Dynamical preparation of a topological state and out-of-equilibrium bulk-boundary correspondence in a Su-Schrieffer-Heeger chain under periodic driving

Souvik Bandyopadhyay and Amit Dutta

Phys. Rev. B 100, 144302 – Published 3 October 2019

Abstract

Exploiting the possibility of temporal variation of the winding number, we have prepared a Su-Schrieffer-Heeger (SSH) chain in its stroboscopic topological state, starting from the trivial one, by application of a periodic perturbation. The periodic driving we employ here is adiabatically switched on to break the particle-hole symmetry and generate a chiral mass term in the effective Floquet Hamiltonian; consequently, the Floquet Hamiltonian also gets deformed without crossing the gapless quantum critical point. The particle-hole symmetry is subsequently restored in the Floquet Hamiltonian by adiabatically switching off a part of the periodic potential. Thereafter, the Floquet Hamiltonian develops a symmetry-protected nontrivial topological winding number. Furthermore, we also observe stroboscopic topologically protected localized edge states in a long open chain and show that a bulk-boundary correspondence survives a unitary nonequilibrium situation in one-dimensional BDI Hamiltonians. Moreover, considering an extended SSH chain with higher neighbor hoppings, we dynamically prepare the system in a stroboscopic out-of-equilibrium topological insulator state starting from a metallic regime. At the same time, we establish the dynamical preparation of higher winding phases in an extended SSH chain with stroboscopic bulk-boundary correspondence in the nonequilibrium state of the system.

Received 20 May 2019
Revised 4 September 2019

DOI:https://doi.org/10.1103/PhysRevB.100.144302

Physics Subject Headings (PhySH)

Quantum phase transitions

Nonequilibrium systems Topological materials

Bloch-Floquet theorem

Condensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Souvik Bandyopadhyay ^* and Amit Dutta

Department of Physics, Indian Institute of Technology, Kanpur, Kanpur 208016, India

^*souvik@iitk.ac.in

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Vol. 100, Iss. 14 — 1 October 2019

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Images

Figure 1
Temporal variation of the winding number [45] as a function of stroboscopic time period for a periodic boundary condition. The winding number assumes a constant value (unity) after the $P$ symmetry have been completely restored in the Floquet Hamiltonian at $t = 2 τ$ , with $τ = 10^{4} T$ . The corresponding change in the bulk polarization current is shown in the Appendix. The initial parameters are $v = 1.55, w = 1.50$ , and $L = 1000$ ; in addition, $ω = 100.0$ and the amplitudes $V_{0} = 0.1$ and $V_{1} = 7.0$ are from Eqs. (11) and (13).
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Figure 2
Overlap of the evolved state $| ψ_{k} (2 τ) 〉$ at $t = 2 τ$ with the Floquet eigenstate at the same time $| ψ_{0}^{F} (k) 〉$ for all $k \in [- π, π]$ . For a sufficiently large $τ$ ( $τ = 10^{4} T$ ), the instantaneous state of the system has a significant overlap with the Floquet topological ground state in the high-frequency limit. The parameters of the initial state are the same as those in Fig. 1.
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Figure 3
Under open boundary conditions, at $t = 2 τ$ , the stroboscopic time-evolved state shows localization at the ends of the chain after the BDI symmetries have been restored in the topological Floquet Hamiltonian. The energy of the localized state $E = 0.003$ for a finite chain with 100 lattice sites. The initial state (same as in Fig. 1) is subjected to a unitary drive as in Eq. (9) with $V_{0} = 0.1, V_{1} = 2.0, τ = 500 T$ , and $ω = 100.0$ , where $i$ denotes the lattice site.
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Figure 4
(a) Under the time-periodic drive (as discussed in Sec. 5a), the renormalized second-neighbor hopping strength $v_{2}^{'}$ as in Eq. (24) (shown in blue) of the extended SSH chain eventually vanishes as the drive is gradually switched on. As a result, the BDI symmetries are restored in the Floquet Hamiltonian. The dynamical winding number (shown in red) however remains quantized stroboscopically to unity throughout. For the initial Hamiltonian $H^{E} (k)$ in Eq. (18), the parameters are $v = 0.15, w = 0.22, v_{2} = 0.15, v_{3} = 0$ , and $w_{3} = 0.015$ ; further, we have chosen $ω = 10^{8}, τ = 10^{5} T, A_{0} = 1.2$ , and a system size of $L = 1000$ . (b) The corresponding stroboscopic state $| ψ_{0} (τ) 〉$ of the system eventually hosts zero-energy states (under OBCs with a system size $L = 60$ ) localized at the ends of the chain (shown in black) after the restoration of the BDI symmetries in the Floquet Hamiltonian for $t \geq τ$ . This establishes that, starting from a delocalized eigenstate $| ψ_{0} (0) 〉$ (shown in orange) of the extended SSH chain, the vanishing of the stroboscopically renormalized second hopping restores the stroboscopic bulk-boundary correspondence in the dynamical state of the system.
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Figure 5
(a) Temporal variation of the stroboscopically observed dynamical winding number (under the unitary driving protocol discussed in Sec. 5b), starting from an eigenstate of the BDI symmetric extended SSH Hamiltonian with the second-neighbor hopping $v_{2} = 0$ in Eq. (18). We further choose $v = 1.0, w = 2.0, v_{3} = 0$ , and $w_{3} = 0.5$ so that the equilibrium winding number of the initial state is unity. In the presence of the time-dependent drive as in Eq. (25) with $V_{0} = - 1.0, V_{1} = 7.0, τ = 10^{4} T$ , and $ω = 100$ , the dynamical winding number eventually assumes a stroboscopically quantized value of 2. The winding number remains invariant thereafter for $t \geq 2 τ$ , i.e., when the $P$ symmetry has been completely restored in the Floquet Hamiltonian. The system size $L = 1000$ . (b) Under OBCs, the out-of-equilibrium state of the driven extended SSH chain hosts two zero-energy states localized at different sites of the chain (shown in blue and orange). Starting from an eigenstate of the Hamiltonian in Eq. (17) with $v = 0.5, w = 0.1, v_{2} = 0, w_{3} = 0.25, v_{3} = 0$ , and a system size $L = 100$ , we follow the corresponding unitary protocol specified in Sec. 5b. The drive is chosen such that $V_{0} = - 0.5, V_{1} = 2.0, τ = 10^{3} T$ , and $ω = 100$ . This establishes a stroboscopic bulk-boundary correspondence once the $P$ symmetry has been completely restored in the Floquet Hamiltonian even for higher dynamical winding number state.
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Figure 6
As a result of the variation of the winding number, the stroboscopic bulk polarization current vanishes for $t > 2 τ$ . The initial state is the ground state of a BDI symmetric SSH model with the parameters $v = 1.55, w = 1.50$ , and $L = 1000 .$ We have chosen $ω = 100.0$ and amplitudes $V_{0} = 0.1$ and $V_{1} = 7.0$ as in Eq. (9). The variation of the polarization current perfectly conforms to the variation of the dynamical winding number presented in Fig. 1 and vanishes when it reaches the value unity.
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