Composite Fermions Research Articles

Particle–hole symmetries in condensed matter

The term “particle–hole symmetry” is beset with conflicting meanings in contemporary physics. Conceived and written from a condensed-matter standpoint, the present paper aims to clarify and sharpen the terminology. In that vein, we propose to define the operation of “particle–hole conjugation” as the tautological algebra automorphism that simply swaps single-fermion creation and annihilation operators, and we construct its invariant lift to the Fock space. Particle–hole symmetries then arise for gapful or gapless free-fermion systems at half filling, as the concatenation of particle–hole conjugation with one or another involution that reverses the sign of the first-quantized Hamiltonian. We illustrate that construction principle with a series of examples including the Su–Schrieffer–Heeger model and the Kitaev–Majorana chain. For an enhanced perspective, we contrast particle–hole symmetries with the charge-conjugation symmetry of relativistic Dirac fermions. We go on to present two major applications in the realm of interacting electrons. For one, we offer a heuristic argument that the celebrated Haldane phase of antiferromagnetic quantum spin chains is adiabatically connected to a free-fermion topological phase protected by a particle–hole symmetry. For another, we review the recent proposal by Son [Phys. Rev. X 5, 031027 (2015)] for a particle–hole conjugation symmetric effective field theory of the half-filled lowest Landau level, and we comment on the emerging microscopic picture of the composite fermion.

Elementary excitations in fractional quantum Hall effect from classical constraints

Classical constraints on the reduced density matrix of quantum fluids in a single Landau level termed as local exclusion conditions (LECs) (Yang 2019 Phys. Rev. B 100 241302), have recently been shown to characterize the ground state of many fractional quantum Hall (FQH) phases. In this work, we extend the LEC construction to build the elementary excitations, namely quasiholes and quasielectrons, of these FQH phases. In particular, we elucidate the quasihole counting, categorize various types of quasielectrons, and construct their microscopic wave functions. Our extensive numerical calculations indicate that the undressed quasielectron excitations of the Laughlin state obtained from LECs are topologically equivalent to those obtained from the composite fermion theory. Intriguingly, the LEC construction unveils interesting connections between different FQH phases and offers a novel perspective on exotic states such as the Gaffnian and the Fibonacci state.

Non-Abelian fermionization and the landscape of quantum Hall phases

The recent proposal of non-Abelian boson-fermion dualities in 2+1 dimensions, which morally relate $U(k)_N$ to $SU(N)_{-k}$ Chern-Simons-matter theories, presents a new platform for exploring the landscape of non-Abelian quantum Hall states accessible from theories of Abelian composite particles. Here we focus on dualities relating theories of Abelian quantum Hall states of bosons or fermions to theories of non-Abelian "composite fermions" partially filling Landau levels. We show that these dualities predict special filling fractions where both Abelian and non-Abelian composite fermion theories appear capable of hosting distinct topologically ordered ground states, one Abelian and the other a non-Abelian, $U(k)_2$ Blok-Wen state. Rather than being in conflict with the duality, we argue that these results indicate unexpected dynamics in which the infrared and lowest Landau level limits fail to commute across the duality. In such a scenario, the non-Abelian topological order can be destabilized in favor of the Abelian ground state, suggesting the presence of a phase transition between the Abelian and non-Abelian states that is likely to be first order. We also generalize these constructions to other non-Abelian fermion-fermion dualities, in the process obtaining new derivations of a variety of paired composite fermion phases using duality, including the anti-Pfaffian state. Finally, we describe how, in multilayer constructions, excitonic pairing of the composite fermions across $N$ layers can also generate the family of Blok-Wen states with $U(k)_2$ topological order.

Noncommutative field theory and composite Fermi liquids in some quantum Hall systems

Composite Fermi liquid metals arise at certain special filling fractions in the quantum Hall regime and play an important role as parent states of gapped states with quantized Hall response. They have been successfully described by the Halperin-Lee-Read (HLR) theory of a Fermi surface of composite fermions coupled to a $U(1)$ gauge field with a Chern-Simons term. However, the validity of the HLR description when the microscopic system is restricted to a single Landau has not been clear. Here for the specific case of bosons at filling $\nu = 1$, we build on earlier work from the 1990s to formulate a low energy description that takes the form of a {\em non-commutative} field theory. This theory has a Fermi surface of composite fermions coupled to a $U(1)$ gauge field with no Chern-Simons term but with the feature that all fields are defined in a non-commutative spacetime. An approximate mapping of the long wavelength, small amplitude gauge fluctuations yields a commutative effective field theory which, remarkably, takes the HLR form but with microscopic parameters correctly determined by the interaction strength. Extensions to some other composite fermi liquids, and to other related states of matter are discussed.

Radial Spin Texture of the Weyl Fermions in Chiral Tellurium.

