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arXiv:2506.22948 · Chaos An Interdisciplinary Journal of Nonlinear Science

Ising spin-1/2 XXZ chain’s quantum problems beyond the spinon paradigm

J. M. P. Carmelo, Pedro D Sacramento

Published Jul 1, 2024• 1 citation
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Abstract

Spin chains are correlated quantum models of great interest in quantum systems and materials exhibiting quasi-one-dimensional magnetic properties. Here, we review results on quantum problems associated with spin chains that are beyond the usual spinon paradigm. Alternatively, we use a representation valid in the thermodynamic limit, N→∞, in terms of the N spin-1/2 physical spins of the spin-1/2XXZ chain in its whole Hilbert space. It was originally introduced for the isotropic point in Carmelo et al. [Phys. Rev. B 92, 165133 (2015)], co-authored by David, and more recently extended to spin anisotropies Δ>1 in Carmelo et al. [Phys. Rev. Res. 5, 043058 (2023)] and J. M. P. Carmelo and P. D. Sacramento [Nucl. Phys. B 974, 115610 (2022); Nucl. Phys. B 997, 116385 (2023) (Corrigendum)]. The physical-spins representation accounts for the spin-1/2XXZ chain's continuous SUq(2) symmetry parameterized by q=Δ+Δ2-1∈]1,∞] and associated with q-spin Sq. Specifically, in this review we consider two quantum problems that are beyond the spinon representation: (a) Spin Bethe strings of length n that have no spinon representation, contribute to the dynamical properties of the spin-1/2XXZ chain with anisotropy Δ>1 and for n=1,2,3 were experimentally identified and realized in the zigzag materials SrCo2V2O8 and BaCo2V2O8; (b) The spin stiffness associated with ballistic spin transport at arbitrary finite temperature, which involves a huge number of energy eigenstates, many of which are generated in the thermodynamic limit from ground states by an infinite number of elementary processes. As found in Carmelo et al. [Phys. Rev. Res. 5, 043058 (2023)] and J. M. P. Carmelo and P. D. Sacramento [Nucl. Phys. B 974, 115610 (2022); Nucl. Phys. B 997, 116385 (2023) (Corrigendum)], the use of the continuous SUq(2) symmetry reveals that for anisotropy Δ>1 the Bethe strings of length n=1,2,3,… describe a number n of physical-spins Sq=0 singlet pairs that for n>1 are bound within a Sq=0 singlet configuration. Their contribution to the spin dynamical structure factor of both the spin-1/2XXZ chain in a longitudinal magnetic field and the spin chains in SrCo2V2O8 is one of the issues addressed in this paper. In addition, the SUq(2) symmetry imposes that only 2Sq out of the N physical spins are the spin carriers. We also review recent results of J. M. P. Carmelo and P. D. Sacramento ["Diffusive spin transport of the spin-1/2 XXZ chain in the Ising regime at zero magnetic field and finite temperature," (submitted) (2024)] concerning the vanishing of the contributions to finite-temperature ballistic spin transport at zero magnetic field. Within the physical-spins representation, this merely follows from the absolute value of the elementary spin currents carried by the M=2Sq spin carriers of all finite-Sq states that contribute to the spin stiffness being finite. Finally, we discuss deviations of the zigzag materials BaCo2V2O8 and SrCo2V2O8 from the one-dimensional physics described the spin-1/2XXZ chain due to selective interchain couplings.

