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

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 ${\tilde{\mathrm{\Delta }}}_{MN}=cos(\pi M/N)$ with $N,M\in {\mathbb{Z}}^{+}$ coprimes and $N>M$, 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 data from the system sizes and anisotropies ${\tilde{\mathrm{\Delta }}}_{MN}$ (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 $\mathrm{\Delta }$ 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 $0<\mathrm{\Delta }<1$ in both ensembles. In contrast, at the isotropic point and in the gapped phase ($\mathrm{\Delta }\ge 1$) $D$ 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 ($\mathrm{\Delta }<1$) 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 ${\tilde{\mathrm{\Delta }}}_{MN}$ anisotropies are lifted.

Figure 1

$XXZ$ 20-site chain level-spacing distribution for states within the sector of zero total magnetization for two different anisotropies. The distribution is averaged over each spin-inversion sector $z$, for every crystal momentum, as well as over parities for $K=\{0,\pi \}$. Note the peak at $\delta E=0$ for $\mathrm{\Delta }=0.5$ signaling the presence of additional degeneracies.

Figure 2

Finite size scaling of the Drude weight for different values of the anisotropy $\mathrm{\Delta }$, calculated within both canonical (CE) and grand canonical ensemble (GCE), and in the presence and absence of a magnetic flux. The CE computations are carried out for states with zero total magnetization. Symbol coding: $▴$ GCE with flux, $▵$ GCE without flux, $\blacksquare$ CE with flux, and $\square$ CE without flux. The dashed lines correspond to second order polynomial fits and are to be taken as a guide to the eye. Top panels show results for the gapless phase, with the crosses representing exact results in the thermodynamic limit [16, 17], and the type of current carrying states that contribute to the finite-$LD$ values labeled as nondegenerate ($\mathrm{NDg}$), degenerate ($\mathrm{Dg}$), and $\mathrm{\Lambda }$-degenerate states (${\mathrm{Dg}}_{\mathrm{\Lambda }}$); the dominant contributor is highlighted in bold. Bottom panels show results for $\mathrm{\Delta }\ge 1$, consistent with vanishing Drude weight for $L\to \infty$, independent of ensemble or symmetry.

Figure 3

Nonfluxed GCE ratios of the contribution to the Drude weight coming from degenerate/nondegenerate states only $D_{\text{(dg,ndg)}}$ to the total Drude weight $D$ (i.e., including both degenerate and nondegenerate states). Filled (empty) symbols correspond to nondegenerate (degenerate) contributions. Squares indicate data for $\mathrm{\Delta }={\tilde{\mathrm{\Delta }}}_{1,3}$, circles for $\mathrm{\Delta }={\tilde{\mathrm{\Delta }}}_{2,5}$, and triangles for $\mathrm{\Delta }=0.25$. Triangles also correspond to any other incommensurate value or commensurate value with $N>L/2$, as, e.g., ${\tilde{\mathrm{\Delta }}}_{13,31}\approx 0.25$. The dashed lines are second order polynomial fits included to guide the eye.

Figure 4

Finite size scaling of the Drude weight in the canonical ensemble for two $\mathrm{\Delta }$ values, calculated in the presence (full symbols) and absence (empty symbols) of a magnetic flux, for different magnetization sectors. Crosses indicate $m_z=0$ values in the presence of a flux. As $m_z$ and $L$ increase the effect of flux disappears for both anisotropies; note however that for larger $\mathrm{\Delta }$ the $m_z=0$ states contribute the least, whereas it is the opposite for smaller $\mathrm{\Delta }$. This explains why, when the weighted average is taken, $m_z=0$ CE data is above (below) the GCE data as seen in the upper (lower) right panel of Fig. 2 for these two anisotropies. Note that our $D$ computation is for fixed $m_z$ (rather than $m_z/L$) and hence the tendency to extrapolate to zero, as well as the nonmonotonicity at large $L$ seen in the top figure.

Figure 5

Finite size scaling of the Drude weight computed in both fluxed CE ($\blacksquare$) and GCE ($▴$) within the gapless phase. The crosses represent the exact lower bound results, Eq. (9), and the stars the thermodynamic Bethe ansatz results, Eq. (10).

Figure 6

Finite size scaling of the Drude weight for additional commensurate anisotropies ${\tilde{\mathrm{\Delta }}}_{MN}$, ordered by decreasing $N$ and computed within both canonical (CE) and grand canonical ensemble (GCE) and in the presence and absence of a magnetic flux. Symbol coding: $▴$ GCE with flux, $▵$ GCE without flux, $\blacksquare$ CE with flux, and $\square$ CE without flux. The dashed lines correspond to second order polynomial fits and are to be taken as a guide to the eye. The red vertical lines indicate the system size at which the $\mathrm{\Lambda }$ degeneracies start taking place. The crosses represent the exact lower bound results, Eq. (9), and the stars the thermodynamic Bethe ansatz results, Eq. (10).