One-dimensional Bose-gas One-dimensional Heisenberg magnet Massive Thirring model Classical r-matrix Fundamentals of inverse scattering method Algebraic Bethe ansatz Quantum field theory integral models on a lattice Theory of scalar products Form factors Mean value of operator Q Assymptotics of correlation functions Temperature correlation functions Appendices References.

Quantum Inverse Scattering Method and Correlation Functions

This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kac interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas.

Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction

We provide a new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model. The proof applies to infinite-range models on arbitrary locally finite transitive infinite graphs. For Bernoulli percolation, we prove finiteness of the susceptibility in the subcritical regime $${\beta < \beta_c}$$
 , and the mean-field lower bound $${\mathbb{P}_\beta[0\longleftrightarrow \infty ]\ge (\beta-\beta_c)/\beta}$$
 for $${\beta > \beta_c}$$
 . For finite-range models, we also prove that for any $${\beta < \beta_c}$$
 , the probability of an open path from the origin to distance n decays exponentially fast in n. For the Ising model, we prove finiteness of the susceptibility for $${\beta < \beta_c}$$
 , and the mean-field lower bound $${\langle \sigma_0\rangle_\beta^+\ge \sqrt{(\beta^2-\beta_c^2)/\beta^2}}$$
 for $${\beta > \beta_c}$$
 . For finite-range models, we also prove that the two-point correlation functions decay exponentially fast in the distance for $${\beta < \beta_c}$$
 .

A New Proof of the Sharpness of the Phase Transition for Bernoulli Percolation and the Ising Model

We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their random-cluster representations. More precisely, we prove that 
1. For the Potts model on transitive graphs, correlations decay exponentially fast for $\beta<\beta_c$. 
2. For the random-cluster model with cluster weight $q\geq1$ on transitive graphs, correlations decay exponentially fast in the subcritical regime and the cluster-density satisfies the mean-field lower bound in the supercritical regime. 
3. For the random-cluster models with cluster weight $q\geq1$ on planar quasi-transitive graphs $\mathbb{G}$, 
$$\frac{p_c(\mathbb{G})p_c(\mathbb{G}^*)}{(1-p_c(\mathbb{G}))(1-p_c(\mathbb{G}^*))}~=~q.$$ As a special case, we obtain the value of the critical point for the square, triangular and hexagonal lattices (this provides a short proof of the result of Beffara and Duminil-Copin [Probability Theory and Related Fields, 153(3-4):511--542, 2012]). 
These results have many applications for the understanding of the subcritical (respectively disordered) phase of all these models. The techniques developed in this paper have potential to be extended to a wide class of models including the Ashkin-Teller model, continuum percolation models such as Voronoi percolation and Boolean percolation, super-level sets of massive Gaussian Free Field, and random-cluster and Potts model with infinite range interactions.

Sharp phase transition for the random-cluster and Potts models via decision trees

We discuss walking behavior in gauge theories and weak first-order phase transitions in statistical physics. Despite appearing in very different systems (QCD below the conformal window, the Potts model, deconfined criticality) these two phenomena both imply approximate scale invariance in a range of energies and have the same RG interpretation: a flow passing between pairs of fixed point at complex coupling. We discuss what distinguishes a real theory from a complex theory and call these fixed points complex CFTs. By using conformal perturbation theory we show how observables of the walking theory are computable by perturbing the complex CFTs. This paper discusses the general mechanism while a companion paper [1] will treat a specific and computable example: the two-dimensional Q-state Potts model with Q > 4. Concerning walking in 4d gauge theories, we also comment on the (un)likelihood of the light pseudo-dilaton, and on non-minimal scenarios of the conformal window termination.

https://link.springer.com/content/pdf/10.1007/JHEP10(2018)108.pdf

Walking, Weak first-order transitions, and Complex CFTs

We prove that the $q$-state Potts model and the random-cluster model with cluster weight $q>4$ undergo a discontinuous phase transition on the square lattice. More precisely, we show 
- Existence of multiple infinite-volume measures for the critical Potts and random-cluster models, 
- Ordering for the measures with monochromatic (resp. wired) boundary conditions for the critical Potts model (resp. random-cluster model), and 
- Exponential decay of correlations for the measure with free boundary conditions for both the critical Potts and random-cluster models. 
The proof is based on a rigorous computation of the Perron-Frobenius eigenvalues of the diagonal blocks of the transfer matrix of the six-vertex model, whose ratios are then related to the correlation length of the random-cluster model. 
As a byproduct, we rigorously compute the correlation lengths of the critical random-cluster and Potts models, and show that they behave as $\exp(\pi^2/\sqrt{q-4})$ as $q$ tends to 4.

Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi$ and energy $\Lambda$, which satisfy $V \psi = \Lambda \psi$, where $V$ is the the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights $a= b=1$ and $c > 0$. We also show that the same vector $\psi$ satisfies $H \psi = E \psi$, where $H$ is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value $E$ computed explicitly. 
Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors that amounts to proving that the random cluster model on $\mathbb{Z}^2$ with cluster weight $q >4$ exhibits a first-order phase transition.

The Bethe ansatz for the six-vertex and XXZ models: an exposition

We consider a general class of two-dimensional spin systems, with continuous but not necessarily smooth, possibly long-range, O(N)-symmetric interactions, for which we establish algebraically decaying upper bounds on spin-spin correlations under all infinite-volume Gibbs measures.

/pdf/upper-bound-on-the-decay-of-correlations-in-a-general-class-511kylcrym.pdf

Upper Bound on the Decay of Correlations in a General Class of O(N)-Symmetric Models

We consider a general class of two-dimensional spin systems, with continuous but not necessarily smooth, possibly long-range, $O(N)$-symmetric interactions, for which we establish algebraically decaying upper bounds on spin-spin correlations under all infinite-volume Gibbs measures. 
As a by-product, we also obtain estimates on the effective resistance of a (possibly long-range) resistor network in which randomly selected edges are shorted.

Maxime Gagnebin

Papers

Discontinuity of the phase transition for the planar random-cluster and Potts models with $q>4$

The Bethe ansatz for the six-vertex and XXZ models: an exposition

Upper Bound on the Decay of Correlations in a General Class of O(N)-Symmetric Models

Upper bound on the decay of correlations in a general class of O(N)-symmetric models