Composite Fermion States Research Articles

Quantum mechanics of composite fermions

We establish the quantum mechanics of composite fermions based on the dipole picture initially proposed by Read. It comprises three complimentary components: a wave equation for determining the wave functions of a composite fermion in ideal fractional quantum Hall states and when subjected to external perturbations, a wave-function for mapping a many-body wave function of composite fermions to a physical wave function of electrons, and a microscopic approach for determining the effective Hamiltonian of the composite fermion. The wave equation resembles the ordinary Schrödinger equation but has drift velocity corrections that are not present in the Halperin-Lee-Read theory. The wave-function constructs a physical wave function of electrons by projecting a state of composite fermions onto a half-filled bosonic Laughlin state of vortices. Remarkably, Jain's wave-function can be reinterpreted as the new in an alternative wave-function representation of composite fermions. The dipole picture and the effective Hamiltonian can be derived from the microscopic model of interacting electrons confined in a Landau level, with all parameters determined. In this framework, we can construct the physical wave function of a fractional quantum Hall state deductively by solving the wave equation and applying the wave-function , based on the effective Hamiltonian derived from first principles, rather than relying on intuition or educated guesses. For ideal fractional quantum Hall states in the lowest Landau level, the approach reproduces the well-established results of the standard theory of composite fermions. We further demonstrate that the reformulated theory of composite fermions can be easily generalized for flat Chern bands. Published by the American Physical Society 2024

Conformal field theory approach to parton fractional quantum Hall trial wave functions

We show that all lowest Landau-level projected and unprojected chiral parton type fractional quantum Hall ground and edge-state trial wave functions, which take the form of products of integer quantum Hall wave functions, can be expressed as conformal field theory (CFT) correlation functions, where we can associate a chiral algebra to each parton state such that the CFT defined by the algebra is the “smallest” such CFT that can generate the corresponding ground and edge-state trial wave functions (assuming that the corresponding chiral algebra does indeed define a physically “sensible” CFT). A field-theoretic generalization of Laughlin's plasma analogy, known as generalized screening, is formulated for these states. If this holds, along with an additional assumption, we argue that the inner products of edge-state trial wave functions, for parton states where the “densest” trial wave function is unique, can be expressed as matrix elements of an exponentiated local action operator of the CFT, generalizing the result of Dubail [], which implies the equality between edge-state and entanglement level counting to state counting in the corresponding CFT. We numerically test this result in the case of the unprojected ν=2/5 composite fermion state and the bosonic ν=1ϕ22 parton state. We discuss how Read's arguments [] still apply, implying that conformal blocks of the CFT defined by the corresponding chiral algebra are valid quasihole trial wave functions, with the adiabatic braiding statistics given by the monodromy of these functions, assuming the existence of a quasiparticle trapping Hamiltonian. Generalizations of these constructions are discussed, with particular attention given to simple current constructions. It is shown that all chiral composite fermion wave functions can be expressed as CFT correlation functions without explicit symmetrization or antisymmetrization and that the ground, edge, and certain quasihole trial wave functions of the ϕnm parton states can be expressed as the conformal blocks of the U(1)⊗SU(n)m WZW models. Finally, we discuss the relation of the ϕ2k series with the Read-Rezayi series, where several examples of quasihole braiding statistics calculations are given. Published by the American Physical Society 2024

Next-generation even-denominator fractional quantum Hall states of interacting composite fermions.

The discovery of the fractional quantum Hall state (FQHS) in 1982 ushered a new era of research in many-body condensed matter physics. Among the numerous FQHSs, those observed at even-denominator Landau level filling factors are of particular interest as they may host quasiparticles obeying non-Abelian statistics and be of potential use in topological quantum computing. The even-denominator FQHSs, however, are scarce and have been observed predominantly in low-disorder two-dimensional (2D) systems when an excited electron Landau level is half filled. An example is the well-studied FQHS at filling factor [Formula: see text] 5/2 which is believed to be a Bardeen-Cooper-Schrieffer-type, paired state of flux-particle composite fermions (CFs). Here, we report the observation of even-denominator FQHSs at [Formula: see text] 3/10, 3/8, and 3/4 in the lowest Landau level of an ultrahigh-quality GaAs 2D hole system, evinced by deep minima in longitudinal resistance and developing quantized Hall plateaus. Quite remarkably, these states can be interpreted as even-denominator FQHSs of CFs, emerging from pairing of higher-order CFs when a CF Landau level, rather than an electron or a hole Landau level, is half-filled. Our results affirm enhanced interaction between CFs in a hole system with significant Landau level mixing and, more generally, the pairing of CFs as a valid mechanism for even-denominator FQHSs, and suggest the realization of FQHSs with non-Abelian anyons.

