The adaptive interpolation method for proving replica formulas. Applications to the Curie-Weiss and Wigner spike models

In this contribution we give a pedagogic introduction to the newly introduced adaptive interpolation method to prove in a simple and unified way replica formulas for Bayesian optimal inference problems. Many aspects of this method can already be explained at the level of the simple Curie-Weiss spin system. This provides a new method of solution for this model which does not appear to be known. We then generalize this analysis to a paradigmatic inference problem, namely rank-one matrix estimation, also refered to as the Wigner spike model in statistics. We give many pointers to the recent literature where the method has been succesfully applied

Holographic Geometry of Entanglement Renormalization in Quantum Field Theories

We study a conjectured connection between the AdS/CFT and a real-space quantum renormalization group scheme, the multi-scale entanglement renormalization ansatz (MERA). By making a close contact with the holographic formula of the entanglement entropy, we propose a general definition of the metric in the MERA in the extra holographic direction, which is formulated purely in terms of quantum field theoretical data. Using the continuum version of the MERA (cMERA), we calculate this emergent holographic metric explicitly for free scalar boson and free fermions theories, and check that the metric so computed has the properties expected from AdS/CFT. We also discuss the cMERA in a time-dependent background induced by quantum quench and estimate its corresponding metric.Comment: 42pages, 9figures, reference added, minor chang

The Linear Information Coupling Problems

Many network information theory problems face the similar difficulty of single-letterization. We argue that this is due to the lack of a geometric structure on the space of probability distribution. In this paper, we develop such a structure by assuming that the distributions of interest are close to each other. Under this assumption, the K-L divergence is reduced to the squared Euclidean metric in an Euclidean space. In addition, we construct the notion of coordinate and inner product, which will facilitate solving communication problems. We will present the application of this approach to the point-to-point channel, general broadcast channel, and the multiple access channel (MAC) with the common source. It can be shown that with this approach, information theory problems, such as the single-letterization, can be reduced to some linear algebra problems. Moreover, we show that for the general broadcast channel, transmitting the common message to receivers can be formulated as the trade-off between linear systems. We also provide an example to visualize this trade-off in a geometric way. Finally, for the MAC with the common source, we observe a coherent combining gain due to the cooperation between transmitters, and this gain can be quantified by applying our technique.Comment: 27 pages, submitted to IEEE Transactions on Information Theor

Structural changes at the semiconductor-insulator phase transition in the single layered La0.5Sr1.5MnO4 perovskite

The semiconductor-insulator phase transition of the single-layer manganite La0.5Sr1.5MnO4 has been studied by means of high resolution synchrotron x-ray powder diffraction and resonant x-ray scattering at the Mn K edge. We conclude that a concomitant structural transition from tetragonal I4/mmm to orthorhombic Cmcm phases drives this electronic transition. A detailed symmetry-mode analysis reveals that condensation of three soft modes -Delta_2(B2u), X1+(B2u) and X1+(A)- acting on the oxygen atoms accounts for the structural transformation. The Delta_2 mode leads to a pseudo Jahn-Teller distortion (in the orthorhombic bc-plane only) on one Mn site (Mn1) whereas the two X1+ modes produce an overall contraction of the other Mn site (Mn2) and expansion of the Mn1 one. The X1+ modes are responsible for the tetragonal superlattice (1/2,1/2,0)-type reflections in agreement with a checkerboard ordering of two different Mn sites. A strong enhancement of the scattered intensity has been observed for these superlattice reflections close to the Mn K edge, which could be ascribed to some degree of charge disproportion between the two Mn sites of about 0.15 electrons. We also found that the local geometrical anisotropy of the Mn1 atoms and its ordering originated by the condensed Delta_2 mode alone perfectly explains the resonant scattering of forbidden (1/4,1/4,0)-type reflections without invoking any orbital ordering.Comment: 3 tables and 10 figures; accepted in Phys. Rev.