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Theoretical and Mathematical Physics

First integrals of the one-dimensional quantum ising model with transverse magnetic field

Vadim V. Anshelevich, Е. В. Гусев

Published May 1, 1981• 4 citations
Publisher
DOI
10.1007/bf01086396

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We analytically investigated the phase diagrams of one-dimensional Ising and XY models. A second-order phase transition occurs in both models only with the transverse magnetic field. In this study, the fully connected ferromagnetic and anti-ferromagnetic XX interactions are applied in addition to the transverse magnetic field. The ferromagnetic XX interaction causes a first-order phase transition in both of the one-dimensional Ising and XY models. In contrast, by applying the anti-ferromagnetic one, the second-order phase transition can be avoided in the one-dimensional Ising model, and the critical amplitude becomes smaller in the one-dimensional XY model even though the second-order phase transition remains and the critical exponent does not change. Because the anti-ferromagnetic XX interaction works to destroy the order of the transverse magnetization, it begins to decrease at a larger strength of the transverse magnetic field. As a result, the susceptibility of the transverse magnetization with respect to the strength of the magnetic field becomes small. It is conjectured that the anti-ferromagnetic XX interaction is generally preferable to the ferromagnetic interaction to improve the performance of quantum annealing.

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We analytically investigated the phase diagrams of one-dimensional Ising and XY models. A second-order phase transition occurs in both models only with the transverse magnetic field. In this study, the fully connected ferromagnetic and anti-ferromagnetic XX interactions are applied in addition to the transverse magnetic field. The ferromagnetic XX interaction causes a first-order phase transition in both of the one-dimensional Ising and XY models. In contrast, by applying the anti-ferromagnetic one, the second-order phase transition can be avoided in the one-dimensional Ising model, and the critical amplitude becomes smaller in the one-dimensional XY model even though the second-order phase transition remains and the critical exponent does not change. Because the anti-ferromagnetic XX interaction works to destroy the order of the transverse magnetization, it begins to decrease at a larger strength of the transverse magnetic field. As a result, the susceptibility of the transverse magnetization with respect to the strength of the magnetic field becomes small. It is conjectured that the anti-ferromagnetic XX interaction is generally preferable to the ferromagnetic interaction to improve the performance of quantum annealing.
Abstract Quantum integrability provides a unique and powerful framework for accurately understanding quantum magnetism. In this review, we focus specifically on several quantum integrable low-dimensional quantum Ising models. We begin with the transverse field Ising chain (TFIC) at quantum critical point and examine how it evolves under perturbations, such as an applied longitudinal field or weak coupling to another quantum critical TFIC. These perturbations reveal a wealth of emergent quantum integrable field theories with exotic many-body excitations, elegantly characterized by conformal invariance and the E 8 and D 8 ( 1 ) Lie algebras, respectively. In exploring these models, we also delve into the framework of exact scattering matrices, which is related to determining spin dynamics within these systems. Finally, we show how the emergent phenomena in these integrable quantum Ising models find experimental realization in Co-based quasi-one-dimensional quantum magnetic materials. The substantial theoretical and experimental advancements in these systems highlight the profound connections between quantum integrable field theory, statistical field theory, and condensed matter physics.
This review reports a series of theoretical and experimental progress on researches of the transverse field Ising chain (TFIC) and the quantum E8 integrable model. For the TFIC, on one hand, a unique exotic quantum critical behavior of Grüneisen ratio (a ratio from magnetic or thermal expansion coefficient to specific heat) is theoretically established; on the other hand microscopic models can accommodate the TFIC universality class are substantially expanded. These progresses successfully promote a series of experiments collaborations to first-time realize the TFIC universality class in quasi one-dimensional anti-ferromagnetic material BaCo2V2O8 and SrCo2V2O8. For the quantum E8 integrable model, the low temperature local dynamics and the dynamical structure factor with zero transfer momentum of this system are analytically determined, where a cascade of edge singularities with power-law divergences are obtained in the continuum region of the dynamical structure factor. After combining with detailed quantum critical scaling behaviors analysis and large scale iTEBD calculation, it successfully facilitates a series of experiments, including THz spectrum measurements, inelastic neutron scattering and NMR experiments, to realize the quantum E8 integrable model in BaCo2V2O8 for the first time. The experimental realization of the quantum E8 integrable model substantially extends the frontiers of studying quantum integrable models in real materials. The series of progress and developments on the TFIC and the quantum E8 integrable model lay down a concrete ground to go beyond quantum integrability, and can inspire studies in condensed matter systems, cold atom systems, statistical field theory and conformal field theory.