Parametrization and thermodynamic scaling of pair correlation functions for the fractional quantum Hall effect

The calculation of pair correlations and density profiles of quasiholes are routine steps in the study of proposed fractional quantum Hall states. Nevertheless, the field has not adopted a standard way to present the results of such calculations in an easily reproducible form. We develop a polynomial expansion that allows for easy quantitative comparison between different candidate wavefunctions, as well as reliable scaling of correlation and quasihole profiles to the thermodynamic limit. We start from the well-known expansion introduced by Girvin [PRB, 30 (1984)] (see also [Girvin, MacDonald and Platzman, PRB, 33 (1986)]), which is physically appealing but, as we demonstrate, numerically unstable. We orthogonalize their basis set to obtain a new basis of modified Jacobi polynomials, whose coefficients can be stably calculated. We then apply our expansion to extract pair correlation expansion coefficients and quasihole profiles in the thermodynamic limit for a wide range of fractional quantum Hall wavefunctions. These include the Laughlin series, composite fermion states with both reverse and direct flux attachment, the Moore-Read Pfaffian state, and BS hierarchy states. The expansion procedure works for both abelian and non-abelian quasiholes, even when the density at the core is not zero. We find that the expansion coefficients for all quantum Hall states considered can be fit remarkably well using a cosine oscillation with exponentially decaying amplitude. The frequency and the decay length are related in an intuitive, but not elementary way to the filling fraction. Different states at the same filling fraction can have distinct values for these parameters. Finally, we also use our scaled correlation functions to calculate estimates for the magneto-roton gaps of the various states.

Sp(2N) Lattice Gauge Theories and Extensions of the Standard Model of Particle Physics

We review the current status of the long-term programme of numerical investigation of Sp(2N) gauge theories with and without fermionic matter content. We start by introducing the phenomenological as well as theoretical motivations for this research programme, which are related to composite Higgs models, models of partial top compositeness, dark matter models, and in general to the physics of strongly coupled theories and their approach to the large-N limit. We summarise the results of lattice studies conducted so far in the Sp(2N) Yang–Mills theories, measuring the string tension, the mass spectrum of glueballs and the topological susceptibility, and discuss their large-N extrapolation. We then focus our discussion on Sp(4), and summarise the numerical measurements of mass and decay constant of mesons in the theories with fermion matter in either the fundamental or the antisymmetric representation, first in the quenched approximation, and then with dynamical fermions. We finally discuss the case of dynamical fermions in mixed representations, and exotic composite fermion states such as the chimera baryons. We conclude by sketching the future stages of the programme. We also describe our approach to open access.

Real-space entanglement spectra of lowest Landau level projected fractional quantum Hall states using Monte Carlo methods

Real-space entanglement spectrum (RSES) of a quantum Hall (QH) wavefunction gives a natural route to infer the nature of its edge excitations. Computation of RSES becomes expensive with an increase in the number of particles and included Landau levels (LL). RSES can be efficiently computed using Monte Carlo (MC) methods for trial states that can be written as products of determinants such as the composite fermion (CF) and parton states. This computational efficiency also applies to the RSES of lowest Landau level (LLL) projected CF and parton states; however, LLL projection to be used here requires approximations that generalize the Jain Kamilla (JK) projection. This work is a careful study of how this approximation should be made. We identify the approximation closest in spirit to JK projection and perform tests of the approximations involved in the projection by comparing the MC results with the RSES obtained from computationally expensive but exact methods. We present the techniques and use them to calculate the exact RSES of the exact LLL projected bosonic Jain $2/3$ state in bipartition of systems of sizes up to $N=24$ on the sphere. For the lowest few angular momentum sectors of the RSES, we present evidence to show that MC results closely match the exact spectra. We also discuss other plausible projection schemes. We also calculate the exact RSES of the unprojected fermionic Jain $2/5$ state obtained from the exact diagonalization of the Trugman-Kivelson Hamiltonian in the two lowest LLs on the sphere. By comparing with the RSES of the unprojected $2/5$ state from Monte Carlo methods, we show that the latter is practically exact.

