using engineered dissipation with hopping ampli-

tudes w±κ, local pumping Γ and local loss λ+κ. (b) Normalized relaxation time τL,b∆b(c.f. Eq. (20)) for bosons, for various

Γ in the highly non-reciprocal limit w= 1, κ = 0.999 leading to a ﬁxed localization length ξloc = 0.26. The EVec prediction

τb∆b∼L/ξloc gets the linear in Lscaling correct. However, as indicated by the dashed line in the ﬁgure, the EVec prediction

diverges as ξloc →0 (corresponding to κ→w). Hence, any prefactor ﬁtted to the actual scaling must be dramatically small,

and will fail given even inﬁnitesimal changes in κ. (c) Same as (b) but for fermions. Here, the linear-in-Lscaling of the EVec

prediction is qualitatively wrong, with τL,b∆bsaturating with L. As discussed in the main text, these strong deviations from

the EVec prediction arise from an interference eﬀect. (d) τL,f/b∆f/bas a function of local loss for w= 1, κ= 0.999, Γ = 0.05.

Changing λdoes not change ξloc (the NHSE is unaﬀected), but strongly modiﬁes τL,f/b; this again strongly disagrees with the

EVec prediction. For large λ, we recover the behaviour of a reciprocal chain with the dissipative gap correctly predicting the

relaxation time.

(setting ℏ= 1)

∂tˆρ=−iw

L−1

j=1

[ˆc†

j+1ˆcj+ h.c.,ˆρ] + κ

L−1

j=1 D[ˆcj−iˆcj+1]ˆρ

+ 2

j=1

λjD[ˆcj]ˆρ+ 2Γ

j=1 D[ˆc†

j]ˆρ≡ Lˆρ, (2)

where we consider an L-site lattice, open boundary con-

ditions, and λj=λ+κ(δj,1+δj,L) [28]. This dissipative

quantum realization of the Hatano-Nelson model has a

variety of interesting properties, and is the subject of

several recent studies [5,27,29–32]. Here wis the coher-

ent hopping strength, D[ˆ

X]ˆρ≡ˆ

Xˆρˆ

X†− { ˆ

X†ˆ

X, ˆρ}/2 is

the usual dissipator, Lis the Liouvillian superoperator,

and ˆcjare fermionic or bosonic annihilation operators for

site j; we analyze both cases simultaneously. The degree

of non-reciprocity can be tuned by varying the correlated

loss κ. This becomes explicit in the equations of motion

for the covariance matrix:

i∂t⟨ˆc†

nˆcm⟩=

j=1 (HHN)mj ⟨ˆc†

nˆcj⟩ − (H†

HN)jn⟨ˆc†

jˆcm⟩+i2Γδnm,(3)

with HHN is the ﬁrst-quantized Hatano-Nelson Hamilto-

nian

HHN ≡

L−1

j=1 w+κ

2|j+ 1⟩⟨j|+w−κ

2|j⟩⟨j+ 1|

−i(κ+λ±Γ)

j=1 |j⟩⟨j|.

(4)

Throughout, we will use the upper and lower sign ±for

fermions and bosons respectively in addition to ﬁxing

w≥0 and w > κ, ensuring that the left-to-right hopping

amplitude is stronger than right-to-left hopping. Note

that the local loss proportional to κcannot be avoided

if we want non-reciprocity. Without it, we could violate

Pauli exclusion or the Heisenberg uncertainty principle

[33]. The extra local loss λthus allows us to vary the

degree of non-reciprocity without changing the total lo-

cal loss. Incoherent pumping Γ ensures the existence of

a steady state with a ﬁnite density of particles with a

non-trivial spatial proﬁle [27]. Finally, note that HHN is

distinct from the no-jump Hamiltonian [34].

