Journal of Statistical Physics manuscript No.
(will be inserted by the editor)
Kardar-Parisi-Zhang physics in integrable rotationally
symmetric dynamics on discrete space-time lattice
ˇ
Ziga Krajnik · Tomaˇz Prosen
Accepted version of the manuscript, 19 February 2020
Abstract We introduce a deterministic SO(3) invariant dynamics of classical
spins on a discrete space-time lattice and prove its complete integrability by ex-
plicitly finding a related non-constant (baxterized) solution of the set-theoretic
quantum Yang-Baxter equation over the 2-sphere. Equipping the algebraic struc-
ture with the corresponding Lax operator we derive an infinite sequence of con-
served quantities with local densities. The dynamics depend on a single continuous
spectral parameter and reduce to a (lattice) Landau-Lifshitz model in the limit of
a small parameter which corresponds to the continuous time limit. Using quasi-
exact numerical simulations of deterministic dynamics and Monte Carlo sampling
of initial conditions corresponding to a maximum entropy equilibrium state we de-
termine spin-spin spatio-temporal (dynamical) correlation functions with relative
accuracy of three orders of magnitude. We demonstrate that in the equilibrium
state with a vanishing total magnetization the correlation function precisely follow
Kardar-Parisi-Zhang scaling hence the spin transport belongs to the universality
class with dynamical exponent z = 3/2, in accordance to recent related simulations
in discrete and continuous time quantum Heisenberg spin 1/2 chains.
Keywords Integrable systems · Classical spin chains · Transport · Space-time
duality · KPZ universality class
1 Introduction
Identifying exactly solvable cases of universal physical phenomena is one of the
central goals of statistical physics. While this endeavour has matured in equilib-
rium physics it is still at its very early stage for non-equilibrium phenomena such as
transport. Considering interacting particle models without any hidden degrees of
freedom, meaning that the underlying microscopic equations of motion are deter-
ministic and reversible
1
, establishing macroscopic transport laws, such as Fick’s or
Physics Department, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska
19, SI-1000, Ljubljana, Slovenia
E-mail: tomaz.prosen@fmf.uni-lj.si
1
Excluding stochastic systems or any external sources of noise.
2
ˇ
Ziga Krajnik, Tomaˇz Prosen
Ohm’s law is particularly hard, while only very recently fully explicit and rigorous
results started to emerge (see e.g. [1]). On a more heuristic level, distinct types of
transport phenomena, related e.g. to two important universality classes given by
the diffusion equation and the Kardar-Parisi-Zhang [2] (KPZ) equation, have been
explained via nonlinear fluctuating hydrodynamics (NFH) which crucially depends
on the number of conserved fields (such as mass, energy or momentum densi-
ties) and nonlinear coupling relations among their currents. With this heuristic
theory, one can predict either diffusive or anomalous broadening of the (mov-
ing) sound-peaks and (static) sound-peak in the space-time resolved dynamical
response functions and their precise asymptotic scaling profiles [3,4,5]. Whenever
three appropriately coupled conserved fields have been identified, such as in the
Femi-Pasta-Ulam problem [6], the mean-field description of Bose gasses [7], or
classical XXZ spin chains at low temperatures [8], the broadenings have been ex-
plained in terms of the KPZ scaling x ∼ t
1/z
with dynamical exponent z = 3/2, in
distinction to diffusive and ballistic universality classes with exponents z = 2 and
z = 1 which, respectively, typically emerge in ‘more generic’ chaotic or integrable
models.
Very recently, however, a new kind of incarnation of KPZ physics in deter-
ministic statistical systems has been suggested. Specifically, studying dynamical
correlations in quantum Heisenberg (XXX) chain of spins 1/2 at vanishing mag-
netization (or zero magnetic field), it has been demonstrated that the sole contri-
bution to transport comes from the heat-peak with the sound-peaks being absent,
but that the former broadens with a perfect KPZ scaling [9] over several orders
of magnitude. This result is consistent with earlier observations [10] of z ∼ 1.5
in the integrable lattice Landau-Lifshitz (LLL) chain of classical spins [11] which
can be thought of as an integrable classical version of the XXX model. More
recently, these numerical experiments have been refined, confirming also the pre-
cise KPZ scaling profile of the heat-peak [12]. Such a behaviour seem to crucially
depend on the complete integrability and on rotational (SO(3)) symmetry, but
not on its quantum or classical nature. While hydrodynamics has recently been
generalized to integrable systems with infinite number of conservation laws [13,
14] (so-called GHD), where diffusion and anomalous diffusion could be included
with some heuristics [16,17], it is not clear at present how and if one can ex-
plain the observed behavior within NFH. From empirical observations, it seems
that for this type of KPZ scaling one needs two ingredients: (i) complete integra-
bility and (ii) existence of a global non-abelian symmetry (such as SU(2)) with
non-commuting conserved generators. See Ref. [15] for additional data corrobo-
rating this conjecture for higher spin integrable quantum models with higher rank
nonabelian symmetries. Since low energy regimes of non-integrable lattice models
are often described by integrable field theories, one could then apply such results
also to non-integrable SO(3) symmetric lattice models at low temperatures [18] or
integrable SO(3) symmetric field theories [19].
