Partial yet definite emergence of the Kardar-Parisi-Zhang class in isotropic spin chains
Kazumasa A. Takeuchi,1, 2, ∗ Jacopo De Nardis,3 Ofer Busani,4 Patrik L. Ferrari,5 and Romain Vasseur6

arXiv:2406.07150v1 [cond-mat.stat-mech] 11 Jun 2024

2

1
Department of Physics, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
Institute for Physics of Intelligence, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
3
Laboratoire de Physique Théorique et Modélisation, CNRS UMR 8089,
CY Cergy Paris Université, 95302 Cergy-Pontoise Cedex, France
4
School of Mathematics, University of Edinburgh, James Clerk Maxwell Building,Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK
5
Institute for Applied Mathematics, Bonn University, Endenicher Allee 60, 53115 Bonn, Germany
6
Department of Physics, University of Massachusetts, Amherst, MA 01003, USA(Dated: June 12, 2024)

Integrable spin chains with a continuous non-Abelian symmetry, such as the one-dimensional
isotropic Heisenberg model, show superdiffusive transport with little theoretical understanding. Although recent studies reported a surprising connection to the Kardar-Parisi-Zhang (KPZ) universality class in that case, this view was most recently questioned by discrepancies in full counting statistics. Here, by combining extensive numerical simulations of the Landau-Lifshitz one-dimensional
magnet, with a framework developed by exact studies of the KPZ class, we characterize various
two-point quantities that remain hitherto unexplored in spin chains, and find full agreement withKPZ scaling laws. This establishes the partial emergence of the KPZ class in isotropic spin chains.
Moreover, we reveal that the KPZ scaling laws are intact in the presence of an energy current, under
the appropriate Galilean boost required by the propagation of spacetime correlation.

Characterization of transport properties in quantum
many-body systems, in particular those of integrable systems with non-diffusive transport, is one of the frontiers of condensed matter physics. Integrability typically results in ballistic transport, as successfully described by the framework of the generalized hydrodynamics [1–3], but it is faced with challenges when ballistic contributions are canceled by symmetry or othermechanisms [4, 5]. Paradigmatic is the situation witha continuous non-Abelian symmetry, in particular theisotropic Heisenberg spin chain, which was reported
to show superdiffusive transport with the characteristic
length ξ(t) ∼ t2/3 [6–8]. Surprisingly, this superdiffusive
exponent was associated with an apparently unrelated
universality class established mainly for classical nonequilibrium systems, namely the Kardar-Parisi-Zhang(KPZ) universality class for fluctuations of growing interfaces and related phenomena [9]. Key evidence [10] was
the precise agreement of the equilibrium two-point spincorrelation function with Prähofer and Spohn’s exact solution for the KPZ class [11], often denoted by fKPZ (·).
On the one hand, this alleged manifestation of the KPZclass is deemed universal [4, 5, 12], as confirmed in various isotropic integrable spin chains, whether quantum[13] or classical [14], and also supported by a few experimental investigations [15, 16]. On the other hand,
it is clear from the symmetry of spins that the magnetization transfer (integrated spin current) must show asymmetric distribution unless the symmetry is explicitly
broken by the initial condition or an external field [17],while for KPZ the corresponding quantity, namely the interface height increment, is intrinsically asymmetric [9].
Recent computational studies [18, 19] further backed thisview, by showing clear evidence that even the kurtosis

