Hessian potential for Fefferman-Graham metric

The Fefferman-Graham metric is frequently used for derivation of the first law of the entanglement thermodynamics. On ther other hand, the entanglement thermodynamics is well formulated by the Hessian geometry. The aim of this work is to relate them with each other by finding the corresponding Hessian potential. We find that the deformation of the bulk Hessian potential for the pure AdS spacetime behaves as a source potential of the boundary Fisher metric, and the deformation coincides with the Fefferman-Graham metric. A peculiar feature different from related works is that we need not to use the Ryu-Takayanagi formula for the above derivation. The canonical parameter space in the Hessian geometry is a kind of the model parameter space, rather than the real classical spacetime in the usual setup of the AdS/CFT correspondence. However, the underlying mathematical structure is the same as that of the AdS/CFT correspondence. This suggests the presence of more global class of holographic transformation.Comment: 13 page

Emergent General Relativity from Fisher Information Metric

We derive the Einstein tensor from the Fisher information metric that is defined by the probability distribution of a statistical mechanical system. We find that the tensor naturally contains essential information of the energy-momentum tensor of a classical scalar field, when the entropy data or the spectrum data of the system are embedded into the classical field as the field strength. Thus, we can regard the Einstein equation as the equation of coarse-grained states for the original microscopic system behind the classical field theory. We make some remarks on quantization of gravity and various quantum-classical correspondences.Comment: 20 page

Correspondence between causality in flat Minkowski spacetime and entanglement in thermofield-double state: Hessian-geometrical study

We examine the Hessian potential that derives the flat Minkowski spacetime in $(1+1)$-dimension. The entanglement thermodynamics by the Hessian geometry enables us to obtain the entanglement entropy of a corresponding quantum state by means of holography. We find that the positivity of the entropy leads to the presence of past and future causal cones in the Minkowski spacetime. We also find that the quantum state is equivalent to the thermofield-double state, and then the entropy is proportional to the temperature. The proportionality is consistent with previous holographic works. The present Hessian geometrical approach captures that the causality in the classical side is converted into quantum entanglement inherent in the thermofield dynamics.Comment: 11 page

Geodesic Distance in Fisher Information Space and Holographic Entropy Formula

In this short note, we examine geodesic distance in Fisher information space in which the metric is defined by the entanglement entropy in CFT_(1+1). It is obvious in this case that the geodesic distance at a constant time is a function of the entropy data embedded into the information space. In a special case, the geodesic equation can be solved analytically, and we find that the distance agrees well with the Ryu-Takayanagi formula. Then, we can understand how the distance looks at the embeded quantum information. The result suggests that the Fisher metric is an efficient tool for constructing the holographic spacetime.Comment: 3 page

Topology and Geometric Structure of Branching MERA Network

We examine a bulk-edge correspondence of branching MERA networks at finite temperatures in terms of algebraic and differential topology. By using homeomorphic mapping, we derive that the networks are nonorientable manifolds such as a M$\ddot{\rm o}$bius strip and a Klein bottle. We also examine the stability of the branch in connection with the second law of black hole thermodynamics. Then, we prove that the MERA network for one-dimensional quantum critical systems spontaneously separates into multiple branches in the IR region of the network. On the other hand, the branch does not occur in more than two dimensions. The result illustrates dimensionality dependence of spin-charge separation / coupling. We point out a role of twist of the surfaces on the phase string between spinon and holon excitations.Comment: 10 pages, 9 figure

Entanglement Entropy and Entanglement Spectrum for Two-Dimensional Classical Spin Configuration

In quantum spin chains at criticality, two types of scaling for the entanglement entropy exist: one comes from conformal field theory (CFT), and the other is for entanglement support of matrix product state (MPS) approximation. They indicates that the matrix dimension of the MPS represents a length scale of spin correlation. On the other hand, the quantum spin-chain models can be mapped onto two-dimensional (2D) classical ones. Motivated by the scaling and the mapping, we introduce new entanglement entropy for 2D classical spin configuration as well as entanglement spectrum, and examine their basic properties in Ising and 3-state Potts models on the square lattice. They are defined by the singular values of the reduced density matrix for a Monte Carlo snapshot. We find scaling relations concerned with length scales in the snapshot at $T_{c}$. There, the spin configuration is fractal, and various sizes of ordered clusters coexist. Then, the singular values automatically decompose the original snapshot into a set of images with different length scale. This is the origin of the scaling. In contrast to the MPS scaling, long-range spin correlation can be described by only few singular values. Furthermore, we find multiple gaps in the entanglement spectrum, and in contrast to standard topological phases, the low-lying entanglement levels below the gap represent spontaneous symmetry breaking. Based on these observations, we discuss about the amount of information contained in one snapshot in a viewpoint of the CFT scaling.Comment: 13 pages, 14 figure

BTZ Black Hole in Fisher Information Spacetime

We examine whether we can make a black hole in Fisher information spacetime and what kind of quantum states produce the black hole solution in terms of the anti-de Sitter spacetime/conformal field theory correspondence. Here we focus on the Banados-Teitelboim-Zanelli black hole. There exists a mathematical representation of entanglement spectra that define the Fisher geometry as the black hole spacetime. We find that this representation is quite similar to the entanglement spectra in a conformal field theory at finite temperature except for minor corrections, and then the inverse temperature corresponds to the position of the event horizon in the Poincare coordinate.Comment: 3 page

VBS/CFT Correspondence and Thermal Tensor Network

It has been recently observed that the reduced density matrix of a two-dimensional (2D) valence bond solid state can be mapped onto the thermal density matrix of a 1D Heisenberg quantum spin chain. Motivated by the observation, I propose a very simple phenomenological theory for this type of correspondence based on a finite-temperature tensor network formalism recently developed. I adress close relationship and sharp difference between the present correspondence and multiscale entanglement renormalization ansatz in terms of network geometry.Comment: 4 pages, 4 figure

Multiscale Entanglement Renormalization Ansatz for Kondo Problem

We derive the multiscale entanglement renormalization ansatz (MERA) for the single impuity Kondo model. We find two types of hidden quantum entanglement: one comes from a finite-temperature effect on the geometry of the MERA network, and the other represents screening of the impurity by conduction electrons. As the latter starts to dominate the electronic state, the Kondo physics emerges. The present result is a simple and beautiful example of a holographic dual of a boundary conformal field theory.Comment: 5 pages, 4 figure

Inverse Mellin Transformation of Continuous Singular Value Decomposition: A Route to Holographic Renormalization

We examine holographic renormalization by the singular value decomposition (SVD) of matrix data generated by the Monte Carlo snapshot of the 2D classical Ising model at criticality. To take the continuous limit of the SVD enables us to find the mathematical form of each SVD component by the inverse Mellin transformation as well as the power-law behavior of the SVD spectrum. We find that each SVD component is characterized by the two-point spin correlator with a finite correlation length. Then, the continuous limit of the decomposition index in the SVD corresponds to the inverse of the correlation length. These features strongly suggest that the SVD contains mathematical structure the same as the holographic renormalization.Comment: 7 pages, 2 figure