Trigonal tellurium, a small-gap semiconductor with pronounced magneto-electric and magneto-optical responses, is among the simplest realizations of a chiral crystal. We have studied by spin- and angle-resolved photoelectron spectroscopy its unconventional electronic structure and unique spin texture. We identify Kramers-Weyl, composite, and accordionlike Weyl fermions, so far only predicted by theory, and show that the spin polarization is parallel to the wave vector along the lines in k space connecting high-symmetry points. Our results clarify the symmetries that enforce such spin texture in a chiral crystal, thus bringing new insight in the formation of a spin vectorial field more complex than the previously proposed hedgehog configuration. Our findings thus pave the way to a classification scheme for these exotic spin textures and their search in chiral crystals.

Quantum impurity model for anyons

One of the hallmarks of quantum statistics, tightly entwined with the concept of topological phases of matter, is the prediction of anyons. Although anyons are predicted to be realized in certain fractional quantum Hall systems, they have not yet been unambiguously detected in experiment. Here we introduce a simple quantum impurity model, where bosonic or fermionic impurities turn into anyons as a consequence of their interaction with the surrounding many-particle bath. A cloud of phonons dresses each impurity in such a way that it effectively attaches fluxes/vortices to it and thereby converts it into an Abelian anyon. The corresponding quantum impurity model, first, provides a new approach to the numerical solution of the many-anyon problem, along with a new concrete perspective of anyons as emergent quasiparticles built from composite bosons or fermions. More importantly, the model paves the way towards realizing anyons using impurities in crystal lattices as well as ultracold gases. In particular, we consider two heavy electrons interacting with a two-dimensional lattice crystal in a magnetic field, and show that when the impurity-bath system is rotated at the cyclotron frequency, impurities behave as anyons as a consequence of the angular momentum exchange between the impurities and the bath. A possible experimental realization is proposed by identifying the statistics parameter in terms of the mean square distance of the impurities and the magnetization of the impurity-bath system, both of which are accessible to experiment. Another proposed application are impurities immersed in a two-dimensional weakly interacting Bose gas.

Hall viscosity of the composite-fermion Fermi seas for fermions and bosons

The Hall viscosity has been proposed as a topological property of incompressible fractional quantum Hall states and can be evaluated as Berry curvature. This paper reports on the Hall viscosities of composite-fermion Fermi seas at $\nu=1/m$, where $m$ is even for fermions and odd for bosons. A well-defined value for the Hall viscosity is not obtained by viewing the $1/m$ composite-fermion Fermi seas as the $n\rightarrow \infty$ limit of the Jain $\nu=n/(nm\pm 1)$ states, whose Hall viscosities $(\pm n+m)\hbar \rho/4$ ($\rho$ is the two-dimensional density) approach $\pm \infty$ in the limit $n\rightarrow \infty$. A direct calculation shows that the Hall viscosities of the composite-fermion Fermi sea states are finite, and also relatively stable with system size variation, although they are not topologically quantized in the entire $\tau$ space. I find that the $\nu=1/2$ composite-fermion Fermi sea wave function for a square torus yields a Hall viscosity that is expected from particle-hole symmetry and is also consistent with the orbital spin of $1/2$ for Dirac composite fermions. I compare my numerical results with some theoretical conjectures.

Hall resistance anomalies in the integer and fractional quantum Hall regime

Experimental evidence of resistance anomalies in the high-mobility two-dimensional electron gas (2DEG) formed in the GaAs/AlGaAs heterostructure, in the integer and fractional quantized Hall regime, is shown. The data complement to a good approximation the semianalytic calculations used to describe the formation of integral and fractional incompressible strips. The widths of current-carrying channels were calculated by incorporating the screening properties of the 2DEG and the effect of a magnetic field in the perpendicular mode. The many-body effects of the composite fermions are taken into consideration for the energy gap for the fractional states. It is shown that incompressible strips at the edges for both integer and fractional filling factors coexist in their evanescent phase for a particular range of magnetic fields, resulting in overshoot effects at the Hall resistance. Specifically, anomalous Hall resistances were noticed for filling factors $\ensuremath{\nu}$ = $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\frac{8}{3}$, 3, $\frac{10}{3}$, $\frac{7}{2}$, and 5. This effect is explained and discussed using the screening theory.

Construction of a series of new ν=2/5 fractional quantum Hall wave functions by conformal field theory

In this paper, a series of $\nu=2/5$ fractional quantum Hall wave functions are constructed from conformal field theory(CFT). They share the same topological properties with states constructed by Jain's composite fermion approach. Upon exact lowest Landau level(LLL) projection, some of Jain composite fermion states would not survive if constraints on Landau level indices given in the appendices of this paper were not satisfied. By contrast, states constructed from CFT always stay in LLL. These states are characterized by different topological shifts and multibody relative angular momenta. As a by-product, in the appendices we prove the necessary conditions for general $ \nu=p/(2p+1) $ composite fermion states to have nonvanishing LLL projection.