Sequencing the Entangled DNA of Fractional Quantum Hall Fluids

We introduce and prove the “root theorem”, which establishes a condition for families of operators to annihilate all root states associated with zero modes of a given positive semi-definite k-body Hamiltonian chosen from a large class. This class is motivated by fractional quantum Hall and related problems, and features generally long-ranged, one-dimensional, dipole-conserving terms. Our theorem streamlines analysis of zero-modes in contexts where “generalized” or “entangled” Pauli principles apply. One major application of the theorem is to parent Hamiltonians for mixed Landau-level wave functions, such as unprojected composite fermion or parton-like states that were recently discussed in the literature, where it is difficult to rigorously establish a complete set of zero modes with traditional polynomial techniques. As a simple application, we show that a modified V1 pseudo-potential, obtained via retention of only half the terms, stabilizes the ν=1/2 Tao–Thouless state as the unique densest ground state.

Entanglement action for the real-space entanglement spectra of chiral Abelian quantum Hall wave functions

We argue and numerically substantiate that the real-space entanglement spectrum (RSES) of chiral Abelian quantum Hall states is given by the spectrum of a local boundary perturbation of a $(1+1)$-dimensional conformal field theory, which describes an effective edge dynamics along the real-space cut. The cut-and-glue approach suggests that the low-lying RSES is equivalent to the low-lying modes of some effective edge action. The general structure of this action is deduced by mapping to a boundary critical problem, generalizing the work of Dubail, Read, and Rezayi [Phys. Rev. B 85, 115321 (2012)]. Using trial wave functions, we numerically test our model of the RSES for the $\ensuremath{\nu}=2/3$ bosonic composite fermion state.

Locally Incompressible Spin State of the Fractional Quantum Hall Effect when v = 3/2

Neutral excitations in a two-dimensional electron system with an orbital and spin quantum number of 1 in the vicinity of filling factor v = 3/2 are studied experimentally. It is found that the rate of excitation relaxation to the ground state slows very strongly at v = 3/2, even though the number of vacancies in the ground state suitable for the relaxation of excitations is macroscopically large. It is shown that neutral excitations with orbital and spin quantum numbers of 1 in the state of v = 3/2 are an example of topologically protected excitation with different spin orderings in the ground and excited states. The state of v = 3/2 is an example of a locally incompressible fractional state of the quantum Hall effect, which is neither a Laughlin liquid nor an integer state of composite fermions.

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.

Fractional quantum Hall effect in CVD-grown graphene

We show the emergence of fractional quantum Hall states in graphene grown by chemical vapor deposition (CVD) for magnetic fields from below 3 T to 35 T where the CVD-graphene was dry-transferred. Effective composite-fermion filling factors up to ν * = 4 are visible and higher order composite-fermion states (with four flux quanta attached) start to emerge at the highest fields. Our results show that the quantum mobility of CVD-grown graphene is comparable to that of exfoliated graphene and, more specifically, that the p/3 fractional quantum Hall states have energy gaps of up to 30 K, well comparable to those observed in other silicon-gated devices based on exfoliated graphene.

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.

Near-field scanning magneto-photoluminescence of composite fermions in In(Ga)P/GaInP quantum dots

We measured spatial and magnetic field dependence of photoluminescence (PL) spectrum of In(Ga)P/GaInP quantum dots (QDs) having up to 8 electrons and quantum confinement ħω 0 down to 2 meV using low temperature (10K) near-field scanning optical microscope (NSOM) providing spatial resolution ∼20 nm and external magnetic field up to B0=10 T. Using these measurements we identified QDs having Wigner-Seitz radius up to 2 and internal magnetic field B int up to 15 T, corresponding to Landau level filling factor v at zero field ν 0 3, 5/2, 2/5 and 1/5. In magneto-PL spectra we observed features related to integer and fractional quantum Hall states and identified composite fermion (CF) states corresponding to Wigner molecule isomers (1, 5) and (0, 6). Our results demonstrate perspectives of using In(Ga)P/GaInP QD structures, providing deterministic localization of CFs in zero external magnetic field, in “magnetic-field-free” topological quantum gates.

