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Closed hierarchy of Heisenberg equations in integrable models with Onsager algebra

by Oleg Lychkovskiy

Submission summary

As Contributors: Oleg Lychkovskiy
Arxiv Link: (pdf)
Date submitted: 2021-03-22 09:29
Submitted by: Lychkovskiy, Oleg
Submitted to: SciPost Physics
Academic field: Physics
  • Mathematical Physics
  • Condensed Matter Physics - Theory
Approach: Theoretical


Dynamics of a quantum system can be described by coupled Heisenberg equations. In a generic many-body system these equations form an exponentially large hierarchy that is intractable without approximations. In contrast, in an integrable system a small subset of operators can be closed with respect to commutation with the Hamiltonian. As a result, the Heisenberg equations for these operators can form a smaller closed system amenable to an analytical treatment. We demonstrate that this indeed happens in a class of integrable models where the Hamiltonian is an element of the Onsager algebra. We explicitly solve the system of Heisenberg equations for operators from this algebra. Two specific models are considered as examples: the transverse field Ising model and the superintegrable chiral 3-state Potts model.

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Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2021-4-19 Invited Report


1- explicit expressions for the time-dependent expectation values of observables corresponding to operators spanning the Onsager algebra
2- exact out-of-equilibrium dynamics for the transverse field Ising model in certain translational invariant initial states of product form
3- approximate out-of-equilibrium dynamics of the polarization in the 3-state chiral Potts model for an fully polarized initial state.


1- limited to Hamiltonians and observables being elements of the Onsager algebra
2- method relies on explicit representation of the Onsager algebra related to a specific model (which is known only for the Ising model)
3- site-resolution e.g. for initial states without translational invariance requires significant extension of the approach


The development of methods providing analytical results for the dynamics of out-of-equilibrium many-body systems is attacting a lot of activity. In this paper the author addresses this problem by identifying cases where the Heisenberg equations of motion close to a finite set for certain classes of observables (as in models of non-interacting fermions or bosons). Specifically, he considers (superintegrable) models where the Hamiltonian is an element of the Onsager algebra.

In this case, the time evolution of the operators $A^n$, $G^n$ spanning this algebra is given by a system of linear equations of motion. Moreover, representations of the Onsager algebra related to spin chains of given length are finite dimensional [26] which allows for a truncation of the system to a finite one. These equations are solved giving explicit expressions for the time-dependent (Heisenberg) operators spanning the Onsager algebra for which the thermodynamic limit can be taken.

The author applies this general result to two specific models. For the transverse field Ising model closed forms for the $A^n$, $G^n$ are known. For a class of translational invariant initial states of product form this allows to obtain compact expressions for the time-dependent expectations values for these operators, extending previous results obtained in the fermionic formulation of the Ising model [34].

The second example considered in the manuscript is the 3-state chiral Potts model. Here the representation of the $A^n$, $G^n$ in terms of local spin operators is not known for general $n$. The author observes, however, that taking into account the first few of these operators into account leads to rapidly converging approximations to the exact expectation value. This is demonstrated for the time dependent local polarization in a completely polarized pure initial state.

In summary, the use of the Onsager algebra provides a straightforward way to study the dynamics of the corresponding superintegrable models with sufficiently simple, in particular translationally invariant initial states.
Adding strictly local operators to the Onsager algebra may well allow to address problems without translational invariance thereby complementing existing approaches. This appears feasible but will require a significant extension of the approach.

I recommend publication of this manuscript in SciPost Physics.

  • validity: top
  • significance: high
  • originality: high
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  • grammar: excellent

Anonymous Report 1 on 2021-4-18 Invited Report


The paper present the solution of the Heisenberg equation for integrable models related to the Onsager algebra. This result allows to study Out-of-equilibrium dynamics of the Ising model and the 3-state Potts model.


No discussion are given about the fact that most of the integrable models are not related to Lie algebra but to quantum groups (XXZ spin chain as exemple). In that case solving the Heisenberg equation could be more complex.


I think that this that the paper is of interest and deserve to be published. If the author can comment about the case of models related to quantum groups It could be a nice addition.

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

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