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.
Due to its extremely rich phase diagram, the two-dimensional electron gas exposed to perpendicular magnetic field has been the subject of intense and sustained study. One particularly interesting problem in this system is that of the half-filled Landau level, where the Fermi sea of composite fermions, a fractional quantum Hall state arising from a pairing instability of the composite fermions, and the quantum Hall nematic were observed in the half-filled $N=0$, $N=1$, and $N \geq 2$ Landau levels, respectively. Thus different ground states developed in different half-filled Landau levels. This…
In spite of its ubiquity in strongly correlated systems, the competition of paired and nematic ground states remains poorly understood. Recently such a competition was reported in the two-dimensional electron gas at filling factor $ν=5/2$. At this filling factor a pressure-induced quantum phase transition was observed from the paired fractional quantum Hall state to the quantum Hall nematic. Here we show that the pressure induced paired-to-nematic transition also develops at $ν=7/2$, demonstrating therefore this transition in both spin branches of the second orbital Landau level. However, we f…
Half-filled Landau levels form a zoo of strongly correlated phases. These include non-Fermi liquids (NFL), fractional quantum Hall (FQH) states, nematic phases, and FQH nematic phases. This diversity poses the question: what keeps the balance between the seemingly unrelated phases? The answer is elusive because the Halperin-Lee-Read (HLR) description that offers a natural departure point is inherent strongly coupled. But the observed nematic phases suggest nematic fluctuations play an important role. To study this possibility, we apply a recently formulated controlled double expansion approach…
We present a theory of spontaneous Fermi surface deformations for half-filled Landau levels (filling factors of the form $ν=2n+1/2$). We assume the half-filled level to be in a compressible, Fermi liquid state with a circular Fermi surface. The Landau level projection is incorporated via a modified effective electron-electron interaction and the resulting band structure is described within the Hartree-Fock approximation. We regulate the infrared divergences in the theory and probe the intrinsic tendency of the Fermi surface to deform through Pomeranchuk instabilities. We find that the correspo…
We compare the energies of different electron solids, such as bubble crystals with triangular and square symmetry and stripe phases, to those of correlated quantum liquids in partially filled intermediate Landau levels. Multiple transitions between these phases when varying the filling of the top-most partially filled Landau level explain the observed reentrance of the integer quantum Hall effect. The phase transitions are identified as first-order. This leads to a variety of measurable phenomena such as the phase coexistence between a Wigner crystal and a two-electron bubble phase in a Landau…
The fractional quantum Hall state at Landau level (LL) filling factor 5/2 is extremely interesting because it is likely the first non-Abelian state, but its precise nature remains unclear after decades of study. We demonstrate this can be resolved by studying the chirality of its graviton excitations, using circularly polarized Raman scattering. We discuss the advantage of this bulk probe over the existing edge probes.
The quantum Hall phase diagram of the half-filled bilayer system in the second Landau level is studied as a function of tunneling and layer separation using exact diagonalization. We make the striking prediction that bilayer structures would manifest two distinct branches of incompressible fractional quantum Hall effect (FQHE) corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the quantum phase transition between them w…
There exists a widely believed opinion, that the many-body ground state of a two-dimensional electron system at a completely filled lowest Landau level (the filling factor $ν=1$) is described by the so-called Hartree-Fock wave function, and that this solution is the unique, exact eigenstate of the system at $ν=1$. I show that this opinion is erroneous, construct an infinite number of other variational many-body wave functions, and discuss the properties of a few states which have the energy substantially lower than the energy of the Hartree-Fock state.
