Matrix Field Theory Research Articles

Vertex models on Feynman diagrams

We discuss the statistical mechanics of vertex models on both generic (“thin”) and planar (“fat”) random graphs. Such models can be formulated as the N → 1 and N → ∞ limits of N × N complex matrix models, respectively. From the graph theoretic perspective one is using matrix model and field theory inspired methods to count various classes of directed graphs. For the thin random graphs we use saddle point methods to solve the models in the thermodynamic, large number of vertices limit and note that, as in the case of the eight-vertex model on the square lattice, various other models such as the Ising model appear as particular limits. The generic solution of the fat graph model is rather more elusive, but we show that for several choices of the couplings the models can be reduced to eigenvalue integrals and their critical behaviour deduced.

DLCQ spectrum ofN=(8,8)super Yang-Mills theory

We consider the 1+1 dimensional N = (8,8) supersymmetric matrix field theory obtained from a dimensional reduction of ten dimensional N = 1 super Yang-Mills. The gauge groups we consider are U(N) and SU(N), where N is finite but arbitrary. We adopt light-cone coordinates, and choose to work in the light-cone gauge. Quantizing this theory via Discretized Light-Cone Quantization (DLCQ) introduces an integer, K, which restricts the light-cone momentum-fraction of constituent quanta to be integer multiples of 1/K. Solutions to the DLCQ bound state equations are obtained for K=2,3 and 4 by discretizing the light-cone super charges, which preserves supersymmetry manifestly. We discuss degeneracies in the massive spectrum that appear to be independent of the light-cone compactification, and are therefore expected to be present in the decompactified limit K -> infinity. Our numerical results also support the claim that the SU(N) theory has a mass gap.

Master Partitions for Large N Matrix Field Theories

We introduce a systematic approach for treating the large N limit of matrix field theories.

Theory of many-fermion systems

A general field-theoretical description of many-fermion systems, both with and without quenched disorder, is developed. Starting from the basic Grassmannian action for interacting fermions, we first bosonize the theory by introducing composite matrix variables that correspond to two-fermion excitations and integrating out the fermion degrees of freedom. The saddle-point solution of the resulting matrix field theory reproduces a disordered Hartree-Fock approximation, and an expansion to Gaussian order about the saddle point corresponds to a disordered random-phase approximation (RPA)-like theory. In the clean limit they reduce to the ordinary Hartree-Fock and random-phase approximations. We first concentrate on disordered systems, and perform a symmetry analysis that allows for a systematic separation of the massless modes from the massive ones. By treating the massive modes in a simple approximation, one obtains a technically satisfactory derivation of the generalized nonlinear $\ensuremath{\sigma}$ model that has been used in the theory of metal-insulator transitions. The theory also allows for the treatment of other phase transitions in the disordered Fermi liquid. We further use renormalization group techniques to establish the existence of a disordered Fermi-liquid fixed point, and show that it is stable for all dimensions $d>2$. The so-called weak-localization effects can be understood as corrections to scaling near this fixed point. The general theory also allows for studying the clean limit. For that case we develop a loop expansion that corresponds to an expansion in powers of the screened Coulomb interaction, and that represents a systematic improvement over RPA. We corroborate the existence of a Fermi-liquid fixed point that is stable for all dimensions $d>1$, in agreement with other recent field-theoretical treatments of clean Fermi liquids.

Quantum critical behavior in gauged Yukawa matrix field theories with quenched disorder

We use the Wilson-Fisher ϵ expansion to study quantum critical behavior in gauged Yukawa matrix field theories with weak quenched disorder. We find that the resulting quantum critical behavior is in the universality class of the pure system. As in the pure system, the phase transition is typically first order, except for a limited range of parameters where it can be second order with computable critical exponents. Our results apply to the study of two-dimensional quantum antiferromagnets with weak quenched disorder and provide an example for fluctuation-induced first order phase transitions in circumstances where naively none are expected.