Dynamics from symmetries in chiral SU(N) gauge theories

The symmetries and dynamics of simple chiral SU(N) gauge theories, with matter Weyl fermions in a two-index symmetric tensor and N + 4 anti-fundamental representations, are examined, by taking advantage of the recent developments involving the ideas of generalized symmetries, gauging of discrete center 1-form symmetries and mixed ’t Hooft anomalies. This class of models are particularly interesting because the conventional ’t Hooft anomaly matching constraints allow a chirally symmetric confining vacuum, with no condensates breaking the U(1) × SU(N + 4) flavor symmetry, and with certain set of massless baryonlike composite fermions saturating all the associated anomaly triangles. Our calculations show that in such a vacuum the UV-IR matching of some 0-form−1-form mixed ’t Hooft anomalies fails. This implies, for the theories with even N at least, that a chirally symmetric confining vacuum contemplated earlier in the literature actually cannot be realized dynamically. In contrast, a Higgs phase characterized by some gauge-noninvariant bifermion condensates passes our improved scrutiny.

Berry phase in the composite Fermi liquid

We derive the definition of the Berry phase for the adiabatic transport of a composite fermion (CF) in a half-filled composite Fermi-liquid (CFL). It is found to be different from that adopted in previous investigations by Geraedts et al. For the standard CFL wave function, we analytically show that the Berry curvature is uniformly distributed in the momentum space. For the Jain-Kamilla wave function, we numerically show that its Berry curvature has a continuous distribution inside the Fermi sea and vanishes outside. We conclude that the CF with respect to both the microscopic wave-functions is not a massless Dirac particle.

Bloch ferromagnetism of composite fermions

In 1929 Felix Bloch suggested that the paramagnetic Fermi sea of electrons should make a spontaneous transition to a fully-magnetized state at very low densities, because the exchange energy gained by aligning the spins exceeds the enhancement in the kinetic energy. We report here the observation of an abrupt, interaction-driven transition to full magnetization, highly reminiscent of Bloch ferromagnetism that has eluded experiments for the last ninety years. Our platform is the exotic two-dimensional Fermi sea of composite fermions at half-filling of the lowest Landau level. Via quantitative measurements of the Fermi wavevector, which provides a direct measure of the spin polarization, we observe a sudden transition from a partially-spin-polarized to a fully-spin-polarized ground state as we lower the composite fermions' density. Our detailed theoretical calculations provide a semi-quantitative account of this phenomenon.

Fractional quantum Hall effect at ν=2+4/9

Motivated by two independent experiments revealing a resistance minimum at the Landau level (LL) filling factor $\nu=2+4/9$, characteristic of the fractional quantum Hall effect (FQHE) and suggesting electron condensation into a yet unknown quantum liquid, we propose that this state likely belongs in a parton sequence, put forth recently to understand the emergence of FQHE at $\nu=2+6/13$. While the $\nu=2+4/9$ state proposed here directly follows three simpler parton states, all known to occur in the second LL, it is topologically distinct from the Jain composite fermion (CF) state which occurs at the same $\nu=4/9$ filling of the lowest LL. We predict experimentally measurable properties of the $4/9$ parton state that can reveal its underlying topological structure and definitively distinguish it from the $4/9$ Jain CF state.

Collective excitations of quantum Hall states under tilted magnetic field

We study the neutral excitations of fractional quantum Hall states in electronic systems of finite width where an external anisotropy is introduced by tilting the magnetic field. As in the isotropic case, the neutral collective excitation can be worked out through the conserving method of composite fermions in the Hamiltonian theory because the interaction potential has a natural cutoff due to the quantum well width. We show how such a computation can be carried out perturbatively for an anisotropic interaction. We find that unlike the charge gap, the neutral collective gap is much more sensitive to the tilt and can thus, for certain fractional quantum Hall states, be easily destroyed by the parallel component of the magnetic field. We also discuss the convergence of the collective spectrum to the activation gap in the large-momentum limit.

Precise Experimental Test of the Luttinger Theorem and Particle-Hole Symmetry for a Strongly Correlated Fermionic System.