and other ratios of even-order cumulants, which are unaffected by the spin’s up-down symmetry, are markedly
different from the predictions of the KPZ class, callingfor a new universality class to describe this class of systems. After all, all pieces of evidence for KPZ reported so
far have been rather weak, being the scaling exponents,
which are simple rational numbers such as 2/3, and theagreement with Prähofer and Spohn’s solution fKPZ (·),
which has been compared with arbitrarily fitted scalingcoefficients.
Here we clarify the fate of the KPZ universality inisotropic integrable spin chains. First we remark that the
deep body of knowledge gained by mathematical studies
on the 1D KPZ class [9] has not been fully utilized. It dictates, for example, the mutual relation between scalingcoefficients. They contain universal quantifiers, which arelost if treated as free fitting parameters. Moreover, thePrähofer-Spohn function is not the only two-point correlator with an exact solution [9]; other two-point functions, such as the equal-time spatial correlator [20] andequal-position two-time correlator [21, 22] have also been
dealt with. The purpose of the present Letter is to makefull use of these results to carry out a comprehensive
test of the KPZ universality in isotropic integrated spin
chains.
In the following, we mainly present results obtained forthe isotropic version of the integrable lattice model introduced by Krajnik et al. [17, 23], based on the LandauLifshitz (LL) magnet [24]. The time evolution is given
conceptually by∂Sj= Sj × (Sj+1 + Sj−1 ),∂t(1)but implemented in a specific manner on a brick-wall

2

101
100

2

`=9(t)

3(d)

9 t2=3

102
101
0

10

10-1

0.4

Baik-Rains

0.2
0
100

100

t

100

102

t

104

104

t

102

104

(b)

1.4

Ct (t1 ; t2 )=Ct (t2 ; t2 )

1

1.2
1
0.8
0.6
100

t

102

104

FIG. 1. The two-point function and the magnetization transfer cumulants for the LL model. (a) Rescaled two-point function C̃2 (ℓ, t) ≡ ξ(t)C2 (ℓ, t) against ℓ/ξ(t), compared withΩ
the Prähofer-Spohn exact solution fKPZ (·). (b) Correlation
length ξ(t). (c) Variance (main panel) and skewness (inset)of the magnetization transfer h. The dashed line in the inset
indicates the skewness of the Baik-Rains distribution. (d) Ratio of the coefficients α1 (t) and α2 (t) evaluated from Eqs. (2)and (3), respectively.

space-time lattice [17, 23] to preserve the integrability of
the LL magnet in continuous space-time. Here, unlessotherwise stated, we started from infinite-temperatureequilibrium states and obtained N = 104 independent
realizations with system size L = 40, 000 and the periodic boundary condition, with time step 0.1. The zcomponent of the spins, Sjz (t), is our magnetization field,
denoted by m(x, t) with x = j hereafter. Another quantity of interest is the integrated
R t spin current, or the magnetization transfer, h(x, t) ≡ 0 J(x, t′ )dt′ , with spin current J(x, t). The magnetization transfer h(x, t) corresponds to the height increment of the growing interfaces,
which is central in the studies of the KPZ class.
First we verify KPZ behavior of the LL model throughthe standard quantities. Figure 1(a) displays the twopoint function C2 (ℓ, t) ≡ ⟨m(x + ℓ, t)m(x, 0)⟩, showing agreement with the Prähofer-Spohn exact solution fKPZ (·). Here the normalized function C̃2 (ℓ, t) ≡Rξ(t)C2 (ℓ, t)dℓ is conΩ C2 (ℓ, t) is shown, where Ω ≡served asRa result of the conservation of the total magnetization m(x, t)dx,
lengthR 2and ξ(t) is the 2correlation
1
2
determined
by
ℓ
C
(ℓ,
t)dℓ
=
σ
ξ(t)
with
σ2 ≡
2
Ω
R 2
u fKPZ (u)du ≈ 0.51. This correlation length is confirmed to show the characteristic power law ξ(t) ∼t2/3 of the KPZ class [Fig. 1(b)]. We also measure
the variance of the magnetization transfer and find thepower-law growth with the KPZ characteristic exponent,Var[h(x, t)] ∼ t2/3 [Fig. 1(c) main panel]. On the other
hand, the skewness is zero and far from the value for the

-1.5

0

t = 10t = 106:8
t = 995:4
KPZ

0.8

-1

0.6

-2
0

0.4

1

2

dC~s
du (0)1dC~sdu (u)

10

C~s (u) 2 Cs (x; t)=,t2=3

9t

2=3

2

0.2

Skew[h]Var[h]

(a)

10

9(t)

0.4

0
0

(c)