Matrix theories from reduced SU( N) Yang-Mills with adjoint fermions

We consider a dimensional reduction of 3+1 dimensional SU( N) Yang-Mills theory coupled to adjoint fermions to obtain a class of 1 + 1 dimensional matrix field theories. We derive the quantized light-cone Hamiltonian in the light-cone gauge A_ = 0 and large- N limit, and then solve for the masses, wavefunctions and structure functions of the color singlet boson and fermion bound states. Among the states we study are many massless string-like states that can be solved for exactly.

THE MASTER FIELD FOR LARGE-N MATRIX MODELS AND QUANTUM GROUPS

In recent works by Singer, Douglas, Gopakumar and Gross an application of results of Voiculescu from noncommutative probability theory to constructions of the master field for large-N matrix field theories have been suggested. It turns out that this master field becomes the Boltzmann field in the free Fock space. In this note we consider interrelations between the master field and quantum semigroups. We define the master field algebra and observe that it is isomorphic to the algebra of functions on the quantum semigroup SU q(2) for q=0. The master field becomes a central element of the quantum group bialgebra. The quantum Haar measure on the SU q(2) for any q gives the Wigner semicircle distribution for the master field. Coherent states on SU q(2) become coherent states in the master field theory.

C > 1 non-critical strings and large- N matrix field theory

Motivated by a possible relativistic string description of hadrons we use a discretised light-cone quantisation and Lanczos algorithm to investigate the phase structure of η 3 matrix field theory in the large N limit. In 1 + 1 dimensions we confirm the existence of Polyakov's non-critical string theory at the boundary between parton-like and string-like phases, finding critical exponents for longitudinal oscillations equal to or consistent with those given by a mean field argument. The excitation spectrum is finite, possibly discrete. We calculate light-cone structure functions and find evidence that the probability Q( x) of a parton in the string carrying longitudinal momentum fraction between x and x + dx has support on all 0 < x < 1, despite the average number of partons being infinite.

CRITICAL BEHAVIOR IN c=1 MATRIX MODEL WITH BRANCHING INTERACTIONS

In order to understand the phase structure of d&gt;1 strings we investigate the c=1 matrix model with g′(tr M(t)2)2 interaction which is the simplest approximation of the large-N reduced model of odd-dimensional matrix field theory. We find three distinct phases: (i) an ordinary c=1 gravity phase, (ii) a branched polymer phase and (iii) an intermediate phase, and compute the disk and cylinder amplitudes in the phases (i) and (iii). Further we also analyze the model with slightly generalized [Formula: see text][Formula: see text] interaction. As a result the multi-critical versions of the phase (ii) are found.

A transfer matrix study of layering transitions in Ising model thin films

We study the layering transitions for a spin 1/2 Ising model under uniform and variable magnetic field in a film of finite thickness limited by a single substrate. Using finite size transfer matrix and mean field theory, we show that the layering transitions occurs only at a temperature higher than T L . The latter depends on the nature of the surface field h s and on its strength H s . In the case where the surface field is short-range, the transfer matrix predicts a splitting of a single multi-layer transition into several single-layer transitions whereas mean field theory does not

Light cone quantization of the c=2 matrix model

We study the large N limit of an interacting two-dimensional matrix field theory, whose perturbative expansion generates the sum over planar random graphs embedded in two-dimensions. In the light cone quantization the theory possesses closed string excitations which become free as N→∞. If the longitudinal momenta are discretized, then the calculation of the free string spectrum reduces to finite matrix diagonalization, the size of the matrix growing as the cut-off is removed. Our numerical results suggest that, for a critical coupling, the light cone string spectrum becomes continuous. This would indicate the massless dynamics of the Liouville mode of two-dimensional gravity, which would constitute a third dimension of the string theory.

