Particle-hole symmetry, many-body localization, and topological edge modes

We study the excited states of interacting fermions in one dimension with particle-hole symmetric disorder (equivalently, random-bond XXZ chains) using a combination of renormalization group methods and exact diagonalization. Absent interactions, the entire many-body spectrum exhibits infinite-randomness quantum critical behavior with highly degenerate excited states. We show that though interactions are an irrelevant perturbation in the ground state, they drastically affect the structure of excited states: Even arbitrarily weak interactions split the degeneracies in favor of thermalization (weak disorder) or spontaneously broken particle-hole symmetry, driving the system into a many-body localized spin glass phase (strong disorder). In both cases, the quantum critical properties of the noninteracting model are destroyed, either by thermal decoherence or spontaneous symmetry breaking. This system then has the interesting and counterintuitive property that edges of the many-body spectrum are less localized than the center of the spectrum. We argue that our results rule out the existence of certain excited state symmetry-protected topological orders.

Figure 1

Phase diagram of the random-bond XXZ chain at energy density $\epsilon =0.5$ from exact diagonalization results. The quantum critical behavior of the free model ($\mathrm{\Delta }=0$) is destroyed by interactions, giving rise to either an ergodic phase at weak disorder where all spins are highly entangled, or to a many-body localized phase with spin glass order at strong disorder. The phase boundary is estimated from finite-size crossings at constant $W$ (blue symbols) or constant $\mathrm{\Delta }$ (green symbols). The excited states in the spin glass phase consist of effective superspins (green spins) showing a random pattern of frozen magnetization varying from eigenstate to eigenstate.

Figure 2

Ergodic to spin glass (MBL) transition. At weak disorder ($W=0.5$) our data are consistent with an ergodic to spin glass (MBL) transition as $\mathrm{\Delta }$ is increased. Top: Ratio of consecutive level spacings showing a transition from GOE to Poisson statistics. Middle: Scaling of ${\chi }_{\mathrm{EA}}$ showing a divergence with system size in the localized phase. Inset: Extrapolations of $m_{\mathrm{EA}}$ with $L^{-1}$ finite-size corrections (see text) are consistent with spin glass order in the MBL phase. Bottom: Finite-size scaling of the entanglement entropy.

Figure 3

Strong disorder spin glass phase ($W=2.0$). Left: Poisson statistics of the level spacings. Inset: When not restricted to a given ${\mathbb{Z}}_2$ sector, the gap ratio $r^{\prime }$ decreases with system size (well below the Poisson value), signaling pairing of the excited eigenstates. Middle: Subextensive scaling of the entanglement entropy. Right: Extrapolations of the spin glass order parameter $m_{\mathrm{EA}}={\chi }_{\mathrm{EA}}/L$ are consistent with nonvanishing values in the thermodynamic limit for all values of $\mathrm{\Delta }>0$, indicating spin glass order. Extrapolations are performed using $1/L^2$ (1) and $1/L$ (2) finite-size corrections. We note that the small dip around $\mathrm{\Delta }\approx 1$ is naturally accounted for by the enhanced probability of local resonances ${\mathrm{\Delta }}_i\approx 1$ [22] (see text). Inset: Linear scaling of ${\chi }_{\mathrm{EA}}$ with system size, consistent with spin glass order.

Figure 4

Ergodic to spin glass (MBL) transition as a function of $\mathrm{\Delta }$, for $W=0$ (top: uniform $J_i=1$) and $W=1$ (bottom). Left: Ratio of consecutive level spacings showing a transition from GOE to Poisson statistics. Middle: Scaling of ${\chi }_{\mathrm{EA}}$ showing a divergence with system size in the localized phase. Right: Finite-size entanglement crossover.

Figure 5

Crossover between volume- and area-law scaling of the entanglement entropy as a function of $\mathrm{\Delta }$ for $W=0.5$.

Figure 6

Ergodic to spin glass (MBL) transition as a function of $W$ at fixed $\mathrm{\Delta }=2.0$.

Figure 7

Strong disorder regime ($W=2.0$) with uniform anisotropy ${\mathrm{\Delta }}_i=\mathrm{\Delta }$. Away from the pathological SU(2)-symmetric point $\mathrm{\Delta }=1$, the results are qualitatively similar to the random ${\mathrm{\Delta }}_i$ case, consistent with a many-body localized phase with spontaneously broken particle-hole symmetry at all values of $\mathrm{\Delta }\ne 1$.