(b) 3

t = 995:4
t = 106:8
t = 10fKPZ,2 (t)=,1 (t)C~2 (`; t)(a) 0.6

3

-2

-2.5

u

0.2
0
0

0.5

1

1.5

100

2

u 2 `=9(t)

t

102

2.5

104

3

1
t2 = 10t2 = 106:8
t2 = 995:4
t2 = 9283:2
KPZ

0.8
0.6
0.4
0.2
0
0

0.1

0.2

0.3

0.4

0.5

t1 =t2

0.6

0.7

0.8

0.9

1

FIG. 2. The spatial (a) and temporal (b) correlation functions of the magnetization transfer for the LL model. (a) The
rescaled spatial correlation function C̃s (u) = Cs (ℓ, t)/αt2/3C̃s(main panel) and its slope ddu(u) (left inset) as functions ofu ≡ ℓ/ξ(t). The dashed lines show the curves for the KPZclass, numerically obtained by TASEP simulations. The rightC̃s(0) at u = 0 with our exact result
inset compares the slope ddu
dC̃s
for the KPZ class, du (0) = −2. (b) The rescaled temporalcorrelation function Ct (t1 , t2 )/Ct (t2 , t2 ) against t1 /t2 , compared with the Ferrari-Spohn exact solution [Eq. (6)] for the
KPZ class.

Baik-Rains distribution [25] expected for the KPZ stationary state (inset). Although the data shown so far
are reproduction of known results [17, 18], we can scrutinize nontrivial relationship underlying these quantities.
According to KPZ scaling laws [9, 11], we have
2αt2/3ℓ
C2 (ℓ, t) ≃
fKPZ
,
(2)
ξ(t)2
ξ(t)
Var[h(x, t)] ≃ αt2/3 Var[BR],(3)

where Var[BR] ≈ 1.15 is the variance of the Baik-Rains
distribution and α is a coefficient. Since these equations
are not guaranteed to describe spin chains, here we evaluate α from data of C2 (x, t) and Var[h(x, t)] independently
and denote them by α1 (t) and α2 (t), respectively. Then,
remarkably, we find α1 (t) = α2 (t) [Fig. 1(d)], substantiating the validity of the KPZ scaling laws (2) and (3)in spin chains. Therefore, interestingly, the Baik-Rains
variance turns out to be in the formula (3), even though
the Baik-Rains distribution itself does not appear.
We further test the validity of KPZ scaling laws in spin
chains through other two-point quantities. First we study
the spatial correlation of the magnetization transfer:
Cs (ℓ, t) ≡ ⟨h(x, t)h(x + ℓ, t)⟩ − ⟨h(x, t)⟩2 .

(4)

Figure 2(a) shows it in the rescaled units, C̃s (u) ≡Cs (ℓ, t)/αt2/3 against u ≡ ℓ/ξ(t). For the KPZ class,Ct (t1 , t2 ) ≡ ⟨h(x, t1 )h(x, t2 )⟩ − ⟨h(x, t1 )⟩⟨h(x, t2 )⟩. (5)The results in Fig. 2(b) show excellent agreement withthe exact solution for the KPZ class obtained by Ferrari
and Spohn [21, 22]:i
Ct (t1 , t2 )1h≃
1 + τ 2/3 − (1 − τ )2/3 ,Ct (t2 , t2 )2

(6)with τ ≡ t1 /t2 .
Now that we verified the validity of the KPZ scalinglaws in various two-point quantities, we test its robustness under different situations. First, we consider thecase with a non-vanishing energy current. This is particularly tempting in view of the hydrodynamic description
proposed by De Nardis et al. [32], which predicts that leftmoving and right-moving giant quasiparticles contributeequally to the magnetization, and this is why the distribution of h becomes symmetric. Therefore, it is important to clarify what happens if the left-right symmetryis broken, e.g., by the presence of a finite energy current.
We prepared such an initial condition by Monte Carlo−λJE
sampling, using the statisticalwith toP weight ∝ e
tal energy current JE ≡ − j Sj · (Sj+1 × Sj+2 ) [33]and λ = 1. Thereby, we indeed realize a situation where
the energy current reaches a constant finite value after ashort transient [Fig. 3(a) inset].
Figure 3(a) shows the two-point function C̃2 (ℓ, t) inthis case. Interestingly, now we find the peak positionof the correlation function moving at a constant velocity,ℓpeak = vpeak t with vpeak = 0.5522 [Fig. 3(b)(c)]. Apartfrom this, the form of the two-point function turns out