QUANTUM DISCRETE GAUGE MODELS WITH BOSONIC AND FERMIONIC DEGREES OF FREEDOM

A general approach to quantizing discrete models (i.e. having a finite number of coordinates) with quadratic first-class constraints is presented in the framework of gauging linear canonical symmetries. Also, it is proposed how a natural superextension of matrix field theories (viz. orthosymplectic "zero-dimensional" matrix field theories) might emerge in this approach.

Matrix field theory for disordered local pairing superconductors with two glass order parameters and spin-flip pairbreaking processes

We derive the exact matrix field theory for a replicated grassmannian representation of a local pairing superconducting disorder ensemble including three superconducting order parameters and the spin-flip pairbreaking mechanism. Disorder is assumed to be gaussian distributed. We find by exactly solving the saddle-point equation the criterion for a vanishing gap 〈Δ〉≦τσ−1+τΔ−1, where 〈Δ〉 denotes the averaged superconducting order parameter, τσ−1 the spin-flip scattering rate, and τΔ−1 the scattering rate corresponding to correlations of Re(Δ−〈Δ〉). Taken at Δ=0, our field theory, which is exact in all orders of τσ−1, contains new terms in addition to those of theO(τσ−1) model derived by Efetov et al. Our formulation transfers correctly to all orders the invariances of the action into symmetries of the matrix field theory. The saddle point approximation is outlined and it is shown how singular corrections to the saddle point density of states arise atEF in a gapless superconductor. Finally singular corrections in the two particle propagator, the density correlation function and the conductivity are calculated for 〈Δ〉=0 in one loop order. It turns out that these corrections can be entirely expressed by those of the single particle density of states.

Matrix field theory for disordered electron systems: A useful formalism fo perturbative expansions

Matrix field theories (MFT) have recently been used to describe the metal-insulator transition for interacting disordered electrons. With a suitable parametrization of the field we show a correspondence between these models and the usual many-body perturbation theory. By use of this correspondence it is easier to perform two-loop calculations of the physical quantities. For some of these quantities this technique has already been used but not explicitly described. Our task is to put forward in an explanatory way the main aspects of this formalism. As an application we explicitly perform the two-loop expansion of the single-particle density of states (DOS) in the particular case when magnetic impurities are present. Such calculations turn out to be consistent with the prediction of the first order renormalization group (RG) scaling equations.

Dynamics of disordered interacting quantum systems

Starting from the Liouville-von Neumann equation for the state operator, a functional integral representation of the generating functional for time-temperature dependent Green's functions in interacting disordered quantum systems is constructed. In the framework of this method, quenched averages can be performed without introducing additional, unphysical degrees of freedom, like, e.g., in then-replica method. For interaction-free systems, the dynamical origin of the Schafer-Wegner symmetry is pointed out. For interacting systems we derive a matrix field theory with a single matrix field, which includes all interaction effects without approximations.

Renormalizations in a matrix field theory model in the 1/N expansion

Renormalizations in a matrix field theory model in the 1/N expansion

Inhomogeneity correlations in dirty superconductors, I

It is shown that fluctuations δ‡ around the impurity averaged superconducting order parameter 〈Δ〉 can be large due to interplay of disorder and (decoupled) interaction. Correlations 〈(δΔ)2〉 exceeding a critical value render superconductors with nonmagnetic impurities gapless. In addition to a mixed phase, 〈Δ〉≠0 andMΔ≠0, a superconducting glass phase with 〈Δ〉=0 andMΔ≠0 is derived. In general, correlations 〈(δΔ)2〉 emerge from nonperturbative, saddle point solutions of a replicated matrix field theory. Replica symmetry, soft modes, and other order parameters are discussed.

Introduction to Feynman Diagrams

S M Bilenky Oxford: Pergamon Press 1974 pp viii + 186 price £6 The author's aim in this book, he says in the foreword, is to present the basic principles of S matrix theory and field theory in such a way that physicists (primarily experimental physicists), and advanced students specializing in elementary particle physics, can see easily how the Feynman rules for relativistic perturbation theory arise. The book begins with the elements of scattering theory in the interaction representation, and classical field theory.