# The equilibrium landscape of the Heisenberg spin chain

### Submission summary

 As Contributors: Enej Ilievski · Eoin Quinn Arxiv Link: https://arxiv.org/abs/1904.11975v3 Date accepted: 2019-07-25 Date submitted: 2019-07-22 Submitted by: Ilievski, Enej Submitted to: SciPost Physics Discipline: Physics Subject area: Mathematical Physics Approach: Theoretical

### Abstract

We characterise the equilibrium landscape, the entire manifold of local equilibrium states, of an interacting integrable quantum model. Focusing on the isotropic Heisenberg spin chain, we describe in full generality two complementary frameworks for addressing equilibrium ensembles: the functional integral Thermodynamic Bethe Ansatz approach, and the lattice regularisation transfer matrix approach. We demonstrate the equivalence between the two, and in doing so clarify several subtle features of generic equilibrium states. In particular we explain the breakdown of the canonical Y-system, which reflects a hidden structure in the parametrisation of equilibrium ensembles.

### Ontology / Topics

See full Ontology or Topics database.

Published as SciPost Phys. 7, 033 (2019)

Dear editor,

We have revised the manuscript by following the suggestions by the referees.
We hope that the manuscript is ready for publication in SciPost.

The authors.

### List of changes

- To better clarify the context of our work, we have rephrased and extended the introduction.
We mention in particular the approach based on the Quantum Transfer Matrix non-linear integral equations (inclunding a reference to Destri and de Vega), and connections to previous work connecting QTM and TBA.
- We highlight $J>0$ corresponds to the ferromagnetic spin chain.
- We softened our statement concerning the `string hypothesis'.
- We added the definition for the scalar convolution integral in Eq. (2.14).
- We clarified the introduction of the infinitesimal regulator $\epsilon\equiv0^+$, and why we include it in our definition of the physical strip.
- In the end of Section 2, we improved the explanations regarding the meaning of the regulator $\epsilon$ for both the local
charges $\mathbf{X}_{j}(v)$ and the mode density operators $\boldsymbol{\rho}_{j}(u)$.
- In the introduction to Section 3, we now mention that local correlation functions are functional of the quasi-particle densities and cite a few relevant papers.
- We have added clarifying remarks after Eq. (4.15).
- In Section 5 we comment that the general vertex model can be regarded as a fused 6-vertex model.
- In Section 6 we have added a footnote on naming conventions for the column transfer matrix.
- In Section 7, we added a footnote regarding the regulator $\epsilon$ and write out explicitly the TBA source terms of the canonical Gibbs state, matching that of ref. [26].
- A summary of the explanation for the breakdown of the canonical $Y$-system is added to the Conclusion in Section 8.

### Submission & Refereeing History

Resubmission 1904.11975v3 on 22 July 2019
Submission 1904.11975v2 on 21 May 2019