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Higher angular momentum band inversions in two dimensions
Abstract
We study a special class of topological phase transitions in two dimensions described by the inversion of bands with relative angular momentum higher than 1. A band inversion of this kind, which is protected by rotation symmetry, separates the trivial insulator from a Chern insulating phase with higher Chern number, and thus generalizes the quantum Hall transition described by a Dirac fermion. Higher angular momentum band inversions are of special interest, as the nonvanishing density of states at the transition can give rise to interesting many-body effects. Here we introduce a series of minimal lattice models which realize higher angular momentum band inversions. We then consider the effect of interactions, focusing on the possibility of electron-hole exciton condensation, which breaks rotational symmetry. An analysis of the excitonic insulator mean field theory further reveals that the ground state of the Chern insulating phase with higher Chern number has the structure of a multicomponent integer quantum Hall state. We conclude by generalizing the notion of higher angular momentum band inversions to the class of time-reversal invariant systems, following the scheme of Bernevig-Hughes-Zhang (BHZ). Such band inversions can be viewed as transitions to a topological insulator protected by rotation and inversion symmetry, and provide a promising venue for realizing correlated topological phases such as fractional topological insulators.
... The condition m > 1 requires additional symmetry to forbid smaller m. For example, m = 3 could arise due the inversion of s and f states in a crystal with C 6 rotational symmetry [29]. ...
The search for fractional quantized Hall phases in the absence of a magnetic field has primarily targeted flat-band systems that mimic the features of a Landau level. In an alternative approach, the fractional excitonic insulator (FEI) has been proposed as a correlated electron-hole fluid that arises near a band inversion between bands of different angular momentum with strong interactions. It remains an interesting challenge to find Hamiltonians with realistic interactions that stabilize this state. Here, we describe composite boson and composite fermion theories that highlight the importance of excitonic pairing in stabilizing FEIs in a class of band inversion models. We predict a sequence of Jain-like and Laughlin-like FEI states, the simplest of which has the topological order of the bosonic fractional quantized Hall state. We discuss implications for recent numerical studies on a chiral spin liquid phase in interacting Chern insulator models.
... Perhaps most unusual has been the preparation of a particle-hole exciton condensate state in a photonics quantum computer [342]. Lastly, the interplay of topology and excitonic insulators is beginning to be explored [343,344], with our old friend Ta 2 NiSe 5 suggested as one possible topological excitonic insulator [126]. These present fertile grounds for further study and for applying the theoretical tools we developed in this work. ...
Electron-hole pairs, known as excitons, are fundamental excitations of solids, and are expected to realise a rich phase diagram of quantum phases in- and out-of-equilibrium: a hypothesized pairing instability leads to the macroscopic condensation of “excitonic insulators” in small-bandgap materials, with quantum coherence and possible super-transport persisting up to room temperature. Alternatively, in large-bandgap semiconductors, the Bose-Einstein condensation of optically active metastable excitons opens the door to nonlinear optical processes, coherent light generation, and exciton lasing. Though first predicted in the 1960s, these excitonic quantum phases have proven frustratingly difficult to implement and identify. In the last decade, novel materials and experimental methods have revitalized this pursuit. This signals an urgent need for theory to propose experimental fingerprints that will distinguish between quantum-coherent excitonic states, uncondensed excitons, and normal host materials. I address this need by focusing on a key common feature: a spontaneously broken U(1) symmetry and the resulting gapless Goldstone collective excitations, which generate unique distinguishing signatures. I first develop a theory for disordered excitonic insulators, which are classified by the symmetries of the introduced impurities. I demonstrate that the Goldstone modes are robust to disorder scattering, and thus dominate low-energy long-range dynamics. This is corroborated by ballistic transport measurements in candidate material Ta₂NiSe₅. I subsequently consider photopumped exciton BECs in twisted heterobilayers of transition metal dichalcogenides such as MoSe₂/WSe₂. I show that despite a momentum-indirect bandgap, strong interactions enable emission from the condensate via a spontaneous production of collective excitations. I predict this “leaky emission” dominates at low temperatures, with a unique spectrum and density dependence.
... Their regular pattern of occurrences at commensurate fillings of the weakly dispersing moiré bands with bandwidths as small as 10meV [6] indicates an enhanced role of interaction effects, including strong-coupling Mott physics complemented by other complex electron phenomena such as superconductivity [1][2][3][4][5], linear-in-temperature resistivity [5], correlated electron states observed in scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS) measurements [7][8][9][10], ferromagnetism and quantum hall physics [11]. Though vast theoretical efforts were made to model the electronic structure [12][13][14][15][16][17][18][19], as well as to explain the superconducting pairing mechanism [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] and the insulating states [15,[35][36][37][38][39][40][41][42], a comprehensive understanding of the insulating states for variable carrier concentrations is lacking. This is obviously important if one wants to identify the mechanism of superconductivity in these systems. ...
In this work, we determine states of electronic order of small-angle twisted bilayer graphene. Ground states are determined for weak and strong couplings which are representatives for varying distances of the twist-angle from its magic value. In the weak-coupling regime, charge density waves emerge which break translational and C 3 -rotational symmetry. In the strong coupling-regime, we find rotational and translational symmetry breaking Mott insulating states for all commensurate moiré band fillings. Depending on the local occupation of superlattice sites hosting up to four electrons, global spin-(ferromagnetic) and valley symmetries are also broken which may give rise to a reduced Landau level degeneracy as observed in experiments for commensurate band fillings. The formation of those particular electron orders is traced back to the important role of characteristic non-local interactions which connect all localized states belonging to one hexagon formed by the AB- and BA-stacked regions of the superlattice.
