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 $L^{d-1}$ 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 $d$, $S\sim c(\partial\Gamma,\partial\Omega)\cdot L^{d-1}\log L$ as the size of a subsystem $L\to\infty$, where $\partial\Gamma$ is the Fermi surface and $\partial\Omega$ is the boundary of the region in real space. The expression for the constant $c(\partial\Gamma,\partial\Omega)$ 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 $S$.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 $\theta$. We first find the basic constraints on $z, \theta$ 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 $\theta$. At short distance, the two-point functions become power-law, with a universal form for $\theta > 0$. 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 $\theta \neq 0$ metrics into full string theory models -- these metrics characterize large regions of the parameter space of Dp-brane metrics for $p\neq 3$. For instance, the theory of N D2-branes in IIA supergravity has z=1 and $\theta = -1/3$ over a wide range of scales, at large $g_s N$.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 $AdS_2\times R^2$ 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, $A$, like $A \log(A)$ 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 $AdS_2\times R^2$ 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