Composite Fermi Liquid Research Articles

Nematic quantum phase transition of composite Fermi liquids in half-filled Landau levels and their geometric response

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 has $z = 3$ dynamical exponent due to Landau damping effects. We show that there is a Berry phase type term which governs the effective dynamics of the nematic order parameter fluctuations, which can be interpreted as a non-universal "Hall viscosity" of the dynamical metric. We show that the effective field theory 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 computation of the Hall viscosity and also show that this term is also obtained by a Ward identity. We show that the disclination of the nematic fluid, 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.

Half-filled Landau level, topological insulator surfaces, and three-dimensional quantum spin liquids

We synthesize and partly review recent developments relating the physics of the half-filled Landau level in two dimensions to correlated surface states of topological insulators in three dimensions. The latter are in turn related to the physics of certain three dimensional quantum spin liquid states. The resulting insights provide an interesting answer to the old question of how particle-hole symmetry is realized in composite fermion liquids. Specifically the metallic state at filling $\nu = \frac{1}{2}$ - described originally in pioneering work by Halperin , Lee, and Read as a liquid of composite fermions - was proposed recently by Son to be described by a particle-hole symmetric effective field theory distinct from that in the prior literature. We show how the relation to topological insulator surface states leads to a physical understanding of the correctness of this proposal. We develop a simple picture of the particle-hole symmetric composite fermion through a modification of older pictures as electrically neutral "dipolar" particles. We revisit the phenomenology of composite fermi liquids (with or without particle-hole symmetry), and show that their heat/electrical transport dramatically violates the conventional Wiedemann-Franz law but satisfies a modified one. We also discuss the implications of these insights for finding physical realizations of correlated topological insulator surfaces.

Mirror symmetry and the half-filled Landau level

We study the dynamics of the half-filled zeroth Landau level of Dirac fermions using mirror symmetry, a supersymmetric duality between certain pairs of $2+1$-dimensional theories. We show that the half-filled zeroth Landau level of a pair of Dirac fermions is dual to a pair of Fermi surfaces of electrically-neutral composite fermions, coupled to an emergent gauge field. Thus, we use supersymmetry to provide a derivation of flux attachment and the emergent Fermi liquid-like state for the lowest Landau level of Dirac fermions. We find that in the dual theory the Coulomb interaction induces a dynamical exponent $z=2$ for the emergent gauge field, making the interactions classically marginal. This enables us to map the problem of $2+1$-dimensional Dirac fermions in a finite transverse magnetic field, interacting via a strong Coulomb interaction, into a perturbatively controlled model. We analyze the resulting low-energy theory using the renormalization group and determine the nature of the BCS interaction in the emergent composite Fermi liquid.

Particle-hole symmetry and the composite Fermi liquid

The half-filled Landau level is widely believed to be described by the Halperin-Lee-Read theory of the composite Fermi liquid (CFL). In this paper, we develop a theory for the particle-hole conjugate of the CFL, the Anti-CFL, which we argue to be a distinct phase of matter as compared with the CFL. The Anti-CFL provides a possible explanation of a recent experiment [Kamburov et. al., Phys. Rev. Lett. 113, 196801 (2014)] demonstrating that the density of composite fermions in GaAs quantum wells corresponds to the electron density when the filling fraction $\nu < 1/2$ and to the hole density when $\nu > 1/2$. We introduce a local field theory for the CFL and Anti-CFL in the presence of a boundary, which we use to study CFL - Insulator - CFL junctions, and the interface between the Anti-CFL and CFL. We show that the CFL - Anti-CFL interface allows partially fused boundary phases in which "composite electrons" can directly tunnel into "composite holes," providing a non-trivial example of transmutation between topologically distinct quasiparticles. We discuss several observable consequences of the Anti-CFL, including a predicted resistivity jump at a first order transition between uniform CFL and Anti-CFL phases. We also present a theory of a continuous quantum phase transition between the CFL and Anti-CFL. We conclude that particle-hole symmetry requires a modified view of the half-filled Landau level, in the presence of strong electron-electron interactions and weak disorder, as a critical point between the CFL and the Anti-CFL.