... On the other hand, previous investigation shows that an important mechanism to stabilize the aforementioned higher-order band crossing is the discrete m-fold rotational symmetry of the underlying lattice structure [23,24]. Through investigating 2D and 3D multiple Weyl points and related systems [25][26][27][28][29][30][31][32][33][34][35], the rotational eigenvalues are shown to be in one to one correspondence with the order of band crossing in the lowenergy sector of these materials, which are described by higher-order Dirac or Weyl models [23,24]. The inclusion of additional nonspatial symmetry further constrains the rotational eigenvalues and hence the band crossing [24]. ...
In topological insulators and topological superconductors, the discrete jump of the topological invariant upon tuning a certain system parameter defines a topological phase transition. A unified framework is employed to address the quantum criticality of the topological phase transitions in one to three spatial dimensions, which simultaneously incorporates the symmetry classification, order of
band crossing,m-fold rotational symmetry, correlation functions, critical exponents, scaling laws, and renormalization group approach. We first classify higher-order Dirac models according to the time-reversal, particle-hole, and chiral symmetries, and determine the even-oddness of the order of band crossing in each symmetry class. The even–oddness further constrains the rotational symmetry permitted in a symmetry class. Expressing the topological invariant in terms of a momentum space integration over a curvature function, the order of band crossing determines the critical exponent of the curvature function, as well as that of the Wannier state correlation function introduced through the Fourier transform of the curvature function. The conservation of topological invariant further yields a scaling law between critical exponents. In addition, a renormalization group approach based on deforming the curvature function is demonstrated for all dimensions and symmetry classes. Through clarification of how the critical quantities, including the jump of the topological invariant
and critical exponents, depend on the nonspatial and the rotational symmetry, our work introduces the notion of universality class into the description of topological phase transitions.
We study the exciton condensation in the heterostructures where the electron layer and the hole layer formed by gapped chiral fermion (GCF) systems are separately gated. High angular momentum such as p- and d-wave-like excitonic pairing may emerge when the gap of the GCF systems is small compared to the Fermi energy, and the chiral winding numbers of the electrons and holes are the same. This is a result of the nontrivial band geometry and can be linked to the Berry curvature when projected onto the Fermi surface. In realistic systems, we propose that staggered graphene and surface states of magnetically doped topological insulators or intrinsic axion insulators are promising candidates for realizing p-wave exciton superfluid, and anomalous Hall conductivity can be used as a signature in experiments.
We study a class of multiorbital models based on those proposed by Venderbos et al. [Phys. Rev. B 98, 235160 (2018)] which exhibit an interplay of topology, interactions, and fermion incoherence. In the noninteracting limit, these models exhibit trivial and Chern insulator phases with Chern number C≥1 bands as determined by the relative angular momentum of the participating orbitals. These quantum anomalous Hall insulator phases are separated by topological transitions protected by crystalline rotation symmetry, featuring Dirac or quadratic band-touching points. Here we study the impact of Sachdev-Ye-Kitaev (SYK) type interactions on these lattice models. Given the random interactions, these models display average symmetries upon disorder averaging, including a charge conjugation symmetry, so they behave as interacting models in topological class D enriched by crystalline rotation symmetry. The phase diagram of this model features a non-Fermi liquid at high temperature and an exciton condensate with nematic transport at low temperature. We present results from the free energy, spectral functions, and the anomalous Hall resistivity as a function of temperature and tuning parameters. Our results are broadly relevant to correlated topological matter in multiorbital systems, and may also be viewed, with a suitable particle hole transformation, as an exploration of strong interaction effects on mean-field topological superconductors.
We study the effect of interelectron Coulomb interactions on the displacement field induced topological phase transition in transition metal dichalcogenide moiré heterobilayers. We find a nematic excitonic insulator phase that breaks the moiré superlattice’s threefold rotational symmetry and preempts the topological phase transition in both AA and AB stacked heterobilayers when the interlayer tunneling is weak, or when the Coulomb interaction is not strongly screened. The nematicity originates from the frustration between the nontrivial spatial structure of the interlayer tunneling, which is crucial to the existence of the topological Chern band, and the interlayer coherence induced by the Coulomb interaction that favors uniformity in layer pseudospin orientations. We construct a unified effective two-band model that captures the physics near the band inversion and applies to both AA and AB stacked heterobilayers. Within the two-band model the competition between the nematic excitonic insulator phase and the Chern insulator phase can be understood as the switching of the energetic order between the s-wave and the p-wave excitons upon increasing the interlayer tunneling.
We construct new many-body invariants for 2D Chern and 3D chiral hinge insulators characterizing quantized pumping of bulk dipole and quadrupole moments. The many-body invariants are written entirely in terms of many-body ground state wave functions on a torus geometry with twisted boundary conditions and a set of unitary operators. We present a number of supporting arguments for the invariants via topological field theory interpretation, adiabatic pumping argument, and direct mapping to free-fermion band indices. Therefore, the invariants explicitly encircle several different pillars of theoretical descriptions of topological phases. Furthermore, our many-body invariants are written in forms which can be directly employed in various numerics including the exact diagonalization and the density-matrix renormalization group simulations. We finally confirm our invariants by numerical computations including an infinite density-matrix renormalization group on quasi-one-dimensional systems.
