We study the robustness of the quantization of the Hall conductivity in the Harper-Hofstadter model towards the details of the protocol with which a longitudinal uniform driving force Fx(t) is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching on of complex time-dependent hopping amplitudes e-iAx(t) in the ◯ direction such that ∂tAx(t)=Fx(t). The switching on can be sudden, Fx(t)=θ(t)F, where F is the steady driving force, or more generally smooth Fx(t)=f(t/t0)F, where f(t/t0) is such that f(0)=0 and f(1)=1. We investigate how the time-averaged (steady-state) particle current density jy in the ŷ direction deviates from the quantized value jyh/F=n due to the finite value of F and the details of the switching-on protocol. Exploiting the time periodicity of the Hamiltonian Ĥ(t), we use Floquet techniques to study this problem. In this picture the (Kubo) linear response F→0 regime corresponds to the adiabatic limit for Ĥ(t). In the case of a sudden quench jyh/F shows F2 corrections to the perfectly quantized limit. When the switching on is smooth, the result depends on the switch-on time t0: For a fixed t0 we observe a crossover force F∗ between a quadratic regime for F<F∗ and a nonanalytic exponential e-γ/|F| for F>F∗. The crossover F∗ decreases as t0 increases, eventually recovering the topological robustness. These effects are in principle amenable to experimental tests in optical lattice cold atomic systems with synthetic gauge fields.

Quantization of the Hall conductivity in the Harper-Hofstadter model / Wauters, Matteo Michele; Santoro, Giuseppe E.. - In: PHYSICAL REVIEW. B. - ISSN 2469-9950. - 98:20(2018), pp. 1-13. [10.1103/PhysRevB.98.205112]

Quantization of the Hall conductivity in the Harper-Hofstadter model

Wauters, Matteo Michele;Santoro, Giuseppe E.
2018

Abstract

We study the robustness of the quantization of the Hall conductivity in the Harper-Hofstadter model towards the details of the protocol with which a longitudinal uniform driving force Fx(t) is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching on of complex time-dependent hopping amplitudes e-iAx(t) in the ◯ direction such that ∂tAx(t)=Fx(t). The switching on can be sudden, Fx(t)=θ(t)F, where F is the steady driving force, or more generally smooth Fx(t)=f(t/t0)F, where f(t/t0) is such that f(0)=0 and f(1)=1. We investigate how the time-averaged (steady-state) particle current density jy in the ŷ direction deviates from the quantized value jyh/F=n due to the finite value of F and the details of the switching-on protocol. Exploiting the time periodicity of the Hamiltonian Ĥ(t), we use Floquet techniques to study this problem. In this picture the (Kubo) linear response F→0 regime corresponds to the adiabatic limit for Ĥ(t). In the case of a sudden quench jyh/F shows F2 corrections to the perfectly quantized limit. When the switching on is smooth, the result depends on the switch-on time t0: For a fixed t0 we observe a crossover force F∗ between a quadratic regime for FF∗. The crossover F∗ decreases as t0 increases, eventually recovering the topological robustness. These effects are in principle amenable to experimental tests in optical lattice cold atomic systems with synthetic gauge fields.
98
20
1
13
205112
https://journals.aps.org/prb/abstract/10.1103/PhysRevB.98.205112
Wauters, Matteo Michele; Santoro, Giuseppe E.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/87921
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