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Physical Review Letters

Fermi-liquid-like state in a half-filled Landau level

Edward H. Rezayi, N. Read

Published Feb 7, 1994• 224 citations
Publisher

Abstract

A system of electrons in a half-filled Landau level is investigated in spherical geometry. For systems of size from N=1 to 14 electrons with flux ${\mathit{N}}_{\mathrm{\ensuremath{\varphi}}}$=2(N-1) the angular momentum of the ground state is as predicted by Hund's second rule for composite fermions of one electron and two vortices at zero magnetic field. Low lying excitations also fit this interpretation and trial wave functions give excellent overlaps. The two-particle correlation function shows a significant correlation hole at short distances and suggests an asymptotically oscillating form at long distances, as in a Fermi liquid.

DOI
10.1103/physrevlett.72.900

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A system of electrons in a half-filled Landau level is investigated in spherical geometry. For systems of size from N=1 to 14 electrons with flux N\ensuremathφ{\mathit{N}}_{\mathrm{\ensuremath{\varphi}}}N\ensuremathφ​=2(N-1) the angular momentum of the ground state is as predicted by Hund's second rule for composite fermions of one electron and two vortices at zero magnetic field. Low lying excitations also fit this interpretation and trial wave functions give excellent overlaps. The two-particle correlation function shows a significant correlation hole at short distances and suggests an asymptotically oscillating form at long distances, as in a Fermi liquid.
Here, the authors investigate the possibility that particle-hole symmetry in the lowest Landau level at half filling arises as an emergent low-energy symmetry. Specifically, they start with the traditional approach to the half-filled Landau level: an effective field theory known as the composite Fermi liquid pioneered by Halperin, Lee, and Read. In this theory, composite fermions arise when flux is attached to electrons in a half-filled Landau level. However, an equivalent starting point is one where one starts with a filled Landau level, depletes electrons (or equivalently, adds holes) to the point of half filling. One then attaches flux to the holes, and the resulting composite hole description is another viable description of the half-filled Landau level. The authors construct a particle-hole symmetric theory by studying a system consisting of alternating quasi-one-dimensional strips of composite Fermi liquid (CFL) and composite hole liquid (CHL), both of which break particle-hole symmetry. Gauge invariance requires the existence of gapless edge modes at the interface between the CFL and CHL. Out of these modes, they construct a neutral Dirac fermion coupled to an emergent U(1) gauge field. Thus, at distances long compared to the strip width, they demonstrate that the system is described by a Dirac fermion coupled to an emergent gauge field, with an anti-unitary particle-hole symmetry, precisely the form conjectured recently by Son to be the low-energy description of a particle-hole symmetric half-filled Landau level.
In a two-dimensional electron gas under a strong magnetic field, correlations generate emergent excitations distinct from electrons. It has been predicted that "composite fermions"--bound states of an electron with two magnetic flux quanta--can experience zero net magnetic field and form a Fermi sea. Using infinite-cylinder density matrix renormalization group numerical simulations, we verify the existence of this exotic Fermi sea, but find that the phase exhibits particle-hole symmetry. This is self-consistent only if composite fermions are massless Dirac particles, similar to the surface of a topological insulator. Exploiting this analogy, we observe the suppression of 2k(F) backscattering, a characteristic of Dirac particles. Thus, the phenomenology of Dirac fermions is also relevant to two-dimensional electron gases in the quantum Hall regime.
We report finite-size studies of composite fermions (CF's) at even-denominator filling fractions. These show that, in the absence of strong Landau level (LL) mixing effects, the CF liquid cannot form for a half-filled second LL. We relate this to the absence of a hierarchy of incompressible states for filling fractions $2+1/3<\ensuremath{\nu}<2+2/3$. We find that a polarized CF liquid should exist for a quarter-filled second LL. We also show that, for CF states, the variation with system size of the ground state energy of interacting electrons follows that for noninteracting particles in zero magnetic field. We use this to estimate the CF effective masses. We suggest an explanation of these results based on the Rezayi-Read trial wave function and present estimates for the flux-binding energy.
We study the two-dimensional electron gas in a magnetic field at filling fraction ν=12ν=\frac{1}{2}ν=21​. At this filling the system is in a gapless state which can be interpreted as a Fermi liquid of composite fermions. We construct trial wave functions for the system on a torus, based on this idea, and numerically compare these to exact wave functions for small systems found by exact diagonalization. We find that the trial wave functions give an excellent description of the ground state of the system, as well as its charged excitations, in all momentum sectors. We analyze the dispersion of the composite fermions and the Berry phase associated with dragging a single fermion around the Fermi surface and comment on the implications of our results for the current debate on whether composite fermions are Dirac fermions.
We propose a particle-hole symmetric theory of the Fermi-liquid ground state of a half-filled Landau level. This theory should be applicable for a Dirac fermion in the magnetic field at charge neutrality, as well as for the v = 1/2 quantum Hall ground state of nonrelativistic fermions in the limit of negligible inter-Landau-level mixing. We argue that when particle-hole symmetry is exact, the composite fermion is a massless Dirac fermion, characterized by a Berry phase of π around the Fermi circle. We write down a tentative effective field theory of such a fermion and discuss the discrete symmetries, in particular, C P. The Dirac composite fermions interact through a gauge, but non-Chern-Simons, interaction. The particle-hole conjugate pair of Jain-sequence states at filling factors n/(2n + 1) and (n + 1)/(2n + 1), which in the conventional composite fermion picture corresponds to integer quantum Hall states with different filling factors, n and n + 1, is now mapped to the same half-integer filling factor n + 1/2 of the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are interpreted as d -wave Bardeen-Cooper-Schrieffer paired states of the Dirac fermion with orbital angular momentum of opposite signs, while s -wave pairing would give rise to a particle-hole symmetric non-Abelian gapped phase. When particle-hole symmetry is not exact, the Dirac fermion has a C P -breaking mass. The conventional fermionic Chern-Simons theory is shown to emerge in the nonrelativistic limit of the massive theory.
While an ordinary Fermi sea is perturbatively robust to interactions, the paradigmatic composite-fermion (CF) Fermi sea arises as a nonperturbative consequence of emergent gauge fields in a system where there was no Fermi sea to begin with. A mean-field picture suggests two Fermi seas, of composite fermions made from electrons or holes in the lowest Landau level, which occupy different areas away from half filling and thus appear to represent distinct states. Using the microscopic theory of composite fermions, which satisfies particle-hole symmetry in the lowest Landau level to an excellent approximation, we show that the Fermi wave vectors at filling factors ν and 1-ν are equal when expressed in units of the inverse magnetic length, and are generally consistent with the experimental findings of Kamburov et al. [Phys. Rev. Lett. 113, 196801 (2014)]. Our calculations suggest that the area of the CF Fermi sea may slightly violate the Luttinger area rule.