DOI
10.1063/5.0204689

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Spin chains are correlated quantum models of great interest in quantum systems and materials exhibiting quasi-one-dimensional magnetic properties. Here, we review results on quantum problems associated with spin chains that are beyond the usual spinon paradigm. Alternatively, we use a representation valid in the thermodynamic limit, N→∞, in terms of the N spin-1/2 physical spins of the spin-1/2XXZ chain in its whole Hilbert space. It was originally introduced for the isotropic point in Carmelo et al. [Phys. Rev. B 92, 165133 (2015)], co-authored by David, and more recently extended to spin anisotropies Δ>1 in Carmelo et al. [Phys. Rev. Res. 5, 043058 (2023)] and J. M. P. Carmelo and P. D. Sacramento [Nucl. Phys. B 974, 115610 (2022); Nucl. Phys. B 997, 116385 (2023) (Corrigendum)]. The physical-spins representation accounts for the spin-1/2XXZ chain's continuous SUq(2) symmetry parameterized by q=Δ+Δ2-1∈]1,∞] and associated with q-spin Sq. Specifically, in this review we consider two quantum problems that are beyond the spinon representation: (a) Spin Bethe strings of length n that have no spinon representation, contribute to the dynamical properties of the spin-1/2XXZ chain with anisotropy Δ>1 and for n=1,2,3 were experimentally identified and realized in the zigzag materials SrCo2V2O8 and BaCo2V2O8; (b) The spin stiffness associated with ballistic spin transport at arbitrary finite temperature, which involves a huge number of energy eigenstates, many of which are generated in the thermodynamic limit from ground states by an infinite number of elementary processes. As found in Carmelo et al. [Phys. Rev. Res. 5, 043058 (2023)] and J. M. P. Carmelo and P. D. Sacramento [Nucl. Phys. B 974, 115610 (2022); Nucl. Phys. B 997, 116385 (2023) (Corrigendum)], the use of the continuous SUq(2) symmetry reveals that for anisotropy Δ>1 the Bethe strings of length n=1,2,3,… describe a number n of physical-spins Sq=0 singlet pairs that for n>1 are bound within a Sq=0 singlet configuration. Their contribution to the spin dynamical structure factor of both the spin-1/2XXZ chain in a longitudinal magnetic field and the spin chains in SrCo2V2O8 is one of the issues addressed in this paper. In addition, the SUq(2) symmetry imposes that only 2Sq out of the N physical spins are the spin carriers. We also review recent results of J. M. P. Carmelo and P. D. Sacramento ["Diffusive spin transport of the spin-1/2 XXZ chain in the Ising regime at zero magnetic field and finite temperature," (submitted) (2024)] concerning the vanishing of the contributions to finite-temperature ballistic spin transport at zero magnetic field. Within the physical-spins representation, this merely follows from the absolute value of the elementary spin currents carried by the M=2Sq spin carriers of all finite-Sq states that contribute to the spin stiffness being finite. Finally, we discuss deviations of the zigzag materials BaCo2V2O8 and SrCo2V2O8 from the one-dimensional physics described the spin-1/2XXZ chain due to selective interchain couplings.
Introduction to Integrable Many-Body Systems II This is the second part of a three-volume introductory course about integrable systems of interacting bodies. The models of interest are quantum spin chains with nearest-neighbor interactions between spin operators, in particular Heisenberg spin-1/2 models. The Ising model in a transverse field, expressible as a quadratic fermion form by using the Jordan-Wigner transformation, is the subject of Sect. 12. The derivation of the coordinate Bethe ansatz for the XXZ Heisenberg chain and the determination of its absolute ground state in various regions of the anisotropy parameter are presented in Sect. 13. The magnetic properties of the ground state are explained in Sect. 14. Sect. 15 concerns excited states and the zero-temperature thermodynamics of the XXZ model. The thermodynamics of the XXZ Heisenberg chain is derived on the basis of the string hypothesis in Sect. 16; the thermodynamic Bethe ansatz equations are analyzed in high-temperature and low-temperature limits. An alternative derivation of the thermodynamics without using strings, leading to a non-linear integral equation determining the free energy, is the subject of Sect. 17. A nontrivial application of the Quantum Inverse Scattering method to the fully anisotropic XYZ Heisenberg chain is described in Sect. 18. Sect. 19 deals with integrable cases of isotropic spin chains with an arbitrary spin.
The studies of this paper on the spin- 1 / 2 XXZ chain at finite temperatures T > 0 have two complementary goals. The first is to identify the spin carriers of all its S q > 0 energy eigenstates and to show that their spin elementary currents fully control the spin-transport quantities. Here, S q is the q -spin of the continuous S U q ( 2 ) symmetry of the model for anisotropy Δ > 1 . To achieve this goal, our studies rely on a suitable exact valid in throughout the Hilbert space and associated with normal spin operators, rather than fractionalized particles such as spinons. Both the spin stiffness and the zero-field spin-diffusion constant are expressed in terms of thermal expectation values of the square of the elementary currents carried by the spin carriers. Our second goal is to confirm that the zero-field and finite-temperature spin transport is normal diffusive for Δ > 1 . We use two complementary methods that rely on an inequality for the T > 0 spin stiffness and the above thermal expectation values, respectively, to show that the contributions to ballistic spin transport vanish. Complementarily, for T > 0 and Δ > 1 , the spin-diffusion constant is found to be finite and enhanced upon lowering T , reaching its largest yet finite values at low temperatures. Evidence suggests that it diverges in the Δ → 1 limit for T > 0 , consistent with T > 0 anomalous superdiffusive spin transport at Δ = 1 and zero field.