0.1

0.3

0.08

0.2

0.06
0.04

2
0

JE

0.1
0

(b)
t
100

0.6

0.02
0
-5

0

#104

(c) -2102

vpeak

C2 (`; t) # t2=3

(a)

5

10

15

`=t2=3

t

0.4
0.2
0
0

20

(d) 0.6

0.5

1.5

t

2
#105

9 t2=30.4

9(t)C~2 (`; t)

1(e) 104

0.5
0.3

102

0.2
0.1
-1

0

1

(` ! `peak (t))=9(t)(f) 104
102

t

102

104

102

104

0
-0.01

Kurt[h]original
comoving
102

t

0.01

(h)

9 t2=3
100

100

(g)

9t

100

100

2

Skew[h]0-2

Var[h]

the multi-point equal-time height correlation has beencharacterized intensively and described in terms of afamily of stochastic processes called the Airy processes[20, 26–30]. For the stationary state, a process called theAirystat process has been considered [31] (see also a review [20]), but it describes the height measured in theabsolute frame (say, h0 (x, t)) instead of the height increment h(x, t) = h0 (x, t) − h0 (x, 0) considered here. We
therefore introduce here the limiting process A0 (u) forthe height increment h(x, t), in other words the stationary version of the Airystat process, and call it the Airy0
process. We evaluate the covariance of the Airy0 process
C0 (u) ≡ ⟨A0 (u)A0 (0)⟩ by numerical simulations of thetotally asymmetric simple exclusion process (TASEP), a
paradigmatic model in the KPZ class, and find it in excellent agreement with the data for the LL model [Fig. 2(a)].
Furthermore, we consider C0 (u) for small u analytically0and prove dCdu (0) = −2 (see Supplementary Material),
which is a characteristic distinct from the other known
Airy processes for which the slope of the covariance at
u = 0 is −1. This is confirmed by our numerical data forboth the LL model and the TASEP [insets of Fig. 2(a)].
Finally, we also investigate the temporal correlation of
the magnetization transfer

`peak

3

104

0.1

t

0
-0.1
100

t

FIG. 3. Results for the case with a finite energy current.
(a) Rescaled two-point function C̃2 (ℓ, t) ≡ ξ(t)C2 (ℓ, t) againstΩℓ/ξ(t) for different times, t = 3000, 8000, 15000, 23000, 32000
from left to right. The displayed data are smoothed and vertically shifted to have C̃2 (ℓ, t) = 0 for ℓ far from the peak.
Inset: total energy current JE (t). (b)(c) The location of
the peak of C̃2 (ℓ, t), ℓpeak (t) (b), and its propagation speed
vpeak (c). The dashed line in (b) shows ℓpeak (t) = vt withv = 0.5522, wrapped by the periodic boundary. (d) Rescaled
two-point function centered at ℓ = ℓpeak (t) (symbols, samecolors as (a)), compared with the Prähofer-Spohn exact solution fKPZ (·) (dashed line). (e) Correlation length ξ(t). The
black dots are the data for the case without energy current,
shown in Fig. 1(b) (f) Variance of the magnetization transfer
h(x, t), measured in the original and comoving frames (bluecircles and red squares, respectively). The thick dashed line
indicates the KPZ growth law Eq. (3) with the value of αdetermined from C2 (ℓ, t). (g)(h) Skewness (g) and kurtosis (h) of the magnetization transfer h(x, t) in the comoving
frame. The values for the Baik-Rains distribution are 0.359
and 0.289, respectively [34], which are far from the data.

to be unchanged, i.e., it is the Prähofer-Spohn function
fKPZ (·) [Fig. 3(d)], and so does the growth of the correlation length, ξ(t) ∼ t2/3 [Fig. 3(e)]. Therefore, the KPZphysics remains intact in the presence of a finite energy
current. The propagation of the space-time correlation
revealed in Fig. 3(a)-(c) is analogous to the case of growing tilted interfaces [35, 36] and nonlinear fluctuating hydrodynamics for unharmonic chains [37]. An importantlesson from these past studies is that one should measure
the magnetization transfer h(x, t) in the frame comov-

