We show that at electronic filling factor $ν=1/2$, the lowest Landau level constraint implies that at any temperature, and in the presence of {\it any} amount of particle-hole symmetric disorder, the {\it composite fermion} Hall conductivity is precisely $-e^2/2h$. (The electronic Hall conductivity is $e^2/2h$.) This is inconsistent with the response of a composite Fermi liquid in {\it zero} effective magnetic field. We also examine perturbatively the nature of the putative composite Fermi liquid in the case in which the bare particles are anyons with ``nearly'' Fermi statistics.
The Universal Composite Fermion Hall Conductance at ν=1/2
We show that at electronic filling factor ν=1/2, the lowest Landau level constraint implies that at any temperature, and in the presence of {\it any} amount of particle-hole symmetric disorder, the {\it composite fermion} Hall conductivity is precisely −e2/2h. (The electronic Hall conductivity is e2/2h.) This is inconsistent with the response of a composite Fermi liquid in {\it zero} effective magnetic field. We also examine perturbatively the nature of the putative composite Fermi liquid in the case in which the bare particles are anyons with ``nearly'' Fermi statistics.
Motivated by the issue of particle-hole symmetry for the composite fermion Fermi sea at the half filled Landau level, Dam T. Son has made an intriguing proposal [Phys. Rev. X {\bf 5}, 031027 (2015)] that composite fermions are Dirac particles. We ask what features of the Dirac-composite fermion theory and its various consequences may be reconciled with the well established microscopic theory of the fractional quantum Hall effect and the 1/2 state, which is based on {\em non-relativistic} composite fermions. Starting from the microscopic theory, we derive the assertion of Son that the particle-hole transformation of electrons at filling factor ν=1/2 corresponds to an effective time reversal transformation (i.e. {kj}→{−kj}) for composite fermions, and discuss how this connects to the absence of 2kF backscattering in the presence of a particle-hole symmetric disorder. By considering bare holes in various composite-fermion Λ levels (analogs of electronic Landau levels) we determine the Λ level spacing and find it to be very nearly independent of the Λ level index, consistent with a parabolic dispersion for the underlying composite fermions. Finally, we address the compatibility of the Chern-Simons theory with the lowest Landau level constraint, and find that the wave functions of the mean-field Chern-Simons theory, as well as a class of topologically similar wave functions, are surprisingly accurate when projected into the lowest Landau level. These considerations lead us to introduce a normal form for the unprojected wave functions of the n/(2pn−1) states that correctly capture the topological properties even without lowest Landau level projection.
We consider the Hall conductivity of composite fermions in the theory of Halperin, Lee, and Read (HLR) [B. I. Halperin, P. A. Lee, and N. Read, Phys. Rev. B 47, 7312 (1993)]. We present a fully quantum-mechanical numerical calculation that shows, under suitable conditions, the HLR theory exhibits a particle-hole symmetric dc electrical Hall response in the presence of quenched disorder. Remarkably, this response of the HLR theory remains robust even when the disorder range is of the order of the Fermi wavelength. We find that deviations from the particle-hole symmetric response can appear in the ac Hall conductivity at frequencies sufficiently large compared to the inverse system size. Our results agree with a recent semiclassical analysis by Wang et al. [C. Wang, N. R. Cooper, B. I. Halperin, and A. Stern, Phys. Rev. X 7, 031029 (2017)] and complement the arguments based on the fully quantum-mechanical model by Kumar et al. [P. Kumar, M. Mulligan, and S. Raghu, Phys. Rev. B 98, 115105 (2018)]. These results provide further evidence that the HLR theory is compatible with an emergent particle-hole symmetry.
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.
The physics of two-dimensional electron gas in a perpendicular magnetic field, i.e., the quantum Hall system, is remarkably rich. At half filling of the lowest Landau level, it has been predicted that "composite fermions"---emergent quasiparticles consisting of an electron attached to two magnetic flux quanta---experience zero net magnetic field and form a Fermi sea, dubbed composite Fermi liquid (CFL). However, despite its seemingly simple appearance, CFL is a strongly correlated quantum many-body state in disguise, and solving it in a controlled manner is extremely difficult, to the extent that the thermodynamic properties of CFL remain largely unknown. In this work, we perform state-of-the-art thermal tensor network simulations of the ν=1/2 Landau level system and observe low-temperature power-law behaviour of the specific heat, signaling the gapless nature of CFL. More importantly, the power is extracted to be close to 2/3, clearly deviating from the ordinary linear-T behaviour of Fermi liquid, suggesting coupling between the CFs and the dynamical emergent gauge field and thereby revealing the quantum many-body nature of the CFL state.
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.
The theory of composite fermion provides a simple and unified picture to understand a vast amount of phenomenology in the quantum Hall regime. However it has remained challenging to formulate this concept properly within a single Landau level, which provides the relevant degrees of freedom in the limit of strong magnetic field. Recently a low-energy noncommutative field theory for bosons at Landau-level filling factor $\ensuremath{\nu}=1$ has been formulated by Dong and Senthil [Z. Dong and T. Senthil, Phys. Rev. B 102, 205126 (2020)]. In the limit of long-wavelength and small-amplitude gauge fluctuation, they found it reduces to the celebrated Halperin-Lee-Read theory of a composite fermion liquid. In this work we consider a Bose-Fermi mixture at total filling factor $\ensuremath{\nu}=1$. Different from previous work, the number density of composite fermions in the mixture and corresponding Fermi momentum can be tuned by changing the filling factor of bosons, ${\ensuremath{\nu}}_{b}=1\ensuremath{-}{\ensuremath{\nu}}_{f}$. This tunability enables us to study the dilute limit ${\ensuremath{\nu}}_{b}\ensuremath{\ll}1$, which allows for a controlled and asymptotically exact calculation of the energy dispersion and effective mass of composite fermions. Furthermore, the approximation of the low-energy description by a commutative field theory is manifestly justified. Most importantly, we demonstrate that gauge fluctuations acquire a Higgs mass due to the presence of a composite boson condensate, as a result of which the system behaves like a genuine Landau- Fermi liquid. Combined with the irrelevance of four-fermion interaction in the dilute limit, we are able to obtain asymptotically exact properties of this composite fermion Fermi liquid. In the opposite limit of ${\ensuremath{\nu}}_{f}\ensuremath{\ll}1$, the Higgs mass goes to zero and we find crossover between Fermi liquid and non-Fermi liquid as temperature increases. Observing these properties either experimentally or numerically provides unambiguous evidence of not only the composite fermions and the Fermi surface they form, but also the presence of emergent gauge fields and their fluctuations due to strong correlation.