Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement. arXiv:1712.03557v2 [cond-mat.stat-mech] 21 Dec 2017 Entanglement hamiltonian and contour in inhomogeneous 1D critical systems 2

The single-parameter scaling hypothesis predicts the absence of delocalized states for noninteracting quasiparticles in low-dimensional disordered systems. We show analytically, using a supersymmetric method combined with a renormalization group analysis, as well as numerically that extended states may occur in the one-and two-dimensional Anderson model with a nonrandom hopping falling off as some power of the distance between sites. The different size scaling of the bare level spacing and the renormalized magnitude of the disorder seen by the quasiparticles finally results in the delocalization of states at one of the band edges of the quasiparticle energy spectrum. DOI: 10.1103/PhysRevLett.90.027404 PACS numbers: 78.30.Ly, 36.20.Kd, 71.30.+h, 71.35.Aa Localization of noninteracting quasiparticles in random media with time-reversal symmetry and finiterange hopping have been extensively studied since the seminal paper by Anderson [1]. The hypothesis of singleparameter scaling, introduced in Ref.[2], led to the general belief that all eigenstates of noninteracting quasiparticles were exponentially localized in one (1D) and two (2D) dimensions (see Refs. [3,4] for a comprehensive review) and that localization-delocalization transitions no longer exist in the thermodynamic limit. Even though models with finite-range hopping work nicely in describing a variety of materials, long-range hopping is often found in different physical systems (e.g., Frenkel excitons). Random long-range hopping was found to give rise to delocalization of states not only in threedimensional systems [1] but also in any dimension [5][6][7][8]. Recent studies [9] revised the validity of the singleparameter scaling hypothesis even within the original 1D Anderson model with nearest-neighbor coupling, although did not question the statement that all eigenstates in 1D random systems are localized.In this Letter, we present analytical and numerical proofs that a localization-delocalization transition may occur in 1D and 2D systems with diagonal disorder and nonrandom intersite coupling which falls off according to a powerlike law. Apart from the importance of this finding from a general point of view, it may be relevant for several physical systems. As an example, let us mention dipolar Frenkel excitons on 2D regular lattices where molecules are subjected to randomness due to a disordered environment [10]. Biological light-harvesting antenna systems represent a realization of the model we are dealing with [11,12]. Magnons in 1D and 2D disordered spin systems provide one more example of interest.We consider the Anderson Hamiltonian on a d-dimensional (d 1;2) simple lattice with N N d sites:where jni is the ket-vector of the state localized at site n, and f" n g are random site energies, assumed to be uncorrelated for different sites and distributed uniformly within an interval ÿ=2;=2, thus having zero mean and standard deviation = 12 p . The hopping integrals between lattice sites m and n will be taken in the form J mn J=jm ÿ nj J mm 0, where J ...

An exponential deformation of 1D critical Hamiltonians gives rise to ground states whose entanglement entropy satisfies a volume-law. This effect is exemplified in the XX and Heisenberg models. In the XX case we characterize the crossover between the critical and the maximally entangled ground state in terms of the entanglement entropy and the entanglement spectrum.

In one dimension the area law for the entanglement entropy is violated maximally by the ground state of a strong inhomogeneous spin chain, the so called concentric singlet phase (CSP), that looks like a rainbow connecting the two halves of the chain. In this paper we show that, in the weak inhomogeneity limit, the rainbow state is a thermofield double of a conformal field theory with a temperature proportional to the inhomogeneity parameter.This result suggests some relation of the CSP with black holes. Finally, we propose an extension of the model to higher dimensions.

We propose to simulate a Dirac field near an event horizon using ultracold atoms in an optical lattice. Such a quantum simulator allows for the observation of the celebrated Unruh effect. Our proposal involves three stages: (1) preparation of the ground state of a massless 2D Dirac field in Minkowski spacetime; (2) quench of the optical lattice setup to simulate how an accelerated observer would view that state; (3) measurement of the local quantum fluctuation spectra by one-particle excitation spectroscopy in order to simulate a De Witt detector. According to Unruh's prediction, fluctuations measured in such a way must be thermal. Moreover, following Takagi's inversion theorem, they will obey the Bose-Einstein distribution, which will smoothly transform into the Fermi-Dirac as one of the dimensions of the lattice is reduced.Comment: v4 17 pages, 12 figures, minor changes, similar to the published versio

Abstract. The rainbow chain is an inhomogenous exactly solvable local spin model that, in its ground state, displays a half-chain entanglement entropy growing linearly with the system size. Although many exact results about the rainbow chain are known, the structure of the underlying quantum field theory has not yet been unraveled. Here we show that the universal scaling features of this model are captured by a massless Dirac fermion in a curved spacetime with constant negative curvature R = −h 2 (h is the amplitude of the inhomogeneity). This identification allows us to use recently developed techniques to study inhomogeneous conformal systems and to analytically characterise the entanglement entropies of more general bipartitions. These results are carefully tested against exact numerical calculations. Finally, we study the entanglement entropies of the rainbow chain in thermal states, and find that there is a nontrivial interplay between the rainbow effective temperature T R and the physical temperature T .arXiv:1611.08559v1 [cond-mat.str-el]

We study the low-energy states of the 1D random-hopping model in the strong disordered regime. The entanglement structure is shown to depend solely on the probability distribution for the length of the effective bonds P (l b ), whose scaling and finite-size behavior are established using renormalization-group arguments and a simple model based on random permutations. Parity oscillations are absent in the von Neumann entropy with periodic boundary conditions, but appear in the higher moments of the distribution, such as the variance. The particle-hole excited states leave the bond-structure and the entanglement untouched. Nonetheless, particle addition or removal deletes bonds and leads to an effective saturation of entanglement at an effective block size given by the expected value for the longest bond.

Berry and Keating conjectured that the classical Hamiltonian H = xp is related to the Riemann zeros. A regularization of this model yields semiclassical energies that behave, on average, as the nontrivial zeros of the Riemann zeta function. However, the classical trajectories are not closed, rendering the model incomplete. In this Letter, we show that the Hamiltonian H = x(p + ℓ(p)²/p) contains closed periodic orbits, and that its spectrum coincides with the average Riemann zeros. This result is generalized to Dirichlet L functions using different self-adjoint extensions of H. We discuss the relation of our work to Polya's fake zeta function and suggest an experimental realization in terms of the Landau model.