Molecular States of Composite Fermions in Self-Organized InP/GaInP Quantum Dots in Zero Magnetic Field

The size and positions of regions of line localization and the magnetic-field (0–10 T) dependence of the low-temperature (10 K) photoluminescence spectra of single InP/GaInP quantum dots with a number of electrons of N = 5–7 and a Wigner–Seitz radius of ~2.5 are determined using a near-field scanning optical microscope. The formation of composite fermion molecules with a size coinciding with that of localization regions and bond lengths of ~30 and 50 nm, respectively, at a Landau-level filling factor from 1/2 to 2/7 in zero magnetic field is established. At N = 6, the pairing and rearrangement of composite fermions under photoexcitation are found, which offers opportunities for the use of InP/GaInP quantum dots to create a magnetic-field-free topological quantum gate.

Effective Abelian theory from a non-Abelian topological order in the ν=2/5 fractional quantum Hall effect

Topological phases of matter are distinguished by topological invariants, such as Chern numbers and topological spins, that quantize their response to electromagnetic currents and changes of ambient geometry. Intriguingly, in the $\ensuremath{\nu}=2/5$ fractional quantum Hall effect, prominent theoretical approaches---the composite fermion theory and conformal field theory---have constructed two distinct states, the Jain composite fermion (CF) state and the Gaffnian state, for which many of the topological indices coincide and even the microscopic ground-state wave functions have high overlap with each other in system sizes accessible to numerics. At the same time, some aspects of these states are expected to be very different; e.g., their elementary excitations should have either Abelian (CF) or non-Abelian (Gaffnian) statistics. In this paper we investigate the close relationship between these two states by considering not only their ground states, but also the low-energy charged excitations. We show that the low-energy physics of both phases is spanned by the same type of quasielectrons of the neighboring Laughlin phase. The main difference between the two states arises due to an implicit assumption of short-range interaction in the CF approach, which causes a large splitting of the variational energies of the Gaffnian excitations. We thus propose that the Jain phase emerges as an effective Abelian low-energy description of the Gaffnian phase when the Hamiltonian is dominated by two-body interactions of sufficiently short range.

Composite fermion state of graphene as a Haldane-Chern insulator

Graphene in the presence of a strong external magnetic field is a unique attraction for investigations of the fractional quantum Hall (fQH) states with odd and even denominators of the fraction. Most of the attempts to understand Graphene in the strong-field regime were made through exploiting the universal low-energy effective description of Dirac fermions emerging from the nearest neighbor hopping model of electrons on a honeycomb lattice. We highlight that accounting for the next-nearest-neighbor hopping terms in doped Graphene can lead to a unique redistribution of magnetic fluxes within the unit cell of the lattice. While this affects all the fQH states, it has a striking effect at a half-filled Landau-level state: it leads to a composite fermion state that is equivalent to the doped topological Chern insulator on a honeycomb lattice. At energies comparable to the Fermi energy, this state possesses a Haldane gap in the bulk proportional to the next-nearest-neighbor hopping and density of dopants. We argue that this microscopically derived energy gap survives the projection to the lowest band. We also conjecture that the gap should be present in a microscopic theory giving the recently proposed particle-hole symmetric Dirac composite fermion scenario of the half-filled Landau-level. The proposed gap is lower than the chemical potential, and is predicted to be parametrically separated from the Dirac point in the latter description. Finally we conclude by proposing experiments to detect this gap; the associated boundary mode; and encourage cold-atom setups to test other predictions of the theory.