We construct new many-body invariants for 2d Chern and 3d chiral hinge insulators, which are characterized by quantized pumping of dipole and quadrupole moments. The invariants that we devise are written entirely in terms of many-body ground state wavefunctions on a torus geometry with a set of unitary operators. We provide a number of supporting evidences for our invariants via topological field theory interpretation, adiabatic pumping argument, and direct mapping to free-fermion band indices. We finally confirm our invariants by numerical computations including infinite density matrix renormalization group on a quasi-one-dimensional system. The many-body invariants therefore explicitly encircle several different pillars of theoretical descriptions of topological phases.
Since graphene has been successfully exfoliated, two-dimensional (2D) materials constitute a vibrant research field and open vast perspectives in high-performance applications. Among them, bismuthene and 2D bismuth (Bi) are unique with superior properties to fabricate state-of-the-art energy saving, storage and conversion devices. The largest experimentally determined bulk gap, even larger than those of stanene and antimonene, allows 2D Bi to be the most promising candidate to construct room-temperature topological insulators. Moreover, 2D Bi exhibits cyclability for high-performance sodium-ion batteries, and the enlarged surface together with the good electrochemical activity renders it an efficient electrocatalyst for energy conversion. Also, the air-stability of 2D Bi is better than that of silicene, germanene, phosphorene and arsenene, which could enable more practical applications. This review aims to thoroughly explore the fundamentals of 2D Bi and its improved fabrication methods, in order to further bridge gaps between theoretical predictions and experimental achievements in its energy-related applications. We begin with an introduction of the status of 2D Bi in the 2D-material family, which is followed by descriptions of its intrinsic properties along with various fabrication methods. The vast implications of 2D Bi for high-performance devices can be envisioned to add a new pillar in energy sciences. In addition, in the context of recent pioneering studies on moiré superlattices of other 2D materials, we hope that the improved manipulation techniques of bismuthene, along with its unique properties, might even enable 2D Bi to play an important role in future energy-related twistronics.
The interplay between symmetry and topology leads to a rich variety of electronic topological phases, protecting states such as topological insulators and Dirac semimetals. Here we focus on a different aspect of this interplay, where symmetry unambiguously indicates the presence of a topological band structure. Our study is comprehensive, encompassing all 230 space groups, and is built on two key advances. First, we find an efficient way to represent band structures in terms of elementary basis states, which are readily computed for each space group. Next, we isolate topological band structures by removing the set that represent simple atomic insulators, corresponding to localized Wannier orbitals. Special cases of our theory include the Fu-Kane parity criterion for inversion symmetric topological insulators, and the previously described filling-enforced quantum band insulators. We describe applications arising from new results.
We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure. The algorithm applies to crystals with broken time-reversal, particle-hole, and chiral symmetries in any dimension. The presented results match the mathematical structure underlying the topological classification of these crystals in terms of K-theory, and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify the allowed topological phases in any possible two-dimensional crystal in class A. We also show how the same procedure can be used to classify the allowed phases for any three-dimensional space group. Employing these classifications, we study transitions between topological phases within class A that are driven by band inversions at high symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure, and identify whether or not they give rise to intermediate Weyl semimetallic phases.
Trial wavefunctions that can be represented by summing over locally-coupled
degrees of freedom are called tensor network states (TNSs); they have seemed
difficult to construct for two-dimensional topological phases that possess
protected gapless edge excitations. We show it can be done for chiral states of
free fermions, using a Gaussian Grassmann integral, yielding
and Chern insulator states, in the sense that the fermionic excitations live in
a topologically non-trivial bundle of the required type. We prove that any
strictly short-range quadratic parent Hamiltonian for these states is gapless;
the proof holds for a class of systems in any dimension of space. The proof
also shows, quite generally, that sets of compactly-supported Wannier-type
functions do not exist for band structures in this class. We construct further
examples of TNSs that are analogs of fractional (including non-Abelian) quantum
Hall phases; it is not known whether parent Hamiltonians for these are also
gapless.
Weyl semimetals are gapless quasi-topological materials with a set of
isolated nodal points forming their Fermi surface. They manifest their
quasi-topological character in a series of topological electromagnetic
responses including the anomalous Hall effect. Here we study the effect of
disorder on Weyl semimetals while monitoring both their nodal/semi-metallic and
topological properties through computations of the localization length and the
Hall conductivity. We examine three different lattice tight-binding models
which realize the Weyl semimetal in part of their phase diagram and look for
universal features that are common to all of the models, and interesting
distinguishing features of each model. We present detailed phase diagrams of
these models for large system sizes and we find that weak disorder preserves
the nodal points up to the diffusive limit, but does affect the Hall
conductivity. We show that the trend of the Hall conductivity is consistent
with an effective picture in which disorder causes the Weyl nodes move within
the Brillouin zone along a specific direction that depends deterministically on
the properties of the model and the neighboring phases to the Weyl semimetal
phase. We also uncover an unusual (non-quantized) anomalous Hall insulator
phase which can only exist in the presence of disorder.
Topological insulators have emerged as a major topic of research in condensed
matter physics with several novel applications proposed. Although there are now
a number of established experimental examples of materials in this class, all
of them can be described by theories based on electronic band structure, which
implies that they do not possess electronic correlations strong enough to
fundamentally change this theoretical description. Here, we review recent
theoretical progress on models of correlated topological insulators beyond an
electronic band structure description, and summarize some of the experimental
scenarios that would facilitate their detection.
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.
We consider a model of spinful fermions on the triangular lattice which
exhibits a C=2 Chern insulator (CI) phase with a quantized anomalous Hall
effect, sandwiched between two normal insulator (NI) phases. The first NI-CI
quantum phase transition is driven by simultaneous mass inversion of a pair of
Dirac fermions, with short range interactions being perturbatively irrelevant.