4
(a)

(b)

100

10 2

60

Var[h]

h 0 (x; t)

80

40

10 0

1.5

9 t1=2Var[h]=,t2=3 vs 73 t

BR
GOE

1

20

0.5
0

0
0

9 t2=3

7 = 0:1
7=1
7 = 10

1

2

3

x

4

10 -2

10

10 0

0

10 2

10

2

10 4

10 6

10 4

t

FIG. 4. Results for the flat initial condition. (a) Snapshots of the height h0 (x, t) = h(x, t) + h0 (x, 0) at t =0, 2000, 4000, · · · , 10000 from bottom to top, for µ = 1. For
visibility, every subsequent snapshot is shifted upward by
20. (b) Variance of the magnetization transfer h for different µ. The dashed line is the KPZ growth law (3) with αdetermined from the equilibrium simulations [Fig. 1(d)]. The
dashed-dotted line is a guide for the eyes showing Var[h] ∼t1/2 . Inset: rescaled variance Var[h]/αt2/3 against µ3 t, forµ = 0.1, 0.5, 1, 2. The bold solid line displays the behavior for
KPZ interfaces [38] (with arbitrary horizontal shift), showing
crossover from the Baik-Rains (BR) distribution (dotted line)
to the characteristic distribution for flat interfaces, namely the
GOE Tracy-Widom distribution (dashed line). Simulation
parameters were L = 40, 000 and N = 1, 000 for µ = 0.1, 0.5, 1
and L = 400, 000 and N = 33 for µ = 2, 10.

ing with the space-time correlator, which amounts to thefollowing expression:
Z t
Z x
′
′
h(x, t) ≡J(x, t )dt −m(x′ , 0)dx′
(7)
0

x−vt

with v = vpeak . With this appropriate definition of
the magnetization transfer, we indeed confirm the KPZ
growth of the variance, Var[h(x, t)] ∼ t2/3 (more specifically, Eq. (3)) [Fig. 3(f) red squares], whereas the naı̈veRtdefinition h(x, t) = 0 J(x, t′ )dt′ results in the apparent
loss of the KPZ exponent (blue circles). On the other
hand, even with the definition (7) without left-right symmetry, we do not find any indication of asymmetric distribution, as evidenced by vanishing skewness [Fig. 3(g)].
The value of the kurtosis also remains far from that of
the Baik-Rains distribution [Fig. 3(h)], just like the casewithout energy current [18, 19]. To summarize, the presence of a finite energy current only necessitates considering the comoving frame; otherwise, it seems to have noeffect on relevant statistical quantities, as long as they
are measured in the comoving frame.
Finally, we study the effect of the initial condition.
Universal statistical properties of the authentic KPZclass are known to depend on the initial condition, thethree representative cases being the domain wall (curved
interface), flat, and stationary initial conditions [9]. It is
important to assess whether KPZ scaling laws for nonstationary cases can describe spin chains under the corresponding, non-equilibrium settings. Among these, thedomain wall initial condition has been extensively studied

