939 research outputs found
Emergent Many-Body Translational Symmetries of Abelian and Non-Abelian Fractionally Filled Topological Insulators
The energy and entanglement spectrum of fractionally filled interacting
topological insulators exhibit a peculiar manifold of low energy states
separated by a gap from a high energy set of spurious states. In the current
manuscript, we show that in the case of fractionally filled Chern insulators,
the topological information of the many-body state developing in the system
resides in this low-energy manifold. We identify an emergent many-body
translational symmetry which allows us to separate the states in
quasi-degenerate center of mass momentum sectors. Within one center of mass
sector, the states can be further classified as eigenstates of an emergent (in
the thermodynamic limit) set of many-body relative translation operators. We
analytically establish a mapping between the two-dimensional Brillouin zone for
the Fractional Quantum Hall effect on the torus and the one for the fractional
Chern insulator. We show that the counting of quasi-degenerate levels below the
gap for the Fractional Chern Insulator should arise from a folding of the
states in the Fractional Quantum Hall system at identical filling factor. We
show how to count and separate the excitations of the Laughlin, Moore-Read and
Read-Rezayi series in the Fractional Quantum Hall effect into two-dimensional
Brillouin zone momentum sectors, and then how to map these into the momentum
sectors of the Fractional Chern Insulator. We numerically check our results by
showing the emergent symmetry at work for Laughlin, Moore-Read and Read-Rezayi
states on the checkerboard model of a Chern insulator, thereby also showing, as
a proof of principle, that non-Abelian Fractional Chern Insulators exist.Comment: 32 pages, 9 figure
Correlation Lengths and Topological Entanglement Entropies of Unitary and Non-Unitary Fractional Quantum Hall Wavefunctions
Using the newly developed Matrix Product State (MPS) formalism for
non-abelian Fractional Quantum Hall (FQH) states, we address the question of
whether a FQH trial wave function written as a correlation function in a
non-unitary Conformal Field Theory (CFT) can describe the bulk of a gapped FQH
phase. We show that the non-unitary Gaffnian state exhibits clear signatures of
a pathological behavior. As a benchmark we compute the correlation length of
Moore-Read state and find it to be finite in the thermodynamic limit. By
contrast, the Gaffnian state has infinite correlation length in (at least) the
non-Abelian sector, and is therefore gapless. We also compute the topological
entanglement entropy of several non-abelian states with and without quasiholes.
For the first time in FQH the results are in excellent agreement in all
topological sectors with the CFT prediction for unitary states. For the
non-unitary Gaffnian state in finite size systems, the topological entanglement
entropy seems to behave like that of the Composite Fermion Jain state at equal
filling.Comment: 5 pages, 5 figures, and supplementary material. Published versio
D-Algebra Structure of Topological Insulators
In the quantum Hall effect, the density operators at different wave-vectors
generally do not commute and give rise to the Girvin MacDonald Plazmann (GMP)
algebra with important consequences such as ground-state center of mass
degeneracy at fractional filling fraction, and W_{1 + \infty} symmetry of the
filled Landau levels. We show that the natural generalization of the GMP
algebra to higher dimensional topological insulators involves the concept of a
D-algebra formed by using the fully anti-symmetric tensor in D-dimensions. For
insulators in even dimensional space, the D-algebra is isotropic and closes for
the case of constant non-Abelian F(k) ^ F(k) ... ^ F(k) connection (D-Berry
curvature), and its structure factors are proportional to the D/2-Chern number.
In odd dimensions, the algebra is not isotropic, contains the weak topological
insulator index (layers of the topological insulator in one less dimension) and
does not contain the Chern-Simons \theta form (F ^ A - 2/3 A ^ A ^ A in 3
dimensions). The Chern-Simons form appears in a certain combination of the
parallel transport and simple translation operator which is not an algebra. The
possible relation to D-dimensional volume preserving diffeomorphisms and
parallel transport of extended objects is also discussed.Comment: 5 page
Matrix Product State Description and Gaplessness of the Haldane-Rezayi State
We derive an exact matrix product state representation of the Haldane-Rezayi
state on both the cylinder and torus geometry. Our derivation is based on the
description of the Haldane-Rezayi state as a correlator in a non-unitary
logarithmic conformal field theory. This construction faithfully captures the
ten degenerate ground states of this model state on the torus. Using the
cylinder geometry, we probe the gapless nature of the phase by extracting the
correlation length, which diverges in the thermodynamic limit. The numerically
extracted topological entanglement entropies seem to only probe the Abelian
part of the theory, which is reminiscent of the Gaffnian state, another model
state deriving from a non-unitary conformal field theory.Comment: Corrected labels in Fig.
Extracting Excitations From Model State Entanglement
We extend the concept of entanglement spectrum from the geometrical to the
particle bipartite partition. We apply this to several Fractional Quantum Hall
(FQH) wavefunctions on both sphere and torus geometries to show that this new
type of entanglement spectra completely reveals the physics of bulk quasihole
excitations. While this is easily understood when a local Hamiltonian for the
model state exists, we show that the quasiholes wavefunctions are encoded
within the model state even when such a Hamiltonian is not known. As a
nontrivial example, we look at Jain's composite fermion states and obtain their
quasiholes directly from the model state wavefunction. We reach similar
conclusions for wavefunctions described by Jack polynomials.Comment: 5 pages, 7 figures, updated versio
fractional topological insulators in two dimensions
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 .Comment: 15 pages, 18 figure
Eta-Pairing in Hubbard Models: From Spectrum Generating Algebras to Quantum Many-Body Scars
We revisit the -pairing states in Hubbard models and explore their
connections to quantum many-body scars to discover a universal scars mechanism.
-pairing occurs due to an algebraic structure known as a Spectrum
Generating Algebra (SGA), giving rise to equally spaced towers of eigenstates
in the spectrum. We generalize the original -pairing construction and
show that several Hubbard-like models on arbitrary graphs exhibit SGAs,
including ones with disorder and spin-orbit coupling. We further define a
Restricted Spectrum Generating Algebra (RSGA) and give examples of
perturbations to the Hubbard-like models that preserve an equally spaced tower
of the original model as eigenstates. The states of the surviving tower exhibit
a sub-thermal entanglement entropy, and we analytically obtain parameter
regimes for which they lie in the bulk of the spectrum, showing that they are
exact quantum many-body scars. The RSGA framework also explains the equally
spaced towers of eigenstates in several well-known models of quantum scars,
including the AKLT model.Comment: 13 pages v2: typos corrected, references adde
Matrix Product States for Trial Quantum Hall States
We obtain an exact matrix-product-state (MPS) representation of a large
series of fractional quantum Hall (FQH) states in various geometries of genus
0. The states in question include all paired k=2 Jack polynomials, such as the
Moore-Read and Gaffnian states, as well as the Read-Rezayi k=3 state. We also
outline the procedures through which the MPS of other model FQH states can be
obtained, provided their wavefunction can be written as a correlator in a 1+1
conformal field theory (CFT). The auxiliary Hilbert space of the MPS, which
gives the counting of the entanglement spectrum, is then simply the Hilbert
space of the underlying CFT. This formalism enlightens the link between
entanglement spectrum and edge modes. Properties of model wavefunctions such as
the thin-torus root partitions and squeezing are recast in the MPS form, and
numerical benchmarks for the accuracy of the new MPS prescription in various
geometries are provided.Comment: 5 pages, 1 figure, published versio
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