[Show abstract][Hide abstract] ABSTRACT: Recently, it was argued that the braiding and statistics of anyons in a
two-dimensional topological phase can be extracted by studying the quantum
entanglement of the degenerate ground-states on the torus. This construction
either required a lattice symmetry (such as $\pi/2$ rotation) or tacitly
assumed that the `minimum entanglement states' (MESs) for two different
bipartitions can be uniquely assigned quasiparticle labels. Here we describe a
procedure to obtain the modular $\mathcal S$ matrix, which encodes the braiding
statistics of anyons, which does not require making any of these assumptions.
Our strategy is to compare MESs of three independent entanglement bipartitions
of the torus, which leads to a unique modular $\mathcal S$. This procedure also
puts strong constraints on the modular $\mathcal T$ matrix without requiring
any symmetries, and in certain special cases, completely determines it. Our
method applies equally to Abelian and non-Abelian topological phases.
Physical Review B 12/2014; 91(3). DOI:10.1103/PhysRevB.91.035127 · 3.74 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: Quantum Spin Liquids (QSLs) are phases of interacting spins that do not order
even at the absolute zero temperature, making it impossible to characterize
them by a local order parameter. In this article, we review the unique view
provided by the quantum entanglement on QSLs. We illustrate the crucial role of
Topological Entanglement Entropy in diagnosing the non-local order in QSLs,
using specific examples such as the Chiral Spin Liquid. We also demonstrate the
detection of anyonic quasiparticles and their braiding statistics using quantum
entanglement. In the context of gapless QSLs, we discuss the detection of
emergent fermionic spinons in a bosonic wavefunction, by studying the size
dependence of entanglement entropy.
New Journal of Physics 02/2013; 15(2). DOI:10.1088/1367-2630/15/2/025002 · 3.56 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: We use entanglement entropy signatures to establish non-Abelian topological
order in projected Chern-insulator wavefunctions. The simplest instance is
obtained by Gutzwiller projecting a filled band with Chern number C=2, whose
wavefunction may also be viewed as the square of the Slater determinant of a
band insulator. We demonstrate that this wavefunction is captured by the
$SU(2)_2$ Chern Simons theory coupled to fermions. This is established most
persuasively by calculating the modular S-matrix from the candidate ground
state wavefunctions, following a recent entanglement entropy based approach.
This directly demonstrates the peculiar non-Abelian braiding statistics of
Majorana fermion quasiparticles in this state. We also provide microscopic
evidence for the field theoretic generalization, that the Nth power of a Chern
number C Slater determinant realizes the topological order of the $SU(N)_C$
Chern Simons theory coupled to fermions, by studying the $SU(2)_3$ (Read-Rezayi
type state) and the $SU(3)_2$ wavefunctions. An advantage of our projected
Chern insulator wavefunctions is the relative ease with which physical
properties, such as entanglement entropy and modular S-matrix can be
numerically calculated using Monte Carlo techniques.
[Show abstract][Hide abstract] ABSTRACT: Topologically ordered phases are gapped states, defined by the
properties of excitations when taken around each other. By calculating
Topological Entanglement Entropy (TEE) of a disc shaped partition using
a Monte Carlo technique, we establish the existence of topological order
in SU(2) symmetric gapped spin liquids and lattice Laughlin states
obtained by the Gutzwiller projection technique. On the other hand, the
TEE of partitioning the torus into two cylinders is generally different
and depends on the chosen ground state. We demonstrate a method to
extract the statistics and braiding of excitations, given just the set
of ground state wave-functions on a torus. Central to our scheme is the
identification of groundstates with minimum entanglement entropy, which
reflect the quasi-particle excitations. We demonstrate our method by
extracting the modularS matrix of an SU(2) symmetric chiral spin liquid,
and prove that its quasi-particles obey semionic statistics. This method
offers a route to a nearly complete determination of the topological
order in several cases.
[Show abstract][Hide abstract] ABSTRACT: Topologically ordered phases are gapped states, defined by the properties of excitations when taken around one another. Here we demonstrate a method to extract the statistics and braiding of excitations, given just the set of ground-state wave functions on a torus. This is achieved by studying the topological entanglement entropy (TEE) upon partitioning the torus into two cylinders. In this setting, general considerations dictate that the TEE generally differs from that in trivial partitions and depends on the chosen ground state. Central to our scheme is the identification of ground states with minimum entanglement entropy, which reflect the quasiparticle excitations of the topological phase. The transformation of these states allows for the determination of the modular S and U matrices which encode quasiparticle properties. We demonstrate our method by extracting the modular S matrix of a chiral spin liquid phase using a Monte Carlo scheme to calculate the TEE and prove that the quasiparticles obey semionic statistics. This method offers a route to nearly complete determination of the topological order in certain cases.
