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arXiv:1206.6530 · Physical Review B

Continuous transitions between composite Fermi liquid and Landau Fermi liquid: A route to fractionalized Mott insulators

Maissam Barkeshli, John McGreevy

Published Aug 21, 2012• 39 citations
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Abstract

One of the most successful theories of a non-Fermi-liquid metallic state is the composite Fermi-liquid (CFL) theory of the half-filled Landau level. In this paper, we study continuous quantum phase transitions out of the CFL state and into a Landau Fermi liquid, in the limit of no disorder and fixed particle number. This transition can be induced by tuning the bandwidth of the Landau level relative to the interaction energy, for instance through an externally applied periodic potential. We find a transition to the Landau Fermi liquid through a gapless Mott insulator with a Fermi surface of neutral fermionic excitations. In the presence of spatial symmetries, we also find a direct continuous transition between the CFL and the Landau Fermi liquid. The transitions have a number of characteristic observable signatures, including the presence of two crossover temperature scales, resistivity jumps, and vanishing compressibility. When the composite fermions are paired instead, our results imply quantum critical points between various non-Abelian topological states, including the $\ensuremath{\nu}=1/2$ Moore-Read Pfaffian [Ising $\ifmmode\times\else\texttimes\fi{}$ $U(1)$ topological order], a version of the Kitaev B phase (Ising topological order), and paired electronic superconductors. To study such transitions, we use a projective construction of the CFL, which goes beyond the conventional framework of flux attachment to include a broader set of quantum fluctuations. These considerations suggest a possible route to fractionalized Mott insulators by starting with fractional quantum Hall states and tuning the Landau-level bandwidth.

