We show that boundaries of 3D weak topological insulators can
become gapped by strong interactions while preserving all symmetries,
leading to Abelian surface topological order. The anomalous nature of
weak topological insulator surfaces manifests itself in a nontrivial
action of symmetries on the quasiparticles; most strikingly, translations
change the anyon types in a manner impossible in strictly 2D systems
with the same symmetry. As a further consequence, screw dislocations
form non-Abelian defects that trap Z4 parafermion zero modes.
We introduce exotic gapless states—“composite Dirac liquids”—that can
appear at a strongly interacting surface of a three-dimensional
electronic topological insulator. Composite Dirac liquids exhibit a gap
to all charge excitations but nevertheless feature a single massless
Dirac cone built from emergent electrically neutral fermions. These
states thus comprise electrical insulators that, interestingly, retain
thermal properties similar to those of the noninteracting topological
insulator surface. A variety of novel fully gapped phases naturally
descend from composite Dirac liquids. Most remarkably, we show that
gapping the neutral fermions via Cooper pairing—which crucially does
not violate charge conservation—yields symmetric non-Abelian
topologically ordered surface phases captured in several recent works.
Other (Abelian) topological orders emerge upon alternatively gapping the
neutral Dirac cone with magnetism. We establish a hierarchical
relationship between these descendant phases and expose an appealing
connection to paired states of composite Fermi liquids arising in the
half filled Landau level of two-dimensional electron gases. To
controllably access these states we exploit a quasi-1D deformation of
the original electronic Dirac cone that enables us to analytically
address the fate of the strongly interacting surface. The algorithm we
develop applies quite broadly and further allows the construction of
symmetric surface topological orders for recently introduced bosonic
topological insulators.
We use the method of bulk-boundary correspondence of topological
invariants to show that disordered topological insulators have at
least one delocalized state at their boundary at zero energy. Those
insulators which do not have chiral (sublattice) symmetry have in
addition the whole band of delocalized states at their boundary,
with the zero energy state lying in the middle of the band. This
result was previously conjectured based on the anticipated
properties of the supersymmetric (or replicated) sigma models with
WZW-type terms, as well as verified in some cases using numerical
simulations and a variety of other arguments. Here we derive this
result generally, in arbitrary number of dimensions, and without
relying on the description in the language of sigma models.
Topologically ordered phases of matter, in particular so-called
symmetry-enriched topological phases, can exhibit quantum number
fractionalization in the presence of global symmetry. In Z2
topologically ordered states in two dimensions, fundamental translations
Tx and Ty acting on anyons can either commute or
anticommute. This property, crystal momentum fractionalization, can be
seen in a periodicity of the excited-state spectrum in the Brillouin
zone. We present a numerical method to detect the presence of this form
of symmetry enrichment given a projected entangled pair state; we study
the minima of the spectrum of correlation lengths of the transfer matrix
for a cylinder. As a benchmark, we demonstrate our method using a
modified toric code model with perturbation. An enhanced periodicity in
momentum clearly reveals the nontrivial anticommutation relation
{Tx,Ty}=0 for the corresponding quasiparticles in
the system.
We consider gapped ℤ2 spin liquids, where spinon
quasiparticles may carry fractional quantum numbers of space-group
symmetry. In particular, spinons can carry fractional crystal momentum.
We show that such quantum number fractionalization has dramatic,
spectroscopically accessible consequences, namely, enhanced periodicity
of the two-spinon density of states in the Brillouin zone, which can be
detected via inelastic neutron scattering. This effect is a sharp
signature of certain topologically ordered spin liquids and other
symmetry-enriched topological phases. Considering square lattice
space-group and time-reversal symmetry, we show that exactly four
distinct types of spectral periodicity are possible.
We employ quantum Monte Carlo techniques to calculate the
Z2 topological invariant in a two-dimensional model of
interacting electrons that exhibits a quantum spin Hall topological
insulator phase. In particular, we consider the parity invariant for
inversion-symmetric systems, which can be obtained from the bulk's
imaginary-time Green's function after an appropriate continuation to
zero frequency. This topological invariant is used here in order to
study the trivial-band to topological-insulator transitions in an
interacting system with spin-orbit coupling and an explicit bond
dimerization. We discuss the accessibility and behavior of this
topological invariant within quantum Monte Carlo simulations.
We calculate a topological invariant, whose value would coincide with
the Chern number in case of integer quantum Hall effect, for fractional
quantum Hall states. In case of Abelian fractional quantum Hall states,
this invariant is shown to be equal to the trace of the K-matrix. In
case of non-Abelian fractional quantum Hall states, this invariant can
be calculated on a case by case basis from the conformal field theory
describing these states. This invariant can be used, for example, to
distinguish between different fractional Hall states numerically even
though, as a single number, it cannot uniquely label distinct
states.