An effective field theory of composite Dirac fermions was proposed by Son [Phys. Rev. X 5, 031027 (2015)] as a theory of the half-filled Landau level with explicit particle-hole symmetry. We compute the electromagnetic response of this Son-Dirac theory on the level of the random phase approximation (RPA), where we pay particular attention to the effect of an additional composite-fermion dipole term that is needed to restore Galilean invariance. We find that once this dipole correction is taken into account, spurious interband transitions and collective modes that are present in the response of…
We describe an experimental technique to measure the chemical potential, $μ$, in atomically thin layered materials with high sensitivity and in the static limit. We apply the technique to a high quality graphene monolayer to map out the evolution of $μ$ with carrier density throughout the N=0 and N=1 Landau levels at high magnetic field. By integrating $μ$ over filling factor, $ν$, we obtain the ground state energy per particle, which can be directly compared with numerical calculations. In the N=0 Landau level, our data show exceptional agreement with numerical calculations over the whole Lan…
We study the effects of disorder on the quantum Hall stripe phases in half-filled high Landau levels using exact numerical diagonalization. We show that, in the presence of weak disorder, a compressible, striped charge density wave, becomes the true ground state. The projected electron density profile resembles that of a smectic liquid. With increasing disorder strength W, we find that there exists a critical value, W_c \sim 0.12 e^2/εl, where a transition/crossover to an isotropic phase with strong local electron density fluctuations takes place. The many-body density of states are qualitativ…
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…
It is well known that there is a particle-hole symmetry for spin-polarized electrons with two-body interactions in a partially filled Landau level, which becomes exact in the limit where the cyclotron energy is large compared to the interaction strength, so one can ignore mixing between Landau levels. This symmetry is explicit in the description of a half-filled Landau level recently introduced by D. T. Son, using Dirac fermions, but it was thought to be absent in the older fermion-Chern- Simons approach, developed by Halperin, Lee, and Read and subsequent authors. We show here, however, that …
We examine the quantum phase diagram of the fractional quantum Hall effect (FQHE) in the lowest two Landau levels in half-filled bilayer structures as a function of tunneling strength and layer separation, i.e., we revisit the lowest Landau level filling factor 1/2 bilayer problem and make new predictions involving bilayers in the half-filled second Landau level (i.e., filling factor 5/2). Using numerical exact diagonalization we investigate the important question of whether this system supports a FQHE described by the non-Abelian Moore-Read Pfaffian state in the strong tunneling regime. In th…
We analyze the experimental observation of metastable anisotropy resistance orientation at half filled quantum Hall fluids by means of a model of a quantum nematic liquid in an explicit symmetry breaking potential. We interpret the observed ``rotation'' of the anisotropy axis as a process of nucleation of nematic domains and compute the nucleation rate within this model. By comparing with experiment, we are able to predict the critical radius of nematic bubbles, $R_c\sim 2.6 μm $. Each domain contains about $10^4$ electrons.
We report on numerical study of the Dirac fermions in partially filled N=3 Landau level (LL) in graphene. At half-filling, the equal-time density-density correlation function displays sharp peaks at nonzero wavevectors $\pm {\bf q^{*}}$. Finite-size scaling shows that the peak value grows with electron number and diverges in the thermodynamic limit, which suggests an instability toward a charge density wave. A symmetry broken stripe phase is formed at large system size limit, which is robust against purturbation from disorder scattering. Such a quantum phase is experimentally observable throug…
Coulomb drag between two parallel two dimensional electron gases has been measured at high magnetic fields in samples with two different layer spacings. As the layer filling factor deviates from nu=1/2, we find that the magnitude of the drag is enhanced quadratically with Delta nu = nu - 1/2, and the curvature of the enhancement is insensitive to both the sign of Delta nu and the spacing between the layers. Our results suggest that the enhancement is not due to non-perturbative interlayer correlations.
We investigate theoretically the fractional quantum Hall effect at half-filling in the lowest Landau level observed in asymmetric wide quantum wells. The asymmetry can be achieved by a potential bias applied between the two sides of the well. Within exact-diagonalization calculations in the spherical geometry, we find that the incompressible state is described in terms of a two-component wave function. Its overlap with the ground state can be optimized with the help of a rotation in the space of the pseudospin, which mimics the lowest two electronic subbands.
We examine the quantum phase diagram of the fractional quantum Hall effect in the lowest Landau level in half-filled bilayer structures as a function of tunneling strength and layer separation. Using numerical exact diagonalization we investigate the important question of whether this system supports a fractional quantum Hall effect described by the non-Abelian Moore-Read Pfaffian state in the strong tunneling regime. We find that, although it is in principle possible, it is unlikely that the non-Abelian FQHE exists in the lowest Landau level. We establish that all so far observed FQHE states …
Half-filled Landau levels host an emergent Fermi-liquid which displays an instability towards pairing, culminating in a gapped even-denominator fractional quantum Hall ground state. While this pairing may be probed by tuning the polarization of carriers in competing orbital and spin degrees of freedom, sufficiently high quality platforms offering such tunability remain few. Here we explore the ground states at filling factor $ν$ = 5/2 in ZnO-based two-dimensional electron systems through a forced intersection of opposing spin branches of Landau levels taking quantum numbers $N$ = 1 and 0. We r…
We report on exact-diagonalization studies of correlated many-electron states in the half-filled Landau levels of graphene, including pseudospin (valley) degeneracy. We demonstrate that the polarized Fermi sea of non-interacting composite fermions remains stable against a pairing transition in the lowest two Landau levels. However, it undergoes spontaneous depolarization, which is unprotected owing to the lack of single-particle pseudospin splitting. These results suggest the absence of the Pfaffian phase in graphene.
