Finite-size anomalies of the Drude weight: role of symmetries and ensembles

Abstract

We revisit the subtelties of computing the high temperature spin stiffness $D$ of the spin-$1/2$ XXZ chain using exact diagonalization to analyze its dependence on system symmetries and ensemble. Within the canonical ensemble and for states with zero magnetization, we find $D$ vanishes exactly due to spin-inversion symmetry for all but the anisotropies ${\tilde{\Delta }}_{MN}=\cos (\pi M/N)$ with $N>M$ and coprime, provided system sizes $L\ge 2N$, for which states with different spin-inversion signature become degenerate due to the underlying $s{l}_2$ loop algebra symmetry. All these loop-algebra degenerate states carry finite currents which we conjecture [based on $L$ and anisotropies ${\tilde{\Delta }}_{MN}$ (with $N<L/2$) available to us] to dominate the grand-canonical ensemble evaluation of $D$ in the thermodynamic limit. Including a magnetic flux not only breaks spin-inversion in the zero magnetization sector but also lifts the loop-algebra degeneracies in all symmetry sectors --- this effect is more pertinent at smaller $\Delta$ due to the larger contributions to $D$ coming from the low-magnetization sectors which are more sensitive to the system's symmetries. Thus we generically find a finite $D$ for fluxed rings and arbitrary $0<\Delta <1$ in both ensembles. In contrast, at the isotropic point and in the gapped phase ($\Delta \ge 1$) $D$ is found to vanish in the thermodynamic limit, independent of symmetry or ensemble. Our analysis demonstrates how convergence to the thermodynamic limit within the gapless phase ($\Delta <1$) may be accelerated and the finite-size anomalies overcome: $D$ extrapolates nicely in the thermodynamic limit to either the recently computed lower-bound or the Thermodynamic Bethe Ansatz result provided both spin-inversion is broken and the additional degeneracies at the ${\tilde{\Delta }}_{MN}$ anisotropies are lifted.

9 + epsilon pages, 6 figures, 2 tables; v2 provides stronger support for origin of these anomalies (version accepted in Phys. Rev. B)

References in corpus (6)

• Open XXZ spin chain: Nonequilibrium steady state and strict bound on ballistic transport
• Spin transport in a one-dimensional anisotropic Heisenberg model
• Microscopic origin of ideal conductivity in integrable quantum models
• Drude weight in systems with open boundary conditions
• Hubbard-to-Heisenberg crossover (and efficient computation) of Drude weights at low temperatures
• Anomalous nonequilibrium current fluctuations in the Heisenberg model

Cited by in corpus (7)

• Finite-temperature transport in one-dimensional quantum lattice models
• Super-diffusion in one-dimensional quantum lattice models
• Superdiffusion in spin chains
• Anomalous low-frequency conductivity in easy-plane XXZ spin chains
• Asymmetry in energy versus spin transport in certain interacting, disordered systems
• Anomalous and regular transport in spin 1/2 chains: AC conductivity
• Coexistence of diffusive and ballistic transport in integrable quantum lattice models