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Fractional Excitonic Insulator
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Abstract
We argue that a correlated fluid of electrons and holes can exhibit a fractional quantum Hall effect at zero magnetic field analogous to the Laughlin state at filling 1/m. We introduce a variant of the Laughlin wavefunction for electrons and holes and show that for m=1 it is the exact ground state of a free fermion model that describes excitonic pairing. For we develop a simple composite fermion mean field theory, and we present evidence that our wavefunction correctly describes this phase. We derive an interacting Hamiltonian for which our wavefunction is the exact ground state, and we present physical arguments that the m=3 state can be realized in a system in which energy bands with angular momentum that differ by 3 cross at the Fermi energy. This leads to a gapless state with excitonic pairing, which we argue is conducive to forming the fractional excitonic insulator in the presence of interactions. Prospects for numerics on model systems and band structure engineering to realize this phase in real materials are discussed.
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We propose a particle-hole symmetric theory of the Fermi-liquid ground state
of a half-filled Landau level. This theory should be applicable for a Dirac
fermion in magnetic field at charge neutrality, as well as for the
quantum Hall ground state of nonrelativistic fermions in the
limit of negligible inter-Landau-level mixing. We argue that when particle-hole
symmetry is exact, the composite fermion is a massless Dirac fermion,
characterized by a Berry phase of around the Fermi circle. We write down
a tentative effective field theory of such a fermion and discuss the discrete
symmetries, in particular . The Dirac composite fermions
interact through a gauge, but non-Chern-Simons, interaction. The particle-hole
conjugate pair of Jain-sequence states at filling factors and
, which in the conventional composite fermion picture
corresponds to integer quantum Hall states with different filling factors, n
and n+1, is now mapped to the same half-integer filling factor of
the Dirac composite fermion. The Pfaffian and anti-Pfaffian states are
interpreted as d-wave Bardeen-Cooper-Schrieffer paired states of the Dirac
fermion with orbital angular momentum of opposite signs, while s-wave pairing
would give rise to a novel particle-hole symmetric nonabelian gapped phase.
When particle-hole symmetry is not exact, the Dirac fermion has a breaking mass. The conventional fermionic Chern-Simons theory is
shown to emerge in the nonrelativistic limit of the massive theory.
We review various features of interacting Abelian topological phases of
matter in two spatial dimensions, placing particular emphasis on fractional
Chern insulators (FCIs) and fractional topological insulators (FTIs). We
highlight aspects of these systems that challenge the intuition developed from
quantum Hall physics - for instance, FCIs are stable in the limit where the
interaction energy scale is much larger than the band gap, and FTIs can possess
fractionalized excitations in the bulk despite the absence of gapless edge
modes.
Topological insulators and their intriguing edge states can be understood in a single-particle picture and can as such be exhaustively classified. Interactions significantly complicate this picture and can lead to entirely new insulating phases, with an altogether much richer and less explored phenomenology. Most saliently, lattice generalizations of fractional quantum Hall states, dubbed fractional Chern insulators, have recently been predicted to be stabilized by interactions within nearly dispersionless bands with nonzero Chern number, C. Contrary to their continuum analogues, these states do not require an external magnetic field and may potentially persist even at room temperature, which make these systems very attractive for possible applications such as topological quantum computation. This review recapitulates the basics of tight-binding models hosting nearly flat bands with nontrivial topology, C≠0, and summarizes the present understanding of interactions and strongly correlated phases within these bands. Emphasis is made on microscopic models, highlighting the analogy with continuum Landau level physics, as well as qualitatively new, lattice specific, aspects including Berry curvature fluctuations, competing instabilities as well as novel collective states of matter emerging in bands with |C|>1. Possible experimental realizations, including oxide interfaces and cold atom implementations as well as generalizations to flat bands characterized by other topological invariants are also discussed.