We investigate broken rotational symmetry (BRS) states for the fractional quantum Hall effect (FQHE) at 1/3-filling of the valence Landau level (LL). Recent Monte Carlo calculations by Musaelian and Joynt [J. Phys.: Condens.\ Matter {\bf 8}, L105 (1996)] suggest that Laughlin's state becomes unstable to a BRS state for some critical finite thickness value. We study in detail the properties of such state by performing a hypernetted-chain calculation that gives results in the thermodynamic limit, complementing other methods which are limited to a finite number of particles. Our results indicate …
We have measured the diagonal conductivity in the microwave regime of an ultrahigh mobility two dimensional electron system. We find a sharp resonance in Re[sigma_{xx}] versus frequency when nu > 4 and the partial filling of the highest Landau level, nu^*, is ~ 1/4 or 3/4 and temperatures < 0.1 K. The resonance appears for a range of nu^* from 0.20 to 0.37 and again from 0.62 to 0.82. the peak frequency, f_{pk} changes from ~ 500 to ~ 150 as nu^* = 1/2 is approached. This range of f_{pk} shows no dependence on nu where the resonance is observed. The quality factor, Q, of the resonance is maxim…
The transresistance (i.e. the voltage induced in one layer by a current in another) between composite fermions in double-layers of half-filled Landau levels is shown to be dominated by scattering due to singular gauge field fluctuations arising from the antisymmetric combination of density fluctuations in two layers. The drag rate is found to be proportional to $T^{4/3}$ for $T$ less than an energy scale $Ω_{cr}$ which depends on the spacing, $d$, between the layers. An observation of a $T^{4/3}$ temperature dependence in the transresistance would be a strong evidence for the existence of the …
We investigate broken rotational symmetry (BRS) states at half-filling of the valence Landau level (LL). We generalize Rezayi-Read's (RR) composite fermion (CF) trial wavefunctions to include anisotropic coupling of the flux quanta to electrons, thus generating a nematic order in the underlying CF liquid. Using the Fermi hypernetted-chain (FHNC) method which readily gives results in the thermodynamic limit, we determine in detail the properties of these states. By using the anisotropic pair distribution and static structure functions we determine the correlation energy and find that, as expect…
We analyze a two-dimensional Pomeranchuk-Nematic instability, trigger by the Landau parameter $F_2<0$, in the presence of a small magnetic field. Using Landau Fermi liquid theory in the isotropic phase, we analyze the collective modes near the quantum critical point $F_2=-1,ω_c=0$ (where $ω_c$ is the cyclotron frequency). We focus on the effects of parity symmetry breaking on the Fermi surface deformation. We show that, for studying the critical regime, the linear response approximation of the Landau-Silin equation is not sufficient and it is necessary to compute corrections at least of order …
Nematicity in quantum Hall systems has been experimentally well established at excited Landau levels. The mechanism of the symmetry breaking, however, is still unknown. Pomeranchuk instability of Fermi liquid parameter $F_{\ell} \le -1$ in the angular momentum $\ell=2$ channel has been argued to be the relevant mechanism, yet there are no definitive theoretical proofs. Here we calculate, using the variational Monte Carlo technique, Fermi liquid parameters $F_\ell$ of the composite fermion Fermi liquid with a finite layer width. We consider $F_{\ell}$ in different Landau levels $n=0,1,2$ as a f…
We discuss monolayer and bilayer quantum Hall systems in which each layer is a half-filled Landau level (LL) system. In the mean field approximation of the Son's formalism there is a common pairing structure that underlines the possibilities for paired ground states in both systems. We argue that the particle-hole (PH) Pfaffian state in the (particle-hole symmetric) half-filled LL of a monolayer, and analogous state in the PH symmetric bilayer (in which each layer is half-filled LL) can be considered as {\em critical states} i.e. states that cannot describe a phase under PH symmetry. We point …
We consider a non-Abelian candidate state at filling factor $ν=3/7$ state belonging to the parton family. We find that, in the second Landau level of GaAs (i.e. at filling factor $ν=2+3/7$), this state is energetically superior to the standard Jain composite-fermion state and also provides a very good representation of the ground state found in exact diagonalization studies of finite systems. This leads us to predict that \emph{if} a fractional quantum Hall effect is observed at $ν=3/7$ in the second Landau level, it is likely to be described by this new non-Abelian state. We enumerate experim…
A recent proposal argues that an alternate description of the half-filled Landau level is a theory of massless Dirac fermions. We examine the possibility of pairing of these Dirac fermions by numerically solving the {\em coupled} Eliashberg equations unlike our previous calculation [Wang and Chakravarty, Phys. Rev. B {\bf 94}, 165138 (2016)]. In addition, vertex corrections are calculated to be zero from the Ward identity. We find that pairing is possible in non-zero angular momentum channels; the only differences are minor numerical shifts. As before, the pairing leads to the gapped Pfaffian …
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 dyn…
