Compression of Correlation Matrices and an Efficient Method for Forming Matrix Product States of Fermionic Gaussian States

Here we present an efficient and numerically stable procedure for compressing a correlation matrix into a set of local unitary single-particle gates, which leads to a very efficient way of forming the matrix product state (MPS) approximation of a pure fermionic Gaussian state, such as the ground state of a quadratic Hamiltonian. The procedure involves successively diagonalizing subblocks of the correlation matrix to isolate local states which are purely occupied or unoccupied. A small number of nearest neighbor unitary gates isolates each local state. The MPS of this state is formed by applying the many-body version of these gates to a product state. We treat the simple case of compressing the correlation matrix of spinless free fermions with definite particle number in detail, though the procedure is easily extended to fermions with spin and more general BCS states (utilizing the formalism of Majorana modes). We also present a DMRG-like algorithm to obtain the compressed correlation matrix directly from a hopping Hamiltonian. In addition, we discuss a slight variation of the procedure which leads to a simple construction of the multiscale entanglement renormalization ansatz (MERA) of a fermionic Gaussian state, and present a simple picture of orthogonal wavelet transforms in terms of the gate structure we present in this paper. As a simple demonstration we analyze the Su-Schrieffer-Heeger model (free fermions on a 1D lattice with staggered hopping amplitudes).Comment: 15 pages, 17 figure

Probabilistic analysis of a differential equation for linear programming

In this paper we address the complexity of solving linear programming problems with a set of differential equations that converge to a fixed point that represents the optimal solution. Assuming a probabilistic model, where the inputs are i.i.d. Gaussian variables, we compute the distribution of the convergence rate to the attracting fixed point. Using the framework of Random Matrix Theory, we derive a simple expression for this distribution in the asymptotic limit of large problem size. In this limit, we find that the distribution of the convergence rate is a scaling function, namely it is a function of one variable that is a combination of three parameters: the number of variables, the number of constraints and the convergence rate, rather than a function of these parameters separately. We also estimate numerically the distribution of computation times, namely the time required to reach a vicinity of the attracting fixed point, and find that it is also a scaling function. Using the problem size dependence of the distribution functions, we derive high probability bounds on the convergence rates and on the computation times.Comment: 1+37 pages, latex, 5 eps figures. Version accepted for publication in the Journal of Complexity. Changes made: Presentation reorganized for clarity, expanded discussion of measure of complexity in the non-asymptotic regime (added a new section

Oil vulnerability in Melbourne

Peak Oil and Climate Change present serious challenges to governments and planners. The sprawling auto based city, which is the model upon which Australian cities have grown is particularly unsuited to a situation of decreasing oil availability and a need to reduce carbon emissions. The aim of this study is to investigate and expose possible variations in the spatial distribution of oil vulnerability in Melbourne. This study assesses vehicle ownership and usage characteristics by local government area (LGA), using data collected by the Victorian Department of Transport's Victorian Integrated Survey of Travel and Activity (VISTA) analysis. An Oil Vulnerability Index has been created and its application suggests that the fast growing outer suburbs of Melbourne are particularly vulnerable to oil price rises. Outer Suburban LGAs were found to have lower average incomes and travel by car more frequently and for longer distances. Future petrol price increases are likely to place stress on household expenditure, mobility and even the long-term viability of some suburbs

An X-ray shadowgraph to locate transient high-energy celestial sources

A new technique has been developed to locate strong, transient X-ray sources such as the recently discovered gamma ray bursts. The instrument, termed a shadowgraph, locates sources by detecting the X-ray shadow cast by a large occulting mask pattern on an imaging detector. Angular resolutions of from 2 to 10 arc minutes are obtainable while essentially full sky coverage is maintained. The optimum energy range of operation is between 20 keV and 100 keV. The high efficiency X-ray imaging detectors, which make it possible to locate bursts with intensities down to approximately 10 photons/sq cm sec, are capable of detecting single 20 keV photons with a spatial resolution of approximately 0.2 mm. The detectors consist of an X-ray to optical conversion phosphor, a multistage image intensifier, and a CCD image readout

Faster Methods for Contracting Infinite 2D Tensor Networks

We revisit the corner transfer matrix renormalization group (CTMRG) method of Nishino and Okunishi for contracting two-dimensional (2D) tensor networks and demonstrate that its performance can be substantially improved by determining the tensors using an eigenvalue solver as opposed to the power method used in CTMRG. We also generalize the variational uniform matrix product state (VUMPS) ansatz for diagonalizing 1D quantum Hamiltonians to the case of 2D transfer matrices and discuss similarities with the corner methods. These two new algorithms will be crucial to improving the performance of variational infinite projected entangled pair state (PEPS) methods.Comment: 20 pages, 5 figures, V. Zauner-Stauber previously also published under the name V. Zaune

Double Exchange in a Magnetically Frustrated System

This work examines the magnetic order and spin dynamics of a double-exchange model with competing ferromagnetic and antiferromagnetic Heisenberg interactions between the local moments. The Heisenberg interactions are periodically arranged in a Villain configuration in two dimensions with nearest-neighbor, ferromagnetic coupling $J$ and antiferromagnetic coupling $-\eta J$. This model is solved at zero temperature by performing a $1/\sqrt{S}$ expansion in the rotated reference frame of each local moment. When $\eta$ exceeds a critical value, the ground state is a magnetically frustrated, canted antiferromagnet. With increasing hopping energy $t$ or magnetic field $B$, the local moments become aligned and the ferromagnetic phase is stabilized above critical values of $t$ or $B$. In the canted phase, a charge-density wave forms because the electrons prefer to sit on lines of sites that are coupled ferromagnetically. Due to a change in the topology of the Fermi surface from closed to open, phase separation occurs in a narrow range of parameters in the canted phase. In zero field, the long-wavelength spin waves are isotropic in the region of phase separation. Whereas the average spin-wave stiffness in the canted phase increases with $t$ or $\eta$, it exhibits a more complicated dependence on field. This work strongly suggests that the jump in the spin-wave stiffness observed in Pr$_{1-x}$Ca$_x$MnO$_3$ with $0.3 \le x \le 0.4$ at a field of 3 T is caused by the delocalization of the electrons rather than by the alignment of the antiferromagnetic regions.Comment: 28 pages, 12 figure