In what follows, we will be interested in the relaxation

time of local observables. Given that the non-reciprocal

hopping in Eq. (4) favors rightward propagation and we

wish to consider the eﬀects of system size on relaxation

dynamics, it is natural to consider the occupation of the

rightmost site ⟨ˆnL(t)⟩. We deﬁne the normalized non-

equilibrium population

δnL(t) = |⟨ˆnL(t)⟩−⟨ˆnL(∞)⟩|

⟨ˆnL(∞)⟩(5)

and for concreteness assume that the chain starts in vac-

uum: δnL(0) = 1. To deﬁne a local relaxation time τL,

we ﬁx a relaxation threshold e−1, and say that site Lhas

relaxed at time τLwhen δnL(τL) = e−1. The essential

physics we discuss is not speciﬁc to the local observable

of interest, initial state or choice of reasonable relaxation

threshold. See App. Efor a discussion of other initial

conditions, and App. Dfor a discussion of the full time-

dependent decay curve.

III. EXPONENTIALLY LARGE COEFFICIENTS

To solve Eq. (3) and extract τL, we ﬁrst diagonalize

HHN. The eigenvalues Eαand biorthonormal left and

right eigenvectors ⟨ψl(α)|ψr(β)⟩=δαβ of the Hatano-

Nelson model are well known [22,23]. We have

Eα=Jcos kα−i(κ+λ±Γ),(6)

and

⟨j|ψr(l)(α)⟩ ≡ ψr(l)

j(α) = r2

L+ 1e(−)j/ξloc sin kαj, (7)

with

J≡p(w+κ)(w−κ), e1/ξloc ≡rw+κ

w−κ.(8)

Here, kα=απ/(L+ 1), α = 1, . . . , L is a quantized

standing-wave momentum. The parameter Jshould be

thought of as a renormalized hopping amplitude, whereas

ξloc is not only the localization length of the eigenvectors

but serves as a measure of non-reciprocity. We stress that

fermions and bosons have the same eigenvectors, and the

only diﬀerence between the two is the uniform decay rate

∆f/b≡ −ImEα=κ+λ±Γ.(9)

Having diagonalized HHN, we can use third quantization

to also diagonalize the full Liouvillian [35–37], analogous

to how one diagonalizes a second-quantized quadratic

Hamiltonian from its single-particle data. The eigen-

values of Lare fully determined by Eαand its eigen-

modes are simply related to the left and right eigen-

vectors of HHN and the steady-state correlation matrix

(S)nm ≡ ⟨ˆc†

nˆcm⟩ss.

We can now use the spectral decomposition of HHN to

formally express ⟨ˆnm(t)⟩as a sum over eigenvectors

⟨ˆnm(t)⟩= 2Γ

j=1 Zt

dt′⟨m|e−iHHN(t−t′)|j⟩

=2Γ

j=1 Zt

dt′X

ψr

m(α)(ψl

j(α))∗e−iEα(t−t′)

.(10)

This solves Eq. (3) for the vacuum initial condition

⟨ˆnm(0)⟩= 0 for all m, hence picking out the homoge-

neous part of the solution [38]. Eq. (10) provides a simple

physical picture of the dynamics. At some time t′≤t,

a pump bath attached to site jinjects a particle, which

then propagates to a site min a time t−t′contributing

P(m, j;t−t′)≡ |⟨m|e−iHHN (t−t′)|j⟩|2(11)

to the density at that site. With each bath adding parti-

cles at a rate 2Γ, and considering we started in vacuum

at time t= 0, summing over all sites jand integrating

over all intermediate times t′gives the total particle num-

ber on site m. Henceforth we will focus mainly on the

rightmost site m=L.

With the propagator ⟨L|e−iHHNt|j⟩written in its spec-

tral representation, from Eq. (7) it seems apparent that

for strong non-reciprocity κ→w,ξloc ≪1, only parti-

cles injected on the ﬁrst site j= 1 which propagate to

site Lmatter, since in this limit the e2(L−1)/ξloc factor

in this term (coming from the exponential localization of

the right eigenvectors) will dominate over all others. To

estimate τL, one might then argue that we must compare

this large term with e−∆f/bt, the temporal decay stem-

ming from the imaginary part of the mode energies. τL

is then roughly the time it takes for the temporal decay

factor to cancel the exponentially-large expansion coeﬃ-

cient. We would thus approximately have

τL,f/b

∼L

ξloc∆f/b

.(12)

This is essentially the argument put forward in Refs. [10,

11] – as we show explicitly in App. C, the exponentially-

large terms in Eq. (10) are related to expansion coeﬃ-

cients in the basis of Liouvillian eigenmodes.