Recent magneto-transport experiments have provided compelling evidence for the presence of an energetically isolated threefold Q-valley degeneracy in few-layer transition metal dichalcogenides. We study the flavor SU(3) symmetry breaking when each Landau level triplet is one-third filled or empty and predict that a pure flavor nematic phase and a flavorless charge-density-wave phase will occur respectively below and above a critical magnetic field. Surprisingly, electrons carry flavor-dependent electric dipole moments even at zero magnetic field, rendering the nematics ferroelectric, allowing …
We provide an effective description of a particle-hole symmetric state of electrons in a half-filled Landau level, starting from the traditional approach pioneered by Halperin, Lee and Read. Specifically, we study 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. When the CFL and CHL strips are identical in size, the resulting state is manifestly invariant under the combined action of a particle-hole transformation with respect to a single Landau level (which interchanges …
We study the interplay of particle-hole symmetry and fermion-vortex duality in multicomponent half-filled Landau levels, such as quantum Hall gallium arsenide bilayers and graphene. For the $ν{=}1/2{+}1/2$ bilayer, we show that particle-hole-symmetric interlayer Cooper pairing of composite fermions leads to precisely the same phase as the electron exciton condensate realized in experiments. This equivalence is easily understood by applying the recent Dirac fermion formulation of $ν{=}1/2$ to two components. It can also be described by Halperin-Lee-Read composite fermions undergoing interlayer …
We construct model wavefunctions for the half-filled Landau level parameterized by "composite fermion occupation-number configurations" in a two-dimensional momentum space which correspond to a Fermi sea with particle-hole excitations. When these correspond to a weakly-excited Fermi sea, they have large overlap with wavefunctions obtained by exact diagonalization of lowest-Landau-level electrons interacting with a Coulomb interaction, allowing exact states to be identified with quasiparticle configurations. We then formulate a many-body version of the single-particle Berry phase for adiabatic …
We study theoretically the magnetoresistance oscillations near a half-filled lowest Landau level ($ν= 1/2$) that result from the presence of a periodic one-dimensional electrostatic potential. We use the Dirac composite fermion theory of Son [Phys. Rev. X 5 031027 (2015)], where the $ν=1/2$ state is described by a $(2+1)$-dimensional theory of quantum electrodynamics. We extend previous work that studied these oscillations in the mean-field limit by considering the effects of gauge field fluctuations within a large flavor approximation. A self-consistent analysis of the resulting Schwinger--Dy…
We present a Landau-Ginzburg theory for a fractional quantized Hall nematic state and the transition to it from an isotropic fractional quantum Hall state. This justifies Lifshitz-Chern-Simons theory -- which is shown to be its dual -- on a more microscopic basis and enables us to compute a ground state wave function in the symmetry-broken phase. In such a state of matter, the Hall resistance remains quantized while the longitudinal DC resistivity due to thermally-excited quasiparticles is anisotropic. We interpret recent experiments at Landau level filling factor ν=7/3 in terms of our theory.
We develop a vortex metal theory for partial filled Landau Level at $ν=\frac{1}{2n}$, whose ground state contains a composite Fermi surface(FS) formed by the vortex of electrons. In the projected Landau Level limit, the composite Fermi surface contains $\frac{-π}{n}$ Berry phase. Such fractional Berry phase is a consequence of LL projection which produces the GMP guiding center algebra and embellishes an anomalous velocity to the equation of motion for the vortex metal. Further, we investigate a particle-hole symmetric bilayer system with $ν_1=\frac{1}{2n}$ and $ν_2=1-\frac{1}{2n}$ at each lay…
We study theoretically the destruction of spin nematic order due to quantum fluctuations in quasi-one dimensional spin-1 magnets. If the nematic ordering is disordered by condensing disclinations then quantum Berry phase effects induce dimerization in the resulting paramagnet. We develop a theory for a Landau-forbidden second order transition between the spin nematic and dimerized states found in recent numerical calculations. Numerical tests of the theory are suggested.
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 $π$ Berry phase. In this paper, we start from the bilayer Half-filled Landau Level system where the two composite Fermi su…
Pairing of composite fermions in half-filled Landau level state is reexamined by solving the BCS gap equation with full frequency dependent current-current interactions. Our results show that there can be a \emph{continuous} transition from the Halperin-Lee-Read state to a chiral odd angular momentum Cooper pair state for short-range contact interaction. This is at odds with the previously established conclusion of first order pairing transition, in which the low frequency effective interaction was assumed for the entire frequency range. We find that even if the low frequency effective interac…
Using a spin-charge separation of the gluon field in the Landau gauge we show that the SU(2) Yang-Mills theory in the low-temperature phase can be considered as a nematic liquid crystal. The ground state of the nematic crystal is characterized by the A^2 condensate of the gluon field. The liquid crystal possesses various topological defects (instantons, monopoles and vortices) which are suggested to play a role in non-perturbative features of the theory.