In order to prepare the stage for a rigorous case study, as well as to sharpen
numerical evidence as much as possible, we are here defining and studying arguably
the simplest dynamical system satisfying the required conditions and demonstrate
that it indeed exhibits KPZ scaling.
In the first part of the paper (section 2) we introduce a discrete time dynamical
system of N normalized angular momenta (classical spins), each taking values on a
2-sphere S
2
, which is generated by a simple symplectic (canonical) transformation
KPZ physics in integrable SO(3) symmetric dynamics on discrete space-time lattice 3
over (S
2
)
×N
. This many-body map has the form of a classical Floquet circuit
built from a simple local 2-spin mapping which is a simple non-linear rational and
rotationally symmetric bijective transformation of a pair of unit 3-vectors. Such
a dynamical system should be of interest in its own right, since we demonstrate
that the 2-spin mapping satisfies a baxterized set theoretic quantum Yang-Baxter
equation where the spectral parameter plays the role of the ‘integration time-step’.
Moreover, we introduce an appropriate Lax matrix and prove the corresponding set
theoretic quantum RLL relation which allows us to compute an extensive family of
conserved fields of the model. The model therefore represents the simplest known
rotationally (SO(3)) symmetric integrable dynamics in discrete-space time and
due to its efficient simulability provides a perfect playground for testing the above
phenomenological conjecture on the KPZ scaling. Moreover, we show that our
dynamics exhibits a remarkable space-time symmetry, namely it is generated by
essentially the same deterministic and reversible many-body map if one flips the
time and space axes. In other words, knowing the value of a fixed spins at all
moments in time, we can find (via ‘space dynamics’) unique values of all other
spins at all time steps.
In the second part of the paper (section 3) we then numerically explore dy-
namical spin-spin correlation functions in the simplest separable invariant state
(which can be understood as an infinite temperature/maximum entropy state at
fixed average magnetization) and demonstrate that it obeys a clean KPZ scaling
for vanishing magnetization. When changing the magnetization parameter we then
demonstrate a crossover to a ballistic scaling which could be captured within GHD.
Moreover, when slightly breaking integrability of the discrete-time mapping while
keeping the same continuous time limit (namely, the LLL model), we demonstrate
an immediate drift of dynamical exponents towards the diffusive value z = 2.
2 Integrable SO(3) invariant dynamics on a discrete space-time lattice
2.1 Definition of the model
Let S
1
, S
2
denote a pair of three-dimensional unit vectors, S
1
· S
1
= S
2
· S
2
= 1.
We define a one parameter family of rational nonlinear maps Φ
τ
between a pair
of 2-spheres Φ
τ
: S
2
× S
2
→ S
2
× S
2
, as:
Φ
τ
(S
1
, S
2
) =
1
σ
2
+ τ
2
σ
2
S
1
+ τ
2
S
2
+ τS
1
× S
2
, σ
2
S
2
+ τ
2
S
1
+ τS
2
× S
1
, (1)
σ
2
:=
1
2
1 + S
1
· S
2
,
where τ ∈ R is a real parameter, which will later be interpreted as the discretization
time step. A simple calculation shows that the map Φ
τ
preserves the unit norm
of the pair of vectors and is invertible, thus it represents a bijection on S
2
× S
2
which is clearly invariant under rotations:
(S
0
1
, S
0
2
) = Φ
τ
(S
1
, S
2
) ⇔ (RS
0
1
, RS
0
2
) = Φ
τ
(RS
1
, RS
2
), R ∈ SO(3). (2)
Eq. (1) defines the elementary two-body propagator of our model. Let us now
proceed to a definition of a discrete-time dynamics for a lattice (chain) of an even
4
ˇ
Ziga Krajnik, Tomaˇz Prosen
number N ∈ 2N of unit vectors:
S
t
x
∈ S
2
, x ∈ Z
N
, t ∈ Z, (3)
which we define as follows:
(S
2t+1
2x
, S
2t+1
2x+1
) = Φ
τ
(S
2t
2x
, S
2t
2x+1
), (S
2t+2
2x−1
, S
2t+2
2x
) = Φ
τ
(S
2t+1
2x−1
, S
2t+1
2x
), (4)
for integer space-time indices x ∈ Z
N/2
, t ∈ Z (see a schematic depiction in Fig. 1).
This prescription can be understood as a discrete-time, deterministic, reversible
dynamical system generated by an invertible dynamical map Ψ
τ
: M → M over
a product of N 2-spheres, M = (S
2
)
×N
, which is defined as a composition of an
even and odd half-time step propagators:
(S
2t+2
0
, S
2t+2
1
, . . . , S
2t+2
N−1
) = Ψ
τ
(S
2t
0
, S
2t
1
, . . . , S
2t
N−1
), (5)
Ψ
τ
= Ψ
odd
τ
◦ Ψ
even
τ
,
Ψ
even
τ
= Φ
⊗N/2
τ
,
Ψ
odd
τ
= η
−1
◦ Ψ
even
τ
◦ η.