Transport of a sliding Wigner crystal in the four flux composite fermion regime

In two-dimensional (2D) electron systems, Wigner crystals (WC) and fractional quantum Hall effect (FQHE) liquids are competing ground states under low temperatures $(T)$ and high magnetic fields $(B)$. Here we report differential conductivity results demonstrating the reentrant insulating phase around $\ensuremath{\nu}=1$/5 in a 2D hole system in AlGaAs/GaAs quantum wells and unexpected features in the solid-liquid phase transition between WC and FQHE liquids in ultrahigh magnetic fields up to 45 T. Remarkably, the electric field $(E)$ plays an equivalent role as the temperature does in our phase diagram. From the $E\ensuremath{-}T$ ``duality'' analysis, a characteristic length of 450 nm is derived, which can be understood as the phase-coherent domain size of WC. Moreover, evidence shows that with weak disorder the insulating phase and composite fermion liquid could be coexisting around $\ensuremath{\nu}=$ 1/5, pointing to the possibility that the insulating phase is the four flux quantum Wigner crystal, as proposed by theories.

Density Functional Theory of Composite Fermions**Supported by the National Basic Research Program of China under Grant No 2012CB921304, and the National Natural Science Foundation of China under Grant No 11325416.

We construct a density functional theory for two-dimensional electron (hole) gases subjected to both strong magnetic fields and external potentials. In particular, we are focused on regimes near even-denominator filling factors, in which the systems form composite fermion liquids. Our theory provides a systematic and rigorous approach to determine the properties of ground states in a fractional quantum Hall regime that is modified by artificial structures. We also propose a practical way to construct an approximated functional.

Composite Dirac Liquids: Parent States for Symmetric Surface Topological Order

We introduce exotic gapless states---`composite Dirac liquids'---that can appear at a strongly interacting surface of a three-dimensional electronic topological insulator. Composite Dirac liquids exhibit a gap to all charge excitations but nevertheless feature a single massless Dirac cone built from emergent electrically neutral fermions. These states thus comprise electrical insulators that, interestingly, retain thermal properties similar to those of the non-interacting topological insulator surface. A variety of novel fully gapped phases naturally descend from composite Dirac liquids. Most remarkably, we show that gapping the neutral fermions via Cooper pairing---which crucially does not violate charge conservation---yields symmetric non-Abelian topologically ordered surface phases captured in several recent works. Other (Abelian) topological orders emerge upon alternatively gapping the neutral Dirac cone with magnetism. We establish a hierarchical relationship between these descendant phases and expose an appealing connection to paired states of composite Fermi liquids arising in the half-filled Landau level of two-dimensional electron gases. To controllably access these states we exploit a quasi-1D deformation of the original electronic Dirac cone that enables us to analytically address the fate of the strongly interacting surface. The algorithm we develop applies quite broadly and further allows the construction of symmetric surface topological orders for recently introduced bosonic topological insulators.

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

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 \nu = 1/2 Moore-Read Pfaffian (Ising x 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 FQH states and tuning the Landau level bandwidth.

Correlation-Hole Induced Paired Quantum Hall States in the Lowest Landau Level

A theory is developed for the paired even-denominator fractional quantum Hall states in the lowest Landau level. We show that electrons bind to quantized vortices to form composite fermions, interacting through an exact instantaneous interaction that favors chiral p-wave pairing. There are two canonically dual pairing gap functions related by the bosonic Laughlin wave function (Jastrow factor) due to the correlation holes. We find that the ground state is the Moore-Read Pfaffian in the long-wavelength limit for weak Coulomb interactions, a new Pfaffian with an oscillatory pairing function for intermediate interactions, and a Read-Rezayi composite Fermi liquid beyond a critical interaction strength. Our findings are consistent with recent experimental observations of the 1/2 and 1/4 fractional quantum Hall effects in asymmetric wide quantum wells.

