34 research outputs found
Szego limit theorem for operators with discontinuous symbols and applications to entanglement entropy
The main result in this paper is a one term Szego type asymptotic formula
with a sharp remainder estimate for a class of integral operators of the
pseudodifferential type with symbols which are allowed to be non-smooth or
discontinuous in both position and momentum. The simplest example of such
symbol is the product of the characteristic functions of two compact sets, one
in real space and the other in momentum space. The results of this paper are
used in a study of the violation of the area entropy law for free fermions in
[18]. This work also provides evidence towards a conjecture due to Harold
Widom.Comment: 18 pages, major revision, to appear in Int. Math. Res. No
Entanglement entropy of fermions in any dimension and the Widom conjecture
We show that entanglement entropy of free fermions scales faster then area
law, as opposed to the scaling for the harmonic lattice, for example.
We also suggest and provide evidence in support of an explicit formula for the
entanglement entropy of free fermions in any dimension , as the size of a subsystem
, where is the Fermi surface and
is the boundary of the region in real space. The expression for the constant
is based on a conjecture due to H. Widom. We
prove that a similar expression holds for the particle number fluctuations and
use it to prove a two sided estimates on the entropy .Comment: Final versio
Universality for orthogonal and symplectic Laguerre-type ensembles
We give a proof of the Universality Conjecture for orthogonal (beta=1) and
symplectic (beta=4) random matrix ensembles of Laguerre-type in the bulk of the
spectrum as well as at the hard and soft spectral edges. Our results are stated
precisely in the Introduction (Theorems 1.1, 1.4, 1.6 and Corollaries 1.2, 1.5,
1.7). They concern the appropriately rescaled kernels K_{n,beta}, correlation
and cluster functions, gap probabilities and the distributions of the largest
and smallest eigenvalues. Corresponding results for unitary (beta=2)
Laguerre-type ensembles have been proved by the fourth author in [23]. The
varying weight case at the hard spectral edge was analyzed in [13] for beta=2:
In this paper we do not consider varying weights.
Our proof follows closely the work of the first two authors who showed in
[7], [8] analogous results for Hermite-type ensembles. As in [7], [8] we use
the version of the orthogonal polynomial method presented in [25], [22] to
analyze the local eigenvalue statistics. The necessary asymptotic information
on the Laguerre-type orthogonal polynomials is taken from [23].Comment: 75 page
Lower order terms in Szego type limit theorems on Zoll manifolds
This is a detailed version of the paper math.FA/0212273. The main motivation
for this work was to find an explicit formula for a "Szego-regularized"
determinant of a zeroth order pseudodifferential operator (PsDO) on a Zoll
manifold. The idea of the Szego-regularization was suggested by V. Guillemin
and K. Okikiolu. They have computed the second term in a Szego type expansion
on a Zoll manifold of an arbitrary dimension. In the present work we compute
the third asymptotic term in any dimension. In the case of dimension 2, our
formula gives the above mentioned expression for the Szego-redularized
determinant of a zeroth order PsDO. The proof uses a new combinatorial
identity, which generalizes a formula due to G.A.Hunt and F.J.Dyson. This
identity is related to the distribution of the maximum of a random walk with
i.i.d. steps on the real line. The proof of this combinatorial identity
together with historical remarks and a discussion of probabilistic and
algebraic connections has been published separately.Comment: 39 pages, full version, submitte
Aspects of holography for theories with hyperscaling violation
We analyze various aspects of the recently proposed holographic theories with
general dynamical critical exponent z and hyperscaling violation exponent
. We first find the basic constraints on from the gravity
side, and compute the stress-energy tensor expectation values and scalar
two-point functions. Massive correlators exhibit a nontrivial exponential
behavior at long distances, controlled by . At short distance, the
two-point functions become power-law, with a universal form for .
