Typicality approach to the optical conductivity in thermal and many-body localized phases

We study the frequency dependence of the optical conductivity $\text{Re}\sigma \left(\omega \right)$ of the Heisenberg spin-$\frac{1}{2}$ chain in the thermal and near the transition to the many-body localized phase induced by the strength of a random $z$-directed magnetic field. Using the method of dynamical quantum typicality, we calculate the real-time dynamics of the spin-current autocorrelation function and obtain the Fourier transform $\text{Re}\sigma \left(\omega \right)$ for system sizes much larger than accessible to standard exact-diagonalization approaches. We find that the low-frequency behavior of $\text{Re}\sigma \left(\omega \right)$ is well described by $\text{Re}\sigma \left(\omega \right)\approx {\sigma }_{\text{dc}}+a{\left|\omega \right|}^{\alpha }$, with $\alpha \approx 1$ in a wide range within the thermal phase and close to the transition. We particularly detail the decrease of ${\sigma }_{\text{dc}}$ in the thermal phase as a function of increasing disorder for strong exchange anisotropies. We further find that the temperature dependence of ${\sigma }_{\text{dc}}$ is consistent with the existence of a mobility edge.

Figure 1

Dynamical phase diagram (sketch) of disordered spin-$\frac{1}{2}$ $XXZ$ chains. Issues studied in this Rapid Communication: Scaling of dc conductivity ${\sigma }_{\text{dc}}$ and low-frequency exponent $\alpha$ for strong interactions $\mathrm{\Delta }\ge 1$ and disorders $W\ge 0$ up to the MBL transition; temperature dependence and existence of mobility edge; typicality in finite systems with $W>0$.

Figure 2

Comparison of DQT $\left(tJ\le 40\right)$ and FTLM $\left(M=400\right)$: $\text{Re}\sigma \left(\omega \right)$ at $\beta \to 0,\mathrm{\Delta }=1$, and $W/J=2$ for $L=22$ and $N=200$. The excellent agreement clearly shows the validity of typicality. Such agreement is also found for other values of $W$ (see Ref. [82]).

Figure 3

$\text{Re}\sigma \left(\omega \right)$ at $\beta \to 0$ for $\mathrm{\Delta }=1.0$ (upper row) and $\mathrm{\Delta }=1.5$ (lower row) in the transition from small $W/J=0.5$ (left-hand side) to strong $W/J=4$ (right-hand side) for the ensemble $\langle S^z\rangle =0$, as obtained numerically for $L=14$ using ED and $L>14$ using DQT $(tJ\le 40$; $L<26$: $N=200$; $L=26$: $N=20)$. For $W=4,L=20$ data are shown for $N=10000$ and $tJ\le 120$ (insets), reducing statistical errors and increasing frequency resolution. In all cases (a)–(h), the low-$\omega$ behavior is well described by $\text{Re}\sigma \left(\omega \right)\approx {\sigma }_{\text{dc}}+a\left|\omega \right|$ (lines). In (e) the perturbative result of Ref. [47] for $W\to 0$ is depicted [82].

Figure 4

(a) Log-log plot of $\text{Re}\sigma \left(\omega \right)-c$, with $c=0$ and $c={\sigma }_{\text{dc}}$, at $W=2.5,\mathrm{\Delta }=1$, and $\beta \to 0$ $\left(\langle S^z\rangle =0,L=20,tJ\le 400,N=1000\right)$ as well as a power-law fit with the exponent $\alpha =0.93$ being close to 1. (b), (c) Disorder dependence of ${\sigma }_{\text{dc}}$, the maximum ${\sigma }_{\text{max}}$, and its position ${\omega }_{\text{max}}$ at $\mathrm{\Delta }=1.0,1.5$ and $\beta \to 0$ $\left(\langle S^z\rangle =0,L=24,tJ\le 40,N=200\right)$. For $W=0$, also the $\mathrm{\Delta }=1.5$ result of, e.g., Ref. [47], is indicated (green square).

Figure 5

(a) $\text{Re}\sigma \left(\omega \right)$ at intermediate $W/J=2$ and various $\beta J\le 2$ for $\mathrm{\Delta }=1$ $\left(\langle S^z\rangle =0,L=24,tJ\le 40,N=1000\right)$. (b) Temperature dependence of ${\sigma }_{\text{dc}}$ for different $L\le 24$. (Small error bars for the two largest $L=20$ and 24 indicate the difference between $N=500$ and 1000.) This temperature dependence is consistent with a mobility edge located at $E-E_{\text{min}}\sim 2J$.