We consider a number of strongly correlated quantum Hall states that are likely to be realized in bilayer quantum Hall systems at total Landau level filling fraction νT=1. One state, the (3,3,-1) state, can occur as an instability of a compressible state in the large d/lB limit, where d and lB are the interlayer distance and magnetic length, respectively. This state has a hierarchical descendent that is interlayer coherent. Another interlayer coherent state, which is expected in the small d/lB limit is the well-known Halperin (1,1,1) state. Using the concept of composite fermion pairing, we discuss the wave functions that describe these states. We construct a phase diagram using the Chern-Simons Landau-Ginzburg theory and discuss the transitions between the various phases. We propose that the longitudinal and Hall-drag resistivities can be used together with interlayer tunneling to experimentally distinguish these different quantum Hall states. Our work indicates the bilayer νT=1 quantum Hall phase diagram to be considerably richer than that assumed so far in the literature.
Applications of conformal field theory to the theory of fractional quantum Hall systems are discussed. In particular, Laughlin's wave function and its cousins are interpreted as conformal blocks in certain rational conformal field theories. Using this point of view a hamiltonian is constructed for electrons for which the ground state is known exactly and whose quasihole excitations have nonabelian statistics; we term these objects “nonabelions”. It is argued that universality classes of fractional quantum Hall systems can be characterized by the quantum numbers and statistics of their excitations. The relation between the order parameter in the fractional quantum Hall effect and the chiral algebra in rational conformal field theory is stressed, and new order parameters for several states are given.
A new definition of order called topological order is proposed for two-dimensional systems in which no long-range order of the conventional type exists. The possibility of a phase transition characterized by a change in the response of the system to an external perturbation is discussed in the context of a mean field type of approximation. The critical behaviour found in this model displays very weak singularities. The application of these ideas to the xy model of magnetism, the solid-liquid transition, and the neutral superfluid are discussed. This type of phase transition cannot occur in a superconductor nor in a Heisenberg ferromagnet.
We present a class of time-reversal-symmetric fractional topological liquid
states in two dimensions that support fractionalized excitations. These are
incompressible liquids made of electrons, for which the charge Hall conductance
vanishes and the spin Hall conductance needs not be quantized. We then analyze
the stability of edge states in these two-dimensional topological fluids
against localization by disorder. We find a Z_2 stability criterion for whether
or not there exists a Kramers pair of edge modes that is robust against
disorder. We also introduce an interacting electronic two-dimensional lattice
model based on partially filled flattened bands of a Z_2 topological band
insulator, which we study using numerical exact diagonalization. We show
evidence for instances of the fractional topological liquid phase as well as
for a time-reversal symmetry broken phase with a quantized (charge) Hall
conductance in the phase diagram for this model.
Chern insulators are band insulators exhibiting a nonzero Hall conductance
but preserving the lattice translational symmetry. We conclusively show that a
partially filled Chern insulator at 1/3 filling exhibits a fractional quantum
Hall effect and rule out charge-density wave states that have not been ruled
out by previous studies. By diagonalizing the Hubbard interaction in the
flat-band limit of these insulators, we show the following: The system is
incompressible and has a 3-fold degenerate ground state whose momenta can be
computed by postulating an generalized Pauli principle with no more than 1
particle in 3 consecutive orbitals. The ground state density is constant, and
equal to 1/3 in momentum space. Excitations of the system are fractional
statistics particles whose total counting matches that of quasiholes in the
Laughlin state based on the same generalized Pauli principle. The entanglement
spectrum of the state has a clear entanglement gap which seems to remain finite
in the thermodynamic limit. The levels below the gap exhibit counting identical
to that of Laughlin 1/3 quasiholes. Both the 3 ground states and excited states
exhibit spectral flow upon flux insertion. All the properties above disappear
in the trivial state of the insulator - both the many-body energy gap and the
entanglement gap close at the phase transition when the single-particle
Hamiltonian goes from topologically nontrivial to topologically trivial. These
facts clearly show that fractional many-body states are possible in topological
insulators.
We show that a suitable combination of geometric frustration, ferromagnetism, and spin-orbit interactions can give rise to nearly flatbands with a large band gap and nonzero Chern number. Partial filling of the flatband can give rise to fractional quantum Hall states at high temperatures (maybe even room temperature). While the identification of material candidates with suitable parameters remains open, our work indicates intriguing directions for exploration and synthesis.