The second CI-NI transition is driven by a single quadratic band touching point
protected by momentum space topology and C_6 lattice symmetry. A one-loop
renormalization group analysis shows that short range interactions lead to a
single marginally relevant perturbation at this transition. We obtain the mean
field phase diagram of this model incorporating weak repulsive Hubbard
interactions, finding an emergent nematic dome around this CI-NI topological
phase transition. We discuss the crossovers in the anomalous Hall conductivity
at nonzero temperature, and the Landau theory of the quantum and thermal
transitions out of the nematic phase. Our results may be relevant to
ferromagnetic double perovskite films with spin-orbit coupling which have been
proposed to host such a NI-CI transition. Our work provides perhaps the
simplest example of a novel emergent phase near a quantum phase transition.
We present a review of experimental and theoretical studies of the anomalous Hall effect (AHE), focusing on recent developments that have provided a more complete framework for understanding this subtle phenomenon and have, in many instances, replaced controversy by clarity. Synergy between experimental and theoretical work, both playing a crucial role, has been at the heart of these advances. On the theoretical front, the adoption of Berry-phase concepts has established a link between the AHE and the topological nature of the Hall currents which originate from spin-orbit coupling. On the experimental front, new experimental studies of the AHE in transition metals, transition-metal oxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors, have more clearly established systematic trends. These two developments in concert with first-principles electronic structure calculations, strongly favor the dominance of an intrinsic Berry-phase-related AHE mechanism in metallic ferromagnets with moderate conductivity. The intrinsic AHE can be expressed in terms of Berry-phase curvatures and it is therefore an intrinsic quantum mechanical property of a perfect cyrstal. An extrinsic mechanism, skew scattering from disorder, tends to dominate the AHE in highly conductive ferromagnets. We review the full modern semiclassical treatment of the AHE together with the more rigorous quantum-mechanical treatments based on the Kubo and Keldysh formalisms, taking into account multiband effects, and demonstrate the equivalence of all three linear response theories in the metallic regime. Finally we discuss outstanding issues and avenues for future investigation.
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.
Topological band insulators (TBIs) are bulk insulating materials which
feature topologically protected metallic states on their boundary. The existing
classification departs from time-reversal symmetry, but the role of the crystal
lattice symmetries in the physics of these topological states remained elusive.
Here we provide the classification of TBIs protected not only by time-reversal,
but also by crystalline symmetries. We find three broad classes of topological
states: (a) Gamma-states robust against general time-reversal invariant
perturbations; (b) Translationally-active states protected from elastic
scattering, but susceptible to topological crystalline disorder; (c) Valley
topological insulators sensitive to the effects of non-topological and
crystalline disorder. These three classes give rise to 18 different
two-dimensional, and, at least 70 three-dimensional TBIs, opening up a route
for the systematic search for new types of TBIs.
Effective field theories that describes the dynamics of a conserved U(1)
current in terms of "hydrodynamic" degrees of freedom of topological phases in
condensed matter are discussed in general dimension D=d+1 using the
functional bosonization technique. For non-interacting topological insulators
(superconductors) with a conserved U(1) charge and characterized by an integer
topological invariant [more specifically, they are topological insulators in
the complex symmetry classes (class A and AIII) and in the "primary series" of
topological insulators in the eight real symmetry classes], we derive the
BF-type topological field theories supplemented with the Chern-Simons (when D
is odd) or the -term (when D is even). For topological insulators
characterized by a topological invariant (the first and second
descendants of the primary series), their topological field theories are
obtained by dimensional reduction. Building on this effective field theory
description for non-interacting topological phases, we also discuss, following
the spirit of the parton construction of the fractional quantum Hall effect by
Block and Wen, the putative "fractional" topological insulators and their
possible effective field theories, and use them to determine the physical
properties of these non-trivial quantum phases.
We report the theoretical discovery of a systematic scheme to produce
topological flat bands (TFBs) with arbitrary Chern numbers. We find that
generically a multi-orbital high Chern number TFB model can be constructed by
considering multi-layer Chern number C=1 TFB models with enhanced translational
symmetry. A series of models are presented as examples, including a two-band
model on a triangular lattice with a Chern number C=3 and an N-band square
lattice model with C=N for an arbitrary integer N. In all these models, the
flatness ratio for the TFBs is larger than 30 and increases with increasing
Chern number. In the presence of appropriate inter-particle interactions, these
models are likely to lead to the formation of novel Abelian and Non-Abelian
fractional Chern insulators. As a simple example, we test the C=2 model with
hardcore bosons at 1/3 filling and an intriguing fractional quantum Hall state
is observed.
Recent theoretical works have demonstrated various robust Abelian and
non-Abelian fractional topological phases in lattice models with topological
flat bands carrying Chern number C=1. Here we study hard-core bosons and
interacting fermions in a three-band triangular-lattice model with the lowest
topological flat band of Chern number C=2. We find convincing numerical
evidence of bosonic fractional quantum Hall effect at the filling
characterized by three-fold quasi-degeneracy of ground states on a torus, a
fractional Chern number for each ground state, a robust spectrum gap, and a gap
in quasihole excitation spectrum. We also observe numerical evidence of a
robust fermionic fractional quantum Hall effect for spinless fermions at the
filling with short-range interactions.
