Topological phase transition underpinning particle-hole symmetry in the Halperin-Lee-Read theory

Prashant Kumar, Michael Mulligan, and S. Raghu

Phys. Rev. B 98, 115105 – Published 5 September 2018

Abstract

Long wavelength descriptions of a half-filled lowest Landau level ( $ν = 1 / 2$ ) must be consistent with the experimental observation of particle-hole (PH) symmetry. The traditional description of the $ν = 1 / 2$ state pioneered by Halperin, Lee, and Read (HLR) naively appears to break PH symmetry. However, recent studies have shown that the HLR theory with weak quenched disorder can exhibit an emergent PH symmetry. We find that such inhomogeneous configurations of the $ν = 1 / 2$ fluid, when described by HLR mean-field theory, are tuned to a topological phase transition between an integer quantum Hall state and an insulator of composite fermions with a dc Hall conductivity $σ_{x y}^{(cf)} = - \frac{1}{2} \frac{e^{2}}{h}$ . Our observations help explain why the HLR theory exhibits PH symmetric dc response.

Received 8 July 2018

DOI:https://doi.org/10.1103/PhysRevB.98.115105

Physics Subject Headings (PhySH)

Anderson localization Composite fermions Dirac fermions Electrical conductivity Fractional quantum Hall effect Integer quantum Hall effect Quantum criticality Topological phase transition

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Prashant Kumar¹, Michael Mulligan², and S. Raghu^1,3

¹Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA
²Department of Physics and Astronomy, University of California, Riverside, Riverside, California 92511, USA
³SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA

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Vol. 98, Iss. 11 — 15 September 2018

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Images

Figure 1
A schematic for the density of states of (a) down spin and (b) up spin when average magnetic field $b_{0} < 0$ and spatial variations of $b (r)$ are small compared to $| b_{0} |$ . The lowest Landau level of down spins at zero energy does not spread as long as $g = 2$ . Also, according to Eqs. (33) and (34), the higher Landau levels spread more in energy because to a leading order $E_{n}^{↓} \propto n b (x)$ . All states except one are assumed to be localized in each positive energy Landau level. As a result of supersymmetric quantum mechanics, the locations of extended and localized states match for the two spins. Anticipating levitation [33] of extended states as the spatial variations of magnetic field are made larger, we have drawn the extended state at a higher energy than the center of each Landau level.
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Figure 2
A schematic of $〈 x_{k} 〉$ vs $k$ for zero-energy wave functions in Eq. (37). As seen in Eq. (38), $x_{k}$ is a periodic function of $k$ . Or, if $k \to k + | b_{0} | L, x_{k} \to x_{k} - L$ . Let us assume that $k = 0$ through $k = | b_{0} | L - 2 π / L$ are filled initially. After adiabatic threading of one flux quantum, $k \to k + 2 π / L$ . Thus, $k = 2 π / L$ through $k = | b_{0} | L$ are filled now. Equivalently, $k = 0$ mode is now replaced by $k = | b_{0} | L$ and the rest of the modes are unaffected. Since the separation between these two modes is $- L$ , one down-spin fermion got transported across the system in negative $x$ direction.
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Figure 3
Numerical calculation of $σ_{x y}^{↓} - σ_{x y}^{↑}$ for various Fermi energies for Hamiltonians in Eqs. (22) and (23) for $b_{0} = 0$ . We have taken $m = 1$ . The energy and length scales are set by taking $k_{F} = 1$ for the datapoint with highest Fermi energy. The cutoff in momentum space is $Λ = 3.3$ and system dimensions are $160 \times 160$ . In addition, the disorder strength and range defined in Eq. (9) are $σ_{V} = 0.034$ and $R = 6$ . We have taken an average over 25 disorder realizations and calculated standard deviation to quantify the agreement between numerics and Eq. (29) at $b_{0} = 0$ . It is shown as the error bars in the figure which can be seen to be negligible. The horizontal green line represents $σ_{x y}^{↓} - σ_{x y}^{↑} = - 1 / 2 π$ .
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Figure 4
A schematic for density of states of $H_{f}$ in Eq. (14) for (a) $b_{0} < 0$ and (b) $b_{0} > 0$ when spatial variations of magnetic field are large compared to $b_{0}$ . Only (a) has zero-energy states that are shown as a Dirac delta function. The disorder washes out any quantum fluctuations in the DoS due to presence of a nonzero $b_{0}$ so that $ρ = m / 2 π$ at the Fermi-energy $μ$ . We assume that all extended states corresponding to higher Landau levels have levitated up [33] and thus all positive energy states below the Fermi energy are localized.
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