Transport in Almost Integrable Models: Perturbed Heisenberg Chains

The heat conductivity $\kappa (T)$ of integrable models, like the one-dimensional spin-$1/2$ nearest-neighbor Heisenberg model, is infinite even at finite temperatures as a consequence of the conservation laws associated with integrability. Small perturbations lead to finite but large transport coefficients which we calculate perturbatively using exact diagonalization and moment expansions. We show that there are two different classes of perturbations. While an interchain coupling of strength $J_{\perp }$ leads to $\kappa (T)\propto 1/J_{\perp }^2$ as expected from simple golden-rule arguments, we obtain a much larger $\kappa (T)\propto 1/J^{\prime 4}$ for a weak next-nearest-neighbor interaction $J^{\prime }$. This can be explained by a new approximate conservation law of the $J\mathrm{\text{−}}{J}^{\prime }$ Heisenberg chain.

Figure 1

Leading order contribution to the scattering rate $\Gamma (\omega )$ from exact diagonalization of a 20-site Heisenberg chain (thin solid line) with weak nnn coupling $J^{\prime }$ (at $T=\infty$). To this order the scattering rate vanishes at $\omega =0$, implying anomalous transport. The same result is obtained if ${\Gamma }_2(\omega )$ is reconstructed from $N$ moments ($N=10\dots 26$) using the maximum entropy method. The inset shows that ${\Gamma }_2(\omega =0)\to 0$ for $N\to \infty$.

Figure 2

Third and fourth order contributions to the scattering rate for various system sizes (see Fig. 1), the first nonvanishing contribution being of order $g^4$. Note that finite-size effects are small.

Figure 3

Calculated heat conductivity of the anisotropic frustrated chain as a function of temperature for various anisotropies $\Delta ={\Delta }^{\prime }$. Inset: leading order contribution to the scattering rate for the isotropic case (thin solid line) as well as with weak anisotropies for $T\to \infty$.

Figure 4

Leading order (in $J_{\perp }/J$) contribution to the scattering rate of weakly coupled spin chains for $T\to \infty$. The finite value at $\omega =0$ leads to a conductivity $\kappa \approx 0.091(3)J^5/(Z{J}_{\perp }^2{T}^2)$ per chain where $Z$ is the number of nearest-neighbor chains.