Quantum Criticality of Hot Random Spin Chains

We study the infinite-temperature properties of an infinite sequence of random quantum spin chains using a real-space renormalization group approach, and demonstrate that they exhibit nonergodic behavior at strong disorder. The analysis is conveniently implemented in terms of $\mathrm{SU}(2{)}_k$ anyon chains that include the Ising and Potts chains as notable examples. Highly excited eigenstates of these systems exhibit properties usually associated with quantum critical ground states, leading us to dub them “quantum critical glasses.” We argue that random-bond Heisenberg chains self-thermalize and that the excited-state entanglement crosses over from volume-law to logarithmic scaling at a length scale that diverges in the Heisenberg limit $k\to \infty$. The excited state fixed points are generically distinct from their ground state counterparts, and represent novel nonequilibrium critical phases of matter.

Figure 1

RG Results for Anyon Chains, obtained by integrating Eqs. (2) and (3) for an initial distribution $\mathcal{P}(S,0)={\delta }_{S,\frac{1}{2}}$. (a) Fixed-point spin distribution ${\mathcal{P}}^{*}(S)$ as $k\to \infty$ has universal scaling independent of $\mathcal{P}(S,0)$. (b) $r_s(\ell )$ (top) ${\mathrm{\Pi }}_0(\ell )$ (bottom) for different $k$ (the arrows indicate increasing values of $k$). (c) RG trajectories for different $k$, arrow length gives speed of flow.

Figure 2

Excited state entanglement. We calculate the entanglement entropy $S_A$ by counting singlets (top). $S_A$ exhibits a crossover from volume-law ($\sim L$) scaling for $L\ll {\xi }_k^{(1)}$ with ${\xi }_k^{(1)}\gtrsim k$ to logarithmic scaling for $L\gg {\xi }_k^{(2)}\sim e^{\alpha k^3}$ with $\alpha =1/4{\pi }^2$.