Tunneling-driven breakdown of the 331 state and the emergent Pfaffian and composite Fermi liquid phases

We examine the possibility of creating the Moore-Read Pfaffian in the lowest Landau level when the multicomponent Halperin 331 state (believed to describe quantum Hall bilayers and wide quantum wells at the filling factor $\ensuremath{\nu}=1/2$) is destroyed by the increase of tunneling. Using exact diagonalization of the bilayer Hamiltonian with short-range and long-range (Coulomb) interactions in spherical and periodic rectangular geometries, we establish that tunneling is a perturbation that drives the 331 state into a compressible composite Fermi liquid, with the possibility for an intermediate critical state that is reminiscent of the Moore-Read Pfaffian. These results are interpreted in the two-component BCS model for Cauchy pairing with a tunneling constraint. We comment on the conditions to be imposed on a system with fluctuating density in order to achieve the stable Pfaffian phase.

Intrinsic Gap and Exciton Condensation in theνT=1Bilayer System

We investigate the quasiparticle excitation of the bilayer quantum Hall (QH) system at a total filling factor nu{T}=1 in the limit of negligible interlayer tunneling under a tilted magnetic field. We show that the intrinsic quasiparticle excitation is of purely pseudospin origin and solely governed by the inter- and intralayer electron interactions. A model based on exciton formation successfully explains the quantitative behavior of the quasiparticle excitation gap, demonstrating the existence of a link between the excitonic QH state and the composite fermion liquid. Our results provide a new insight into the nature of the phase transition between the two states.

Interlayer Coherent Composite Fermi Liquid Phase in Quantum Hall Bilayers

We introduce an interlayer coherent composite Fermi liquid for nu = 1/2 + 1/2 bilayers, in which interlayer Coulomb repulsion drives exciton condensation of composite fermions. As a result, composite fermions propagate coherently between layers--even though electrons do not--and form bonding and antibonding Fermi seas. This phase is compressible with respect to symmetric currents but quantum Hall-like in the counterflow channel. Quantum oscillations of the composite Fermi seas generate a new series of incompressible states at nu = p/[2(p +/- 1)] per layer (p an integer), which is a bilayer analogue of Jain's sequence.

Paired Composite Fermion Phase of Quantum Hall Bilayers atν=12+12

We provide numerical evidence for composite fermion pairing in quantum Hall bilayer systems at filling nu=1/2+1/2 for intermediate spacing between the layers. We identify the phase as p_(x)+ip_(y) pairing, and construct high accuracy trial wave functions to describe the ground state on the sphere. For large distances between the layers, and for finite systems, a competing "Hund's rule" state, or composite fermion liquid, prevails for certain system sizes.

Spin Phase Transition in a Correlated Composite Fermion Liquid

The composite fermion (CF) model [1] has been very successful in the explanation of incompressibility of a whole family of fractional quantum Hall (FQH) states [2] at the “Jain series” of the Landau level (LL) filling factors νe = n/(2np ± 1). At these fillings, the CF transformation involving attachment of an even number 2p of magnetic flux quanta φ0 = hc/e to each electron converts a partially filled LL of electrons with strong (Coulomb) interaction into a small number νCF ≡ n of completely filled LL’s of nearly noninteracting CF’s (the general relation between electron and CF filling factors being ν−1 CF = ν −1 e − 2p). In filled shells, the weak residual CF–CF interactions play no role and the incompressibility of νe = n/(2np ± 1) states is said to result from a single-particle cyclotron gap of the CF’s in the reduced magnetic field B∗ = B − 2pφ0%, where % is the 2D electron/CF concentration (connected to νe = 2π%λ via magnetic length λ = √ hc/eB). Recently, Pan et al. [3] have observed the FQH effect in a spin-polarized electron gas at a series of filling factors between two neighboring Jain states, i.e. at 1/3 < νe < 2/5, and thus corresponding to the fractional CF filling factors, 1 < νCF < 2. In particular, the incompressible states observed at νe = 4/11, 3/8, 5/13 correspond to νCF = 4/3, 3/2, and 5/3, respectively. Clearly, incompressibility of these states must depend on the CF–CF interaction within a partially filled second CF LL.

Phase diagram for quantum hall bilayers at nu=1.

