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Integrable Matrix Models in Discrete Space-Time

by Žiga Krajnik, Enej Ilievski, Tomaz Prosen

Submission summary

As Contributors: Enej Ilievski
Arxiv Link: (pdf)
Date submitted: 2020-03-16
Submitted by: Ilievski, Enej
Submitted to: SciPost Physics
Discipline: Physics
Subject area: Mathematical Physics
Approaches: Theoretical, Computational


We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic $\sigma$-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau--Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar--Parisi--Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.

Current status:
Editor-in-charge assigned

Reports on this Submission

Anonymous Report 2 on 2020-6-15 Invited Report


1. New results on the role of non-abelian symmetries for superdiffusive transport in 1D.
2. Numerical results are convincing.
3. Construction of a novel class of integrable matrix models.




The authors introduce a (to the best of my knowledge) novel class of
integrable classical matrix models in discrete space time. These can
be viewed as Trotterizations of non-relativistic coset sigma
models. The models are integrable by virtue of a discrete
zero-curvature condition that guarantees the existence of infinitely
many conservation laws. The authors then carry out numerical studies
of transport properties in equilibrium by computing non-equal time
two-point functions of Noether charges. The numerics establishes in a
very convincing fashion that the dynamical exponent is z=3/2, in
agreement with KPZ superdiffusive behavior. They also find scaling
collapse of scaled "dynamical structure factors".

I think this is an excellent piece of work that makes an important
contribution to the current debate on KPZ universality in equilibrium
transport properties in one dimensional quantum and classical
many-body systems. In particular it provides strong support for the
conjecture formulated by the authors, namely that all discrete space-time
models built as Floquet circuits from two-body symplectic Yang-Baxter maps
with dynamical variables living on compact non-abelian symmetric
spaces exhibit superdiffusion of the KPZ type in equilibrium states
with unbroken symmetry. The work paves the way a work in a number of
interesting directions, e.g. the role of anisotropy. The manuscript is
very well written and I recommend publication in its current form.

  • validity: top
  • significance: top
  • originality: top
  • clarity: top
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2020-5-8 Invited Report


A simple class of integrable dynamical systems with reasonably convincing numerical evidence of KPZ universality.


1) Notations are not always consistent, see some examples below, which makes some parts of the paper hard to read without having to guess what the authors meant.

2) It is not clear which parts of the detailed analysis of the integrable structure of the model in section 2 is used for the numerical study of the large scale KPZ behavior in section 3.


The authors introduce a new class of integrable dynamical systems describing the discrete time evolution of matrix valued fields located on a one-dimensional lattice with periodic boundary conditions. The invariant measure of the model and the integrability of the dynamics are studied in details.

The two-point correlation function of the models is then studied numerically for large values of time and number of lattice sites. Starting with equilibrium initial condition, the properly rescaled result is, after fitting amplitude and distance, in reasonable agreement with the stationary two-point function of the KPZ fixed point on the infinite line. Since the central part of the KPZ stationary two-point function is very close to a Gaussian, however, different choices of fitting parameters would presumably almost make it possible to fit the data obtained with a Gaussian too.

Given the simplicity of the models introduced and the distinct possibility that further studies manage to establish analytically that these models belong to KPZ universality, I think that this paper is perfectly suitable for publication in scipost.

Requested changes

1) It is not immediately clear that $L^{(+)}(\lambda, M)$ in (2.2) and $L_{n,m}^{(+)}(\lambda)$ in (2.1) are the same object. It would be useful to write down explicitly the relation between $L^{(+)}(\lambda, M_{l}^{t})$ and $L_{n,m}^{(+)}(\lambda)$.

2) The space $\mathcal{M}_{1}$ before (2.7) is not defined.

3) What is $\mathcal{Z}^{(k,N)}$ in (2.68) ? Is it the same as $\mathcal{Z}$ in (2.64) ?

4) The use of the Duistermaat–Heckman formula in section 2.3.3 should probably be detailed a bit.

5) It would be useful to specify precisely which parts of section 2 are used in section 3. Is the integrability of the model needed at all for the numerics ?

6) fig. 5: (d) $(2,N)=(1,4)$ ?

  • validity: high
  • significance: high
  • originality: high
  • clarity: ok
  • formatting: perfect
  • grammar: perfect

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