Topological quantum quench dynamics carrying arbitrary Hopf and second Chern numbers

Motohiko Ezawa

Phys. Rev. B 98, 205406 – Published 9 November 2018

Abstract

A quantum quench is a nonequilibrium dynamics governed by the unitary evolution. We propose a two-band model whose quench dynamics is characterized by an arbitrary Hopf number belonging to the homotopy group $π_{3} (S^{2}) = Z$ . When we quench a system from an insulator with the Chern number $C_{i} \in π_{2} (S^{2}) = Z$ to another insulator with the Chern number $C_{f}$ , the preimage of the Hamiltonian vector forms links having the Hopf number $C_{f} - C_{i}$ . We also investigate a quantum-quench dynamics for a four-band model carrying an arbitrary second Chern number $N \in π_{4} (S^{4}) = Z$ , which can be realized by quenching a three-dimensional topological insulator having the three-dimensional winding number $N \in π_{3} (S^{3}) = Z$ .

Received 24 August 2018

DOI:https://doi.org/10.1103/PhysRevB.98.205406

Physics Subject Headings (PhySH)

Geometric & topological phases Quantum quench Topological phase transition Topological phases of matter

Topological insulators

Topology

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Motohiko Ezawa

Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, Japan

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Issue

Vol. 98, Iss. 20 — 15 November 2018

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Images

Figure 1
Hamiltonian vector with (a) $N = 1$ , (b) $N = 2$ , and (c) $N = 3$ . They form meron structure with the winding number $N$ at the high-symmetry points $Γ, M, X$ , and $Y$ .
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Figure 2
Topological phase diagram as a function of $t_{2} / t_{1}$ and $m / t_{1}$ . The Chern number is shown in each phase. The phase boundaries are determined by the condition $M_{K} = 0$ , where $K$ stands for the high-symmetry point indicated in the figure. The arrows in red represent the quantum quench processes we have numerically studied.
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Figure 3
Trivial to topological quench. Bird's eye view of the almost zero-energy surface of the Hamiltonian vector with $N = 1$ . They form closed linking structures. We have quenched from the trivial state with the mass $m / t_{1} = 3$ to the topological state with the mass $m / t_{1} = 1$ while keeping $t_{2} / t_{1} = 0$ to draw figures. (a) The preimage of ${\hat{d}}_{y} = 1$ is colored in magenta, while that of ${\hat{d}}_{y} = - 1$ is colored in cyan. (b) The preimage of ${\hat{d}}_{x} = 1$ is colored in magenta, while that of ${\hat{d}}_{x} = - 1$ is colored in cyan. (c) The preimage of ${\hat{d}}_{z} = 1$ is colored in magenta, while that of ${\hat{d}}_{z} = - 1$ is colored in cyan. The $d$ vectors are also shown in (a).
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Figure 4
Trivial to topological quench. Bird's eye view of the almost zero-energy surface of the Hamiltonian vector with (a1) $N = 1$ , (b1) $N = 2$ , and (c1) $N = 3$ . They form closed linked loops. We have quenched from the trivial state with the mass $m / t_{1} = 3$ to the topological state with the mass $m / t_{1} = 1$ while keeping $t_{2} / t_{1} = 0$ to draw figures. The preimage of ${\hat{d}}_{y} = 1$ is colored in magenta, while that of ${\hat{d}}_{y} = - 1$ is colored in cyan. (a2)–(c2) and (a3)–(c3) are the corresponding top and side views.
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Figure 5
Topological to trivial quench. Bird's eye view of the almost zero-energy surface of the Hamiltonian vector with (a1) $N = 1$ , (b1) $N = 2$ , and (c1) $N = 3$ . They form open linked helix. We have quenched from the topological states with the mass $m / t_{1} = 1$ to the trivial states with the mass $m / t_{1} = 3$ while keeping $t_{2} / t_{1} = 0$ to draw figures. The preimage of ${\hat{d}}_{y} = 1$ is colored in magenta, while that of ${\hat{d}}_{y} = - 1$ is colored in cyan. (a2)–(c2) and (a3)–(c3) are the corresponding top and side views.
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Figure 6
Topological to topological quench (a) from a topological insulator with $C_{i} = - 1$ to another topological insulator with $C_{f} = 1$ , (b) from a topological insulator with $C_{i} = - 1$ to another topological insulator with $C_{f} = 2$ , and (c) from a topological insulator with $C_{i} = - 2$ to another topological insulator with $C_{f} = 2$ . The Brillouin zone is indicated by the square. The preimage of ${\hat{d}}_{y} = 1$ is colored in magenta, while that of ${\hat{d}}_{y} = - 1$ is colored in cyan.
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