To check Eq. (12), in Fig. 1b. and 1c. we plot

τLin the large non-reciprocal limit κ= 0.999wgiving

ξloc = 0.26 and, for L= 100, an expansion coeﬃcient

of e2(L−1)/ξloc ∼10330 [39]. In Fig. 1d., we plot the de-

pendence of the relaxation time on the additional local

loss parameter λ. In each case, the EVec prediction of

Eq. (12) falls short, albeit in diﬀerent ways.

First, for bosons (Fig. 1b), Eq. (12) in fact gets the

linear-in-Lscaling correct. The relaxation times for

bosons show some dependence on Γ, which is indepen-

dent of ξloc, but one might argue that this is simply a

prefactor diﬀerence. However, a more serious discrep-

ancy appears when we consider the dependence of relax-

ation times of ξloc. As an example, consider Fig. 2, where

we ﬁx all parameters except κ. Then, for a ﬁxed chain

length, varying κallows us to investigate the dependence

of relaxation time on ξloc, with ξloc →0 when κ→w. In

this case, we observe that the EVec prediction diverges

with 1/ξloc, while the true relaxation time saturates.

Second, for fermions (Fig. 1c), Eq. (12) gets the scaling

qualitatively wrong. The τLrelaxation time only scales

with system size up to a point, and ultimately saturates

for large enough system sizes. Despite the NHSE, for

large system sizes, fermions do not display the predicted

linear-in-Lscaling of Eq. (12).

Finally, we ﬁnd that for a ﬁxed ξloc, the true relax-

ation times can vary drastically (Fig. 1d). This is done

1.5 2.0

1/ξloc

100

120

140

Relax. Time τL,b∆b

(a)

L = 100

Γ =0.01

Γ =0.2

1.5 2.0

1/ξloc

Relax. Time τL,b∆b

(b)

L = 30

Γ =0.01

Γ =0.2

FIG. 2. (a) Normalized relaxation time, as deﬁned in Eq. (5),

for the right-most site τL,b∆bfor bosons, with w= 1,Γ =

0.01,0.2, λ = 0, L = 100 ﬁxed, plotted against 1/ξloc.ξloc

is varied by varying κ. Solid line indicates numerical result,

and dashed line is obtained from the EVec. prediction. The

prefactor for the EVec. prediction is ﬁxed by setting the EVec.

prediction to be equal to the numerical result for κ= 0.9. We

see that the relaxation times do not scale as 1/ξloc. (b) Similar

to (a) but for L= 30. In (a), L≥ξprop for both cases, but

here L<ξprop for Γ = 0.01. In both cases, the discrepancy

remains.

by adding an local loss parameter which is completely

independent of ξloc. In this case, when λis suﬃciently

large, we even recover the usual dissipative gap scaling.

We emphasize that at all points in this plot the NHSE is

still present to the same degree quantiﬁed by ξloc. How-

ever, anomalous relaxation has completely disappeared.

On the one hand, the argument leading to Eq. (12)

seems obvious – surely a drastic sensitivity to bound-

ary conditions must show up in relaxation times. On

the other, we have just demonstrated a range of ways

in which Eq. (12) fails in a minimal model exhibiting

the NHSE. While appealing, the argument leading to

Eq. (12) is clearly misleading; we diagnose and correct

the crucial error in what follows.

