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  • Letter

Limitations and improvements of the relaxation time approximation in the quantum master equation: Linear conductivity in insulating systems

Ibuki Terada, Sota Kitamura, Hiroshi Watanabe, and Hiroaki Ikeda
Phys. Rev. B 109, L180302 – Published 17 May 2024
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    Abstract

    The nonequilibrium steady states of quantum materials have many challenges. Here, we highlight issues with the relaxation time approximation (RTA) for the dc conductivity in insulating systems. The RTA to the quantum master equation (QME) is frequently employed as a simple method, yet this phenomenological approach is exposed as containing a fatal flaw, displaying non-negligible dc conductivity in the linear response regime. We find that the puzzling behavior is caused by the fact that the density matrix in the RTA incompletely incorporates the first order of the external field. To solve this problem, we have derived a calculation scheme based on the QME that ensures correct behavior in low electric fields. Our method reproduces well the overall features of the exact electric currents in the whole field region. It is not time consuming, and its application to lattice systems is straightforward. This method will encourage progress in this research area as a simple way to more accurately describe nonequilibrium steady states.

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    • Received 1 February 2024
    • Revised 26 April 2024
    • Accepted 29 April 2024

    DOI:https://doi.org/10.1103/PhysRevB.109.L180302

    ©2024 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Dissipative dynamicsElectrical conductivityLandau-Zener effectQuantum transport
    1. Physical Systems
    Insulators
    1. Techniques
    Boltzmann theoryQuantum master equation
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Ibuki Terada1, Sota Kitamura2, Hiroshi Watanabe3,4, and Hiroaki Ikeda1

    • 1Department of Physics, Ritsumeikan University, Shiga 525-8577, Japan
    • 2Department of Applied Physics, The University of Tokyo, Hongo, Tokyo, 113-8656, Japan
    • 3Research Organization of Science and Technology, Ritsumeikan University, Shiga 525-8577, Japan
    • 4Department of Liberal Arts and Basic Sciences, College of Industrial Technology, Nihon University, Chiba 275-8576, Japan

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    Issue

    Vol. 109, Iss. 18 — 1 May 2024

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    • Figure 1
      Figure 1

      (a) Field dependence of the dc current JRTA in the RTA at T=0.01δ. The inset depicts JRTA in a wide range. Solid lines denote the RTA currents for several τ−1. The dotted line represents the tunneling probability PLZ. One can see the exponential behavior in the inset, but encounter an unexpected linear behavior at a low E limit in the main panel. (b) Intraband JintraRTA (dashed line) and interband JinterRTA (dotted line) contributions of JRTA at τ−1=0.02δ as a function of E/Eth. JinterRTA shows unphysical linear E behavior.

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    • Figure 2
      Figure 2

      Occupation number of a single electron in the upper band. Solid lines and solid circles denote the results in DPA and the numerically exact results [14], respectively. The schematic figure represents the electron dynamics in the Landau-Zener model. The DPA results are almost consistent with the exact results. The small deviation is due to the damping-induced excitation inherent in open systems.

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    • Figure 3
      Figure 3

      (a) Field dependence of the dc current JDPA in the DPA. The inset depicts JDPA in the wide range. In the main panel, solid lines denote the DPA currents for several τ−1, the dashed line represents the RTA current at τ−1=0.02δ, and the solid circles the result of the numerically exact result. In the inset, one can see that the DPA result reproduces the exact result well. JDPA almost vanishes at E<0.2Eth. We can verify in (b) that JDPA has no linear E terms with an accuracy of less than 10−7δ at E<0.06Eth. At the intermediate region E∼0.4Eth, JDPA deviates from the exact result by ∼10−3δ but roughly increases with the tunneling probability P, implying that the excited carriers are carrying the electric current.

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