TABLE I. Two-point properties of KPZ confirmed in the equilibrium state of the LL magnet. The new results obtained inthis work are marked with *.
exponents
ξ(t) ∼ t2/3 , Var[h(x, t)] ∼ t2/3
two-point function C2 (ℓ, t) Prähofer-Spohn solution fKPZ
*variance amplitude
Eq. (3) with Var[BR] ≈ 1.15
*spatial correlation Cs (ℓ, t) Airy0 covariance, Fig. 2(a)*time correlation Ct (t1 , t2 ) Ferrari-Spohn solution, Eq. (6)(e.g., [6, 16]), and recent simulations suggested that KPZ
may be observed only for finite times, being eventuallyreplaced by the diffusive scaling [17]. This can be argued
to result from the violation of the SU(2) symmetry, due
to the chemical potential µ used to prepare each domainof biased spins [39]. Compared to this, the fate of theflat initial condition, i.e., the initial condition withoutbias of spins, is not clear and has not been studied to ourknowledge, even if some recent simulations of quantum
spin chains hint that KPZ behavior is also visible whenstarting from non-stationary states [40].
We realize a flat initial condition, by drawing eachspin Sj (0) from infinite-temperature equilibrium distribution with a space-dependent vectorial chemical poten|µ |tial µj , ρ(Sj ) = 4π sinhj |µj | eµj ·Sj . The chemical potential is determined as follows: (i) µ1 = 0, (ii) µj≥2 =
Pj−1
tot
tot
tot
′
−µSj−1/|Sj−1| with Sj−1≡
j ′ =1 Sj and µ > 0[41]. This amounts to generating an initial height proPjzfile h0 (j, 0) ≡j ′ =1 Sj ′ that looks like a trajectory ofan Ornstein-Uhlenbeck process [Fig. 4(a) bottom curve]
instead of a Brownian trajectory for the equilibrium caseµ = 0. For KPZ interfaces, such initial conditions are
expected to result asymptotically in the flat KPZ statistics [42], more precisely through a dynamical crossoverfrom the stationary statistics (the Baik-Rains distribution) to the flat one (the GOE Tracy-Widom distribution) [38] without changing the scaling Var[h] ∼ t2/3 , asdemonstrated for TASEP here (Fig. S1). In contrast, for
the LL magnet, we find completely different behavior forµ > 0, showing crossover from the KPZ scaling t2/3 to the
diffusive one t1/2 [Fig. 4(b); see also Fig. 4(a) for the evolution of h(x, t)]. Close scrutiny of the behavior reveals
that this crossover takes place at time scale µ−3 [Fig. 4(b)inset], in agreement with anomalous relaxation discussed
in Ref. [39]. This indicates that the local violation of theisotropy (SU(2) for quantum spins) is sufficient for KPZ
to break down in spin chains.
In summary, using the isotropic LL spin chain, we carried out quantitative tests of KPZ scaling laws for various
two-point quantities that have not been characterized forspin chains so far, and found precise agreement in all ofthem (Table I). Nevertheless, the KPZ scaling laws seemto not describe higher-order quantities, as evidenced byearlier studies [18, 19]. Therefore, the strict KPZ classrules only a subset of statistical properties of isotropic

5
integrable spin chains (and other cases with a continuous
non-Abelian symmetry [12]). It is of primary importanceto clarify the underlying principles of such partial emergence of the KPZ class. It could be explained by the
coupled Burgers equation as proposed in Ref. [32] or by
introducing a larger number of hydrodynamic modes inthe system. Also one cannot exclude that there exists ahitherto unknown observable that can capture full KPZ
statistics. Our finding on the robustness of the KPZ scaling in the presence of energy current, as well as its breakdown by the local violation of isotropy, may also be hints
for probing this mystery, which hangs over such simple
quantum many-body systems as the isotropic Heisenberg
spin chain.
Acknowledgments. The authors acknowledge that
this work deeply benefited from collaboration with
Sarang Gopalakrishnan. K.A.T. also appreciates enlightening discussions with I. Bloch, H. Spohn, K. Takasan,
D. Wei, and J. Zeiher, as well as useful exchanges in
Les Houches School of Physics “Kardar–Parisi–Zhang
equation: new trends in theories and experiments”(2024).
The work is supported in part by KAKENHI from Japan Society for the Promotion of Science(Grant Nos. JP23K17664, JP20H01826, JP19H05800,
JP24K00593) (K.A.T.), by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy - GZ 2047/1, projekt-id
390685813 and by the Projektnummer 211504053 - SFB
1060 (P.L.F.), by ANR-22-CPJ1-0021-01 and ERC Starting Grant 101042293 (HEPIQ) (J.D.N), and by the US
Department of Energy, Office of Science, Basic Energy
Sciences, under award No. DE-SC0023999 (R.V.).

