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arXiv:1605.04768 · Physical review. B./Physical review. B

Quenching in Chern insulators with satellite Dirac points: The fate of edge states

Utso Bhattacharya, Joanna Hutchinson, Amit Dutta

Published Apr 11, 2017• 24 citations
arXivHTMLPDF

Abstract

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 $N3$ 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 $(C)$. 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 $|C|=2$ 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 $|C|=1$ before ending up in the phase $C=0$.

DOI
10.1103/physrevb.95.144304

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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.
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.
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 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.
Topological phases with large Chern numbers have important implications. They were previously predicted to exist by considering fabricated long-range interactions or multilayered materials. Stimulated by recent wide interests in Floquet topological phases, here we propose a scheme to engineer large-Chern-number phases with ease by periodic quenching. Using a two-band system as an example, we theoretically show how a variety of topological phases with widely tunable Chern numbers can be generated by periodic quenching between two simple Hamiltonians that otherwise give low Chern numbers. The obtained large Chern numbers are explained through the emergence of multiple Dirac cones in the Floquet spectra. The transition lines between different topological phases in the two-band model are also explicitly found, thus establishing a class of easily solvable but very rich systems useful for further understandings and applications of topological phases in periodically driven systems.
Chern insulators exhibit fascinating properties, which originate from the topologically nontrivial state characterized by the Chern number. How these properties change if the system is quenched between topologically distinct phases is, however, not fully understood. In this paper, we investigate the quench dynamics of the prototypical massive Dirac model for topological insulators in two dimensions. We consider both dissipationless dynamics and the effect of electron-phonon interactions, and ask how the transient dynamics and nonequilibrium steady states affect simple observables. Specifically, we discuss a time-dependent generalization of the Hall effect and the dichroism of the photoexcitation probability between left and right circularly polarized light. We present optimized schemes based on these observables, which can reveal the evolution of the topological state of the quenched system.
Graphene-based Sierpiński fractals host a zero-energy chiral mode and spin-1 Dirac dispersions within the nearest-neighbor tight-binding model. However, the presence of complex next-nearest neighbor hopping arising from the local flux and the staggered Semenoff mass terms, modeled within the Haldane Hamiltonian, breaks the time-reversal and spatial inversion symmetries, respectively, and makes these flat bands dispersive. Moreover, they introduce rich topological phases in this class of systems that can be characterized by Chern numbers up to ±3\pm 3±3, i.e., beyond the conventional honeycomb lattice. These observations pave the way for the exploration of 2D periodic fractals beyond graphene, where topological phase transitions can be realized through externally applied fields.