We study the quantum anomalous Hall effect described by a class of
two-component Haldane models on square lattices. We show that the latter can be
transformed into a pseudospin triplet p+ip-wave paired superfluid. In the long
wave length limit, the ground state wave function is described by Halperin's
(1,1,-1) state of neutral fermions analogous to the double layer quantum Hall
effect. The vortex excitations are charge e/2 abelian anyons which carry a
neutral Dirac fermion zero mode. The superconducting proximity effect induces
`tunneling' between `layers' which leads to topological phase transitions
whereby the Dirac fermion zero mode fractionalizes and Majorana fermions emerge
in the edge states. The charge e/2 vortex excitation carrying a Majorana zero
mode is a non-abelian anyon. The proximity effect can also drive a conventional
insulator into a quantum anomalous Hall effect state with a Majorana edge mode
and the non-abelian vortex excitations.
We propose an experimental scheme to realize and detect the quantum anomalous
Hall effect in an anisotropic square optical lattice which can be generated
from available experimental set-ups of double-well lattices with minor
modifications. A periodic gauge potential induced by atom-light interaction is
introduced to give a Peierls phase for the nearest-neighbor site hopping. The
quantized anomalous Hall conductivity is investigated by calculating the Chern
number as well as the chiral gapless edge states of our system. Furthermore, we
show in detail the feasability for its experimental detection through light
Bragg scattering of the edge and bulk states with which one can determine the
topological phase transition from usual insulating phase to quantum anomalous
Hall phase.
The anomalous Hall effect is a fundamental transport process in solids arising from the spin-orbit coupling. In a quantum
anomalous Hall insulator, spontaneous magnetic moments and spin-orbit coupling combine to give rise to a topologically nontrivial
electronic structure, leading to the quantized Hall effect without an external magnetic field. Based on first-principles calculations,
we predict that the tetradymite semiconductors Bi2Te3, Bi2Se3, and Sb2Te3 form magnetically ordered insulators when doped with transition metal elements (Cr or Fe), in contrast to conventional dilute
magnetic semiconductors where free carriers are necessary to mediate the magnetic coupling. In two-dimensional thin films,
this magnetic order gives rise to a topological electronic structure characterized by a finite Chern number, with the Hall
conductance quantized in units of e2/h (where e is the charge of an electron and h is Planck’s constant).
We analyze pairing of fermions in two dimensions for fully-gapped cases with
broken parity (P) and time-reversal (T), especially cases in which the gap
function is an orbital angular momentum (l) eigenstate, in particular
(p-wave, spinless or spin-triplet) and (d-wave, spin-singlet). For
, these fall into two phases, weak and strong pairing, which may be
distinguished topologically. In the cases with conserved spin, we derive
explicitly the Hall conductivity for spin as the corresponding topological
invariant. For the spinless p-wave case, the weak-pairing phase has a pair
wavefunction that is asympototically the same as that in the Moore-Read
(Pfaffian) quantum Hall state, and we argue that its other properties (edge
states, quasihole and toroidal ground states) are also the same, indicating
that nonabelian statistics is a {\em generic} property of such a paired phase.
The strong-pairing phase is an abelian state, and the transition between the
two phases involves a bulk Majorana fermion, the mass of which changes sign at
the transition. For the d-wave case, we argue that the Haldane-Rezayi state is
not the generic behavior of a phase but describes the asymptotics at the
critical point between weak and strong pairing, and has gapless fermion
excitations in the bulk. In this case the weak-pairing phase is an abelian
phase which has been considered previously. In the p-wave case with an unbroken
U(1) symmetry, which can be applied to the double layer quantum Hall problem,
the weak-pairing phase has the properties of the 331 state, and with nonzero
tunneling there is a transition to the Moore-Read phase. The effects of
disorder on noninteracting quasiparticles are considered.
