Heat Conductivity of the Heisenberg Spin-$1/2$ Ladder: From Weak to Strong Breaking of Integrability

We investigate the heat conductivity $\kappa$ of the Heisenberg spin-$1/2$ ladder at finite temperature covering the entire range of interchain coupling $J_{\perp }$, by using several numerical methods and perturbation theory within the framework of linear response. We unveil that a perturbative prediction $\kappa \propto J_{\perp }^{-2}$, based on simple golden-rule arguments and valid in the strict limit $J_{\perp }\to 0$, applies to a remarkably wide range of $J_{\perp }$, qualitatively and quantitatively. In the large $J_{\perp }$ limit, we show power-law scaling of opposite nature, namely, $\kappa \propto J_{\perp }^2$. Moreover, we demonstrate the weak and strong coupling regimes to be connected by a broad minimum, slightly below the isotropic point at $J_{\perp }=J_{\parallel }$. Reducing temperature $T$, starting from $T=\infty$, this minimum scales as $\kappa \propto T^{-2}$ down to $T$ on the order of the exchange coupling constant. These results provide for a comprehensive picture of $\kappa (J_{\perp },T)$ of spin ladders.

Figure 1

Thermal conductivity $\kappa$ (per chain) versus $J_{\perp }/J_{\parallel }$ and $\beta$. PT, known perturbative regime. Issues clarified in this Letter are as follows: extent of power-law scaling (dashed line) close to PT and sum-rule (SR) regimes, nonperturbative numerical treatment for entire $J_{\perp }$ range, location of minimal conductivity, and temperature variation.

Figure 2

(a) $t$ and (b) $\omega$ dependence of the autocorrelation $C$ for strong $J_{\perp }/J_{\parallel }\ge 1$, $\beta J_{\parallel }\to 0$, and $N\le 32$, as obtained from DQT. Spectra in (b) are obtained by Fourier transforming finite-$t$ data $t\le 10\tau \sim 20/J_{\parallel }$ (symbols). Inset: Low-$\omega$ limit for $J_{\perp }/J_{\parallel }=1$, the largest $N=32$, and $t\le 5\tau$ (crosses), $10\tau$ (other symbols), $50\tau$ (curves). Main panel: Spectra from Lanczos methods ($N=28$ MCLM of Ref. [61], $N=22$ FTLM) are shown (curves). Note that method-related errors are negligibly small [63].

Figure 3

(a) The $t$ dependence of $C$ for various small $J_{\perp }/J_{\parallel }=0.15,\dots ,0.75$, obtained from DQT for $\beta J_{\parallel }\to 0$ and $N\le 30$. (b) Spectrum for $J_{\perp }/J_{\parallel }=0.25$, obtained by Fourier transforming finite-$t$ data $t\le 5\tau \sim 80/J_{\parallel }$ (symbols). Inset: Low-$\omega$ limit for the largest $N=32$ and $t\le 5\tau$ (diamonds), $10\tau$ (crosses). Main panel: Spectrum from $N=22$ and 28 MCLM and a Lorentzian fit are shown (curves).

Figure 4

(a) Scaling of the conductivity $\kappa$ with $J_{\perp }$, obtained from DQT and finite-$t$ data $t\le 5\tau$ for $\beta J_{\parallel }\to 0$ and $N\le 32$ (closed symbols). Results for the simplified operator $j^{\prime }=j_{\parallel }$ are also depicted at $J_{\perp }/J_{\parallel }\sim 1$ (open symbols). Additionally, power laws $0.097(J_{\perp }/J_{\parallel }{)}^{-2}$ and $0.21(J_{\perp }/J_{\parallel }{)}^2$ are shown (lines). (b) PT for the scattering rate $\gamma$, carried out using DQT. The PT of Ref. [59] is also depicted (bullet).

Figure 5

(a) The $t$ dependence of $C$ for $\beta J_{\parallel }=0.5,\dots ,1.5$, obtained from DQT for $J_{\perp }/J_{\parallel }=1$ and $N=28$. (b) Spectrum for $\beta J_{\parallel }=1$, obtained by Fourier transforming finite-$t$ data $t\le 5\tau \sim 12/J_{\parallel }$ for $N\le 32$. Additionally, a spectrum from $N=22$ FTLM is depicted (curve). Inset: $T$ dependence of the conductivity $\kappa$, calculated by $N=32$ DQT (closed symbols, curve) and $N=22$ FTLM (open symbols).