Energy transport in a disordered spin chain with broken U(1) symmetry: Diffusion, subdiffusion, and many-body localization

We explore the physics of the disordered XYZ spin chain using two complementary numerical techniques: exact diagonalization (ED) on chains of up to 17 spins, and time-evolving block decimation (TEBD) on chains of up to 400 spins. Our principal findings are as follows. First, we verify that the clean XYZ spin chain shows ballistic energy transport for all parameter values that we investigated. Second, for weak disorder there is a stable diffusive region that persists up to a critical disorder strength that depends on the XY anisotropy. Third, for disorder strengths above this critical value, energy transport becomes increasingly subdiffusive. Fourth, the many-body localization transition moves to significantly higher disorder strengths as the XY anisotropy is increased. We discuss these results, and their relation to our current physical picture of subdiffusion in the approach to many-body localization.

Figure 1

(a) A disordered spin-1/2 XYZ chain, with Lindblad driving applied to the pair of spins at each end to impose a temperature gradient. In our time-evolving block decimation (TEBD) studies, we time-evolve such a system until it reaches its nonequilibrium steady state (NESS). We supplement that analysis by exact diagonalization (ED) studies on closed (and much shorter) chains. (b) The resulting “phase diagram” of the disordered spin-1/2 XYZ chain, for an Ising anisotropy of $\mathrm{\Delta }=1.2$. Here, $\eta$ is the XY anisotropy of the exchange interaction between the spins, and $W$ is the strength of the random-field disorder. In the left-hand panel, the dashed line shows the border between diffusive and subdiffusive energy transport determined from our TEBD studies, and the bars are error estimates. The color scale shows the transport exponent $\gamma$ estimated via interpolation between the numerically determined values, which are indicated by gray points. The right-hand panel shows the location of the MBL transition, determined by three different analyses of our ED results: the crossover in level statistics from random matrix to Poissonian $r$; the peak in the standard deviation of the Shannon entropy $\sigma \left(S_{\mathrm{Sh}}\right)$; and the peak in the standard deviation of the von Neumann entropy $\sigma \left(S_{\mathrm{vN}}\right)$.

Figure 2

The energy-transport exponent $\gamma$ in the disordered spin-1/2 XYZ chain at weak to moderate disorder strengths, as determined from our TEBD numerical results. All results reported are for an Ising anisotropy of $\mathrm{\Delta }=1.2$. (a) The exponent $\gamma$ as a function of the XY anisotropy $\eta$, for various values of the disorder strength $W$. $\gamma =1$ corresponds to diffusive energy transport; for $\gamma >1$, energy transport is subdiffusive. (b) The exponent $\gamma$ as a function of the disorder strength $W$, for various values of the XY anisotropy $\eta$. The open symbols in both panels indicate cases in which the chain was not long enough to achieve fully diffusive behavior, and these points should therefore be disregarded (see Fig. 3 and the corresponding text).

Figure 3

Testing whether our chains are long enough for the scaling limit to have been reached. In this example the XY anisotropy parameter $\eta =0.4$. (a) The numerical collapse of the energy current $j^E$ as a function of chain length $L$ onto a single curve under suitable scaling by the disorder strength $W$. Note the typical crossover from ballistic behavior in short chains to diffusive behavior—indicated by the dashed line—in longer ones. (b) The “running exponent” $\gamma \left(x\right)$, determined from tangential power-law fits to the universal curve, showing that $\gamma$ reaches the diffusive value of 1 above the critical length scale $x^{★}\approx 25–30$.

Figure 4

Locating the MBL transition via exact diagonalization. This figure shows the (a) level statistics $r$ parameter, (b) the standard deviation of the entanglement entropy distribution, and (c) the standard deviation of the Shannon entropy distribution, all as a function of disorder strength $W$ for several values of the XY anisotropy parameter $\eta$. These results were obtained from exact diagonalization of the Hamiltonian for a disordered XYZ spin chain of length $L=15$ with periodic boundary conditions. The error bars are smaller than the symbol size.