34 research outputs found
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
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
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
Bulk-Edge Correspondence in the Entanglement Spectra
Li and Haldane conjectured and numerically substantiated that the
entanglement spectrum of the reduced density matrix of ground-states of
time-reversal breaking topological phases (fractional quantum Hall states)
contains information about the counting of their edge modes when the
ground-state is cut in two spatially distinct regions and one of the regions is
traced out. We analytically substantiate this conjecture for a series of FQH
states defined as unique zero modes of pseudopotential Hamiltonians by finding
a one to one map between the thermodynamic limit counting of two different
entanglement spectra: the particle entanglement spectrum, whose counting of
eigenvalues for each good quantum number is identical (up to accidental
degeneracies) to the counting of bulk quasiholes, and the orbital entanglement
spectrum (the Li-Haldane spectrum). As the particle entanglement spectrum is
related to bulk quasihole physics and the orbital entanglement spectrum is
related to edge physics, our map can be thought of as a mathematically sound
microscopic description of bulk-edge correspondence in entanglement spectra. By
using a set of clustering operators which have their origin in conformal field
theory (CFT) operator expansions, we show that the counting of the orbital
entanglement spectrum eigenvalues in the thermodynamic limit must be identical
to the counting of quasiholes in the bulk. The latter equals the counting of
edge modes at a hard-wall boundary placed on the sample. Moreover, we show this
to be true even for CFT states which are likely bulk gapless, such as the
Gaffnian wavefunction.Comment: 20 pages, 6 figure
Haldane Statistics in the Finite Size Entanglement Spectra of Laughlin States
We conjecture that the counting of the levels in the orbital entanglement
spectra (OES) of finite-sized Laughlin Fractional Quantum Hall (FQH) droplets
at filling is described by the Haldane statistics of particles in a
box of finite size. This principle explains the observed deviations of the OES
counting from the edge-mode conformal field theory counting and directly
provides us with a topological number of the FQH states inaccessible in the
thermodynamic limit- the boson compactification radius. It also suggests that
the entanglement gap in the Coulomb spectrum in the conformal limit protects a
universal quantity- the statistics of the state. We support our conjecture with
ample numerical checks.Comment: 4.1 pages, published versio
Real-Space Entanglement Spectrum of Quantum Hall States
We investigate the entanglement spectra arising from sharp real-space
partitions of the system for quantum Hall states. These partitions differ from
the previously utilized orbital and particle partitions and reveal
complementary aspects of the physics of these topologically ordered systems. We
show, by constructing one to one maps to the particle partition entanglement
spectra, that the counting of the real-space entanglement spectra levels for
different particle number sectors versus their angular momentum along the
spatial partition boundary is equal to the counting of states for the system
with a number of (unpinned) bulk quasiholes excitations corresponding to the
same particle and flux numbers. This proves that, for an ideal model state
described by a conformal field theory, the real-space entanglement spectra
level counting is bounded by the counting of the conformal field theory edge
modes. This bound is known to be saturated in the thermodynamic limit (and at
finite sizes for certain states). Numerically analyzing several ideal model
states, we find that the real-space entanglement spectra indeed display the
edge modes dispersion relations expected from their corresponding conformal
field theories. We also numerically find that the real-space entanglement
spectra of Coulomb interaction ground states exhibit a series of branches,
which we relate to the model state and (above an entanglement gap) to its
quasiparticle-quasihole excitations. We also numerically compute the
entanglement entropy for the nu=1 integer quantum Hall state with real-space
partitions and compare against the analytic prediction. We find that the
entanglement entropy indeed scales linearly with the boundary length for large
enough systems, but that the attainable system sizes are still too small to
provide a reliable extraction of the sub-leading topological entanglement
entropy term.Comment: 13 pages, 11 figures; v2: minor corrections and formatting change
Series of Abelian and Non-Abelian States in C>1 Fractional Chern Insulators
We report the observation of a new series of Abelian and non-Abelian
topological states in fractional Chern insulators (FCI). The states appear at
bosonic filling nu= k/(C+1) (k, C integers) in several lattice models, in
fractionally filled bands of Chern numbers C>=1 subject to on-site Hubbard
interactions. We show strong evidence that the k=1 series is Abelian while the
k>1 series is non-Abelian. The energy spectrum at both groundstate filling and
upon the addition of quasiholes shows a low-lying manifold of states whose
total degeneracy and counting matches, at the appropriate size, that of the
Fractional Quantum Hall (FQH) SU(C) (color) singlet k-clustered states
(including Halperin, non-Abelian spin singlet states and their
generalizations). The groundstate momenta are correctly predicted by the FQH to
FCI lattice folding. However, the counting of FCI states also matches that of a
spinless FQH series, preventing a clear identification just from the energy
spectrum. The entanglement spectrum lends support to the identification of our
states as SU(C) color-singlets but offers new anomalies in the counting for
C>1, possibly related to dislocations that call for the development of new
counting rules of these topological states.Comment: 12 pages with supplemental material, 20 figures, published versio