∗

kat@kaztake.org[1] O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura,Emergent hydrodynamics in integrable quantum systems
out of equilibrium, Phys. Rev. X 6, 041065 (2016).
[2] B. Bertini, M. Collura, J. De Nardis, and M. Fagotti,
Transport in out-of-equilibrium XXZ chains: Exact profiles of charges and currents, Phys. Rev. Lett. 117,
207201 (2016).
[3] B. Doyon, Lecture notes on generalised hydrodynamics,
SciPost Phys. Lect. Notes 18, 1 (2020).
[4] V. B. Bulchandani, S. Gopalakrishnan, and E. Ilievski,
Superdiffusion in spin chains, J. Stat. Mech. 2021,
084001 (2021).
[5] S. Gopalakrishnan and R. Vasseur, Superdiffusion fromnonabelian symmetries in nearly integrable systems,
Annu. Rev. Condens. Matter Phys. 15, 159 (2024).
[6] M. Ljubotina, M. Žnidarič, and T. Prosen, Spin diffusionfrom an inhomogeneous quench in an integrable system,
Nat. Commun. 8, 16117 (2017).
[7] S. Gopalakrishnan and R. Vasseur, Kinetic theory of spin
diffusion and superdiffusion in XXZ spin chains, Phys.
Rev. Lett. 122, 127202 (2019).
[8] J. De Nardis, M. Medenjak, C. Karrasch, and E. Ilievski,

Anomalous spin diffusion in one-dimensional antiferromagnets, Phys. Rev. Lett. 123, 186601 (2019).
[9] For reviews, see, e.g., K. A. Takeuchi, An appetizer
to modern developments on the Kardar-Parisi-Zhang
universality class, Physica A 504, 77 (2018); I. Corwin, The Kardar-Parisi-Zhang equation and universality
class, Random Matrices Theory Appl. 1, 1130001 (2012);
S. Prolhac, KPZ fluctuations in finite volume (2004),
arXiv:2401.15016.
[10] M. Ljubotina, M. Žnidarič, and T. Prosen, Kardar-ParisiZhang physics in the quantum Heisenberg magnet, Phys.
Rev. Lett. 122, 210602 (2019).
[11] M. Prähofer and H. Spohn, Exact scaling functions for
one-dimensional stationary KPZ growth, J. Stat. Phys.
115, 255 (2004).
[12] E. Ilievski, J. De Nardis, S. Gopalakrishnan, R. Vasseur,and B. Ware, Superuniversality of superdiffusion, Phys.
Rev. X 11, 031023 (2021).
[13] B. Ye, F. Machado, J. Kemp, R. B. Hutson, and N. Y.
Yao, Universal Kardar-Parisi-Zhang dynamics in integrable quantum systems, Phys. Rev. Lett. 129, 230602(2022).
[14] A. Das, M. Kulkarni, H. Spohn, and A. Dhar, KardarParisi-Zhang scaling for an integrable lattice LandauLifshitz spin chain, Phys. Rev. E 100, 042116 (2019).
[15] A. Scheie, N. E. Sherman, M. Dupont, S. E. Nagler, M. B.
Stone, G. E. Granroth, J. E. Moore, and D. A. Tennant, Detection of Kardar-Parisi-Zhang hydrodynamics
in a quantum heisenberg spin-1/2 chain, Nat. Phys. 17,726 (2021).
[16] D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp,
K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan,
N. Y. Yao, I. Bloch, and J. Zeiher, Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion, Science
376, 716 (2022).
[17] Ž. Krajnik, E. Ilievski, and T. Prosen, Absence of normal
fluctuations in an integrable magnet, Phys. Rev. Lett.
128, 090604 (2022).
[18] Ž. Krajnik, J. Schmidt, E. Ilievski, and T. Prosen, Dynamical criticality of magnetization transfer in integrable
spin chains, Phys. Rev. Lett. 132, 017101 (2024).
[19] E. Rosenberg et al., Dynamics of magnetization at infinite temperature in a Heisenberg spin chain, Science 384,
48 (2024).
[20] J. Quastel and D. Remenik, Airy processes and variational problems, in Topics in Percolative and Disordered Systems, Springer Proceedings in Mathematics & Statistics, Vol. 69, edited by A. Ramı́rez,
G. Ben Arous, P. Ferrari, C. Newman, V. Sidoravicius,and M. Vares (Springer, New York, 2014) pp. 121–171,
arXiv:1301.0750.
[21] P. L. Ferrari and H. Spohn, On time correlations for KPZ
growth in one dimension, SIGMA 12, 074 (2016).
[22] P. L. Ferrari and A. Occelli, Time-time covariance forlast passage percolation with generic initial profile, Math.
Phys. Anal. Geom. 22, 1 (2019).
[23] Ž. Krajnik, E. Ilievski, T. Prosen, and V. Pasquier,Anisotropic Landau-Lifshitz model in discrete spacetime, SciPost Phys. 11, 051 (2021).
[24] M. Lakshmanan, The fascinating world of the landau–lifshitz–gilbert equation: an overview, Phil. Trans.
R. Soc. A 369, 1280 (2011).
[25] J. Baik and E. M. Rains, Limiting distributions for apolynuclear growth model with external sources, J. Stat.

