Power-law banded random matrix ensemble as a model for quantum many-body Hamiltonians

Abstract

We explore interpretations of the power-law banded random matrix (PLBRM) ensemble as Hamiltonians of one-dimensional quantum many-body systems. We introduce and compare a number of labeling schemes for assigning random matrix basis indices to many-body basis vectors. We compare the physical properties of the resulting Hamiltonians, focusing on the half-system eigenstate bipartite entanglement entropy. We show and quantify how the different PLBRM phases (ergodic, weakly ergodic, localized), known from the single-particle interpretation, can be interpreted as entanglement transitions in the quantum many-body interpretation. For the weakly ergodic phase, where spectral edge and bulk eigenstates show distinct behavior, we perform a detailed scaling analysis to provide a quantitative picture of the boundaries between different types of entanglement scaling behaviors. In particular, we identify and characterize an intermediate set of eigenstates whose entanglement entropy have volume law scaling but nonvanishing deviation from the Page value expected for maximally ergodic states.

Figures (11)

• Figure 1: Phase diagram of the power-law banded random matrix ensemble, showing the phases of mid-spectrum eigenstates as a function of the power-law exponent $\alpha$. The panels show the eigenstate entanglement entropy for a single realization as a function of the normalized eigenstate index (random labeling scheme, $L=10$), for the three phases ($\alpha = 0.25$, $\alpha = 0.75$, $\alpha = 1.25$). In the panels, the ground state is on the left of the horizontal axis, while the highest excited eigenstate is on the right. The horizontal dashed line indicates the Page entanglement entropy $S_\text{Page}$. (Ensemble-averaged versions of such plots are discussed later, see Fig. \ref{['fig: full']}.)
• Figure 2: Badness \ref{['eq: badness']} as a function of the system size $L$ for each of the labeling methods. For the random labeling scheme, the sample-to-sample variance is smaller than the marker size.
• Figure 3: Ensemble-averaged eigenstate entanglement entropy for mid-spectrum eigenstates with $\alpha = 1$, computed using the binary labeling scheme (no site randomization). The spin chain is decomposed in subsystems $A$ and $B$ consisting of the first $L_A$ and last $L_B = L - L_A$ sites, respectively. The inset illustrates the decomposition for $L_A = 4$ and $L_B = 6$. The dotted lines give the Page values \ref{['eq: SPage']} for $L=6$, $L=10$, and $L=14$.
• Figure 4: Difference between the ensemble-averaged eigenstate entanglement entropy and the Page value as a function of the eigenstate index $n$ (starting from $n = 1$ and ordered by increasing energy), scaled by the Hilbert space dimension $N$. On the horizontal axes, the ground states are on the left, while the highest excited eigenstates are on the right. Left and right panels use random and Gray code labeling schemes respectively. The three rows correspond to the three phases ($\alpha = 0.25$, $\alpha = 0.75$, $\alpha = 1.25$). The arrows indicate the direction of flow for the bulk eigenstates with increasing system size. Note the very different scales on the vertical axes for the different values of $\alpha$.
• Figure 5: Ensemble-averaged entanglement entropies of mid-spectrum eigenstates, with indices $n$ ranging from $N / 2 - 10$ to $N / 2 + 10$. Top row shows $S_\text{ent}$ and bottom row shows $S_\text{Page}-S_\text{ent}$. The left and right panels use the random and Gray code labeling scheme respectively. In the top row, the horizontal dashed lines give the Page values for the system sizes under consideration.