Symmetry breaking and localization in a random Schwinger model with commensuration

We numerically investigate a lattice-regularized version of quantum electrodynamics in one spatial dimension (Schwinger model). We work at a density where lattice commensuration effects are important and preclude analytic solution of the problem by bosonization. We therefore numerically investigate the interplay of confinement, lattice commensuration, and disorder in the form of a random chemical potential. We begin by pointing out that the ground state at commensurate filling spontaneously breaks the translational symmetry of the lattice. This feature is absent in the conventional lattice regularization, which breaks the relevant symmetry explicitly, but is present in an alternative (symmetric) regularization that we introduce. Remarkably, the spontaneous symmetry breaking survives the addition of a random chemical potential (which explicitly breaks the relevant symmetry), in apparent contradiction of the Imry-Ma theorem, which forbids symmetry breaking in one dimension with this kind of disorder. We identify the long-range interaction as the key ingredient enabling the system to evade Imry-Ma constraints. We examine spatially resolved energy level statistics for the disordered system, which suggest that the low-energy Hilbert space exhibits ergodicity breaking, with level statistics that fail to follow random matrix theory. A careful examination of the structure of the first excited state reveals that disorder-induced localization is responsible for the deviations from random matrix theory and further reveals that the elementary excitations are charge neutral and therefore not long range interacting.

Figure 1

Fourier transform of the density-density correlation function $\langle {\sigma }_x^z{\sigma }_{x+r}^z\rangle$ in the ground state $E_0$ and first five excited states, as obtained using exact diagonalization on a system of 12 spins. The plots in the left-hand column are with the conventional regularization (CR) based on a picture of a two-component plasma and with a two-site unit cell. The plots in the right-hand column are for the alternative symmetric regularization (SR) based on a picture of a one-component plasma and with a one-site unit cell. The first two rows show data obtained with open boundary conditions (OBCs), and the last two rows show periodic boundary conditions (PBCs). The labels $x,\lambda ,\theta$ denote the strength of the kinetic, interaction, and disorder terms, respectively. Remarkably, the choice of lattice regularization (and the presence or absence of disorder) makes little difference to the density correlation function, which has a structure around $k=\pi$, indicating that the ground state has a two-site unit cell.

Figure 2

Fourier-transformed density-density correlations in the ground and first excited states, obtained using DMRG for a chain with $N=100$ spins and open boundary conditions. The first column corresponds to the conventional regularization; the second column corresponds to symmetric regularization. Bond dimension $D$ is indicated on the plots, as is interaction strength $\lambda$ and disorder strength $\theta$. The disorder strength $\theta =0.0,0.5,0.1$ from top to bottom. The peak at $\pi$ indicates a density structure with a two-site unit cell, regardless of the choice of regularization or addition of disorder.

Figure 3

Energy-resolved level statistics ratios for the clean and disordered Schwinger models, obtained using numerical exact diagonalization on $N=16$ spins, with kinetic scale $x=1.0$ and interaction scale $\lambda =0.125$. In the top plot, we consider the clean system in the symmetric regularization with PBCs and further resolve by the symmetry sector. The parity eigenvalue and translation phase angle are indicated in the plot. Even in this case, $\langle r\rangle$ meanders near the Poisson value, suggesting integrability. This is supported by Fig. 4. In the bottom plot, we prepared 100 disorder realizations in OBCs, with $\theta =0.05$. Note that at low energies, the mean level statistics ratio approaches the Poisson value of 0.38.

Figure 4

Eigenenergies of the first 20 states as obtained with the Lanczos algorithm on a system of 16 spins in OBCs. We consider only states with the same parity eigenvalue $P=1$. Note the clear level crossings in the clean system, indicative of residual integrability.

Figure 5

Spatial profile of density imbalance [Eq. (6)] between the ground state and first excited state for a system of $N=100$ spins, conventional regularization, and open boundary conditions treated using DMRG. Here $x=1.0,\lambda =0.2$, and $\theta =0,0,01,0.05,0.1$ indicate the strength of the kinetic, interaction, and disorder terms, respectively. Note that the results are clearly indicative of the first excited state (of a disordered system) containing a localized excitation, the localization length of which decreases as disorder strength is increased. Note also that the localization length is too large to be accessible with exact diagonalization (consistent with analytic predictions [15]), thus necessitating DMRG. Note also that the elementary excitations shown here are charge neutral (the density difference is equally likely to be positive or negative), consistent with general expectations.

Figure 6

Same as Fig. 5, but with the symmetric regularization.

Figure 7

(a) For certain disorder realizations, the density imbalance in the first excited state is bilocalized. In this case we take the localization length to be the mean of the decay lengths for the two peaks. (b) In other cases, there are two overlapping peaks which cannot be cleanly separated. We exclude such configurations from our estimate of localization length.

Figure 8

Localization length $\xi$ (extracted by eye from Figs. 5 and 6) plotted against disorder strength. The dashed lines are the best fit to the scaling prediction $\xi \sim {\theta }^{-2/3}$. The error bars come from sample-to-sample fluctuations and are estimated by averaging over at least ten disorder realizations.