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

Abstract

We revisit the numerical problem of computing the high temperature spin stiffness, or Drude weight, $D$ of the spin-$1/2$ $XXZ$ chain using exact diagonalization to systematically analyze its dependence on system symmetries and ensemble. Within the canonical ensemble and for states with zero total magnetization, we find $D$ vanishes exactly due to spin-inversion symmetry for all but the anisotropies ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}_{MN}=cos(\ensuremath{\pi}M/N)$ with $N,M\ensuremath{\in}{\mathbb{Z}}^{+}$ coprimes and $N>M$, provided system sizes $L\ensuremath{\ge}2N$, 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 data from the system sizes and anisotropies ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}_{MN}$ (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 $\mathrm{\ensuremath{\Delta}}$ 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 $0<\mathrm{\ensuremath{\Delta}}<1$ in both ensembles. In contrast, at the isotropic point and in the gapped phase ($\mathrm{\ensuremath{\Delta}}\ensuremath{\ge}1$) $D$ 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 ($\mathrm{\ensuremath{\Delta}}<1$) 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 ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\mathrm{\ensuremath{\Delta}}}}_{MN}$ anisotropies are lifted.