Asymmetry in energy versus spin transport in certain interacting disordered systems

We study energy transport in disordered $XXZ$ spin-1/2 chains driven to nonequilibrium configurations by thermal reservoirs of different temperatures at the boundaries, using large-scale matrix product simulations. In particular we discuss the transition between diffusive and subdiffusive transport in sectors of zero and finite magnetization at high temperature. At large anisotropies we find that diffusive energy transport prevails over a large range of disorder strengths, which is in contrast to spin transport that is subdiffusive in the same regime for weak disorder. However, at finite magnetization both energy and spin currents decay as a function of the system size with the same exponent. We conclude that diffusion of energy is much more pervasive than that of magnetization in these disordered spin-1/2 systems, and occurs across a significant range of the interaction-disorder parameter phase space. We support the existence of this asymmetry, reminiscent of that in the clean limit, by an analytical estimation of diffusion constants for weak disorder.

Figure 1

(a) Scheme of a disordered $XXZ$ spin-$1/2$ chain driven out of equilibrium by unequal boundary reservoirs, which impose a temperature and/or chemical potential imbalance by inducing different two-site thermal states at each edge. (b) Energy transport phase diagram, indicated by the current scaling exponent $\gamma$ as a function of interaction $\mathrm{\Delta }$ and disorder $h$. The solid black line defines the diffusive-subdiffusive boundary for zero magnetization and the dotted horizontal lines are the associated error bars. The underlying colors are Gaussian extrapolation of data, with brighter colors ($\gamma >1$, orange) indicating subdiffusion and darker colors ($\gamma =1$, green) diffusion. The gray dashed line is the diffusive-subdiffusive crossover for zero-magnetization spin transport [4].

Figure 2

NESS transport properties at zero magnetization for $\mathrm{\Delta }=1$ and various disorder strengths. (a) Energy profiles for different system sizes and $h=0.60$, with scaled axis $x=(k-1)/(L-2)$ for $k=1,...,L-1$. Inset: magnetization profiles for $L=32$ and different disorder strengths, with scaled axis $x=(k-1)/(L-1)$ for $k=1,...,L$. (b) Scaling of energy current; the disorder strength increases from top to bottom. The symbols correspond to the results of the simulations, and the lines are fits to the scaling $j^{\text{E}}\sim L^{-\gamma }$. Inset: scaling exponents $\gamma$ as a function of $h$ for different anisotropies $\mathrm{\Delta }$.

Figure 3

Comparison of energy and spin transport in the strongly interacting case $\mathrm{\Delta }=1.5$ and $h=0.6$, showing diffusive and subdiffusive transport, respectively. This is in qualitative agreement with the conclusions of Ref. [8] that spin and energy transport can be different in this model.

Figure 4

NESS transport at finite magnetization. (a) Scaling of energy and spin currents for $\mathrm{\Delta }=1$ and two different disorder strengths, indicating diffusion ($h=0.5$) and subdiffusion ($h=0.9$). (b) Scaling of energy and spin currents with $\mathrm{\Delta }=1.75$ and $h=0.6$, indicating diffusion. Inset: spin profiles for different system sizes, with rescaled axis $x$.