[Show abstract][Hide abstract] ABSTRACT: Quantum spin liquids are phases of matter whose internal structure is not captured by a local order parameter. Particularly intriguing are critical spin liquids, where strongly interacting excitations control low energy properties. Here we calculate their bipartite entanglement entropy that characterizes their quantum structure. In particular we calculate the Renyi entropy S(2) on model wave functions obtained by Gutzwiller projection of a Fermi sea. Although the wave functions are not sign positive, S(2) can be calculated on relatively large systems (>324 spins) using the variational Monte Carlo technique. On the triangular lattice we find that entanglement entropy of the projected Fermi sea state violates the boundary law, with S(2) enhanced by a logarithmic factor. This is an unusual result for a bosonic wave function reflecting the presence of emergent fermions. These techniques can be extended to study a wide class of other phases.
[Show abstract][Hide abstract] ABSTRACT: We study entanglement properties of candidate wave-functions for SU(2)
symmetric gapped spin liquids and Laughlin states. These wave-functions are
obtained by the Gutzwiller projection technique. Using Topological Entanglement
Entropy \gamma\ as a tool, we establish topological order in chiral spin liquid
and Z2 spin liquid wave-functions, as well as a lattice version of the Laughlin
state. Our results agree very well with the field theoretic result \gamma =log
D where D is the total quantum dimension of the phase. All calculations are
done using a Monte Carlo technique on a 12 times 12 lattice enabling us to
extract \gamma\ with small finite size effects. For a chiral spin liquid
wave-function, the calculated value is within 4% of the ideal value. We also
find good agreement for a lattice version of the Laughlin \nu =1/3 phase with
the expected \gamma=log \sqrt{3}.
[Show abstract][Hide abstract] ABSTRACT: There are many phases of insulators with inversion symmetry (with no
other symmetry required). In particular, certain inversion parities
cannot change unless there is a phase transition. I will show how to use
these parities to classify phases of topological insulators and explain
which combinations of these parities have physical consequences (e.g.
for the magnetoelectric effect). Many of these results can be derived by
pictorial arguments using the entanglement spectrum.
[Show abstract][Hide abstract] ABSTRACT: Spin liquids are exotic quantum states that do not break any symmetry. Though much is known about gapped spin-liquids, critical spin-liquids with strongly interacting gapless excitations in two and three spatial dimensions are less understood. Candidate ground state wave-functions for such states however can be constructed using the Gutzwiller projection method. We use bipartite entanglement entropy, in particular the Renyi entropy S2 to investigate the quantum structure of these wave-functions. Using the Variational Monte-Carlo technique, we calculate the Renyi entropy of a critical spin liquid - the projected Fermi sea state on the triangular lattice. We find a violation of the boundary law, with S2 enhanced by a logarithmic factor, an unusual result for a bosonic wave-function reflecting the presence of emergent spinons that form a Fermi surface. The Renyi entropy for algebraic spin liquids is found to obey the area law, consistent with the presence of emergent Dirac fermions in the system. Projection is found to completely alter the entanglement properties of nested Fermi surface states. These results show that the Renyi entropy calculations could serve as a diagnostic for gapless fractionalized phases.
[Show abstract][Hide abstract] ABSTRACT: Topological band insulators are usually characterized by symmetry-protected surface modes or quantized linear-response functions (like Hall conductance). Here we present a way to characterize them based on certain bulk properties of just the ground-state wave function, specifically, the properties of its entanglement spectrum. We prove that whenever protected surface states exist, a corresponding protected “mode” exists in the entanglement spectrum as well. Besides this, the entanglement spectrum sometimes succeeds better at indicating topological phases than surface states. We discuss specifically the example of insulators with inversion symmetry which is found to act as an antiunitary symmetry on the entanglement spectrum. A Kramers degeneracy can then arise even when time-reversal symmetry is absent. This degeneracy persists for interacting systems. The entanglement spectrum is therefore a promising tool to characterize topological band insulators and superconductors beyond the free-particle approximation.
[Show abstract][Hide abstract] ABSTRACT: We study Aharonov-Bohm (AB) conductance oscillations arising from the surface states of a topological insulator nanowire, when a magnetic field is applied along its length. With strong surface disorder, these oscillations are predicted to have a component with anomalous period Φ(0)=hc/e, twice the conventional period. The conductance maxima are achieved at odd multiples of 1/2Φ(0), implying that a π AB phase for electrons strengthens the metallic nature of surface states. This effect is special to topological insulators, and serves as a defining transport property. A key ingredient, the surface curvature induced Berry phase, is emphasized here. We discuss similarities and differences from recent experiments on Bi2Se3 nanoribbons, and optimal conditions for observing this effect.
