Hall response and edge current dynamics in Chern insulators out of equilibrium

We investigate the transport properties of Chern insulators following a quantum quench between topological and nontopological phases. Recent works have shown that this yields an excited state for which the Chern number is preserved under unitary evolution. However, this does not imply the preservation of other physical observables, as we stressed in our previous work. Here we provide an analysis of the Hall response following a quantum quench in an isolated system, with explicit results for the Haldane model. We show that the Hall conductance is no longer related to the Chern number in the postquench state, in agreement with previous work. We also examine the dynamics of the edge currents in finite-size systems with open boundary conditions along one direction. We show that the late-time behavior is captured by a Generalized Gibbs Ensemble, after multiple traversals of the sample. We discuss the effects of generic open boundary conditions and confinement potentials.

Figure 1

Phase diagram of the Haldane model showing topological ($\nu =\pm 1$) and nontopological phases ($\nu =0$) [22]. The triangle and the square indicate the starting points of quenches along the dashed line, as shown in Figs. 4 and 5.

Figure 2

(a) Band structure of the Haldane model [22]. (b) At low energies, the model reduces to a sum of two Dirac Hamiltonians with a gapped relativistic dispersion relation.

Figure 3

Setup used to evaluate the Hall response in the presence of periodic boundary conditions. An auxiliary time-dependent magnetic flux $\mathrm{\Phi }\left(t\right)$ generates a longitudinal electric field $E_x\left(t\right)=-\frac{\partial \mathrm{\Phi }\left(t\right)}{\partial t}$. Note that this flux does not correspond to the time-reversal symmetry-breaking parameter $\phi$ in the Haldane model. The Hall conductance ${\sigma }_{xy}\left(t\right)$ is obtained from the transverse current that is in phase with $E_x\left(t\right)$.

Figure 4

Hall conductance in units of $q^2/h$ following a quantum quench from the topological phase with $\phi =\pi /3$ and $M=1$ (as indicated by the triangular symbol in Fig. 1) to $\phi ={\phi }^{\prime }$ and $M=1$. The gray dots are numerical results for the Haldane model obtained from Eq. (9). The black solid line is the analytical result for the corresponding quenches in the low-energy Dirac approximation, obtained by summing contributions from Eq. (13). The results are in quantitative agreement for small quenches in the vicinity of $\phi =\pi /3$ and in qualitative agreement for larger quenches. The vertical dashed lines correspond to the boundaries of the topological phases. The Hall conductance remains numerically close to $q^2/h$ for quenches within the same topological phase ($\nu =1$) but does not saturate at $-q^2/h$ when quenching to the other topological phase ($\nu =-1$).

Figure 5

Hall conductance in units of $q^2/h$ following a quantum quench from the nontopological phase with $\phi =\pi$ and $M=1$ (as indicated by the square symbol in Fig. 1) to $\phi ={\phi }^{\prime }$ and $M=1$. The gray dots are numerical results for the Haldane model obtained from Eq. (9). The black solid line is the analytical result for the corresponding quenches in the low-energy Dirac approximation, obtained by summing contributions from Eq. (13). The results are in quantitative agreement for small quenches in the vicinity of $\phi =\pi$ and in qualitative agreement for larger quenches. The vertical dashed lines correspond to the boundaries of the topological phases.

Figure 6

Hall response ${\sigma }_{xy}\left(\omega \right)$ in units of $q^2/h$ evaluated using Eq. (7) for frequencies below the direct band gap $\mathrm{\Delta }\equiv \min \left[{\mathrm{\Delta }}_{b,b^{\prime }}\left(\mathbf{k}\right)\right]$, following a quantum quench from the topological phase with $M=1$ and $\phi =\pi /3$, to the nontopological phase with $M=1$ and $\phi ={\phi }^{\prime }$. The three curves correspond to ${\phi }^{\prime }=7\pi /8$ (circles), ${\phi }^{\prime }=\pi$ (triangles), and ${\phi }^{\prime }=9\pi /8$ (squares), for which $\mathrm{\Delta }\sim 0.67$, 2, and $0.67\times t_1/\hslash$, respectively.