The map:
η(S
0
, S
1
, . . . , S
N−2
, S
N−1
) = (S
1
, S
2
, . . . , S
N−1
, S
0
) (6)
is a periodic translation on a classical spin-ring M. The tensor product of maps
over a cartesian product of their domain sets is defined as (Ω ⊗ Λ)(x, y) ≡
(Ω(x), Λ(y)).
Note that this discrete space-time dynamics is a classical analog of a local
quantum circuit representation of a Trotter decomposition of unitary Hamilto-
nian dynamics. Particularly, since as we will show below, Φ
τ
can be generated
by a suitable 2-spin Hamiltonian and hence the many-body map Ψ is a canoni-
cal transformation which is generated by a suitable (periodically) time-dependent
Hamiltonian. The model can thus also be interpreted as a classical local Floquet
circuit.
2.2 The Hamiltonian structure of the elementary two-body interaction
Before demonstrating the integrability of the model we make a brief detour and
show the Hamiltonian and symplectic character of the building blocks of the model.
We seek a Hamiltonian H(S
1
, S
2
) that will generate the canonical transformation
(1) through the equations of motion after a time specified by τ. Since the two-body
interaction is invariant under SO(3), the Hamiltonian must be a function of the
scalar invariant of a pair of unit vectors, i.e. it should be of the form:
H(S
1
, S
2
) = 2h(σ), σ
2
=
1
2
(1 + S
1
· S
2
). (7)
Since we aim to interpret vectors S
n
, n ∈ {1, 2}, as classical angular momenta
(which we shall simply refer to as ‘spins’) we invoke the SO(3) Poisson bracket
with the canonical relations:
{S
n;a
, S
m;b
} = δ
n,m
X
c
ε
abc
S
n;c
, (8)
KPZ physics in integrable SO(3) symmetric dynamics on discrete space-time lattice 5
S
2t+2
2x
S
2t
2x
S
2t+1
2x
S
2t+3
2x
S
2t
2x+2
S
2t
2x+4
S
2t
2x+1
S
2t
2x+3
S
2t
2x+5
· · ·
.
.
.
t
x
Φ
τ
Φ
τ
Φ
τ
Φ
τ
Φ
τ
Φ
τ
Φ
τ
Φ
τ
Φ
τ
Φ
τ
Φ
τ
Φ
τ
Fig. 1: Classical local symplectic circuit representation of discrete space-time dy-
namics of the model. Lattice spins (taking values on 2-spheres) are denoted by
black circles. A green rectangle represents a two-body propagator (1) applied to a
pair of spins. Time increases along the vertical axis, the spatial lattice runs along
the horizontal axis.
where ε
abc
is the Levi-Civita symbol and S
n;a
, a ∈ {1, 2, 3} denote the three
components of the vector S
n
. Hamilton’s equations of motion then take the form:
˙
S
n
:=
dS
n
dt
= {S
n
, H}, (9)
where the Poisson bracket acts on every component of the vector S
j
. Explicitly,
we have:
˙
S
1
= (S
1
× S
2
)
h
0
(σ)
2σ
,
˙
S
2
= (S
2
× S
1
)
h
0
(σ)
2σ
= −
˙
S
1
. (10)
The pair of Eqs. (10) implies that the sum of the spins and their dot product is
conserved in time. The equations of motion can be rewritten in a form such that
the second vector in the cross product is of unit length Σ = (S
1
+ S
2
)/(2σ):
˙
S
n
= h
0
(σ)S
n
× Σ (11)
from which it is clearly seen that the spins rotate around their conserved sum with
angular velocity h
0
(σ) := dh(σ)/dσ, hence their time evolution can be explicitly
expressed via Rodrigues’ rotation formula:
S
1
(t) =
1
2
h
S
1
(0)(1 + cos h
0
t) + S
2
(0)(1 − cos h
0
t) + S
1
(0) × S
2
(0)
sin h
0
t
σ
i
, (12)
S
2
(t) =
1
2
h
S
2
(0)(1 + cos h
0
t) + S
1
(0)(1 − cos h
0
t) + S
2
(0) × S
1
(0)
sin h
0
t
σ
i
.
Comparing the pair of Eqs. (12) at time t = 1 with Eq. (1) we conclude that the
angular velocity must satisfy the following differential equation:
tan h
0
(σ) =
2στ
τ
2
− σ
2
, (13)
![Fig. 3: Spin autocorrelation function (50) of the integrable space-time discrete dynamics (4) at zero average magnetization µ = 0. Parameters of the simulation: τ = 1, number of lattice sites N = 213, final simulation time tmax = 2 12, averaging over a sample of M = 105 initial spin configurations sampled from distribution (48). Panel (a) shows a density plot of |C(x, t)| in log-scale indicated in the legend. In (b) we plot snapshots of autocorrelation function cross sections rescaled according to (51), at times indicated in the legend. The dotted lines show the best-fit Gaussian and KPZ scaling functions gPS [28]. The KPZ fit is of the form C(x, t)t 2 3 = ag ( bx/t 2 3 ) , with a = 0.024, b = 0.29.](/figures/fig-3-spin-autocorrelation-function-50-of-the-integrable-2e92whw5.webp)