Beyond fractional quantum Hall
Unlike most electronic topological phenomena, the fractional quantum Hall effect requires correlations among electrons. Spanton et al. describe a class of related but even more unusual states, the fractional Chern insulators (see the Perspective by Repellin and Regnault). They observed these states in samples of bilayer graphene, where one of the graphene layers was misaligned by a small angle with respect to an adjoining layer of hexagonal boron nitride. The misalignment created a superlattice potential and topologically nontrivial bands, which had a fractional filling, thanks to strong electronic interactions. The findings expand the class of correlated topological states, which have been predicted to harbor exotic excitations.
Science , this issue p. 62 ; see also p. 31
The Hall conductance of a two-dimensional electron gas has been studied in a uniform magnetic field and a periodic substrate potential U. The Kubo formula is written in a form that makes apparent the quantization when the Fermi energy lies in a gap. Explicit expressions have been obtained for the Hall conductance for both large and small Uℏomegac.
This paper presents theoretical considerations of a new kind of insulating phase which has recently been theoretically predicted but has as yet not been found experimentally. This phase is expected to occur when semiconductors with very small band gap or semimetals with very small band overlap are cooled to a sufficiently low temperature. The present paper first develops a BCS-like theory of the ground state and analyzes the nature of the response to a general perturbation, from which collective modes (of a sound-like nature), response to a static magnetic field, and conductivity are calculated. Finally, some discussion of the possible experimental realization of this new phase is presented.
An indirect band-gap semiconductor may be converted to a semimetal, or vice versa, by application of pressure. At low temperature, an excitonic phase or some other anomaly must occur in the neighborhood of the transition pressure. We also discuss the direct-band-gap case and the case where a band gap is zero by symmetry, as in gray tin.
This letter presents variational ground-state and excited-state wave functions which describe the condensation of a two-dimensional electron gas into a new state of matter.
Measurements of the Hall voltage of a two-dimensional electron gas, realized with a silicon metal-oxide-semiconductor field-effect transistor, show that the Hall resistance at particular, experimentally well-defined surface carrier concentrations has fixed values which depend only on the fine-structure constant and speed of light, and is insensitive to the geometry of the device. Preliminary data are reported.
We present a pedagogical review of the physics of fractional Chern insulators
with a particular focus on the connection to the fractional quantum Hall
effect. While the latter conventionally arises in semiconductor
heterostructures at low temperatures and in high magnetic fields, interacting
Chern insulators at fractional band filling may host phases with the same
topological properties, but stabilized at the lattice scale, potentially
leading to high-temperature topological order. We discuss the construction of
topological flat band models, provide a survey of numerical results, and
establish the connection between the Chern band and the continuum Landau
problem. We then briefly summarize various aspects of Chern band physics that
have no natural continuum analogs, before turning to a discussion of possible
experimental realizations. We close with a survey of future directions and open
problems, as well as a discussion of extensions of these ideas to higher
dimensions and to other topological phases.
We study the bosonization of chiral fermion theories on arbitrary compact Riemann surfaces. We express the fermionic and bosonic correlation functions in terms of theta functions and prove their equality. This is used to obtain explicit expressions for a class of chiral determinants relevant to string theory. The anomaly structure of these determinants and their behaviour on degenerate Riemann surfaces is analysed. We apply these results to multi-loop calculations of the bosonic string.
