The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimension physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer's companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure

Fusion rules in conformal field theory

Several aspects of fusion rings and fusion rule algebras, and of their manifestations in twodimensional (conformal) field theory, are described: diagonalization and the connection with modular invariance; the presentation in terms of quotients of polynomial rings; fusion graphs; various strategies that allow for a partial classification; and the role of the fusion rules in the conformal bootstrap programme.Comment: 68 pages, LaTeX. changed contents of footnote no.

Null Cones and Einstein's Equations in Minkowski Spacetime

If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segr\'{e} classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the generalized eigenvector formalism reduce the configuration space to the causality-respecting part. In this smaller space, gauge transformations do not form a group, but only a groupoid. The flat metric removes the difficulty of defining equal-time commutation relations in quantum gravity and guarantees global hyperbolicity

Niobium hyperfine structure in crystal calcium tungstate

A study of the niobium hyperfine structure in single crystal calcium tungstate was made by the combination of the technique of electron paramagnetic resonance and electron nuclear double resonance (EPR/ENDOR). The microwave frequency was about 9.4 GHz and the radio frequency from 20MHz to 70 MHz. The rare earth ions Nd(3+), U(3+), or Tm(3+) were added as the charge compensator for Nb(5+). To create niobium paramagnetic centers, the sample was irradiated at 77 deg K with a 10 thousand curie Co-60 gamma source for 1 to 2 hours at a dose rate of 200 K rads per hour and then transferred quickly into the cavity. In a general direction of magnetic field, the spectra showed 4 sets of 10 main lines corresponding to 4 nonequivalent sites of niobium with I = 9/2. These 4 sets of lines coalesced into 2 sets of 10 in the ab-plane and into a single set of 10 along the c-axis. This symmetry suggested that the tungsten ions are substituted by the niobium ions in the crystal