An exciting new prospect in condensed matter physics is the possibility of
realizing fractional quantum Hall (FQH) states in simple lattice models without
a large external magnetic field. A fundamental question is whether
qualitatively new states can be realized on the lattice as compared with
ordinary fractional quantum Hall states. Here we propose new symmetry-enriched
topological states, topological nematic states, which are a dramatic
consequence of the interplay between the lattice translation symmetry and
topological properties of these fractional Chern insulators. When a partially
filled flat band has a Chern number N, it can be mapped to an N-layer quantum
Hall system. We find that lattice dislocations can act as wormholes connecting
the different layers and effectively change the topology of the space. Lattice
dislocations become defects with non-trivial quantum dimension, even when the
FQH state being realized is by itself Abelian. Our proposal leads to the
possibility of realizing the physics of topologically ordered states on high
genus surfaces in the lab even though the sample has only the disk geometry.
We perform a complete classification of two-band \bk\cdot\mathbf{p}
theories at band crossing points in 3D semimetals with n-fold rotation
symmetry and broken time-reversal symmetry. Using this classification, we show
the existence of new 3D topological semimetals characterized by
-protected double-Weyl nodes with quadratic in-plane (along )
dispersion or -protected triple-Weyl nodes with cubic in-plane dispersion.
We apply this theory to the 3D ferromagnet HgCrSe and confirm it is a
double-Weyl metal protected by symmetry. Furthermore, if the direction of
the ferromagnetism is shifted away from the [001]- to the [111]-axis, the
double-Weyl node splits into four single Weyl nodes, as dictated by the point
group of that phase. Finally, we discuss experimentally relevant effects
including splitting of multi-Weyl nodes by applying breaking strain and
the surface Fermi arcs in these new semimetals.
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.
In this paper we construct fully symmetric wavefunctions for the
spin-polarized fractional Chern insulators (FCI) and time-reversal-invariant
fractional topological insulators (FTI) in two dimensions using the parton
approach. We show that the lattice symmetry gives rise to many different FCI
and FTI phases even with the same filling fraction (and the same
quantized Hall conductance in FCI case). They have different
symmetry-protected topological orders, which are characterized by different
projective symmetry groups. We mainly focus on FCI phases which are realized in
a partially filled band with Chern number one. The low-energy gauge groups of a
generic FCI wavefunctions can be either SU(m) or
the discrete group , and in the latter case the associated low-energy
physics are described by Chern-Simons-Higgs theories. We use our construction
to compute the ground state degeneracy. Examples of FCI/FTI wavefunctions on
honeycomb lattice and checkerboard lattice are explicitly given. Possible
non-Abelian FCI phases which may be realized in a partially filled band with
Chern number two are discussed. Generic FTI wavefunctions in the absence of
spin conservation are also presented whose low-energy gauge groups can be
either or . The constructed wavefunctions
also set up the framework for future variational Monte Carlo simulations.
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.
Using a functional renormalization group approach, we study
interaction-driven instabilities in quadratic band crossing point two-orbital
models in two dimensions, extending a previous study of Sun et al. [1]. The
wavevector-dependence of the Bloch eigenvectors of the free Hamiltonian causes
interesting instabilities toward spin nematic, quantum anomalous Hall and
quantum spin Hall states. In contrast with other known examples of
interaction-driven topological insulators, in the system studied here, the QSH
state occurs at arbitrarily small interaction strength and for rather simple
intra- and inter-orbital repulsions.
Topological insulators are electronic materials that have a bulk band gap like an ordinary insulator but have protected conducted states on their edge or surface. These states are possible due to the combination of spin-orbit interactions and time-reversal symmetry. The two-dimensional (2D) topological insulator is a quantum spin Hall insulator, which is a close cousin of the integer quantum Hall state. A three-dimensional (3D) topological insulator supports novel spin-polarized 2D Dirac fermions on its surface. In this Colloquium the theoretical foundation for topological insulators and superconductors is reviewed and recent experiments are described in which the signatures of topological insulators have been observed. Transport experiments on HgTe/CdTe quantum wells are described that demonstrate the existence of the edge states predicted for teh quantum spin hall insulator. Experiments on Bi1-xSbx, Bi<2Se3, Bi2Te3 and Sb2Te3 are then discussed that establish these materials as 3D topological insulators and directly probe the topology of their surface states. Exotic states are described that can occur at the surface of a 3D topological insulator due to an induced energy gap. A magnetic gap leads to a novel quantum Hall state that gives rise to a topological magnetoelectric effect. A superconducting energy gap leads to a state that supports Majorana fermions and may provide a new venue for realizing proposals for topological quantum computation. Prospects for observing these exotic states are also discussed, as well as other potential device applications of topological insulators.