We present a phase diagram for a double quantum well bilayer electron gas in the quantum Hall regime at a total filling factor nu=1, based on exact numerical calculations of the topological Chern number matrix and the (interlayer) superfluid density. We find three phases: a quantized Hall state with pseudospin superfluidity, a quantized Hall state with pseudospin "gauge-glass" order, and a decoupled composite Fermi liquid. Comparison with experiments provides a consistent explanation of the observed quantum Hall plateau, Hall drag plateau, and vanishing Hall drag resistance, as well as the zero-bias conductance peak effect, and suggests some interesting points to pursue experimentally.

Two-dimensional electron system in high magnetic fields: Wigner crystal versus composite-fermion liquid.

The two-dimensional system of electrons in a high magnetic field offers an opportunity to investigate a phase transition from a quantum liquid into a Wigner solid. Recent experiments have revealed an incipient composite fermion liquid in a parameter range where theory and many experiments had previously suggested the Wigner crystal phase, thus calling into question our current understanding. This Letter shows how very small quantitative corrections (<1%) in the energy due to the weak interaction between composite fermions can cause a fundamental change in the nature of the ground state, thus providing insight into the puzzling experimental results.

Quasiparticles in the 111 state and its compressible ancestors

We investigate the relationship of the spontaneously inter-layer coherent ``111''state of quantum Hall bilayers at total filling factor \nu=1 to ``mutual'' composite fermions, in which vortices in one layer are bound to electrons in the other. Pairing of the mutual composite fermions leads to the low-energy properties of the 111 state, as we explicitly demonstrate using field-theoretic techniques. Interpreting this relationship as a mechanism for inter-layer coherence leads naturally to two candidate states with non-quantized Hall conductance: the mutual composite Fermi liquid, and an inter-layer coherent charge e Wigner crystal. The experimental behavior of the interlayer tunneling conductance and resistivity tensors are discussed for these states.

Effective action of a compressible quantum Hall state edge: Application to tunneling

The electrodynamical response of the edge of a compressible quantum Hall system affects tunneling into the edge. Using composite Fermi liquid theory, we derive an effective action for the edge modes interacting with tunneling charge. This action generalizes the chiral Luttinger liquid theory of the quantum Hall edge to compressible systems in which transport is characterized by a finite Hall angle. In addition to the standard terms, the action contains a dissipative term. The tunneling exponent is calculated as a function of the filling fraction for several models, including screened and unscreened long-range Coulomb interaction, as well as a short-range interaction. We find that tunneling exponents are robust and to a large extent insensitive to the particular model. We discuss recent tunneling measurements in overgrown cleaved edge systems, and demonstrate that the profile of charge density near the edge is very sensitive to the parameters of the system. In general, the density is nonmonotonic, and can deviate from the bulk value by up to 30%. Implications for the correspondence to chiral Luttinger edge theories are discussed.

Possible composite-fermion liquid as a crossover from Wigner crystal to bubble phase in higher Landau levels

The ground state cohesive energies per electron of the composite fermion (CF) Fermi sea, the Laughlin state and the charge density wave (CDW) at higher Landau levels (LLs) are computed. It is shown that whereas for $n\geq 2$ LL, the CDW state is generally more energetically preferable than those of the CF liquid and the Laughlin liquid, the $\nu =4+1/6$ CF liquid state unexpectedly has lower ground state energy than that of the CDW state. We suggest this CF liquid between the Wigner crystal and the bubble phase may lead to the crossover from the normal integer quantum Hall liquid to the novel re-entrant integer quantum Hall state observed in the recent magneto-transport experiments.

Incompressible paired hall state, stripe order, and the composite fermion liquid phase in half-filled landau levels

We consider the two lowest Landau levels at half filling. In the higher Landau level (nu = 5/2), we find a first-order phase transition separating a compressible striped phase from a paired quantum Hall state, which is identified as the Moore-Read state. The critical point is very near the Coulomb potential and the transition can be driven by increasing the width of the electron layer. We find a much weaker transition (either second-order or a crossover) from pairing to the composite fermion Fermi-liquid behavior. A very similar picture is obtained for the lowest Landau level, but the transition point is not near the Coulomb potential.