Based on a generalized free energy we derive exact thermodynamic Bethe ansatz formulas for the expectation value of the spin current, the spin current-charge, charge-charge correlators, and consequently the Drude weight. These formulas agree with recent conjectures within the generalized hydrodynamics formalism. They follow, however, directly from a proper treatment of the operator expression of the spin current. The result for the Drude weight is identical to the one obtained 20 years ago based on the Kohn formula and TBA. We numerically evaluate the Drude weight for anisotropies \Delta=\cos(\gamma) Δ = cos ( γ ) with \gamma = \pi n/m γ = π n / m , n\leq m n ≤ m integer and coprime. We prove, furthermore, that the high-temperature asymptotics for general \gamma=\pi n/m γ = π n / m —obtained by analysis of the quantum transfer matrix eigenvalues—agrees with the bound which has been obtained by the construction of quasi-local charges.
Based on a generalized free energy we derive exact thermodynamic Bethe ansatz formulas for the expectation value of the spin current, the spin current-charge, charge-charge correlators, and consequently the Drude weight. These formulas agree with recent conjectures within the generalized hydrodynamics formalism. They follow, however, directly from a proper treatment of the operator expression of the spin current. The result for the Drude weight is identical to the one obtained 20 years ago based on the Kohn formula and TBA. We numerically evaluate the Drude weight for anisotropies \Delta=\cos(\gamma) Δ = cos ( γ ) with \gamma = \pi n/m γ = π n / m , n\leq m n ≤ m integer and coprime. We prove, furthermore, that the high-temperature asymptotics for general \gamma=\pi n/m γ = π n / m —obtained by analysis of the quantum transfer matrix eigenvalues—agrees with the bound which has been obtained by the construction of quasi-local charges.
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 Using generalized hydrodynamics (GHD), we exactly evaluate the finite-temperature spin Drude weight at zero magnetic field for the integrable XXZ chain with arbitrary spin and easy-plane anisotropy. First, we construct the fusion hierarchy of the quantum transfer matrices (T-functions) and 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 thermodynamic Bethe ansatz (TBA) equations. Then, employing GHD, we calculate the spin Drude weight at arbitrary finite temperatures. As a result, a characteristic fractal-like structure of the Drude weight is observed at arbitrary spin, 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 high-temperature limit of the Drude weight.
Based on the method of hydrodynamic projections we derive a concise formula for the Drude weight of the repulsive Lieb-Liniger \delta δ -Bose gas. Our formula contains only quantities which are obtainable from the thermodynamic Bethe ansatz. The Drude weight is an infinite-dimensional matrix, or bilinear functional: it is bilinear in the currents, and each current may refer to a general linear combination of the conserved charges of the model. As a by-product we obtain the dynamical two-point correlation functions involving charge and current densities at small wavelengths and long times, and in addition the scaled covariance matrix of charge transfer. We expect that our formulas extend to other integrable quantum models.
Abstract In this note, we discuss the low and high temperature contribution of thermodynamic Bethe ansatz (TBA) dressed excitations in the thermodynamics and energy-magnetization relaxation within the generalized hydrodynamics approach in the linear response regime. In particular, we show how the temperature dependent dispersions of the excitations reproduce well known behavior of the specific heat, magnetic susceptibility, spin and energy Drude weights. In this context, we derive a further formulation of the Drude weights from the finite wavevector relaxation. Furthermore, we contrast the TBA description of thermodynamics and dynamics in terms of a multitude of string excitations to that in terms of a single quasi-particle in low energy effective theories.
The anomalous nature of spin transport in the XXZ quantum spin chain has been a topic of theoretical interest for some time. Here, the integrability of the underlying dynamics leads to a ballistic component of the spin current, characterized by a spin Drude weight, which measures the degree of divergence of the zero-frequency spin conductivity. However, this quantity had previously proven to be beyond the reach of standard Bethe ansatz techniques. Here, the authors show that a recently developed hydrodynamic formalism for quantum integrable models may be used to compute the spin Drude weight. They also propose a numerical scheme to obtain hydrodynamic predictions for finite-time energy transport. This suggests that the hydrodynamic approach captures completely the ballistic component that dominates transport at long times and distances in the gapless regime of the XXZ model.
We study nonlinear Drude weights (NLDWs) for the spin-21 XXZ chain in the critical regime at zero temperature. The NLDWs are generalizations of the linear Drude weight. Via the nonlinear extension of the Kohn formula, they can be read off from higher-order finite-size corrections to the ground-state energy in the presence of a U(1) magnetic flux. The analysis of the ground-state energy based on the Bethe ansatz reveals that the NLDWs exhibit convergence, power-law, and logarithmic divergence, depending on the anisotropy parameter $\mathrm{\ensuremath{\Delta}}$. We determine the convergent and power-law divergent regions, which depend on the order of the response n. Then, we examine the behavior of the NLDWs at the boundary between the two regions and find that they converge for n=0,1,2(mod4), while they show logarithmic divergence for n=3(mod4). Furthermore, we identify particular anisotropies $\mathrm{\ensuremath{\Delta}}=cos(\ensuremath{\pi}r/(r+1))$ (r=1,2,3,...) at which the NLDW at any order n converges to a finite value.