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 give field theory descriptions of the time-reversal invariant quantum spin Hall insulator in 2+1 dimensions and the particle-hole symmetric insulator in 1+1 dimensions in terms of massive Dirac fermions. Integrating out the massive fermions we obtain a low-energy description in terms of a topological field theory, which is entirely determined by anomaly considerations. This description allows us to easily construct low-energy effective actions for the corresponding `fractional' topological insulators, potentially corresponding to new states of matter. We give a holographic realization of these fractional states in terms of a probe brane system, verifying that the expected topologically protected transport properties are robust even at strong coupling. Comment: 13 pages, 1 figure
We present a symmetry-based analysis of competition between different gapped states that have been proposed in bilayer graphene (BLG), which are all degenerate on a mean field level. We classify the states in terms of a hidden SU(4) symmetry, and distinguish symmetry protected degeneracies from accidental degeneracies. One of the states, which spontaneously breaks discrete time reversal symmetry but no continuous symmetry, is identified as a Quantum Anomalous Hall (QAH) state, which exhibits quantum Hall effect at zero magnetic field. We investigate the lifting of the accidental degeneracies by thermal and zero point fluctuations, taking account of the modes softened under RG. Working in a 'saddle point plus quadratic fluctuations' approximation, we identify two types of RG- soft modes which have competing effects. Zero point fluctuations, dominated by 'transverse' modes which are unique to BLG, favor the QAH state. Thermal fluctuations, dominated by 'longitudinal' modes, favor a SU(4) symmetry breaking multiplet of states. We discuss the phenomenology and experimental signatures of the QAH state in BLG, and also propose a way to induce the QAH state using weak external magnetic fields. Comment: minor changes, made to match journal version
Low-energy electronic structure of (unbiased) bilayer graphene is made of two Fermi points with quadratic dispersions, if trigonal-warping and other high order contributions are ignored. We show that as a result of this qualitative difference from single-layer graphene, short-range (or screened Coulomb) interactions are marginally relevant. We use renormalization group to study their effects on low-energy properties of the system, and show that the two quadratic Fermi points spontaneously split into four Dirac points, at zero temperature. This results in a nematic state that spontaneously breaks the six-fold lattice rotation symmetry (combined with layer permutation) down to a two-fold one, with a finite transition temperature. Critical properties of the transition and effects of trigonal warping are also discussed. Comment: 4 pages + references
We analyze generalizations of two-dimensional topological insulators which can be realized in interacting, time reversal invariant electron systems. These states, which we call fractional topological insulators, contain excitations with fractional charge and statistics in addition to protected edge modes. In the case of s(z) conserving toy models, we show that a system is a fractional topological insulator if and only if sigma(sH)/e* is odd, where sigma(sH) is the spin-Hall conductance in units of e/2pi, and e* is the elementary charge in units of e.
We investigate the stability of a quadratic band-crossing point (QBCP) in 2D fermionic systems. At the noninteracting level, we show that a QBCP exists and is topologically stable for a Berry flux +/-2pi if the point symmetry group has either fourfold or sixfold rotational symmetries. This putative topologically stable free-fermion QBCP is marginally unstable to arbitrarily weak short-range repulsive interactions. We consider both spinless and spin-1/2 fermions. Four possible ordered states result: a quantum anomalous Hall phase, a quantum spin Hall phase, a nematic phase, and a nematic-spin-nematic phase.
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 wave function for electrons and holes and show that for m=1 it is the exact ground state of a free fermion model that describes px+ipy excitonic pairing. For m>1 we develop a simple composite fermion mean field theory, and we present evidence that our wave function correctly describes this phase. We derive an interacting Hamiltonian for which our wave function 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 (px+ipy)3 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.
We describe a mechanism by which fermions in topologically trivial bands can form correlated states exhibiting a fractional quantum Hall (FQH) effect upon introduction of strong repulsive interactions. These states are solid-liquid composites, in which a FQH liquid is induced by the formation of charge order (CO), following a recently proposed paradigm of symmetry-breaking topological (SBT) order [Phys. Rev. Lett. 113, 216404 (2014)]. We devise a spinless fermion model on a triangular lattice, featuring a topologically trivial phase when interactions are omitted. Adding strong short-range repulsion, we first establish a repulsion-driven CO phase at density particles per site, then dope the model to higher densities . At (), we observe definitive signatures of both CO and the FQH effect --- sharply peaked static structure factor, gapped and degenerate energy spectrum and fractionally quantized Hall conductivity --- over a range of all model parameters. We thus obtain the first strong evidence for fermionic SBT order of FQH type in topologically trivial bands.
The past decade's apparent success in predicting and experimentally discovering distinct classes of topological insulators (TIs) and semimetals masks a fundamental shortcoming: out of 200,000 stoichiometric compounds extant in material databases, only several hundred of them - a set essentially of measure zero - are topologically nontrivial. Are TIs that esoteric, or does this reflect a fundamental problem with the current piecemeal approach to finding them? Two fundamental shortcomings of the current approach are: the focus on delocalized Bloch wavefunctions - rather than the local, chemical bonding in materials - and a classification scheme based on a collection of seemingly unrelated topological indices. To remedy these issues, we propose a new and complete electronic band theory that assembles the last missing piece - the link between topology and local chemical bonding - with the conventional band theory of electrons. Topological Quantum Chemistry is a description of the universal global properties of all possible band structures and materials comprised of a graph theoretical description of momentum space and a dual group theoretical description in real space. This patches together local dispersions into distinct global groups of energy bands: we classify all the possible bands for all 230 crystal symmetry groups involving or d orbitals on any of the Wyckoff positions of every space group. We show how our topological band theory sheds new light on known TIs, and demonstrate the power of our method to predict a plethora of new TIs.
Topological materials have become the focus of intense research in recent years, since they exhibit fundamentally new physical phenomena with potential applications for novel devices and quantum information technology. One of the hallmarks of topological materials is the existence of protected gapless surface states, which arise due to a nontrivial topology of the bulk wave functions. This review provides a pedagogical introduction into the field of topological quantum matter with an emphasis on classification schemes. We consider both fully gapped and gapless topological materials and their classification in terms of global symmetries, such as time-reversal, as well as spatial symmetries, such as reflection. Furthermore, we survey the classification of gapless modes localized on topological defects. The classification of these systems is discussed by use of homotopy groups, Clifford algebras, K-theory, and non-linear sigma models describing the Ander-son (de-)localization at the surface or inside a defect of the material. Theoretical model systems and their topological invariants are reviewed together with recent experimental results in order to provide a unified and comprehensive perspective of the field. While the bulk of this article is concerned with the topological properties of noninteracting or mean-field Hamiltonians, we also provide a brief overview of recent results and open questions concerning the topological classifications of interacting systems.
