Dynamical localization in ${\mathbb{Z}}_2$ lattice gauge theories

We study quantum quenches in two-dimensional lattice gauge theories with fermions coupled to dynamical ${\mathbb{Z}}_2$ gauge fields. Through the identification of an extensive set of conserved quantities, we propose a generic mechanism of charge localization in the absence of quenched disorder both in the Hamiltonian and in the initial states. We provide diagnostics of this localization through a set of experimentally relevant dynamical measures, entanglement measures, as well as spectral properties of the model. One of the defining features of the models that we study is a binary nature of emergent disorder, related to ${\mathbb{Z}}_2$ degrees of freedom. This results in a qualitatively different behavior in the strong disorder limit compared to typically studied models of localization. For example, it gives rise to a possibility of a delocalization transition via a mechanism of quantum percolation in dimensions higher than 1D. We highlight the importance of our general phenomenology to questions related to dynamics of defects in Kitaev's toric code, and to quantum quenches in Hubbard models. While the simplest models we consider are effectively noninteracting, we also include interactions leading to many-body localizationlike logarithmic entanglement growth. Finally, we consider effects of interactions that generate dynamics for conserved charges, which gives rise to only transient localization behavior, or quasi-many-body localization.

Figure 1

Schematic picture of the model (1). Left panel shows star $\widehat{A}$ and plaquette $\widehat{B}$ operators, with ${\widehat{\sigma }}^x, {\widehat{\sigma }}^z$ operators denoted by crosses and circles, respectively. Fermion hopping is defined by $J$ and the direction of the spin-1/2 on that bond. Center and right panels show the duality transformation to new spins $\widehat{\tau }$. The model can be defined with periodic boundary conditions, as in the center panel. In the case of open boundaries, we define incomplete boundary “stars” as shown in the right panel.

Figure 2

Schematic picture showing the transformation of the initial state into a dual representation. On the left is an initial state with fermions in a charge density wave (filled sites—blue, empty sites—white). The bond spins are polarized along the $z$ axis. The dual state (right panel) has the fermions in the same configuration, but the wave function is an equal superposition of all charge configurations. In the dual representation, the charge sector leads to exact averaging over the binary potential, which is shown in gray.

Figure 3

Results for the time-evolution of the fermion subsystem in 1D. (a) Persistence of a charge density wave measured by the density imbalance between neighboring sites, $\mathrm{\Delta }\rho$, see Eq. (23), for $N=200$ sites with periodic boundary conditions. The inset shows results in the long-time limit, $Jt={10}^9$, as a function of effective disorder strength $h/J$. (b) and (c) Spreading of a domain wall (in the initial configuration) for $h/J=0.5$ and 2, respectively, for the system with $N=20$ sites and open boundary conditions. Local density is shown (filled—yellow, empty—blue). (d) Time evolution of particle occupation numbers on the right-half of the system $N_{\text{half}}$ from an initial domain wall configuration for $N=200$ sites. Inset shows results for the long time limit as a function of $h/J$. All calculations are performed using the determinant method of Appendix pp1.

Figure 4

Time evolution of the bipartite von Neumann entanglement entropy in the 1D case. We start from a charge density wave initial state in a system with open boundary conditions and $h/J=10$. We partition the system into two halves along the central bond and results are computed using exact diagonalization.

Figure 5

Density of states for the 1D chain (15). (a) DOS for different values of $h/J$. Inset shows the DOS for values of $h>J$ (where there is a gap in the DOS). (b) DOS for a very large value of $h/J=500$. The energy is offset by $2h$ and we focus on one of the two sub-bands that form for large $h$. The DOS is computed using the kernel polynomial method (see Appendix pp3), shown in blue. We compare this with the DOS constructed using Eq. (26), shown in red. Inset in (b) shows a comparison of the observed distribution of chain lengths with the corresponding distribution in the thermodynamic limit.

Figure 6

Time evolution of the fermion subsystem in 2D. (a) Density imbalance $\mathrm{\Delta }\rho$ measured along a slice through the center of the system with the initial state described by a charge density wave, see text. Inset shows the long-time limit for the charge density wave, and checkerboard initial states. (b) and (c) Spreading of the domain wall for $h/J=0.5$ and $h/J=2$, respectively, measured along the slice through the center of the system. (d) Number of particles $N_{\text{half}}$ along the center in the initially empty half of the system. In (a)–(c), we use a square lattice with $N=32\times 32$ sites and in (d) $N=30\times 50$. Results are computed using the determinant method of Appendix pp1.