An effective single-band Hamiltonian representing a crystal electron in a uniform magnetic field is constructed from the tight-binding form of a Bloch band by replacing ℏ k[over →] by the operator p[over →]-eA[over →]/c. The resultant Schrödinger equation becomes a finite-difference equation whose eigenvalues can be computed by a matrix method. The magnetic flux which passes through a lattice cell, divided by a flux quantum, yields a dimensionless parameter whose rationality or irrationality highly influences the nature of the computed spectrum. The graph of the spectrum over a wide range of "rational" fields is plotted. A recursive structure is discovered in the graph, which enables a number of theorems to be proven, bearing particularly on the question of continuity. The recursive structure is not unlike that predicted by Azbel', using a continued fraction for the dimensionless parameter. An iterative algorithm for deriving the clustering pattern of the magnetic subbands is given, which follows from the recursive structure. From this algorithm, the nature of the spectrum at an "irrational" field can be deduced; it is seen to be an uncountable but measure-zero set of points (a Cantor set). Despite these-features, it is shown that the graph is continuous as the magnetic field varies. It is also shown how a spectrum with simplified properties can be derived from the rigorously derived spectrum, by introducing a spread in the field values. This spectrum satisfies all the intuitively desirable properties of a spectrum. The spectrum here presented is shown to agree with that predicted by A. Rauh in a completely different model for crystal electrons in a magnetic field. A new type of magnetic "superlattice" is introduced, constructed so that its unit cell intercepts precisely one quantum of flux. It is shown that this cell represents the periodicity of solutions of the difference equation. It is also shown how this superlattice allows the determination of the wave function at nonlattice sites. Evidence is offered that the wave functions belonging to irrational fields are everywhere defined and are continuous in this model, whereas those belonging to rational fields are only defined on a discrete set of points. A method for investigating these predictions experimentally is sketched.
We report the theoretical discovery of a class of 2D tight-binding models containing nearly flatbands with nonzero Chern numbers. In contrast with previous studies, where nonlocal hoppings are usually required, the Hamiltonians of our models only require short-range hopping and have the potential to be realized in cold atomic gases. Because of the similarity with 2D continuum Landau levels, these topologically nontrivial nearly flatbands may lead to the realization of fractional anomalous quantum Hall states and fractional topological insulators in real materials. Among the models we discover, the most interesting and practical one is a square-lattice three-band model which has only nearest-neighbor hopping. To understand better the physics underlying the topological flatband aspects, we also present the studies of a minimal two-band model on the checkerboard lattice.
We present Laughlin-Jastrow wave functions for incompressible fluid states of two-dimensional electrons at Landau-level filling factor 1/m that satisfy periodic boundary conditions. This rederivation of Laughlin-type states emphasizes that it is correct short-distance behavior of the wave functions rather than angular momentum considerations that lie behind the explanation of the fractional quantized effect.
We introduce and analyze a class of model systems to study transitions
in the integer quantum Hall effect (IQHE). Even without disorder our
model exhibits an IQHE transition as a control parameter is varied. We
find that the transition is in the two-dimensional Ising universality
class and compute all associated exponents and critical transport
properties. The fixed point has time-reversal, particle-hole, and parity
invariance. We then consider the effect of quenched disorder on the IQHE
transition and find the following. (i) Randomness in the control
parameter (which breaks all the above symmetries) translates into bond
randomness in the Ising model and is hence marginally irrelevant. The
transition may equally well be viewed as a quantum percolation of edge
states localized on equipotentials. The absence of random-phase factors
for the edge states is responsible for the nongeneric (Ising) critical
properties. (ii) For a random magnetic field (which preserves
particle-hole symmetry in every realization) the model exhibits an
exactly solvable fixed line, described in terms of a product of a
Luttinger liquid and an SU(n) spin chain. While exponents vary
continuously along the fixed line, the longitudinal conductivity is
constant due to a general conformal sum rule for Kac-Moody algebras
(derived here), and is computed exactly. We also obtain a closed
expression for the extended zero-energy wave function for every
realization of disorder and compute its exact multifractal spectrum
f(α) and the exponents of all participation ratios. One point on
the fixed line corresponds to a recently proposed model by Gade and
Wegner. (iii) The model in the presence of a random on-site potential
scales to a strong disorder regime, which is argued to be described by a
symplectic nonlinear-sigma-model fixed point. (iv) We find a plausible
global phase diagram in which all forms of disorder are simultaneously
considered. In this generic case, the presence of random-phase factors
in the edge-state description indicates that the transition is described
by a Chalker-Coddington model, with a so far analytically inaccessible
fixed point.