A three-dimensional (3D) Dirac semimetal is the 3D analog of graphene whose
bulk band shows a linear dispersion relation in the 3D momentum space. Since
each Dirac point with four-fold degeneracy carries a zero Chern number, a Dirac
semimetal can be stable only in the presence of certain crystalline symmetries.
In this work, we propose a general framework to classify stable 3D Dirac
semimetals. Based on symmetry analysis, we show that various types of stable 3D
Dirac semimetals exist in systems having the time-reversal, inversion, and
uniaxial rotational symmetries. There are two distinct classes of stable 3D
Dirac semimetals. In the first class, a pair of 3D Dirac points locate on the
rotation axis, away from its center. Moreover, the 3D Dirac semimetals in this
class have nontrivial topological properties characterized by 2D topological
invariants, such as the invariant or the mirror Chern number. These 2D
topological indices give rise to stable 2D surface Dirac cones, which can be
deformed to Fermi arcs when the surface states couple to the bulk states on the
Fermi level. On the other hand, the second class of Dirac SM phases possess a
3D Dirac point at a time-reversal invariant momentum (TRIM) on the rotation
axis and do not have surface states in general.
It is shown that topological insulating phases driven by interactions can be
realized without the need for spin-orbit coupling or large intersite
interaction in a two-dimensional system of spin-1/2 fermions with a single pair
of parabolically touching bands. Using a weak-coupling, Wilsonian
renormalization group procedure, we show that a quantum anomalous Hall phase is
realized for Hubbard interaction, while a quantum spin Hall phase is favored
for longer-ranged interactions.
A (single flavor) quadratic band crossing in two dimensions is known to have
a generic instability towards a quantum anomalous Hall (QAH) ground state for
infinitesimal repulsive interactions. Here we introduce a generalization of a
quadratic band crossing which is protected only by rotational symmetry. By
focusing on the representative case of a parabolic and flat band touching,
which also allows for a straightforward lattice realization, the interaction
induced nematic phase is found generally to compete successfully with the QAH
insulator, and to become the dominant instability in certain parts of the phase
diagram already at weak coupling. The full phase diagram of the model, together
with its topological properties, is mapped out using a perturbative
renormalization group, strong coupling analysis, the mean-field theory.
Interestingly, the Berry flux varies continuously in the single flavour limit
with various control parameters.
We propose a simple microscopic model to numerically investigate the
stability of a two dimensional fractional topological insulator (FTI). The
simplest example of a FTI consists of two decoupled copies of a Laughlin state
with opposite chiralities. We focus on bosons at half filling. We study the
stability of the FTI phase upon addition of two coupling terms of different
nature: an interspin interaction term, and an inversion symmetry breaking term
that couples the copies at the single particle level. Using exact
diagonalization and entanglement spectra, we numerically show that the FTI
phase is stable against both perturbations. We compare our system to a similar
bilayer fractional Chern insulator. We show evidence that the time reversal
invariant system survives the introduction of interaction coupling on a larger
scale than the time reversal symmetry breaking one, stressing the importance of
time reversal symmetry in the FTI phase stability. We also discuss possible
fractional phases beyond .
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.
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 properties of states in which particle-hole pairs of nonzero angular momentum condense. These states generalize charge- and spin-density-wave states, in which s-wave particle-hole pairs condense. We show that the p-wave spin-singlet state of this type has Peierls ordering, while the d-wave spin-singlet state is the staggered flux state. We discuss model Hamiltonians which favor p- and d-wave density-wave order. There are analogous orderings for pure spin models, which generalize spin-Peierls order. The spin-triplet density-wave states are accompanied by spin-1 Goldstone bosons, but these excitations do not contribute to the spin-spin correlation function. Hence they must be detected with NQR or Raman-scattering experiments. Depending on the geometry and topology of the Fermi-surface, these states may admit gapless fermionic excitations. As the Fermi-surface geometry is changed, these excitations disappear at a transition which is third order in mean-field theory. The singlet d-wave- and triplet p-wave density-wave states are separated from the corresponding superconducting states by zero-temperature O(4)-symmetric critical points.
In a mean-field-theory treatment the ground state of a graphene bilayer spontaneously breaks inversion symmetry for arbitrarily weak electron-electron interactions when trigonal-warping terms in the band structure are ignored. We report on a perturbative renormalization-group calculation, which assesses the robustness of this instability, comparing with the closely related case of the charge-density-wave instability incorrectly predicted by mean-field theory in a one-dimensional electron gas. We conclude that spontaneous inversion symmetry breaking in graphene is not suppressed by quantum fluctuations but that, because of trigonal warping, it may occur only in high quality suspended bilayers.
We survey various quantized bulk physical observables in two- and
three-dimensional topological band insulators invariant under translational
symmetry and crystallographic point group symmetries (PGS). In two-dimensional
insulators, we show that: (i) the Chern number of a -invariant insulator
can be determined, up to a multiple of n, by evaluating the eigenvalues of
symmetry operators at high-symmetry points in the Brillouin zone; (ii) the
Chern number of a -invariant insulator is also determined, up to a
multiple of n, by the eigenvalue of the Slater determinant of a
noninteracting many-body system and (iii) the Chern number vanishes in
insulators with dihedral point groups , and the quantized electric
polarization is a topological invariant for these insulators. In
three-dimensional insulators, we show that: (i) only insulators with point
groups , and PGS can have nonzero 3D quantum Hall
coefficient and (ii) only insulators with improper rotation symmetries can have
quantized magnetoelectric polarization in the term
, the axion term in the electrodynamics of the
insulator (medium).