Next, the calculation of the holographic entanglement entropy reveals the
existence of novel phases which violate the area law. The entropy in these
phases has a behavior that interpolates between that of a Fermi surface and
that exhibited by systems with extensive entanglement entropy. Finally, we
describe microscopic embeddings of some metrics into full
string theory models -- these metrics characterize large regions of the
parameter space of Dp-brane metrics for . For instance, the theory of
N D2-branes in IIA supergravity has z=1 and over a wide range
of scales, at large .Comment: 35 pages; v2: new references added; v3: proper reference [14] added;
v4: minor clarification
Entangled Dilaton Dyons
Einstein-Maxwell theory coupled to a dilaton is known to give rise to
extremal solutions with hyperscaling violation. We study the behaviour of these
solutions in the presence of a small magnetic field. We find that in a region
of parameter space the magnetic field is relevant in the infra-red and
completely changes the behaviour of the solution which now flows to an
attractor. As a result there is an extensive ground state
entropy and the entanglement entropy of a sufficiently big region on the
boundary grows like the volume. In particular, this happens for values of
parameters at which the purely electric theory has an entanglement entropy
growing with the area, , like which is believed to be a
characteristic feature of a Fermi surface. Some other thermodynamic properties
are also analysed and a more detailed characterisation of the entanglement
entropy is also carried out in the presence of a magnetic field. Other regions
of parameter space not described by the end point are also
discussed.Comment: Some comments regarding comparison with weakly coupled Fermi liquid
changed, typos corrected and caption of a figure modifie
Abelian gauge potentials on cubic lattices
The study of the properties of quantum particles in a periodic potential
subject to a magnetic field is an active area of research both in physics and
mathematics; it has been and it is still deeply investigated. In this review we
discuss how to implement and describe tunable Abelian magnetic fields in a
system of ultracold atoms in optical lattices. After discussing two of the main
experimental schemes for the physical realization of synthetic gauge potentials
in ultracold set-ups, we study cubic lattice tight-binding models with
commensurate flux. We finally examine applications of gauge potentials in
one-dimensional rings.Comment: To appear on: "Advances in Quantum Mechanics: Contemporary Trends and
Open Problems", G. Dell'Antonio and A. Michelangeli eds., Springer-INdAM
series 201
Block Spin Density Matrix of the Inhomogeneous AKLT Model
We study the inhomogeneous generalization of a 1-dimensional AKLT spin chain
model. Spins at each lattice site could be different. Under certain conditions,
the ground state of this AKLT model is unique and is described by the
Valence-Bond-Solid (VBS) state. We calculate the density matrix of a contiguous
block of bulk spins in this ground state. The density matrix is independent of
spins outside the block. It is diagonalized and shown to be a projector onto a
subspace. We prove that for large block the density matrix behaves as the
identity in the subspace. The von Neumann entropy coincides with Renyi entropy
and is equal to the saturated value.Comment: 20 page
Holographic Entanglement Entropy and Fermi Surfaces
The entanglement entropy in theories with a Fermi surface is known to produce
a logarithmic violation of the usual area law behavior. We explore the
possibility of producing this logarithmic violation holographically by
analyzing the IR regions of the bulk geometries dual to such theories. The
geometry of Ogawa, Takayanagi, and Ugajin is explored and shown to have a null
curvature singularity for all values of parameters, except for dynamical
critical exponent 3/2 in four dimensions. The results are extended to general
hyperscaling violation exponent. We explore strings propagating through the
singularity and show that they become infinitely excited, suggesting the
singularity is not resolved by stringy effects and may become a full-fledged
"stringularity." An Einstein-Maxwell-dilaton embedding of the nonsingular
geometry is exhibited where the dilaton asymptotes to a constant in the IR. The
unique nonsingular geometry in any given number of dimensions is proposed as a
model to study the T=0 limit of a theory with a Fermi surface.Comment: 20 pages plus appendices, 5 figures; v2 discussion clarified, results
generalized, and acknowledgments update
Schr\"odinger Holography with and without Hyperscaling Violation
We study the properties of the Schr\"odinger-type non-relativistic holography
for general dynamical exponent z with and without hyperscaling violation
exponent \theta. The scalar correlation function has a more general form due to
general z as well as the presence of \theta, whose effects also modify the
scaling dimension of the scalar operator. We propose a prescription for minimal
surfaces of this "codimension 2 holography," and demonstrate the (d-1)
dimensional area law for the entanglement entropy from (d+3) dimensional
Schr\"odinger backgrounds. Surprisingly, the area law is violated for d+1 < z <
d+2, even without hyperscaling violation, which interpolates between the
logarithmic violation and extensive volume dependence of entanglement entropy.
Similar violations are also found in the presence of the hyperscaling
violation. Their dual field theories are expected to have novel phases for the
parameter range, including Fermi surface. We also analyze string theory
embeddings using non-relativistic branes.Comment: 62 pages and 6 figures, v2: several typos in section 5 corrected,
references added, v3: typos corrected, references added, published versio