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On elliptic quantum integrability: vertex models, solid-on-solid models and spin chains

Jules Lamers

Published Jun 1, 2016• 2 citations
OpenAlex

Abstract

In this thesis the topic of quantum integrability is investigated, with the elliptic case constituting the core of the thesis. The first part of the thesis is about the exact computation of the partition function governing a six-vertex or solid-on-solid model on a lattice of arbitrary but fixed size for a specific choice of boundary conditions. Exact expressions for these quantities were already available due to the work of Korepin and Izergin and others. Using the approach put forward by Galleas we study the partition functions from another point of view. We show that this approach contains that of Korepin-Izergin, while offering an algorithm that allows one to construct, rather than guess, a formula for the partition function. This yields a new expression for that function in the case of a reflecting end and domain walls on the three other ends. The shorter second part of the thesis is about the question whether the partially anisotropic (think: XXZ) version of Inozemtsev's elliptic spin chain is exactly solvable. This model, interpolating between the XXZ and Haldane-Shastry spin chains, is very interesting from a theoretical viewpoint. Inozemtsev's exact solution of the original fully isotropic (XXX) model is rather intricate, and it would be very interesting to know whether that is an isolated case or part of a more general pattern. A few other modifications of Inozemtsev's spin chain are known to be exactly solvable, yet, unlike the partially isotropic version, those models are not 'continuously connected' to (deformations of) Inozemtsev's spin chain. Although we have not yet found a satisfying answer this question is very interesting, and our findings so far might be of interest to other researchers. Quantum integrability and exact solvability are beautiful topics in mathematical physics. They come with algebraic and analytic structures that provide powerful tools enabling one to analyse the models in great detail. The unavoidable consequence, however, is that this field can be quite technical, which tends to makes it rather inaccessible. Therefore a fair portion of this thesis is devoted to an introduction aimed at non-experts with the hope of making the remainder more accessible. These pieces may also be useful by themselves as introductions to quantum integrability in vertex models and exact solvability in spin chains.

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In this work, we study an elliptic solid-on-solid model with domain-wall boundaries having the elliptic quantum group Ep,γ[gl2^] as its underlying symmetry algebra. We elaborate on results previously presented in Galleas [Phys. Rev. E 94, 010102 (2016)] and extend our analysis to include continuous families of single determinantal representations for the model’s partition function. Interestingly, our families of representations are parameterized by two continuous complex variables which can be arbitrarily chosen without affecting the partition function.

