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arXiv:1504.01910 · Physical Review Letters

Quantum Quenches in Chern Insulators

M. D. Caio, Nigel R. Cooper, M. J. Bhaseen

Published Dec 2, 2015• 159 citations
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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.

DOI
10.1103/physrevlett.115.236403

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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.
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.
We perform a sudden quench on the Haldane model with long range interactions, more specifically generalizing to the next to next nearest neighbor hopping, referred to as the N3N3N3 model in our work. Such a model possesses both isotropic and multiple anisotropic (satellite) Dirac points which lead to a rich topological phase diagram consisting of phases with higher Chern number (C)(C)(C). Quenches between the topological and the nontopological phases of such an infinite system probe the effect of the presence of the anisotropic Dirac points on the nonequilibrium response of the topological system. Interestingly, the Chern number remains the same before and after the quench for both the quenching protocols, even when the quench of the system is carried out between two different topological phases. However, for a finite system, we establish that the initial edge current asymptotically decays to zero when the system is quenched to the nontopological phase although the Chern number for the corresponding periodically wrapped system remains unaltered; what is remarkable is that when the Hamiltonian is quenched from ∣C∣=2|C|=2∣C∣=2 phase to the nontopological phase the edge current associated with the inner channel decays at a faster rate than the outer channel resembling a situation in which the system passes through the phase with ∣C∣=1|C|=1∣C∣=1 before ending up in the phase C=0C=0C=0.
In a finite-time quantum quench of the Haldane model, the Chern number determining the topology of the bulk remains invariant, as long as the dynamics is unitary. Nonetheless, the corresponding boundary attribute, the edge current, displays interesting dynamics. For the case of sudden and adiabatic quenches the postquench edge current is solely determined by the initial and the final Hamiltonians, respectively. However for a finite-time ($\ensuremath{\tau}$) linear quench in a Haldane nanoribbon, we show that the evolution of the edge current from the sudden to the adiabatic limit is not monotonic in $\ensuremath{\tau}$ and has a turning point at a characteristic time scale $\ensuremath{\tau}={\ensuremath{\tau}}_{0}$. For small $\ensuremath{\tau}$, the excited states lead to a huge unidirectional surge in the edge current of both edges. On the other hand, in the limit of large $\ensuremath{\tau}$, the edge current saturates to its expected equilibrium ground-state value. This competition between the two limits lead to the observed nonmonotonic behavior. Interestingly, ${\ensuremath{\tau}}_{0}$ seems to depend only on the Semenoff mass and the Haldane flux. A similar dynamics for the edge current is also expected in other systems with topological phases.
We study the coherent nonequilibrium dynamics of interacting two-dimensional systems after a quench from a trivial to a topological Chern insulator phase. While the many-body wave function is constrained to remain topologically trivial under local unitary evolution, we find that the Hall response of the system can dynamically approach a thermal value of the postquench Hamiltonian, even though the efficiency of this thermalization process is shown to strongly depend on the microscopic form of the interactions. Quite remarkably, the effective temperature of the steady-state Hall response can be arbitrarily tuned with the quench parameters. Our findings suggest a way of inducing and observing low-temperature topological phenomena in interacting ultracold atomic gases, where the considered quench scenario can be realized in current experimental setups.
We investigate evolution of entanglement spectra of the Haldane model for Chern insulators upon a sudden quench within the same topological phase. In particular, we focus on the location of the entanglement spectrum crossing, which signifies the bulk topology. It is shown that the coplanarity condition for the pseudomagnetic field of the model, which can be used to determine the crossing in the equilibrium case, needs to be relaxed. We analytically derive the non-equilibrium condition with the help of an edge-state wave function ansatz and a dynamically induced length-scale cutoff. With spectral analyses, it is realized that the oscillatory behavior of the crossing is dominated by the interband excitations at the van Hove singularities.
We study the quantum quench of a finite rate across topological quantum transitions in two-dimensional Chern and Z2{Z}_{2}Z2​ topological insulators. We choose the representative Haldane model and the Kane-Mele model to investigate the behavior of excitation density generated by the quench and the impact of disorder on it. For the Haldane model, as long as the spectral gap is not closed by disorder, we find the excitation density at the end of the quench obeys the power-law decay with decreasing quench rate, and the power is consistent with the prediction of the Kibble-Zurek mechanism. By contrast, the Kibble-Zurek scaling of excitation density is absent in the Kane-Mele model once disorder is switched on, which we attribute to the emergence of a disorder-induced gapless region. In particular, the anti-Kibble-Zurek behavior of excitation density, namely, larger excitation density at slower quench, is observed at suitable model parameters. Moreover, we demonstrate that particle's onsite occupation can be used as a local measurable quantity to probe the breakdown of adiabatic evolution. The difference of onsite occupation between the time-evolved state and the instantaneous ground state at the end of the quench can successfully capture the key features of excitation density for both the Haldane and Kane-Mele models under periodic and more realistic open boundary conditions, thus facilitating the experimental characterization of the quench dynamics in these models.