The Hilbert spaces of the edge excitations of several ``paired'' fractional
quantum Hall states, namely the Pfaffian, Haldane-Rezayi and 331 states, are
constructed and the states at each angular momentum level are enumerated. The
method is based on finding all the zero energy states for those Hamiltonians
for which each of these known ground states is the exact, unique, zero-energy
eigenstate of lowest angular momentum in the disk geometry. For each state, we
find that, in addition to the usual bosonic charge-fluctuation excitations,
there are fermionic edge excitations. The edge states can be built out of
quantum fields that describe the fermions, in addition to the usual scalar
bosons (or Luttinger liquids) that describe the charge fluctuations. The
fermionic fields in the Pfaffian and 331 cases are a non-interacting Majorana
(i.e., real Dirac) and Dirac field, respectively. For the Haldane-Rezayi state,
the field is an anticommuting scalar. For this system we exhibit a chiral
Lagrangian that has manifest SU(2) symmetry but breaks Lorentz invariance
because of the breakdown of the spin statistics connection implied by the
scalar nature of the field and the positive definite norm on the Hilbert space.
Finally we consider systems on a cylinder where the fluid has two edges and
construct the sectors of zero energy states, discuss the projection rules for
combining states at the two edges, and calculate the partition function for
each edge excitation system at finite temperature in the thermodynamic limit.
It is pointed out that the conformal field theories for the edge states are
examples of orbifold constructions.
A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization of the Hall conductance sigmaxy in the absence of an external magnetic field. Massless fermions without spectral doubling ccur at critical values of the model parameters, and exhibit the so-called ``parity anomaly'' of (2+1)-dimensional field theories.
Starting directly from the microscopic Hamiltonian, we derive a field-theory model for the fractional quantum Hall effect. By considering an approximate coarse-grained version of the same model, we construct a Landau-Ginzburg theory similar to that of Girvin. The partition function of the model exhibits cusps as a function of density and the Hall conductance is quantized at filling factors =(2k-1)-1 with k an arbitrary integer. At these fractions the ground state is incompressible, and the quasiparticles and quasiholes have fractional charge and obey fractional statistics. Finally, we show that the collective density fluctuations are massive.
The spontaneous interlayer phase coherent (111) state of a bilayer quantum Hall system at filling factor nu = 1 may be viewed as a condensate of interlayer particle-hole pairs or excitons. We show that when the layers are biased in such a way that these excitons are very dilute, they may be viewed as pointlike bosons. We calculate the exciton dispersion relation and show that the exciton-exciton interaction is dominated by the dipole moment they carry. In addition to the phase coherent state, we also find a Wigner crystal/glass phase in the presence/absence of disorder which is an insulating state for the excitons. The position of the phase boundary is estimated and the transition between these two phases is discussed.
We show that the quantum spin Hall (QSH) effect, a state of matter with topological properties distinct from those of conventional insulators, can be realized in mercury telluride-cadmium telluride semiconductor quantum wells. When the thickness of the quantum well is varied, the electronic state changes from a normal to an "inverted" type at a critical thickness d(c). We show that this transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. We also discuss methods for experimental detection of the QSH effect.
Topological insulators are materials with a bulk excitation gap generated by the spin orbit interaction, and which are different from conventional insulators. This distinction is characterized by Z_2 topological invariants, which characterize the groundstate. In two dimensions there is a single Z_2 invariant which distinguishes the ordinary insulator from the quantum spin Hall phase. In three dimensions there are four Z_2 invariants, which distinguish the ordinary insulator from "weak" and "strong" topological insulators. These phases are characterized by the presence of gapless surface (or edge) states. In the 2D quantum spin Hall phase and the 3D strong topological insulator these states are robust and are insensitive to weak disorder and interactions. In this paper we show that the presence of inversion symmetry greatly simplifies the problem of evaluating the Z_2 invariants. We show that the invariants can be determined from the knowledge of the parity of the occupied Bloch wavefunctions at the time reversal invariant points in the Brillouin zone. Using this approach, we predict a number of specific materials are strong topological insulators, including the semiconducting alloy Bi_{1-x} Sb_x as well as \alpha-Sn and HgTe under uniaxial strain. This paper also includes an expanded discussion of our formulation of the topological insulators in both two and three dimensions, as well as implications for experiments. Comment: 16 pages, 7 figures; published version
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