A large number of recent works point to the emergence of intriguing analogs
of fractional quantum Hall states in lattice models due to effective
interactions in nearly flat bands with Chern number C=1. Here, we provide an
intuitive and efficient construction of almost dispersionless bands with higher
Chern numbers. Inspired by the physics of quantum Hall multilayers and
pyrochlore-based transition-metal oxides, we study a tight-binding model
describing spin-orbit coupled electrons in N parallel kagome layers connected
by apical sites forming N-1 intermediate triangular layers (as in the
pyrochlore lattice). For each N, we find finite regions in parameter space
giving a virtually flat band with C=N. We analytically express the states
within these topological bands in terms of single-layer states and thereby
explicitly demonstrate that the C=N wave functions have an appealing structure
in which layer index and translations in reciprocal space are intricately
coupled. This provides a promising arena for new collective states of matter.
We point out the possibility of nearly flat band with Chern number C=2 on the
dice lattice in a simple nearest-neighbor tightbinding model. This lattice can
be naturally formed by three adjacent (111) layers of cubic lattice, which
may be realized in certain thin films or artificial heterostructures, such as
SrTiO/SrIrO/SrTiO trilayer heterostructure grown along (111)
direction. The flatness of two bands is protected by the bipartite nature of
the lattice. Including the Rashba spin-orbit coupling on nearest-neighbor bonds
separate the flat bands with others but maintains their flatness. Repulsive
interaction will drive spontaneous ferromagnetism on the Kramer pair of flat
bands and split them into two nearly flat bands with Chern number . We
thus propose that this may be a route to quantum anomalous Hall effect and
further conjecture that partial filling of the C=2 band may realize exotic
fractional quantum Hall effects.
We study translationally-invariant insulators with inversion symmetry that
fall outside the established classification of topological insulators. These
insulators are not required to have gapless boundary modes in the energy
spectrum. However, they do exhibit protected modes in the entanglement spectrum
localized on the cut between two entangled regions. Their entanglement entropy
cannot be made to vanish adiabatically, and hence the insulators can be called
topological. There is a direct connection between the inversion eigenvalues of
the band structure and the mid-gap states in the entanglement spectrum. The
classification of protected entanglement levels is given by an integer , which is the difference between the negative inversion eigenvalues at
inversion symmetric points in the Brillouin zone, taken in sets of two. When
the Hamiltonian describes a Chern insulator or a non-trivial T-invariant
topological insulator, the entanglement spectrum exhibits spectral flow. If the
Chern number is zero for the former, or T is broken in the latter, the
entanglement spectrum does \emph{not} have spectral flow, but, depending on the
inversion eigenvalues, can still have protected midgap bands. Although spectral
flow is broken, the mid-gap entanglement bands cannot be adiabatically removed,
and the insulator is `topological.' In 1D, we establish a link between the
product of the inversion eigenvalues of all occupied bands at all inversion
momenta and charge polarization. In 2D, we prove a link between the product of
the inversion eigenvalues and the parity of the Chern number. In 3D, we find a
topological constraint on the product of the inversion eigenvalues indicating
that some 3D materials are topological metals, and we show the link between the
inversion eigenvalues and the 3D Quantum Hall Effect and the magnetoelectric
polarization in the absence of T-symmetry.
Topological insulators are new states of quantum matter which can not be
adiabatically connected to conventional insulators and semiconductors. They are
characterized by a full insulating gap in the bulk and gapless edge or surface
states which are protected by time-reversal symmetry. These topological
materials have been theoretically predicted and experimentally observed in a
variety of systems, including HgTe quantum wells, BiSb alloys, and BiTe
and BiSe crystals. We review theoretical models, materials properties
and experimental results on two-dimensional and three-dimensional topological
insulators, and discuss both the topological band theory and the topological
field theory. Topological superconductors have a full pairing gap in the bulk
and gapless surface states consisting of Majorana fermions. We review the
theory of topological superconductors in close analogy to the theory of
topological insulators.
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.
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.
The control of intrinsic anomalous Hall effect (AHE) in metallic ferromagnets by Berry phases was described. The Berry phases, which is a Fermi liquid property, is found to be accumulated by adiabatic motion of quasiparticles on the Fermi surface. It was found that in the absence of BCS pairing processes, breakdown of extra conservation laws occurred only through nonadiabatic impurity or surface scattering. The addition of quasiparticle Berry phases as a topological ingredient to the Landau Fermi-liquid theory was proposed, in the presence of broken inversion or time-reversal symmetry.
We study the effects of spin orbit interactions on the low energy electronic structure of a single plane of graphene. We find that in an experimentally accessible low temperature regime the symmetry allowed spin orbit potential converts graphene from an ideal two-dimensional semimetallic state to a quantum spin Hall insulator. This novel electronic state of matter is gapped in the bulk and supports the transport of spin and charge in gapless edge states that propagate at the sample boundaries. The edge states are nonchiral, but they are insensitive to disorder because their directionality is correlated with spin. The spin and charge conductances in these edge states are calculated and the effects of temperature, chemical potential, Rashba coupling, disorder, and symmetry breaking fields are discussed.
The quantum Hall liquid is a novel state of matter with profound emergent properties such as fractional charge and statistics. The existence of the quantum Hall effect requires breaking of the time reversal symmetry caused by an external magnetic field. In this work, we predict a quantized spin Hall effect in the absence of any magnetic field, where the intrinsic spin Hall conductance is quantized in units of 2(e/4pi). The degenerate quantum Landau levels are created by the spin-orbit coupling in conventional semiconductors in the presence of a strain gradient. This new state of matter has many profound correlated properties described by a topological field theory.