Figure 7

Postquench Hall conductance in units of $q^2/h$ for a single Dirac point ($\alpha =+$), following a quench from $m>0$ to $m^{\prime }$. For the same choice of quench parameters, the other Dirac point ($\alpha =-$) displays the opposite Hall conductance. Note that for quenches in the Haldane model, the changes in mass for each Dirac point may differ, yielding a nonzero Hall response. The dashed lines correspond to the asymptotes ${\sigma }_{xy}\left(0\right)=\pm (\pi /8)q^2/h$, which can be determined analytically from Eq. (13).

Figure 8

Finite-size cylindrical geometry with periodic (open) boundary conditions along the $x$ ($y$) direction. When $\phi \ne 0$ there are generically edge currents, $J_1^x$ and $J_N^x$, flowing along the sample boundaries in opposite directions. After a quantum quench, the edge currents evolve as a function of time, and currents flow into the interior of the sample.

Figure 9

Total edge current $J_1^x\left(t\right)$ in units of $q{t}_{1}a/\hslash$ following a quantum quench within the topological phase with $\nu =-1$. The time $t$ is measured in units of $\hslash /t_1$. We set $\phi =\pi /3$ and quench $M$ from 1.4 to $-1.4$. The solid horizontal (green) line corresponds to the time-averaged total edge current, in the quasistationary regime before the onset of finite-size traversals. Inset: Late-time data after 29 traversals of the sample. The dashed (red) line corresponds to the prediction of the GGE. This agrees with the time average of the late-time data, as shown by the solid horizontal line (blue), to within approximately $3\%$.

Figure 10

Comparison of the time-averaged total edge current in the Haldane model at late times (circles) and the prediction $〈{\widehat{\rho }}_{\mathrm{GGE}}{\widehat{J}}_1^x〉$ of the GGE (triangles) for quantum quenches with $\phi =\pi /3$ held fixed and $M=1.4\to M^{\prime }$. The time averaging is performed over a single traversal period of the finite-size system, following 29 traversals of the sample. The currents are measured in units of $q{t}_{1}a/\hslash$.

Figure 11

Equilibrium properties of the Haldane model in the cylindrical geometry shown in Fig. 8, with an additional harmonic confinement potential $V_n$ given by Eq. (15). We consider the topological phase with $M=0$ and $\phi =\pi /3$ and set $N=70, \mathcal{C}=1.2$, and $\zeta =15$. (a) Particle density ${\rho }_n$ as a function of the row index $n\in 1,...,70$. At half-filling, the potential confines the particles into the region $\zeta <n<N-\zeta$, as indicated by the vertical dashed lines. This imposes an effective system size in which it is possible to observe edge effects. (b) Total longitudinal current $J_n^x$ along the cylinder showing clearly separated edge currents, broadened due to the smooth potential. Hall currents exist in the interior of the sample due to the effective electric field generated by the trap.

Figure 12

Dynamics of the currents $|J_n^x\left(t\right)|$ following a quantum quench from the topological phase with $M=0$ and $\phi =\pi /3$ to the nontopological phase $M=3$ and $\phi =\pi /3$. We set $N=70, \mathcal{C}=1.2$, and $\zeta =15$. The spreading of the currents into the interior (and to a lesser extent the exterior) of the effective sample is visible even in the presence of the harmonic potential. However, the propagation speed departs from the effective speed of light, due to the broadening of the edge currents and the contribution of bulk Hall currents.

Figure 13

(a) Illustration of the potential landscape of a hexagonal optical lattice in the presence of a rotationally symmetric harmonic trap. (b) Circulating equilibrium currents in the resulting disk geometry. Following a quantum quench between the topological and the nontopological phases the edge currents are expected to propagate into the interior (and the exterior) of the sample at the effective speed of light. This will be smoothed out due to the broadening of the edge currents and the presence of the bulk Hall currents.