A fundamental concept in physics is the Fermi surface, the constant-energy surface in momentum space encompassing all the occupied quantum states at absolute zero temperature. In 1960, Luttinger postulated that the area enclosed by the Fermi surface should remain unaffected even when electron-electron interaction is turned on, so long as the interaction does not cause a phase transition. Understanding what determines the Fermi surface size is a crucial and yet unsolved problem in strongly interacting systems such as high-T_{c} superconductors. Here we present a precise test of the Luttinger theorem for a two-dimensional Fermi liquid system where the exotic quasiparticles themselves emerge from the strong interaction, namely, for the Fermi sea of composite fermions (CFs). Via direct, geometric resonance measurements of the CFs' Fermi wave vector down to very low electron densities, we show that the Luttinger theorem is obeyed over a significant range of interaction strengths, in the sense that the Fermi sea area is determined by the density of the minority carriers in the lowest Landau level. Our data also address the ongoing debates on whether or not CFs obey particle-hole symmetry, and if they are Dirac particles. We find that particle-hole symmetry is obeyed, but the measured Fermi sea area differs quantitatively from that predicted by the Dirac model for CFs.

Helical spin liquid in a triangular XXZ magnet from Chern-Simons theory

We propose a finite-temperature phase diagram for the 2D spin-$1/2$ $J_1-J_2$ $XXZ$ antiferromagnet on the triangular lattice. Our analysis, based on a composite fermion representation, yields several phases. This includes a zero-temperature helical spin liquid with $N=6$ {\it anisotropic} Dirac cones, and with nonzero vector chirality implying a broken $\mathbb{Z}_2$ symmetry. It is terminated at $T=0$ by a continuous quantum phase transition to $120^\circ$ ordered state around $J_2/J_1\approx0.089$ in the XX limit; these phases share a double degeneracy, which persists to finite $T$ above the helical spin liquid. By contrast, at $J_2/J_1 \simeq 0.116$, the transition into a stripe phase appears as first order. We further discuss experimental and numerical consequences of the helical order and the anisotropic nature of the Dirac dispersion.

Unification for darkly charged dark matter

We provide a simple UV theory for a Dirac dark matter with a massless Abelian gauge boson. We introduce a single fermion transforming as the $\bf{16}$ representation in the SO(10)$'$ gauge group, which is assumed to be spontaneously broken to SU(5)$'\times$U(1)$'$. The SU(5)$'$ gauge interaction becomes strong at an intermediate scale and then we obtain a light composite Dirac fermion with U(1)$'$ gauge interaction at the low-energy scale. Its thermal relic can explain the observed amount of dark matter consistently with other cosmological and astrophysical constraints. We discuss that a nonzero kinetic mixing between the U(1)$'$ gauge boson and the Hypercharge gauge boson is allowed and the temperature of the visible sector and the dark matter sector can be equal to each other.

Pfaffian paired states for half-integer fractional quantum Hall effect

In this review, the physics of Pfaffian paired states, in the context of fractional quantum Hall effect, is discussed using field-theoretical approaches. The Pfaffian states are prime examples of topological ([Formula: see text]-wave) Cooper pairing and are characterized by non-Abelian statistics of their quasiparticles. Here we focus on conditions for their realization and competition among them at half-integer filling factors. Using the Dirac composite fermion description, in the presence of a mass term, we study the influence of Landau level mixing in selecting a particular Pfaffian state. While Pfaffian and anti-Pfaffian are selected when Landau level mixing is not strong, and can be taken into account perturbatively, the particle–hole (PH) Pfaffian state requires non-perturbative inclusion of at least two Landau levels. Our findings, for small Landau level mixing, are in accordance with numerical investigations in the literature, and call for a non-perturbative approach in the search for PH Pfaffian correlations. We demonstrated that a method based on the Chern–Simons field-theoretical approach can be used to generate characteristic interaction pseudo-potentials for Pfaffian paired states.

Effective p-wave Fermi-Fermi Interaction Induced by Bosonic Superfluids

We study the two-dimensional Bose-Fermi mixture on square lattice at finite temperature by using the determinant quantum Monte Carlo method within the weakly interacting regime. Here we consider the attractive Bose-Hubbard model and free spinless fermions. In the absence of boson-fermion interactions, we obtain the boundary of the collapsed state of the attractive bosons. In the presence of boson-fermion interactions, an effective p-wave interaction between fermions will be induced as far as the bosons are in a superfluid state. Moreover, we find the emergence of the composite fermion pairs at low temperatures.

Unconventional properties of high-temperature superconducting cuprates

In spite of much interest in various unconventional properties of superconducting cuprates such as the anisotropic superconductivity, the pseudogap state, the competing state in the underdoped region, strange metal in normal state, and anomalies in the optical sum rules etc, its microscopic mechanism still remains unsolved issues. Here these properties are considered using our recently proposed theory emphasizing that the electronic state of superconductors can be described by doping-dependent composite fermions. It is found that the anisotropy of the superconductive gap and the pseudogap are evaluated, the competing states is derived by mixed states model, and that T-linearity of resistivity and the anomaly of optical sum rules in the nearly optimal doping can be derived from considering the interplay between the composite fermion bands. It is also suggested that a crossover separating the strange metal and the normal metal in the overdoped region can be explained by the coupling effect of composite fermions