DOI
10.1103/physrevb.86.075136

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One of the most successful theories of a non-Fermi-liquid metallic state is the composite Fermi-liquid (CFL) theory of the half-filled Landau level. In this paper, we study continuous quantum phase transitions out of the CFL state and into a Landau Fermi liquid, in the limit of no disorder and fixed particle number. This transition can be induced by tuning the bandwidth of the Landau level relative to the interaction energy, for instance through an externally applied periodic potential. We find a transition to the Landau Fermi liquid through a gapless Mott insulator with a Fermi surface of neutral fermionic excitations. In the presence of spatial symmetries, we also find a direct continuous transition between the CFL and the Landau Fermi liquid. The transitions have a number of characteristic observable signatures, including the presence of two crossover temperature scales, resistivity jumps, and vanishing compressibility. When the composite fermions are paired instead, our results imply quantum critical points between various non-Abelian topological states, including the \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2 Moore-Read Pfaffian [Ising \ifmmode×\else\texttimes\fi\ifmmode\times\else\texttimes\fi{}\ifmmode×\else\texttimes\fi U(1)U(1)U(1) topological order], a version of the Kitaev B phase (Ising topological order), and paired electronic superconductors. To study such transitions, we use a projective construction of the CFL, which goes beyond the conventional framework of flux attachment to include a broader set of quantum fluctuations. These considerations suggest a possible route to fractionalized Mott insulators by starting with fractional quantum Hall states and tuning the Landau-level bandwidth.
Interlayer coherent composite Fermi liquid in Half-filled Landau Level bilayers, a window toward the hidden πππ Berry phase
For interacting 2D electrons in the presence of magnetic field at half filling, the system forms a `composite Fermi liquid(CFL)' with emergent Fermi surface and exhibits metallic behavior, based on the standard Halperin, Lee, and Read(HLR) field theory. Recently, Son introduces a composite Dirac theory as the low energy effective description of Half-filled Landau Level. Such theory exhibits particle-hole symmetry and the underlying composite Fermi surface displays a robust πππ Berry phase. In this paper, we start from the bilayer Half-filled Landau Level system where the two composite Fermi surface acquires interlayer coherence and forms bonding/anti-bonding composite fermi sea. The corresponding interlayer coherent composite Fermi liquid(ICCFL) phase provides a straightforward landscape to verify the Dirac nature in Son's theory and extract the hidden Berry phase structure of the composite Fermi surface. The ICCFL phase contains two Fermi surfaces which are detached in most regions but adhesive at two hot spots. Such nematic structure is a consequence of the Berry phase encoded in the Dirac Fermi surface which is absent in HLR theory. Due to the nematicity in ICCFL, the system supports half-quantum vortex with deconfined π2\fracπ{2}2π​ gauge flux and the phase transition toward ICCFL contains a Lifshitz criticality with z=3z=3z=3 dynamical exponent. In addition, the exciton order parameter carries topological spin number so the ICCFL contains a unique Wen-Zee term which connects EM response with the background geometry curvature.
Composite Fermi liquids (CFLs) are compressible states that can occur for 2D interacting fermions confined in the lowest Landau level at certain Landau level fillings. They have been understood as Fermi seas formed by composite fermions which are bound states of electromagnetic fluxes and electrons as reported by Halperin, Lee, and Read [Phys. Rev. B 47, 7312 (1993)PRBMDO0163-182910.1103/PhysRevB.47.7312]. At half filling, an explicitly particle-hole symmetric theory based on Dirac fermions was proposed by Son [Phys. Rev. X 5, 031027 (2015)PRXHAE2160-330810.1103/PhysRevX.5.031027] as an alternative low energy description. In this work, we investigate the Berry curvature of CFL model wave functions at a filling fraction of one-quarter, and observe that it is uniformly distributed over the Fermi sea except at the center where an additional π phase was found. Motivated by this, we propose an effective theory which generalizes Son's half filling theory, by internal gauge flux attachment, to all filling fractions in which fermionic CFLs can occur. The numerical results support the idea of internal gauge flux attachment.
We present a theory of the isotropic-nematic quantum phase transition in the composite Fermi liquid arising in half-filled Landau levels. We show that the quantum phase transition between the isotropic and the nematic phase is triggered by an attractive quadrupolar interaction between electrons, as in the case of conventional Fermi liquids. We derive the theory of the nematic state and of the phase transition. This theory is based on the flux attachment procedure, which maps an electron liquid in half-filled Landau levels into the composite Fermi liquid close to a nematic transition. We show that the local fluctuations of the nematic order parameters act as an effective dynamical metric interplaying with the underlying Chern-Simons gauge fields associated with the flux attachment. Both the fluctuations of the Chern-Simons gauge field and the nematic order parameter can destroy the composite fermion quasiparticles and drive the system into a non-Fermi liquid state. The effective-field theory for the isotropic-nematic phase transition is shown to have z=3z=3z=3 dynamical exponent due to the Landau damping of the dense Fermi system. We show that there is a Berry-phase-type term that governs the effective dynamics of the nematic order parameter fluctuations, which can be interpreted as a nonuniversal ``Hall viscosity'' of the dynamical metric. We also show that the effective-field theory of this compressible fluid has a Wen-Zee-type term. Both terms originate from the time-reversal breaking fluctuation of the Chern-Simons gauge fields. We present a perturbative (one-loop) computation of the Hall viscosity and also show that this term is also obtained by a Ward identity. We show that the topological excitation of the nematic fluid, the disclination, carries an electric charge. We show that a resonance observed in radio-frequency conductivity experiments can be interpreted as a Goldstone nematic mode gapped by lattice effects.
We propose that quantum Hall bilayers in the presence of a periodic potential at the scale of the magnetic length can host examples of a deconfined metal-insulator transition (DMIT), where a Fermi-liquid (FL) metal with a generic electronic Fermi surface evolves into a gapped insulator (or an insulator with Goldstone modes) through a continuous quantum phase transition.The transition can be accessed by tuning a single parameter, and its universal critical properties can be understood using a controlled framework.At the transition, the two layers are effectively decoupled, where each layer undergoes a continuous transition from a FL to a generalized composite Fermi liquid (gCFL).The thermodynamic and transport properties of the gCFL are similar to the usual CFL, while its spectral properties are qualitatively different.The FL-gCFL quantum critical point hosts a sharply defined Fermi surface without long-lived electronic quasiparticles.Immediately across the transition, the two layers of gCFL are unstable to forming an insulating phase.We discuss the topological properties of the insulator and various observable signatures associated with the DMIT.
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.
Fluctuations of conserved quantities within a subsystem are nonlocal observables that provide unique insights into quantum many-body systems. In this paper, we study bipartite charge (and spin) fluctuations across interaction-driven “metal-insulator transitions” out of Landau Fermi liquids. The “charge insulators” include a class of non-Fermi-liquid states of fractionalized degrees of freedom, such as compressible composite Fermi liquids (for spinless electrons) and incompressible spin-liquid Mott insulators (for spin- 1 / 2 electrons). We find that charge fluctuations F exhibit distinct leading-order scalings across the transition: F ∼ L log ( L ) in Landau Fermi liquids and F ∼ L in charge insulators, where L is the linear size of the subsystem. In composite Fermi liquids, under certain conditions, we also identify a universal constant term − f ( θ ) | σ x y | / ( 2 π ) when the subsystem geometry contains a sharp corner, where f ( θ ) denotes a function of the corner angle and σ x y is the Hall conductivity. At the critical point, provided the transition is continuous, the leading scaling F ∼ L is accompanied by a subleading universal corner contribution − log ( L ) f ( θ ) C ρ / 2 with the same angle dependence f ( θ ) , and the universal coefficient C ρ is directly related to the predicted universal jumps in longitudinal and Hall resistivities. These results establish fluctuation-transport relations, paving the way for numerical and experimental studies of unconventional quantum criticalities in metals.