IV. INTERFERENCE BETWEEN

EIGENMODES

The failure of Eq. (12) makes it evident that we cannot

simply estimate the magnitude of the propagator using

the localization properties of the eigenvectors. To under-

stand why, ﬁrst note that Eq. (7) implies

⟨m|e−iHHNt|j⟩=e(m−j)/ξloc−∆f/bt⟨m|e−iHJt|j⟩,(13)

where HJis a reciprocal nearest-neighbour tight-binding

Hamiltonian with hopping amplitude Junder open

boundary conditions. Focusing on the impact of this

previously-ignored second factor, we now work in the

strongly non-reciprocal limit κ→wand set λ= 0, just

as in Fig. (1) . In this limit, leftwards propagation is

strongly suppressed. A particle which hits the right-most

edge of the chain essentially cannot be reﬂected back and

aﬀect any other site; the boundary becomes irrelevant.

Estimating the propagator by its fully translationally-

invariant counterpart gives

P(L, j;t)≈e2(L−j)/ξloc−2∆f/btZπ

−π

2πeik(L−j)e−iJ cos kt

(14)

and we show in App. Bthat this is in fact an excellent

approximation in the regime of interest.

Using Eq. (8) we deduce that although the ﬁrst factor

of Eq. (14) is enormous for j≪Land strong reciprocity

(i.e. ξloc ≪1), the second factor in this limit can be very

small, as J→0. In fact, expanding to lowest order in J

and expressing all quantities in terms of wand κ:

P(L, j;t)≈Je1/ξloc 2(L−j)e−2∆f/btt2(L−j)

4L−j([L−j]!)2

= (w+κ)2(L−j)e−2∆f/btt2(L−j)

4L−j([L−j]!)2.(15)

We see explicitly that the exponentially-large contribu-

tion from ξloc has been oﬀset by the small renormalized

hopping amplitude.

The above dramatic cancellation is due to the inter-

ference between the eigenvectors of the Hatano-Nelson

model, as in Eq. (15) we have summed over all αin the

spectral decomposition of the propagator

⟨L|e−iHHNt|j⟩=X

ψr

L(α)(ψl

j(α))∗e−iEαt.(16)

Indeed, before Eq. (12) we incorrectly estimated the mag-

nitude of P(L, j;t) because we focused on a single eigen-

vector. This incorrectly neglects the fact that for J→0,

we have many modes with closely spaced eigenvalues Eα,

and the possibility of destructive interference when we

sum their contributions. In App. A, we show this picto-

rially by plotting side-by-side the size of each term in the

sum, along with their relative phases. While each term

is exponentially large, the phase variation between terms

leads to a near perfect cancellation in the sum over all

eigenvectors.

While this explanation might seem mechanistic, it has

a more general physical underpinning. This cancellation

had to occur to enforce locality: a particle injected at

j≪Lcannot instantaneously propagate to Lwith an

exponentially-large amplitude, and hence cannot instan-

taneously know about the exponential modiﬁcation of

wavefunctions associated with the boundary condition.

We expand on this point in the next section, arguing

that this reasoning also holds for intermediate times and

is not limited to the Hatano-Nelson model.

Despite demonstrating that the localization length

plays no role in determining the relaxation time, it still

does not explain why τLdisplays qualitatively diﬀerent

behavior for fermions and bosons nor why it is sensi-

tive to the small incoherent pumping rate Γ. To uncover

the origin of this eﬀect, for simplicity consider the per-

fectly non-reciprocal case κ=wwhere the dissipative

gap takes the form ∆f/b=w±Γ and Eq. (15) is exact

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Anomalouslylargerelaxationtimesindissipativelatticemodelsbeyondthenon-HermitianskineffectGideonLee,1AlexanderMcDonald,1,2,3andAashishClerk11PritzkerSchoolofMolecularEngineering,TheUniversityofChicago,Chicago,Illinois60637,USA2InstitutQuantique&D´epartementdePhysique,Universit´edeSherbrooke,Sherbrook...

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Anomalously large relaxation times in dissipative lattice models beyond the non-Hermitian skin effect Gideon Lee1Alexander McDonald1 2 3and Aashish Clerk1

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