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

## TL;DR

This paper investigates finite-size effects on the high-temperature spin stiffness in the XXZ chain, highlighting the influence of symmetries and ensembles, and clarifying conditions under which the Drude weight remains finite or vanishes.

## Contribution

It reveals how symmetries and degeneracies affect the Drude weight calculations and proposes methods to improve convergence to the thermodynamic limit in the gapless phase.

## Key findings

- D vanishes due to spin-inversion symmetry except at specific anisotropies
- Including magnetic flux breaks symmetries and yields finite D for 0<Δ<1
- D vanishes at the isotropic point and in the gapped phase regardless of symmetries

## 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 $sl_2$ 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.