Theory for Magnetic-Field-Driven 3D Metal-Insulator Transitions in the Quantum Limit

Peng-Lu Zhao, Hai-Zhou Lu, and X. C. Xie

Phys. Rev. Lett. 127, 046602 – Published 23 July 2021

Abstract

Metal-insulator transitions driven by magnetic fields have been extensively studied in 2D, but a 3D theory is still lacking. Motivated by recent experiments, we develop a scaling theory for the metal-insulator transitions in the strong-magnetic-field quantum limit of a 3D system. By using a renormalization-group calculation to treat electron-electron interactions, electron-phonon interactions, and disorder on the same footing, we obtain the critical exponent that characterizes the scaling relations of the resistivity to temperature and magnetic field. By comparing the critical exponent with those in a recent experiment [F. Tang et al., Nature (London) 569, 537 (2019)], we conclude that the insulating ground state was not only a charge-density wave driven by electron-phonon interactions but also coexisting with strong electron-electron interactions and backscattering disorder. We also propose a current-scaling experiment for further verification. Our theory will be helpful for exploring the emergent territory of 3D metal-insulator transitions under strong magnetic fields.

Received 11 January 2021
Revised 7 April 2021
Accepted 17 June 2021

DOI:https://doi.org/10.1103/PhysRevLett.127.046602

Physics Subject Headings (PhySH)

Charge density waves Critical exponents Critical field Quantum transport Topological insulators

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Peng-Lu Zhao ¹, Hai-Zhou Lu^1,2,*, and X. C. Xie^3,4,5

¹Shenzhen Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology (SUSTech), Shenzhen 518055, China
²Shenzhen Key Laboratory of Quantum Science and Engineering, Shenzhen 518055, China
³International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
⁴CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China
⁵Beijing Academy of Quantum Information Sciences, West Building 3, No. 10, Xibeiwang East Road, Haidian District, Beijing 100193, China

^*Corresponding author. luhz@sustech.edu.cn

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Vol. 127, Iss. 4 — 23 July 2021

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Images

Figure 1
(a) A quantum phase transition can be described by a nonthermal parameter $λ$ that converges to $λ_{c}$ at the quantum critical point. The boundaries (dashed curves) of the quantum critical region are described by $T \propto {| λ - λ_{c} |}^{z ν}$ , where $z$ is the dynamical critical exponent and $ν$ is the correlation length exponent. (b) The magnetic field $B$ can serve as $λ$ , and $z$ and $ν$ can be used to construct a measurable critical exponent $ξ$ that describes the resistivity as a scaling function of the magnetic field and temperature, e.g., $ρ (B) / ρ (B_{c}) = f (| B - B_{c} | / T^{1 / ξ})$ , where $f (x)$ is a scaling function with $f (0) = 1$ .
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Figure 2
A $z$ -direction magnetic field splits the energy spectrum of a 3D insulator or metal (a) into 1D bands of Landau levels dispersing with the wave vector $k_{z}$ (b). The $n \pm$ stand for the $n$ th Landau bands. In the quantum limit, the Fermi energy $E_{F}$ crosses the lowest Landau band $0 +$ at $\pm k_{F}$ , which is a metal phase and the starting point of this work. (c) Correlations between electrons near $- k_{F}$ and $k_{F}$ open a gap of size $2 | ϕ |$ and induce an insulating phase.
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Figure 3
The renormalization-group flow in the $u - β$ plane for (a) $γ = 2$ and (b) $γ = 0.002$ . $u$ and $β$ measure the effective electron-electron and electron-phonon interactions, respectively. $γ = | v_{b} / v_{F} |$ is the ratio of the phonon speed to Fermi velocity. There are two fixed points at ${(i)}_{c}$ $(u_{*}, β_{*}) = (0, 0)$ and ${(ii)}_{c}$ $[0, {(1 + 1 / γ)}^{2}]$ . (c) The renormalization-group flow in the $u − Δ_{b}$ plane, in the absence of electron-phonon coupling. In this case, there are two non-Gaussian fixed points at $(u_{*}, Δ_{b}^{*}) = (0, 1 / 4)$ and $(1 / 2, 1 / 2)$ . The red curve distinguishes a disordered metal with zero electron-electron coupling on the left and a Wigner crystal on the right [39]. (d)–(f) Running of the electron-electron coupling $u$ , electron-phonon coupling $β$ , and $u / β$ with $ℓ$ , for different initial values of $u$ . The initial values of both $Δ_{b}$ and $β$ are 0.1. (d)–(f) share the same legends. The green and red lines overlap in (e).
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Figure 4
(a) The experimental setup for the current-scaling measurement. The longitudinal resistivity $ρ \equiv V / I$ along the $x$ direction as a function of the magnetic field $B$ is measured at different measurement currents $I$ to construct a scaling function $ρ (B) / ρ (B_{c}) = F (| B - B_{c} | / I^{1 / κ})$ in (b), where $F (x)$ is a scaling function with $F (0) = 1$ . The linear fitting of the $\log (d ρ / d B |_{B = B_{c}})$ vs $- \log (I)$ yields $1 / κ$ . With the fitted $ξ$ in Fig. 1 and $κ$ here, the correlation length exponent $ν_{B} = κ - ξ$ and the dynamical critical exponent $z = ξ / (κ - ξ)$ can be found, to provide more experimental connections for our theory to identify the insulating ground state in the experiments.
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