The sixteenfold way and the quantum Hall effect at half-integer filling factors

Fractional quantum Hall states at half-integer filling factors have been observed in many systems beyond the $5/2$ and $7/2$ plateaus in GaAs quantum wells. This includes bilayer states in GaAs, several half-integer plateaus in ZnO-based heterostructures, and quantum Hall liquids in graphene. In all cases, Cooper pairing of composite fermions is believed to explain the plateaus. The nature of Cooper pairing and the topological order on those plateaus are hotly debated. Different orders are believed to be present in different systems. This makes it important to understand experimental signatures of all proposed orders. We review the expected experimental signatures for all possible composite-fermion states at half-integer filling. We address Mach-Zehnder interferometry, thermal transport, tunneling experiments, and Fabry-P\'{e}rot interferometry. For this end, we introduce a uniform description of the topological orders of Kitaev's sixteenfold way in terms of their wave-functions, effective Hamiltonians, and edge theories.

Pairing states of composite fermions in double-layer graphene

Pairing interaction between fermionic particles leads to composite Bosons that condense at low temperature. Such condensate gives rise to long range order and phase coherence in superconductivity, superfluidity, and other exotic states of matter in the quantum limit. In graphene double-layers separated by an ultra-thin insulator, strong interlayer Coulomb interaction introduces electron-hole pairing across the two layers, resulting in a unique superfluid phase of interlayer excitons. In this work, we report a series of emergent fractional quantum Hall ground states in a graphene double-layer structure, which is compared to an expanded composite fermion model with two-component correlation. The ground state hierarchy from bulk conductance measurement and Hall resistance plateau from Coulomb drag measurement provide strong experimental evidence for a sequence of effective integer quantum Hall effect states for the novel two-component composite fermions (CFs), where CFs fill integer number of effective LLs (Lambda-level). Most remarkably, a sequence of incompressible states with interlayer correlation are observed at half-filled Lambda-levels, which represents a new type of order involving pairing states of CFs that is unique to graphene double-layer structure and beyond the conventional CF model.

Pair-Density-Wave Order and Paired Fractional Quantum Hall Fluids

The properties of the isotropic incompressible $\nu=5/2$ fractional quantum Hall (FQH) state are described by a paired state of composite fermions in zero (effective) magnetic field, with a uniform $p_x+ip_y$ pairing order parameter, which is a non-Abelian topological phase with chiral Majorana and charge modes at the boundary. Recent experiments suggest the existence of a proximate nematic phase at $\nu=5/2$. This finding motivates us to consider an inhomogeneous paired state - a $p_x+ip_y$ pair-density-wave (PDW) - whose melting could be the origin of the observed liquid-crystalline phases. This state can viewed as an array of domain and anti-domain walls of the $p_x+i p_y$ order parameter. We show that the nodes of the PDW order parameter, the location of the domain walls (and anti-domain walls) where the order parameter changes sign, support a pair of symmetry-protected counter-propagating Majorana modes. The coupling behavior of the domain wall Majorana modes crucially depends on the interplay of the Fermi energy $E_{F}$ and the PDW pairing energy $E_{\textrm{pdw}}$. The analysis of this interplay yields a rich set of topological states. The pair-density-wave order state in paired FQH system provides a fertile setting to study Abelian and non-Abelian FQH phases - as well as transitions thereof - tuned by the strength of the paired liquid crystalline order.

Quantum Hall hierarchy from coupled wires

The coupled-wire construction provides a useful way to obtain microscopic Hamiltonians for various two-dimensional topological phases, among which fractional quantum Hall states are paradigmatic examples. Using the recently introduced flux attachment and vortex duality transformations for coupled wires, we show that this construction is remarkably versatile to encapsulate phenomenologies of hierarchical quantum Hall states: the Jain-type hierarchy states of composite fermions filling Landau levels and the Haldane-Halperin hierarchy states of quasiparticle condensation. The particle-hole conjugate transformation for coupled-wire models is also given as a special case of the hierarchy construction. We also propose coupled-wire models for the composite Fermi liquid, which turn out to be compatible with a sort of the particle-hole symmetry implemented in a nonlocal way at $\nu=1/2$. Furthermore, our approach shows explicitly the connection between the Moore-Read Pfaffian state and a chiral $p$-wave pairing of the composite fermions. This composite fermion picture is also generalized to a family of the Pfaffian state, including the anti-Pfaffian state and Bonderson-Slingerland hierarchy states.