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Chinese Physics Letters

Quantum Magnetism from Low-Dimensional Quantum Ising Models with Quantum Integrability

Yunjing Gao, Jianda Wu

Published Mar 5, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0256-307x/42/4/047501

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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.
Quantum integrability emerging near a quantum critical point (QCP) is manifested by an exotic excitation spectrum that is organized by the associated algebraic structure. A well-known example is the emergent E8{E}_{8}E8​ integrability near the QCP of a transverse field Ising chain (TFIC), which was long predicted theoretically and initially proposed to be realized in the quasi-one-dimensional (q1D) quantum magnet CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​. However, later measurements on the spin excitation spectrum of this material revealed a series of satellite peaks in low energies that go beyond the description of the E8{E}_{8}E8​ Lie algebra. Motivated by this experimental progress, we hereby revisit the spin excitations of CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​ by combining numerical calculation and analytical analysis. We show that, as effects of strong interchain fluctuations, the spectrum of the system near the 1D QCP is characterized by an emergent D8(1){D}_{8}^{(1)}D8(1)​ Lie algebra instead of the E8{E}_{8}E8​ one. We further show that the D8(1){D}_{8}^{(1)}D8(1)​ spectrum can be realized in a broad class of interacting quantum systems. Our results advance the exploration and manipulation of integrability in quantum critical systems.
Abstract The one dimensional (1D) spin-1/2 Heisenberg–Ising model, a prototype quantum many-body system, has been intensively studied for many years. In this review, after a short introduction on some basic concepts of group theory for the octahedral group, a detailed pedagogical framework is laid down to derive the low-energy effective Hamiltonian for the Co-based materials. The 1D spin-1/2 Heisenberg–Ising model is obtained when applying the analysis to quasi-1D antiferromagnetic materials BaCo 2 V 2 O 8 and SrCo 2 V 2 O 8 . After the preparation, we review the theoretical progresses of a variety of novel magnetic excitations and emergent physics in the 1D spin-1/2 Heisenberg–Ising model, and further summarize their recent experimental realizations.
By appropriately perturbing a critical transverse-field Ising chain away from its critical point, the system can develop a finite correlation length with a characteristic purely massive spectrum, whose ratios and correlations are precisely described by an integrable field theory and an infinite set of integrals of motion corresponding to the E8E_8E8​ Lie algebra. In this work, we report on experimental observation of a characteristic massive spectrum close to transverse field-induced dimensional reduction in a quasi-two-dimensional quantum magnet Cu2_22​(OH)3_33​Br, providing evidence for an emergent E8E_8E8​ symmetry and the corresponding excitations of bound states in the sublattice of its ferromagnetic chains. These results demonstrate the power of integrable field theory in describing emergent many-body quantum critical phenomena in condensed matter systems.
We review quantum phase transitions of spin systems in transverse magnetic fields taking the examples of the spin-1/2 Ising and XY models in a transverse field. Beginning with an overview of quantum phase transitions, we introduce a number of model Hamiltonians. We provide exact solutions in one spatial dimension connecting them to conformal field theoretical studies. We also discuss Kitaev models and some other exactly solvable spin systems. Studies of quantum phase transitions in the presence of quenched randomness and with frustrating interactions are presented in detail. We discuss novel phenomena like Griffiths-McCoy singularities. We then turn to more recent topics like information theoretic measures of the quantum phase transitions in these models such as concurrence, entanglement entropy, quantum discord and quantum fidelity. We then focus on non-equilibrium dynamics of a variety of transverse field systems across quantum critical points and lines. After mentioning rapid quenching studies, we dwell on slow dynamics and discuss the Kibble-Zurek scaling for the defect density following a quench across critical points and its modifications for quenching across critical lines, gapless regions and multicritical points. Topics like the role of different quenching schemes, local quenching, quenching of models with random interactions and quenching of a spin chain coupled to a heat bath are touched upon. The connection between non-equilibrium dynamics and quantum information theoretic measures is presented at some length. We indicate the connection between Kibble-Zurek scaling and adiabatic evolution of a state as well as the application of adiabatic dynamics as a tool of a quantum optimization technique known as quantum annealing. The final section is dedicated to a detailed discussion on recent experimental studies of transverse Ising-like systems.
Quantum phase transitions take place between distinct phases of matter at zero temperature. Near the transition point, exotic quantum symmetries can emerge that govern the excitation spectrum of the system. A symmetry described by the E8 Lie group with a spectrum of eight particles was long predicted to appear near the critical point of an Ising chain. We realize this system experimentally by using strong transverse magnetic fields to tune the quasi-one-dimensional Ising ferromagnet CoNb2O6 (cobalt niobate) through its critical point. Spin excitations are observed to change character from pairs of kinks in the ordered phase to spin-flips in the paramagnetic phase. Just below the critical field, the spin dynamics shows a fine structure with two sharp modes at low energies, in a ratio that approaches the golden mean predicted for the first two meson particles of the E8 spectrum. Our results demonstrate the power of symmetry to describe complex quantum behaviors.
Recently, single-site observables have been shown to be useful for probing critical slowing down in sudden quench dynamics [Da\ifmmode \breve{g}\else \u{g}\fi{} et al., Phys. Rev. B 107, L121113 (2023)]. Here, we demonstrate the potential of single-site magnetization as a probe of quantum phase transitions in integrable and nonintegrable transverse-field Ising chains (TFIC). We analytically prove the requirement of zero modes for the quasistationary regime to emerge at a probe site near the edge, and show how this regime gives rise to a nonanalytic behavior in the dynamical order profiles. Our t\mathit{t}t-DMRG calculations verify the results of the quench mean-field theory for near-integrable TFIC both with finite-size and finite-time scaling analyses. We find that both finite-size and finite-time analyses suggest a dynamical critical point for a strongly nonintegrable and locally connected TFIC. We finally demonstrate the presence of a quasistationary regime in the power-law interacting TFIC, and extract local dynamical order profiles for TFIC in the long-range Ising universality class with algebraic light cones.