Integrability of quantum chains: theory and applications to the spin-1/2 $XXZ$ chain

Abstract

In this contribution we review the theory of integrability of quantum systems in one spatial dimension. We introduce the basic concepts such as the Yang-Baxter equation, commuting currents, and the algebraic Bethe ansatz. Quite extensively we present the treatment of integrable quantum systems at finite temperature on the basis of a lattice path integral formulation and a suitable transfer matrix approach (quantum transfer matrix). The general method is carried out for the seminal model of the spin-1/2 $XXZ$ chain for which thermodynamic properties like specific heat, magnetic susceptibility and the finite temperature Drude weight of the thermal conductivity are derived.

References in corpus (14)

• Transport and conservation laws
• Susceptibility of the Spin 1/2 Heisenberg Antiferromagnetic Chain
• Thermodynamics of the Anisotropic Spin-1/2 Heisenberg Chain and Related Quantum Chains
• Low energy effective Hamiltonian for the XXZ spin chain
• The spin-1/2 Heisenberg chain: thermodynamics, quantum criticality and spin-Peierls exponents
• Thermodynamics of the Spin-1/2 Antiferromagnetic Uniform Heisenberg Chain
• The Hubbard chain at finite temperatures: ab initio calculations of Tomonaga-Luttinger liquid properties
• Continued fraction TBA and functional relations in XXZ model at root of unity
• Quantum Integrals of Motion for the Heisenberg Spin Chain
• Exact thermodynamics and Luttinger liquid properties of the integrable t-J model
• Exact calculation of thermodynamical quantities of the integrable t-J model
• Critical exponents of a multicomponent anisotropic t-J model in one dimension
• Simplification of thermodynamic Bethe-ansatz equations
• Presence of energy flux in quantum spin chains: An experimental signature

Cited by in corpus (56)

• What is an integrable quench?
• T-systems and Y-systems in integrable systems
• Spectral statistics and many-body quantum chaos with conserved charge
• A real-time study of diffusive and ballistic transport in spin-1/2 chains using the adaptive time-dependent density matrix renormalization group method
• Quenching the XXZ spin chain: quench action approach versus generalized Gibbs ensemble
• Generalized Gibbs Ensemble for Heisenberg Spin Chains
• Exact steady states for quantum quenches in integrable Heisenberg spin chains
• Dynamical free energy and the Loschmidt-echo for a class of quantum quenches in the Heisenberg spin chain
• Form factor expansion for thermal correlators
• From the Quantum Transfer Matrix to the Quench Action: The Loschmidt echo in $XXZ$ Heisenberg spin chains
• Spin transport in the XXZ chain at finite temperature and momentum
• Non-analytic behavior of the Loschmidt echo in XXZ spin chains: exact results
• The Drude weight of the spin-1/2 XXZ chain: density matrix renormalization group versus exact diagonalization
• Baxter's Q-operators and operatorial Backlund flow for quantum (super)-spin chains
• Integrable models and quantum spin ladders: comparison between theory and experiment for the strong coupling ladder compounds
• Short distance correlators in the XXZ spin chain for arbitrary string distributions
• Overlaps between eigenstates of the XXZ spin-1/2 chain and a class of simple product states
• Non-Equilibrium Mass Transport in the 1D Fermi-Hubbard Model
• Exact calculation of the magnetocaloric effect in the spin-1/2 XXZ chain
• Low-temperature asymptotics of integrable systems in an external field
• Functional Bethe ansatz methods for the open XXX chain
• Integrable Matrix Product States from boundary integrability
• Spin and thermal conductivity of quantum spin ladders
• Long-distance behavior of temperature correlation functions in the one-dimensional Bose gas
• Magneto-thermal properties of the Heisenberg-Ising orthogonal-dimer chain with triangular XXZ-clusters
• Manipulating energy and spin currents in nonequilibrium systems of interacting qubits
• Exact description of the magnetoelectric effect in the spin-1/2 XXZ-chain with Dzyaloshinskii-Moriya interaction
• A note on the spin-1/2 XXZ chain concerning its relation to the Bose gas
• Integrable quenches in nested spin chains I: the exact steady states
• Alternation of sign of magnetization current in driven XXZ chains with twisted XY boundary gradients
• Magnetic susceptibility of quantum spin systems calculated by sine square deformation: one-dimensional, square lattice, and kagome lattice Heisenberg antiferromagnets
• Lattice distortions in a sawtooth chain with Heisenberg and Ising bonds
• Correlation functions of one-dimensional bosons at low temperature
• Correlation lengths of the repulsive one-dimensional Bose gas
• Thermodynamics, contact and density profiles of the repulsive Gaudin-Yang model
• Introduction to Quantum Integrability
• Finite temperature correlation functions from discrete functional equations
• Thermodynamics, spin-charge separation and correlation functions of spin-1/2 fermions with repulsive interaction
• Finite temperature density matrix and two-point correlations in the antiferromagnetic XXZ chain
• Exact boundary free energy of the open XXZ chain with arbitrary boundary conditions
• Phase diagrams of the Ising-Heisenberg chain with S=1/2 triangular XXZ clusters
• Popcorn Drude weights from quantum symmetry
• Optimal transport and von Neumann entropy in an Heisenberg XXZ chain out of equilibrium
• Thermodynamics, density profiles and correlation functions of the inhomogeneous one-dimensional spinor Bose gas
• Anomalous currents in driven XXZ chain with boundary twisting at weak coupling or weak driving
• Efficient Thermodynamic Description of Multi-Component One-Dimensional Bose Gases
• Excitation Spectrum Gap and Spin-Wave Stiffness of XXZ Heisenberg Chains: Global Renormalization-Group Calculation
• Ground-state phase diagram for a system of interacting, $D(D_3)$ non-Abelian anyons
• Full decoherence induced by local fields in open spin chains with strong boundary couplings
• A Q-operator for the quantum transfer matrix
• Surface energy and elementary excitations of the XXZ spin chain with arbitrary boundary fields
• The propagator of the finite XXZ spin-$\frac{1}{2}$ chain
• Giant caloric effects in spin chain materials
• Low-temperature asymptotic of the transverse dynamical structure factor for a magnetically polarized XX chain
• Specific heat and susceptibility of S=1/2 antiferromagnets on square, triangular, and kagome lattices
• Thermodynamics of an Ising-like XXZ chain in a longitudinal magnetic field in the framework of the Quantum Transfer Matrix approach