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In this thesis the topic of quantum integrability is investigated, with the elliptic case constituting the core of the thesis. The first part of the thesis is about the exact computation of the partition function governing a six-vertex or solid-on-solid model on a lattice of arbitrary but fixed size for a specific choice of boundary conditions. Exact expressions for these quantities were already available due to the work of Korepin and Izergin and others. Using the approach put forward by Galleas we study the partition functions from another point of view. We show that this approach contains that of Korepin-Izergin, while offering an algorithm that allows one to construct, rather than guess, a formula for the partition function. This yields a new expression for that function in the case of a reflecting end and domain walls on the three other ends. The shorter second part of the thesis is about the question whether the partially anisotropic (think: XXZ) version of Inozemtsev's elliptic spin chain is exactly solvable. This model, interpolating between the XXZ and Haldane-Shastry spin chains, is very interesting from a theoretical viewpoint. Inozemtsev's exact solution of the original fully isotropic (XXX) model is rather intricate, and it would be very interesting to know whether that is an isolated case or part of a more general pattern. A few other modifications of Inozemtsev's spin chain are known to be exactly solvable, yet, unlike the partially isotropic version, those models are not 'continuously connected' to (deformations of) Inozemtsev's spin chain. Although we have not yet found a satisfying answer this question is very interesting, and our findings so far might be of interest to other researchers. Quantum integrability and exact solvability are beautiful topics in mathematical physics. They come with algebraic and analytic structures that provide powerful tools enabling one to analyse the models in great detail. The unavoidable consequence, however, is that this field can be quite technical, which tends to makes it rather inaccessible. Therefore a fair portion of this thesis is devoted to an introduction aimed at non-experts with the hope of making the remainder more accessible. These pieces may also be useful by themselves as introductions to quantum integrability in vertex models and exact solvability in spin chains.
The Inozemtsev chain is an exactly solvable interpolation between the short-range Heisenberg and long-range Haldane–Shastry (HS) chains. In order to unlock its potential to study spin interactions with tunable interaction range using the powerful tools of integrability, the model’s mathematical properties require better understanding. As a major step in this direction, we present a new generalisation of the Inozemtsev chain with spin symmetry reduced to \textit{U}(1) 𝑈 ( 1 ) , interpolating between a Heisenberg XXZ chain and the XXZ-type HS chain, and integrable throughout. Underlying it is a new quantum many-body system that extends the elliptic Ruijsenaars system by including spins, contains the trigonometric spin-Ruijsenaars–Macdonald system as a special case, and yields our spin chain by ‘freezing’. Our models have potential applications from condensed-matter to high-energy theory, and provide a crucial step towards a general theory for long-range integrability.
Abstract We consider the construction of quantum-integrable spin chains with q -deformed long-range interactions by ‘freezing’ integrable quantum many-body systems with spins. The input is a (quantum) spin-Ruijsenaars system along with an equilibrium configuration of the underlying spinless classical Ruijsenaars–Schneider system. We focus on the elliptic case. For a distinguished choice of equilibrium, the resulting long-range spin chain has a real spectrum and admits a short-range limit, providing an integrable interpolation from nearest-neighbor to long-range interacting spins. We first define an action of the modular group on the spinless elliptic Ruijsenaars–Schneider system to show that, for a fixed elliptic parameter, it has a whole modular family of classical equilibrium configurations. These typically have constant but nonzero momenta. Then we use the setting of deformation quantization to provide a uniform framework for freezing elliptic spin-Ruijsenaars systems at any classical equilibrium while preserving quantum integrability. As we showed in previous work, the results include the Heisenberg, Inozemtsev and Haldane–Shastry chains along with their xxz -like q -deformations (face type), or the antiperiodic Haldane–Shastry chain of Fukui–Kawakami, its elliptic generalization of Sechin–Zotov, and their completely anisotropic q -deformations due to Matushko–Zotov (vertex type). Finally, we show how freezing fits in the setting of ‘hybrid’ integrable systems.
We consider the construction of quantum-integrable spin chains with q -deformed long-range interactions by ‘freezing’ integrable quantum many-body systems with spins. The input is a (quantum) spin-Ruijsenaars system along with an equilibrium configuration of the underlying spinless classical Ruijsenaars–Schneider system. We focus on the elliptic case. For a distinguished choice of equilibrium, the resulting long-range spin chain has a real spectrum and admits a short-range limit, providing an integrable interpolation from nearest-neighbor to long-range interacting spins. We first define an action of the modular group on the spinless elliptic Ruijsenaars–Schneider system to show that, for a fixed elliptic parameter, it has a whole modular family of classical equilibrium configurations. These typically have constant but nonzero momenta. Then we use the setting of deformation quantization to provide a uniform framework for freezing elliptic spin-Ruijsenaars systems at any classical equilibrium while preserving quantum integrability. As we showed in previous work, the results include the Heisenberg, Inozemtsev and Haldane–Shastry chains along with their xxz -like q -deformations (face type), or the antiperiodic Haldane–Shastry chain of Fukui–Kawakami, its elliptic generalization of Sechin–Zotov, and their completely anisotropic q -deformations due to Matushko–Zotov (vertex type). Finally, we show how freezing fits in the setting of ‘hybrid’ integrable systems.
We review recent progress towards the solution of exactly solved isotropic vertex models with arbitrary toroidal boundary conditions.Quantum space transformations make it possible the diagonalization of the corresponding transfer matrices by means of the quantum inverse scattering method.Explicit expressions for the eigenvalues and Bethe ansatz equations of the twisted isotropic spin chains based on the Bn,B_{n},Bn​, DnD_{n}Dn​ and CnC_{n}Cn​ Lie algebras are presented.The applicability of this approach to the eight vertex model with non-diagonal twists is also discussed.
The algebraic Bethe ansatz is one of the main approaches to solving integrable models in statistical mechanics. This thesis is a detailed introduction to aspects of this technique. In chapter 1, following Takhtajan1, we fully explain all preliminaries that are required by this technique in the context of the 8-vertex model, including the solution of the Yang-Baxter equations in terms of Boltzmann weights that are parameterized by elliptic function. In chapter 2, also following Takhtajan, we restrict our attention to the special case of the 6-vertex model, with Boltzmann weights parameterized by hyperbolic trigonometric functions, formulate the model in detail on a finite lattice with periodic boundary conditions, and show how the algebraic Bethe ansatz can be used to characterize the eigenvectors and eigenvalues, in the homogeneous case (all vertical rapidities are equal, and all horizontal rapidities are also equal). In chapter 3, following Bogoliubov et al2, we consider the 6-vertex model in inhomogeneous case (all rapidities are now independenti variables) but with domain wall boundary conditions. Using the algebraic Bethe ansatz, and Izergin’s result for the partition function, we calculate boundary 1-point functions of the six-vertex model on a finite lattice and represent them in determinant form. Takhatajan, L. J. Introduction to Agebraic Bethe Ansatz, Lecture Notes in Physics, Exactly solvable problems in condensed matter and relativistic field theory 2 Boundary correlation functions of the six-vertex model J. Phys. A: Math. Gen. 35, 5525-5541 (2002)
In this work, we study an elliptic solid-on-solid model with domain-wall boundaries having the elliptic quantum group Ep,γ[gl2^] as its underlying symmetry algebra. We elaborate on results previously presented in Galleas [Phys. Rev. E 94, 010102 (2016)] and extend our analysis to include continuous families of single determinantal representations for the model’s partition function. Interestingly, our families of representations are parameterized by two continuous complex variables which can be arbitrarily chosen without affecting the partition function.