The quasiparticle composition of the eigenstates of quantum spin chains with nearest-neighbor exchange coupling subject to strong axial anisotropy is the focus of this dissertation. Complementary sets of quasiparticles are generated from opposite ends of the spectrum for the systematic generation of all eigenstates. The exclusion statistics of all quasiparticles thus introduced are determined exactly and used, in selected cases, for the exact thermodynamic analysis of the underlying spin chain model. This project starts from the Ising model with spin s = 1/2 and then branches out to the spin s = 1/2 XXZ model in the axial regime and to the spin s = 1 Ising model. In the s = 1/2 Ising case, the two complementary sets of quasiparticles are antiferromagnetic domain walls (solitons) and ferromagnetic domains (strings). In the s = 1 Ising case, they are soliton pairs and nested strings, respectively. The string quasiparticles of the s = 1/2 Ising chains are identified as limiting cases of Bethe ansatz solutions of the axial XXZ model. The thermodynamics of the Ising chain for s 1/2 is shown to be equivalent to that of a system of two species of solitons and for s = 1 to that of a system of six species of soliton pairs. Solitons exist on single bonds but soliton pairs may be spread across many bonds. The thermodynamics of a system of domains spanning up to M lattice sites is amenable to exact analysis and shown to become equivalent, in the limit M → ∞, to the thermodynamics of the s = 1/2 Ising chain. A relation is presented between the solitons in the Ising limit and the spinons in the XX limit of the s = 1/2 XXZ chain. The exclusion statistics is semionic for both spinons and solitons but the spinon vacuum is unique whereas the soliton vacuum is twofold with broken symmetry.
The quasiparticle composition of the eigenstates of quantum spin chains with nearest-neighbor exchange coupling subject to strong axial anisotropy is the focus of this dissertation. Complementary sets of quasiparticles are generated from opposite ends of the spectrum for the systematic generation of all eigenstates. The exclusion statistics of all quasiparticles thus introduced are determined exactly and used, in selected cases, for the exact thermodynamic analysis of the underlying spin chain model. This project starts from the Ising model with spin s = 1/2 and then branches out to the spin s = 1/2 XXZ model in the axial regime and to the spin s = 1 Ising model. In the s = 1/2 Ising case, the two complementary sets of quasiparticles are antiferromagnetic domain walls (solitons) and ferromagnetic domains (strings). In the s = 1 Ising case, they are soliton pairs and nested strings, respectively. The string quasiparticles of the s = 1/2 Ising chains are identified as limiting cases of Bethe ansatz solutions of the axial XXZ model. The thermodynamics of the Ising chain for s 1/2 is shown to be equivalent to that of a system of two species of solitons and for s = 1 to that of a system of six species of soliton pairs. Solitons exist on single bonds but soliton pairs may be spread across many bonds. The thermodynamics of a system of domains spanning up to M lattice sites is amenable to exact analysis and shown to become equivalent, in the limit M → ∞, to the thermodynamics of the s = 1/2 Ising chain. A relation is presented between the solitons in the Ising limit and the spinons in the XX limit of the s = 1/2 XXZ chain. The exclusion statistics is semionic for both spinons and solitons but the spinon vacuum is unique whereas the soliton vacuum is twofold with broken symmetry.
The exclusion statistics of two complementary sets of quasiparticles, generated from opposite ends of the spectrum, are identified for Ising chains with spin s=1/2,1. In the s=1/2 case the two sets are antiferromagnetic domain walls (solitons) and ferromagnetic domains (strings). In the s=1 case they are soliton pairs and nested strings, respectively. The Ising model is equivalent to a system of two species of solitons for s=1/2 and to a system of six species of soliton pairs for s=1. Solitons exist on single bonds but soliton pairs may be spread across many bonds. The thermodynamics of a system of domains spanning up to M lattice sites is amenable to exact analysis and shown to become equivalent, in the limit M -> infinity, to the thermodynamics of the s=1/2 Ising chain. A relation is presented between the solitons in the Ising limit and the spinons in the XX limit of the s=1/2 XXZ chain.
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles. Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplifications occur. The XXZ spectrum can be generated from separate pseudovacua as configurations of sets of quasiparticles with different exclusion statistics. These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set. The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes. In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics. In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics. Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime. In the Ising limit, the solitons become antiferromagnetic domain walls.
The coordinate Bethe ansatz solutions of the XXZ model for a one-dimensional spin-1/2 chain are analyzed with focus on the statistical properties of the constituent quasiparticles.Emphasis is given to the special cases known as XX, XXX, and Ising models, where considerable simplifications occur.The XXZ spectrum can be generated from separate pseudovacua as configurations of sets of quasiparticles with different exclusion statistics.These sets are complementary in the sense that the pseudovacuum of one set contains the maximum number of particles from the other set.The Bethe ansatz string solutions of the XXX model evolve differently in the planar and axial regimes.In the Ising limit they become ferromagnetic domains with integer-valued exclusion statistics.In the XX limit they brake apart into hard-core bosons with (effectively) fermionic statistics.Two sets of quasiparticles with spin 1/2 and fractional statistics are distinguished, where one set (spinons) generates the XXZ spectrum from the unique, critical ground state realized in the planar regime, and the other set (solitons) generates the same spectrum from the twofold, antiferromagnetically ordered ground state realized in the axial regime.In the Ising limit, the solitons become antiferromagnetic domain walls.
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.
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.
The mapping between the fermion and spinon compositions of eigenstates in the one-dimensional spin-1/2 XX model on a lattice with N sites is used to describe the spinon interaction from two different perspectives: (i) For finite N the energy of all eigenstates is expressed as a function of spinon momenta and spinon spins, which, in turn, are solutions of a set of Bethe ansatz equations. The latter are the basis of an exact thermodynamic analysis in the spinon representation of the XX model. (ii) For N -> infinity the energy per site of spinon configurations involving any number of spinon orbitals is expressed as a function of reduced variables representing momentum, filling, and magnetization of each orbital. The spins of spinons in a single orbital are found to be coupled in a manner well described by an Ising-like equivalent-neighbor interaction, switching from ferromagnetic to antiferromagnetic as the filling exceeds a critical level. Comparisons are made with results for the Haldane-Shastry model.