Figure 7

(a) Single-particle density of states for a 2D square lattice for different $h/J$. The DOS is computed using the kernel polynomial method, see Appendix pp3. (b) Localization length in 1D and 2D. In 1D, the localization length is computed using the spectral formula (27) as explained in the main text. In 2D, the localization length is scaled by a factor of 20 and is computed by the transfer matrix method on a strip of width 100 sites and length 250 000 sites.

Figure 8

Schematic picture of the site percolation problem. (a) A fully connected lattice on which fermions can hop in the limit $h/J=0$. [(b) and (c)] When $h/J\gg 1$, the fermions are constrained to hopping between sites with the same effective potential (filled sites and bonds) and the other sites become inaccessible (open circles) leading to a quantum site percolation problem. (b) Connected sites for the bias $p=0.5$ showing that the absence of paths (in the thermodynamic limit), which connect opposite sides of the lattice. (c) The same for the bias $p=0.7$, which is above the percolation threshold $p_c\approx 0.5927$, showing in red connected paths across the system.

Figure 9

(a) Total number of particles making it across a domain wall $N_{\text{half}}$ for various values of the charge distribution bias, $p$. The figure shows the rescaled value $N_{\text{half}}/L$, and the extrapolated limit $L\to \infty$, where $L$ is the linear dimension of the system. Error bars correspond to two standard deviations in the linear fit from finite size scaling. Note that $p_c\approx 0.5927$ corresponds to the classical site percolation threshold for a square lattice. (b) Density of states for different values of the bias $p$ and $h/J=20$. Inset shows the corresponding energy-resolved localization length. The results shown in (a) were obtained using the determinant method (Appendix pp1). We used the KPM (Appendix pp3) for the DOS in (b) and the transfer matrix approach for the localization length (Appendix pp4) for a system of $150\times 250000$ sites.

Figure 10

Quantum quench from an initial charge density wave state, $h/J=20$. (a) Density imbalance $\mathrm{\Delta }\rho \left(t\right)\propto {\sum }_j|\langle 0|{\widehat{n}}_j\left(t\right)-{\widehat{n}}_{j+1}\left(t\right)|0\rangle |$ and the time-averaged value of $\frac{1}{t}{\int }_0^t\text{d}\tau \mathrm{\Delta }\rho \left(\tau \right)$ (dashed lines) after the same quench. (b) Von Neumann entanglement entropy computed using ED for $N=12$ sites (thin, light) for various values of $\mathrm{\Delta }$ shown on a semi-log scale. The spatial bipartition is taken along the central bond. (Inset) The same data on a linear scale for $\mathrm{\Delta }/J=0,0.01$. Dashed lines show fitted linear and logarithmic curves.

Figure 11

(a) Entanglement entropy after a quantum quench for various values of the longitudinal field $B_x$ shown on a semi-log scale. A window of the same data for $B_x/J=0.2,0.5$ is given in the inset. Dashed lines correspond to $ln\left(t\right)$ and $ln(ln(t\left)\right)$ behavior. (b) Density imbalance after a quench from a charge density wave. Results obtained using ED.

Figure 12

Density imbalance after a quantum quench from a CDW with dynamical charges. We study a system with $N=10$ sites with periodic boundary conditions and $h=J$. The light curves are the imbalance $\mathrm{\Delta }\rho$ and the dark thick lines are the time averaged value$\frac{1}{t}{\int }_0^t\text{d}\tau \mathrm{\Delta }\rho \left(\tau \right)$. (a) With an additional transverse field $B_z{\sum }_i{\widehat{\sigma }}_{i,i+1}^z$. (b) Additional fermion hopping $\epsilon {\sum }_{\langle ij\rangle }{\widehat{f}}_i^{†}{\widehat{f}}_j$. (c) Added transverse Ising coupling$h_z{\sum }_i{\widehat{\sigma }}_{j-1,j}^z{\widehat{\sigma }}_{j,j+1}^z$. Dynamics is computed using Krylov subspace methods (Appendix pp2).

Figure 13

Finite size scaling of the asymptotic density imbalance in presence of additional Ising coupling $h_z{\sum }_i{\widehat{\sigma }}_{j-1,j}^z{\widehat{\sigma }}_{j,j+1}^z$. We use system sizes $N=6,8,10$ with $h_z=0.1J$ and $h=J$. Light curves show the density imbalance $\mathrm{\Delta }\rho$ and the dark and thick curves show the time averaged value of $\frac{1}{t}{\int }_0^t\text{d}\tau \mathrm{\Delta }\rho \left(\tau \right)$. See Appendix pp2 for details of the Krylov subspace numerical method used.