Quantum Quenches in Chern Insulators

M. D. Caio, N. R. Cooper, and M. J. Bhaseen

Phys. Rev. Lett. 115, 236403 – Published 2 December 2015

Abstract

We explore the nonequilibrium response of Chern insulators. Focusing on the Haldane model, we study the dynamics induced by quantum quenches between topological and nontopological phases. A notable feature is that the Chern number, calculated for an infinite system, is unchanged under the dynamics following such a quench. However, in finite geometries, the initial and final Hamiltonians are distinguished by the presence or absence of edge modes. We study the edge excitations and describe their impact on the experimentally observable edge currents and magnetization. We show that, following a quantum quench, the edge currents relax towards new equilibrium values, and that there is light-cone spreading of the currents into the interior of the sample.

Received 8 April 2015

DOI:https://doi.org/10.1103/PhysRevLett.115.236403

Authors & Affiliations

M. D. Caio¹, N. R. Cooper², and M. J. Bhaseen¹

¹Department of Physics, King’s College London, Strand, London WC2R 2LS, United Kingdom
²T.C.M. Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom

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Vol. 115, Iss. 23 — 4 December 2015

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Images

Figure 1
(a) Phase diagram of the Haldane model obtained from the low-energy Dirac fermion representation, showing topological ( $ν = \pm 1$ ) and nontopological phases ( $ν = 0$ ) [2]. We consider quantum quenches and sweeps between different regions of the phase diagram, as illustrated by the arrows. (b) The low-energy spectrum of the Haldane model is described by excitations around two Dirac points. After a quench, carriers in the lower band are excited to the upper band.
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Figure 2
Probability of occupying the upper band for a single Dirac point ( $α = - 1$ ) with $c = 1$ , following a quench of $m_{α}$ . Sign-preserving quench, $m_{-} = - 1 \to m_{-}^{'} = - 0.1$ (solid line), and a sign-changing quench, $m_{-} = - 1 \to m_{-}^{'} = 0.1$ (dashed line). In both cases, $b_{-} (\infty) = 0$ , corresponding to the time independence of $ν_{-} (t)$ using Eq. (4). The sign-changing quench yields $| b_{-} (0) |^{2} = 1$ , but the contribution to $ν_{-} (t)$ in Eq. (4) is compensated by the change in sign of $m_{-}$ . As a result, $ν_{-}$ is unchanged from its initial value.
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Figure 3
Energy spectrum of the Haldane model obtained by exact diagonalization on a finite-size strip of width $N = 20$ unit cells with armchair edges. We take periodic (open) boundary conditions along (transverse to) the strip and set $t_{1} = 1$ , $t_{2} = \frac{1}{3}$ , and $M = 1$ . (a) Equilibrium population of the energy levels in the nontopological phase with $φ = π / 6$ . (b) Repopulation of the levels after a quench to the topological phase with $φ = π / 3$ , corresponding to the solid arrow in Fig. 1. The size of the dots is proportional to the probability of finding a particle in the mode. Postquench, the filling of the edge states and the bands is nontrivial.
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Figure 4
Equilibrium properties of the Haldane model on a finite-size strip as used in Fig. 3. (a) Total longitudinal current $J_{n}^{x}$ along the strip as a function of the transverse spatial index $n \in 1, \dots, 20$ , for $M = 0$ and $φ = π / 3$ . (b) Edge currents corresponding to $J_{n}^{x}$ with $n = 1$ (solid) and $n = 20$ (dashed) for $M = 0$ . The edge currents exhibit $π$ periodicity in $φ$ and vanish when $φ = π / 2$ . (c) Orbital magnetization $M$ as a function of $φ$ for $M = 0$ . (d) Intensity plot of $M$ where the dashed lines correspond to the boundaries of the topological phases. Numerically, we observe that $M$ vanishes on the loci $M = \pm \sin φ$ (solid) within the topological phases. The loci are fits to the numerical data (triangles) where $M = 0$ . The magnetization also vanishes on the vertical lines $φ = π / 2, π, 3 π / 2, \dots$ , as follows from symmetry considerations.
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Figure 5
Dynamics of the edge current $J_{N}^{x} (t)$ for $N = 30$ (circles) and $N = 40$ (crosses) following a quantum quench between the topological phase and the nontopological phase with $t_{1} = 1$ , $t_{2} = \frac{1}{3}$ , and fixed $φ = π / 3$ . Quenches from $M = 1.4$ to $M = 1.6$ (main panel) and from $M = 1.4$ to $M = 2.2$ (inset) showing that the edge currents approach new equilibrium values. For the chosen parameters, these are very close to the ground state expectation values of $J_{N}^{x}$ in the final Hamiltonian, as indicated by the horizontal lines.
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Figure 6
Dynamics of the currents $| J_{n}^{x} (t) |$ following a quantum quench from the topological to the nontopological phase for the parameters used in the inset of Fig. 5. The damped oscillations of the edge currents are clearly visible, as is the light-cone spreading of the currents into the interior of the sample, where $c = 3 t_{1} / 2 ℏ = 3 / 2$ is the effective speed of light. The waves propagating from the two edges meet at time $t \sim (N / 2) \sqrt{3} / 2 c \sim 5.77$ , leading to resurgent oscillations in finite-size samples, see the Supplemental Material [21].
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