Subdiffusive dynamics and critical quantum correlations in a disorder-free localized Kitaev honeycomb model out of equilibrium

Disorder-free localization has recently emerged as a mechanism for ergodicity breaking in homogeneous lattice gauge theories. In this work we show that this mechanism can lead to unconventional states of quantum matter as the absence of thermalization lifts constraints imposed by equilibrium statistical physics. We study a Kitaev honeycomb model in a skew magnetic field subject to a quantum quench from a fully polarized initial product state and observe nonergodic dynamics as a consequence of disorder-free localization. We find that the system exhibits a subballistic power-law entanglement growth and quantum correlation spreading, which is otherwise typically associated with thermalizing systems. In the asymptotic steady state the Kitaev model develops volume-law entanglement and power-law decaying dimer quantum correlations even at a finite energy density. Our work sheds light on the potential for disorder-free localized lattice gauge theories to realize quantum states in two dimensions with properties beyond what is possible in an equilibrium context.

Figure 1

(a) Model interactions. Arrows indicate Majorana fermion hopping. Four Majorana fermions interact within every Y junction, shaded in gray. (b) $\langle D^{z\left(x\right)}\left(t\right)\rangle$. (c) $C^{\mu \mu }={\langle D_j^{\mu }{D}_{j+r}^{\mu }\rangle }_c$. The propagating wave front is determined by the threshold $\left|C\right|\ge {10}^{-6}$, subject to a power-law fitting $r\propto {\left(Jt\right)}^{1/z}$ indicated by white lines. Here $z=2.5\left(2\right)$ for $|C^{zz}|$ and $z=2.7\left(3\right)$ for $|C^{xx}|$. (d) Collapse of correlation growth at fixed distance with rescaled time. The parameters are $J_z=J_x=J_y=J$, $\tilde{h}=0.25J$, and $10000$ disorder samples of system size with $60\times 60$ unit cells (7200 spins) have been used.

Figure 2

Dimer correlations in the steady state, averaged over ${10}^{3.6}\lesssim Jt\lesssim {10}^4$. Black lines indicate power-law fitting $r^{-\mathrm{\Delta }}$ with $\mathrm{\Delta }=0.63\left(2\right)$ for both $C^{zz}$ and $C^{xx}$. Insets show correlations at $r=2$ extrapolated to finite values in the thermodynamic limit.

Figure 3

Projective bipartite entanglement entropy, averaged over 1000 disorder samples. The inset shows the entanglement cut. The black line indicates the power-law fitting proportional to $t^{1/z}$ with $z=2.4\left(1\right)$. At late time, entropy saturates to a volume law $S_v(Jt={10}^4)=0.40\left(1\right)Nln2/2$, as shown in the inset.

Figure 4

Tuning the density of $\pi$ fluxes. (a) Projective bipartite entanglement entropy of Majorana fermions. (b) and (c) Steady-state spatial correlation function $|C^{zz\left(xx\right)}|$, averaged in the time window ${10}^3\lesssim Jt\lesssim {10}^4$. The parameters are $J_z=J_x=J_y\equiv J$, $\tilde{h}=0.25J$, $L_x=L_y=50$, and 1000 disorder samples have been used.

Figure 5

(a) The left axis shows the level spacing ratio, for system size $L_x=L_y=60$ with $10000$ disorder samples. The characteristic value for the Poisson ensemble, $2ln2-1\simeq 0.3863$, and the one for the Gaussian unitary ensemble, approximately equal to $0.5996$, are indicated. The right axis shows the localization length obtained by one-parameter scaling for a sequence of quasi-1D long stripes for $L_y=8,16,32,64,128$ and $L_x\le {10}^6$. (b) Chern number for system size $L_x=L_y=40$. Black dots are for 500 disorder samples, while the blue (red) line is for the zero ($\pi$) flux clean system.

Figure 6

(a) $V_{mn}$. The mode index corresponds to the energy in ascending order. (b) Correlation functions of steady state, averaged in the time window ${10}^3\lesssim Jt\lesssim {10}^4$. The power-law-fitting exponent is 0.50(6) for $C^{zz}$ and 0.51(4) for $C^{xx}$. The parameters are $\tilde{h}=0.25J$, $L_x=16$, $L_y=4$, and 200 disorder samples. The $V=0$ case for comparison takes $10000$ disorder samples.