Abstract Using the Bethe ansatz method and the thermodynamic Bethe ansatz equations for the higher-spin integrable XXZ chain, the regular zero frequency contribution to the spin current correlation (spin dc conductivity) is analyzed for the spin-1/2 XXZ chain with the anisotropy 0 ⩽ Δ < 1 . In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly evaluate the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit. We find that the high temperature proportionality constant σ 0 reg of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π / cos − 1 ⁡ Δ ⩾ 2 with the gap increasing larger than the second power of growing magnetization on one quasi-particle. The isotropic Δ = 1 point is exceptional. Close to this point, σ 0 reg slowly increases in proportion to the first power of the one quasi-particle magnetization. On the other hand, σ 0 reg is proportional to the second power of the same when p 0 approaches irrational numbers.
Spin-1/2 chains demonstrate some of the striking effects of interactions and quantum fluctuations in one-dimensional systems. The XXZ model has been used to study the unusual properties of anisotropic spin chains in an external magnetic field. The zero temperature phase diagram for this model exhibits a critical or quasi-long-range-ordered phase which is a realization of a Luttinger liquid. While many static properties of spin-1/2 chains have been explained by combinations of analytical techniques such as bosonization and Bethe ansatz, the standard approach fails in the calculation of some time-dependent correlation functions. I present a study of the longitudinal dynamical structure factor for the XXZ model in the critical regime. I show that an approximation for the line shape of the dynamical structure factor in the limit of small momentum transfer can be obtained by going beyond the Luttinger model and treating irrelevant operators associated with band curvature effects. This approach is able to describe the width of the on-shell peak and the high-frequency tail at finite magnetic field. Integrability is shown to affect the low-energy effective model at zero field, with consequences for the line shape. The power-law singularities at the thresholds of the particle-hole continuum are investigated using an analogy with the X-ray edge problem. Using methods of Bethe ansatz and conformal field theory, I compute the exact exponents for the edge singularities of the dynamical structure factor. The same methods are used to study the long-time asymptotic behavior of the spin self-correlation function, which is shown to be dominated by a high-energy excitation.
Using the Bethe ansatz method and generalized hydrodynamics (GHD), properties of the spin transport is analyzed for the integrable XXZ chain with the anisotropy 0 ≤ ∆ < 1.Our approach starts from reconstructing the thermodynamic Bethe ansatz (TBA) equations for the integrable XXZ chain with arbitrary spin.We obtain the fusion hierarchy of the quantum transfer matrices (T -functions) in the arbitrary spin case, and we derive functional relations (T -and Y -systems) satisfied by the T -functions and certain combinations of them (Y -functions).Through analytical arguments, the Y -system is reduced to a set of non-linear integral equations, equivalent to the TBA equations.At first, we use the TBA equations for some higher-spin chains in order to examine the regular zero frequency contribution to the spin current correlation (spin dc conductivity) for the spin-1/2 XXZ chain.In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly obtain the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit.We find that the high temperature proportionality constant σ reg 0 of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π/ cos -1 ∆ ≥ 2 with the gap increasing larger than the second power of growing magnetization on one quasiparticle.The isotropic ∆ = 1 point is exceptional.Close to this point, σ reg 0 slowly increases in proportion to the first power of the one quasi-particle magnetization.On the other hand, σ reg 0 is proportional to the second power of the same when p 0 approaches irrational numbers.At second, we examine the finite-temperature spin Drude weight for the integrable XXZ chain with arbitrary spin, where GHD methods are employed.We calculate the spin Drude weight at zero magnetic field, and we find a characteristic fractal-like structure of the Drude weight at arbitrary spin across a broad range of temperatures, similar to the spin-1/2 case.In our approach, the solutions to the TBA equations (i.e., the Y -functions) can be explicitly written in terms of the T -functions, thus allowing for a systematic calculation of the hightemperature limit of the Drude weight, as well as the spin dc conductivity in the spin 1/2 case. B Proof of the equivalence between(2.43) and (2.73) C Delivation of the TBA equations for the integrable XXZ chain with arbitrary spin C.1 Free energy . . . . . . . . . .
Abstract The gapped spin- 12{1\over 2}21​ XXZ chain with spin anisotropy Δ>1\Delta > 1Δ>1 is a strongly correlated quantum system of great interest in a variety of physical contexts. It is well established that, at zero magnetic field, the spin-diffusion constant associated with the regular part of the spin conductivity is finite both at very low and at high temperatures. It is therefore expected that the spin-diffusion constant remains finite at all temperatures. However, a direct calculation of the spin-diffusion constant over the full temperature range is typically intractable. In this paper, we review recent results showing that, at zero magnetic field, the spin-diffusion constant is finite for all temperatures when Δ>1\Delta > 1Δ>1 , and diverges in the limit Δ→1\Delta \rightarrow 1Δ→1 . Consequently, at zero magnetic field, finite-temperature spin transport is diffusive for Δ>1\Delta > 1Δ>1 , but becomes superdiffusive as Δ→1\Delta \rightarrow 1Δ→1 . These results are in agreement with predictions from the generalized hydrodynamics framework. However, the precise values of the spin-diffusion constant remains unknown except in certain temperature limits.