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 F-x(t) is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching on of complex time-dependent hopping amplitudes e(-i/(h) over bar Ax(t)) in the (x) over cap direction such that partial derivative(t)A(x)(t) = F-x(t). The switching on can be sudden, F-x(t) = theta(t)F, where F is the steady driving force, or more generally smooth F-x(t) = f (t/t(0))F, where f (t/t(0)) is such that f(0) = 0 and f(1) = 1. We investigate how the time-averaged (steady-state) particle current density j(y) in the (y) over cap direction deviates from the quantized value j(y)h/F = n due to the finite value of F and the details of the switching-on protocol. Exploiting the time periodicity of the Hamiltonian (H) over cap (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 (H) over cap (t). In the case of a sudden quench j(y)h/F shows F-2 corrections to the perfectly quantized limit. When the switching on is smooth, the result depends on the switch-on time t(0): For a fixed t(0) we observe a crossover force F* between a quadratic regime for F < F* and a nonanalytic exponential e(-gamma/vertical bar F vertical bar) for F > F*. The crossover F* decreases as t(0) 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 M.; Santoro, Giuseppe E.. - In: PHYSICAL REVIEW. B. - ISSN 2469-9950. - 98:20(2018). [10.1103/physrevb.98.205112]

Quantization of the Hall conductivity in the Harper-Hofstadter model

Matteo M. Wauters
Primo
;
2018-01-01

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 F-x(t) is turned on. In the vector potential gauge, through Peierls substitution, this involves the switching on of complex time-dependent hopping amplitudes e(-i/(h) over bar Ax(t)) in the (x) over cap direction such that partial derivative(t)A(x)(t) = F-x(t). The switching on can be sudden, F-x(t) = theta(t)F, where F is the steady driving force, or more generally smooth F-x(t) = f (t/t(0))F, where f (t/t(0)) is such that f(0) = 0 and f(1) = 1. We investigate how the time-averaged (steady-state) particle current density j(y) in the (y) over cap direction deviates from the quantized value j(y)h/F = n due to the finite value of F and the details of the switching-on protocol. Exploiting the time periodicity of the Hamiltonian (H) over cap (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 (H) over cap (t). In the case of a sudden quench j(y)h/F shows F-2 corrections to the perfectly quantized limit. When the switching on is smooth, the result depends on the switch-on time t(0): For a fixed t(0) we observe a crossover force F* between a quadratic regime for F < F* and a nonanalytic exponential e(-gamma/vertical bar F vertical bar) for F > F*. The crossover F* decreases as t(0) 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.
2018
20
Wauters, Matteo M.; Santoro, Giuseppe E.
Quantization of the Hall conductivity in the Harper-Hofstadter model / Wauters, Matteo M.; Santoro, Giuseppe E.. - In: PHYSICAL REVIEW. B. - ISSN 2469-9950. - 98:20(2018). [10.1103/physrevb.98.205112]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/400340
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