Spin conductivity of the XXZ chain in the antiferromagnetic massive regime

TL;DR

The paper develops an exact form-factor expansion for the dynamical current-current correlator in the spin-1/2 XXZ chain in the antiferromagnetic massive regime at zero temperature, enabling highly accurate calculations of time-dependent spin transport and the optical conductivity. The expansion sums over even numbers of particle–hole excitations, with the leading one-hole–one-particle (two-spinon) term dominating the long-time, large-distance behavior and admitting a closed-form expression for the associated optical conductivity inside the 2-spinon band. Higher spinon sectors (ℓ≥2) are numerically tractable and contribute progressively less away from the isotropic point, with the f-sum rule showing that ℓ=1 and ℓ=2 capture most spectral weight. The work provides a robust, non-perturbative framework for finite-frequency spin transport in an integrable model and lays the groundwork for finite-temperature extensions and broader operator analyses.

Abstract

We present a series representation for the dynamical two-point function of the local spin current for the XXZ chain in the antiferromagnetic massive regime at zero temperature. From this series we can compute the correlation function with very high accuracy up to very long times and large distances. Each term in the series corresponds to the contribution of all scattering states of an even number of excitations. These excitations can be interpreted in terms of an equal number of particles and holes. The lowest term in the series comprises all scattering states of one hole and one particle. This term determines the long-time large-distance asymptotic behaviour which can be obtained explicitly from a saddle-point analysis. The space-time Fourier transform of the two-point function of currents at zero momentum gives the optical spin conductivity of the model. We obtain highly accurate numerical estimates for this quantity by numerically Fourier transforming our data. For the one-particle, one-hole contribution, equivalently interpreted as a two-spinon contribution, we obtain an exact and explicit expression in terms of known special functions. For large enough anisotropy, the two-spinon contribution carries most of the spectral weight, as can be seen by calculating the f-sum rule.

Figures (6)

• Figure 1: The real and imaginary contributions of the $\ell=1$ term and of the sum of $\ell=1$ and $\ell=2$ terms to $\bigl\langle {\cal J}_1 (t) {\cal J}_{1} \bigr\rangle$ for $\Delta=1.2$ are compared to LCRG results. The insets show that the $\ell=2$ contribution becomes negligible on this scale for $tJ>2$.
• Figure 2: $\langle{\cal J}_1 (t) {\cal J}\bigr\rangle$ for $\Delta=3$, $L_c=349$ and times $0<tJ<5$ (left), $10 < tJ <24$ (right).
• Figure 3: $\langle{\cal J}_1 (t) {\cal J}\bigr\rangle$ for $\Delta=1.5$ and times $0<tJ<5$ (left), $10 < tJ <24$ (right). We used $L_c=219$.
• Figure 4: Comparison of the analytic result (\ref{['sigmaTwospinon']}) and a numerical Fourier transformation of the $\ell=1$ part of $\bigl\langle {\cal J}_1 (t) {\cal J}_{m+1} \bigr\rangle$ for anisotropy $\Delta=3$. For the latter we use $0\le m \le 399$ and $0\le tJ \le 50$.
• Figure 5: $\ell=1$ contribution, Eq. (\ref{['sigmaTwospinon']}), to ${\rm Re}\, \sigma^{(2)}(\omega)$ for various $\Delta$.

Theorems & Definitions (5)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3