We classify distinct types of quantum number fractionalization
occurring in two-dimensional topologically ordered phases, focusing in
particular on phases with Z2 topological order, that is, on
gapped Z2 spin liquids. We find that the fractionalization
class of each anyon is an equivalence class of projective
representations of the symmetry group, corresponding to elements of the
cohomology group H2(G,Z2). This result leads us to
a symmetry classification of gapped Z2 spin liquids, such
that two phases in different symmetry classes cannot be connected
without breaking symmetry or crossing a phase transition. Symmetry
classes are defined by specifying a fractionalization class for each
type of anyon. The fusion rules of anyons play a crucial role in
determining the symmetry classes. For translation and internal
symmetries, braiding statistics plays no role, but can affect the
classification when point group symmetries are present. For square
lattice space group, time-reversal, and SO(3) spin rotation symmetries,
we find 2098176≈221 distinct symmetry classes. Our symmetry
classification is not complete, as we exclude, by assumption,
permutation of the different types of anyons by symmetry operations. We
give an explicit construction of symmetry classes for square lattice
space group symmetry in the toric code model. Via simple examples, we
illustrate how information about fractionalization classes can, in
principle, be obtained from the spectrum and quantum numbers of excited
states. Moreover, the symmetry class can be partially determined from
the quantum numbers of the four degenerate ground states on the torus.
We also extend our results to arbitrary Abelian topological orders
(limited, though, to translations and internal symmetries), and compare
our classification with the related projective symmetry group
classification of parton mean-field theories. Our results provide a
framework for understanding and probing the sharp distinctions among
symmetric Z2 spin liquids and are a first step toward a full
classification of symmetric topologically ordered phases.
We study one-dimensional, interacting, gapped fermionic systems
described by variants of the Peierls-Hubbard model, and we characterize
their phases via a topological invariant constructed out of their
Green's functions. We demonstrate that the existence of topologically
protected, zero-energy states at the boundaries of these systems can be
tied to the value of the topological invariant, just like when working
with the conventional, noninteracting topological insulators. We use a
combination of analytical methods and the numerical density matrix
renormalization group method to calculate the values of the topological
invariant throughout the phase diagrams of these systems, thus deducing
when topologically protected boundary states are present. We are also
able to study topological states in spin systems because, deep in the
Mott insulating regime, these fermionic systems reduce to spin chains.
In this way, we associate the zero-energy states at the end of an
antiferromagnetic spin-1 Heisenberg chain with a topological invariant
equal to 2.
We propose a spin-dependent optical lattice potential that realizes
a three-dimensional antiferromagnetic topological insulator in a gas of
cold, two-state fermions such as alkaline earths, as well as a model
that describes the tight-binding limit of this potential. We discuss the
physically observable responses of the gas that can verify the presence
of this phase, in particular rapid rotation in response to the trap
potential. We also point out how this model can be used to obtain
two-dimensional flat bands with nonzero Chern number.
We revisit the fermionic parton approach to S=1/2 quantum spin
liquids with SU(2) spin-rotation symmetry, and the associated projective
symmetry group (PSG) classification. We point out that the existing PSG
classification is incomplete; upon completing it, we find spin-liquid
states with S=1 and S=0 Majorana fermion excitations coupled to a
deconfined Z2 gauge field. The crucial observation leading us
to this result is that, like space group and time-reversal symmetries,
spin rotations can act projectively on the fermionic partons; that is, a
spin rotation may be realized by simultaneous SU(2) spin and gauge
rotations. We show that there are only two realizations of spin
rotations acting on fermionic partons: the familiar naive realization
where spin rotation is not accompanied by any gauge transformation, and
a single type of projective realization. We discuss the PSG
classification for states with projective spin rotations. To illustrate
these results, we show that there are four such PSGs on the
two-dimensional square lattice. We study the properties of the
corresponding states, finding that one—with gapless Fermi points—is a
stable phase beyond mean-field theory. In this phase, depending on
parameters, a small Zeeman magnetic field can open a partial gap for the
Majorana fermion excitations. Moreover, there are nearby gapped phases
supporting Z2 vortex excitations obeying non-Abelian statistics. We
conclude with a discussion of various open issues, including the
challenging question of where such S=1 Majorana spin liquids may occur
in models and in real systems.
Topological insulators are noninteracting, gapped fermionic systems
which have gapless boundary excitations. They are characterized by
topological invariants, which can be written in many different ways,
including in terms of Green’s functions. Here we show that the existence
of the edge states directly follows from the existence of the
topological invariant written in terms of the Green’s functions, for
all ten classes of topological insulators in all spatial dimensions. We
also show that the resulting edge states are characterized by their own
topological invariant, whose value is equal to the topological invariant
of the bulk insulator. This can be used to test whether a given model
Hamiltonian can describe an edge of a topological insulator. Finally, we
observe that the results discussed here apply equally well to
interacting topological insulators, with certain
modifications.