6
Phys. 100, 523 (2000).
[26] M. Prähofer and H. Spohn, Scale Invariance of the PNG
Droplet and the Airy Process, J. Stat. Phys. 108, 1071
(2002).
[27] T. Sasamoto, Spatial correlations of the 1D KPZ surface
on a flat substrate, J. Phys. A 38, L549 (2005).
[28] A. Borodin, P. L. Ferrari, M. Prähofer, and T. Sasamoto,Fluctuation properties of the TASEP with periodic initialconfiguration, J. Stat. Phys. 129, 1055 (2007).
[29] H. Widom, On asymptotics for the Airy process, J. Stat.
Phys. 115, 1129 (2004).
[30] R. Basu, O. Busani, and P. L. Ferrari, On the exponent governing the correlation decay of the Airy1 process,Commun. Math. Phys. 398, 1171 (2023).
[31] J. Baik, P. L. Ferrari, and S. Péché, Limit process ofstationary TASEP near the characteristic line, Commun.
Pure Appl. Math. 63, 1017 (2010).
[32] J. De Nardis, S. Gopalakrishnan, and R. Vasseur, Nonlinear fluctuating hydrodynamics for Kardar-Parisi-Zhang
scaling in isotropic spin chains, Phys. Rev. Lett. 131,
197102 (2023).
[33] X. Zotos, F. Naef, and P. Prelovsek, Transport and conservation laws, Phys. Rev. B 55, 11029 (1997).
[34] M. Prähofer and H. Spohn, Universal distributions for
growth processes in 1 + 1 dimensions and random matri-ces, Phys. Rev. Lett. 84, 4882 (2000).
[35] P. L. Ferrari, Slow decorrelations in Kardar-Parisi-Zhang
growth, J. Stat. Mech. 2008, P07022 (2008).
[36] I. Corwin, P. L. Ferrari, and S. Péché, Universality of slow
decorrelation in KPZ growth, Ann. Inst. H. Poincaré B
Probab. Statist. 48, 134 (2012).
[37] C. B. Mendl and H. Spohn, Searching for the TracyWidom distribution in nonequilibrium processes, Phys.
Rev. E 93, 060101 (2016).
[38] K. A. Takeuchi, Crossover from growing to stationary
interfaces in the Kardar-Parisi-Zhang class, Phys. Rev.
Lett. 110, 210604 (2013).
[39] S. Gopalakrishnan, R. Vasseur, and B. Ware, Anomalous
relaxation and the high-temperature structure factor of
XXZ spin chains, Proc. Natl. Acad. Sci. USA 116, 16250(2019).
[40] G. Cecile, J. De Nardis, and E. Ilievski, Squeezed ensembles and anomalous dynamic roughening in interacting
integrable chains, Phys. Rev. Lett. 132, 130401 (2024).
[41] AfterP zthis initialization, we shift all spins slightly so thatj Sj (0) = 0 holds strictly.
[42] J. Quastel and D. Remenik, How flat is flat in random interface growth?, Trans. Am. Math. Soc. 371, 6047 (2019).