[Show abstract][Hide abstract] ABSTRACT: We study three dimensional insulators with inversion symmetry, in which other
point group symmetries, such as time reversal, are generically absent. Their
band topology is found to be classified by the parities of occupied states at
time reversal invariant momenta (TRIM parities), and by three Chern numbers.
The TRIM parities of any insulator must satisfy a constraint: their product
must be +1. The TRIM parities also constrain the Chern numbers modulo two. When
the Chern numbers vanish, a magneto-electric response parameterized by "theta"
is defined and is quantized to theta= 0, 2pi. Its value is entirely determined
by the TRIM parities. These results may be useful in the search for magnetic
topological insulators with large theta. A classification of inversion
symmetric insulators is also given for general dimensions. An alternate
geometrical derivation of our results is obtained by using the entanglement
spectrum of the ground state wave-function.
[Show abstract][Hide abstract] ABSTRACT: Dislocations in topological insulators can host a one-dimensional metallic state that is topologically protected. We discuss experimental consequences for Bi0.9Sb0.1 alloys, including an unusual strain-induced conductivity effect. With a view to studying interaction effects, microscopic parameters for the one-dimensional metallic modes are derived, starting from a Liu-Allen tight-binding model. The Luttinger parameter for the one-dimensional metal in Bi0.9Sb0.1 is estimated. A different route to a metallic defect line is found in model systems where SU(2) spin rotation symmetry is spontaneously broken, leading to a topological insulator. Line defects of the order parameter are found to be metallic, if a strong topological insulator is realized. We study models exhibiting this phase on the diamond and ideal wurtzite lattices. Prospects for experimental realizations are discussed.
[Show abstract][Hide abstract] ABSTRACT: How do we uniquely identify a quantum phase, given its ground state
wave-function? This is a key question for many body theory especially when we
consider phases like topological insulators, that share the same symmetry but
differ at the level of topology. The entanglement spectrum has been proposed as
a ground state property that captures characteristic edge excitations. Here we
study the entanglement spectrum for topological band insulators. We first show
that insulators with topological surface states will necessarily also have
protected modes in the entanglement spectrum. Surprisingly, however, the
converse is not true. Protected entanglement modes can also appear for
insulators without physical surface states, in which case they capture a more
elusive property. This is illustrated by considering insulators with only
inversion symmetry. Inversion is shown to act in an unusual way, as an
antiunitary operator, on the entanglement spectrum, leading to this protection.
The entanglement degeneracies indicate a variety of different phases in
inversion symmetric insulators, and these phases are argued to be robust to the
introduction of interactions.
[Show abstract][Hide abstract] ABSTRACT: We study three dimensional systems where strong repulsion leads to an
insulating state via spontaneously generated spin-orbit interactions. We
discuss a microscopic model where the resulting state is topological. Such
topological `Mott' insulators differ from their band insulator counterparts in
that they possess an additional order parameter, a rotation matrix, that
describes the spontaneous breaking of spin-rotation symmetry. We show that line
defects of this order are associated with protected one dimensional modes in
the {\em strong} topological Mott insulator, which provides a bulk
characterization of this phase.
[Show abstract][Hide abstract] ABSTRACT: Topological defects, such as domain walls and vortices, have long fascinated physicists. A novel twist is added in quantum systems such as the B-phase of superfluid helium He3, where vortices are associated with low-energy excitations in the cores. Similarly, cosmic strings may be tied to propagating fermion modes. Can analogous phenomena occur in crystalline solids that host a plethora of topological defects? Here, we show that indeed dislocation lines are associated with one-dimensional fermionic excitations in a `topological insulator', a novel phase of matter believed to be realized in the material Bi0.9Sb0.1. In contrast to fermionic excitations in a regular quantum wire, these modes are topologically protected and not scattered by disorder. As dislocations are ubiquitous in real materials, these excitations could dominate spin and charge transport in topological insulators. Our results provide a novel route to creating a potentially ideal quantum wire in a bulk solid.
[Show abstract][Hide abstract] ABSTRACT: Topological defects, such as domain walls and vortices, have long fascinated physicists. A novel twist is added in quantum systems like the B-phase of superfluid helium He$_3$, where vortices are associated with low energy excitations in the cores. Similarly, cosmic strings may be tied to propagating fermion modes. Can analogous phenomena occur in crystalline solids that host a plethora of topological defects? Here we show that indeed dislocation lines are associated with one dimensional fermionic excitations in a `topological insulator', a novel band insulator believed to be realized in the bulk material Bi$_{0.9}$Sb$_{0.1}$. In contrast to fermionic excitations in a regular quantum wire, these modes are topologically protected like the helical edge states of the quantum spin-Hall insulator, and not scattered by disorder. Since dislocations are ubiquitous in real materials, these excitations could dominate spin and charge transport in topological insulators. Our results provide a novel route to creating a potentially ideal quantum wire in a bulk solid.