We consider antiferromagnets breaking both time-reversal (Θ) and a
primitive-lattice translational symmetry (T1/2) of a crystal
but preserving the combination S=ΘT1/2. The S symmetry leads
to a Z2 topological classification of insulators, separating
the ordinary insulator phase from the “antiferromagnetic topological
insulator” phase. This state is similar to the “strong” topological
insulator with time-reversal symmetry and shares with it such properties
as a quantized magnetoelectric effect. However, for certain surfaces the
surface states are intrinsically gapped with a half-quantum Hall effect
[σxy=e2/(2h)], which may aid experimental
confirmation of θ=π quantized magnetoelectric coupling. Step edges on
such a surface support gapless, chiral quantum wires. In closing we
discuss GdBiPt as a possible example of this topological
class.
Magnetoelectric responses are a fundamental characteristic of
materials that break time-reversal and inversion symmetries (notably
multiferroics) and, remarkably, of “topological insulators” in which
those symmetries are unbroken. Previous work has shown how to compute
spin and lattice contributions to the magnetoelectric tensor. Here we
solve the problem of orbital contributions by computing the
frozen-lattice electronic polarization induced by a magnetic field. One
part of this response (the “Chern-Simons term”) can appear even in
time-reversal-symmetric materials and has been previously shown to be
quantized in topological insulators. In general materials there are
additional orbital contributions to all parts of the magnetoelectric
tensor; these vanish in topological insulators by symmetry and also
vanish in several simplified models without time reversal and inversion
whose magnetoelectric couplings were studied before. We give two
derivations of the response formula, one based on a uniform magnetic
field and one based on extrapolation of a long-wavelength magnetic
field, and discuss some of the consequences of this formula.
Several new features arise in the ground-state phase diagram of a
spin-1 condensate trapped in an optical trap when the magnetic-dipole
interaction between the atoms is taken into account along with
confinement and spin precession. The boundaries between the regions of
ferromagnetic and polar phases move as the dipole strength is varied
and the ferromagnetic phases can be modulated. The magnetization of the
ferromagnetic phase perpendicular to the field becomes modulated as a
helix winding around the magnetic field direction with a wavelength
inversely proportional to the dipole strength. This modulation should be
observable for current experimental parameters in 87Rb. Hence
the much-sought supersolid state with broken continuous translation
invariance in one direction and broken global U(1) invariance, occurs
generically as a metastable state in this system as a result of dipole
interaction. The ferromagnetic state parallel to the applied magnetic
field becomes striped in a finite system at strong dipolar
coupling.
The orbital motion of electrons in a three-dimensional solid can
generate a pseudoscalar magnetoelectric coupling θ, a fact we
derive for the single-particle case using a recent theory of polarization
in weakly inhomogeneous materials. This polarizability θ is the
same parameter that appears in the “axion electrodynamics” Lagrangian
ΔℒEM=(θe2/2πh)E·B, which is
known to describe the unusual magnetoelectric properties of the
three-dimensional topological insulator (θ=π). We compute
θ for a simple model that accesses the topological insulator and
discuss its connection to the surface Hall conductivity. The orbital
magnetoelectric polarizability can be generalized to the many-particle
wave function and defines the 3D topological insulator, like the integer
quantum Hall effect, in terms of a topological ground-state response
function.
The topological insulator is an electronic phase stabilized by
spin-orbit coupling that supports propagating edge states and is not
adiabatically connected to the ordinary insulator. In several ways it
is a spin-orbit-induced analog in time-reversal-invariant systems of
the integer quantum Hall effect (IQHE). This paper studies the
topological insulator phase in disordered two-dimensional systems,
using a model graphene Hamiltonian introduced by Kane and Mele
[Phys. Rev. Lett.95, 226801 (2005)] as an example. The
nonperturbative definition of a topological insulator given here is
distinct from previous efforts in that it involves boundary phase
twists that couple only to charge, does not refer to edge states, and
can be measured by pumping cycles of ordinary charge. In this
definition, the phase of a Slater determinant of electronic states is
determined by a Chern parity analogous to Chern number in the IQHE
case. Numerically, we find, in agreement with recent network model
studies, that the direct transition between ordinary and topological
insulators that occurs in band structures is a consequence of the
perfect crystalline lattice. Generically, these two phases are
separated by a metallic phase, which is allowed in two dimensions when
spin-orbit coupling is present. The same approach can be used to study
three-dimensional topological insulators.
In quantum mechanics a localized attractive potential typically
supports a (possibly infinite) set of bound states, characterized by a
discrete spectrum of allowed energies, together with a continuum of
scattering states, characterized (in one dimension) by an
energy-dependent phase shift. The 1/x2 potential on
0<x<∞confounds all of our intuitions and expectations.
Resolving its paradoxes requires sophisticated theoretical machinery:
regularization, renormalization, anomalous symmetry-breaking, and
self-adjoint extensions. Our goal is to introduce the essential ideas
at a level accessible to advanced undergraduates.