Symmetry classes of disordered fermions

Building upon Dyson's fundamental 1962 article known in random-matrix theory as 'the threefold way', we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds. The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V, a space of endomorphisms in End(V+V*), a bilinear form on V+V* which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V*. Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way. This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.Comment: 52 pages, dedicated to Freeman J. Dyson on the occasion of his 80th birthda

Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model

We generalize the string functions C_{n,r}(tau) associated with the coset ^sl(2)_k/u(1) to higher string functions A_{n,r}(tau) and B_{n,r}(tau) associated with the coset W(k)/u(1) of the W-algebra of the logarithmically extended ^sl(2)_k conformal field model with positive integer k. The higher string functions occur in decomposing W(k) characters with respect to level-k theta and Appell functions and their derivatives (the characters are neither quasiperiodic nor holomorphic, and therefore cannot decompose with respect to only theta-functions). The decomposition coefficients, to be considered ``logarithmic parafermionic characters,'' are given by A_{n,r}(tau), B_{n,r}(tau), C_{n,r}(tau), and by the triplet \mathscr{W}(p)-algebra characters of the (p=k+2,1) logarithmic model. We study the properties of A_{n,r} and B_{n,r}, which nontrivially generalize those of the classic string functions C_{n,r}, and evaluate the modular group representation generated from A_{n,r}(tau) and B_{n,r}(tau); its structure inherits some features of modular transformations of the higher-level Appell functions and the associated transcendental function Phi.Comment: 34 pages, amsart++, times. V2: references added; minor changes; some nonsense in B.3.3. correcte

Chiral Dynamics and Fermion Mass Generation in Three Dimensional Gauge Theory

We examine the possibility of fermion mass generation in 2+1- dimensional gauge theory from the current algebra point of view.In our approach the critical behavior is governed by the fluctuations of pions which are the Goldstone bosons for chiral symmetry breaking. Our analysis supports the existence of an upper critical number of Fermion flavors and exhibits the explicit form of the gap equation as well as the form of the critical exponent for the inverse correlation lenght of the order parameterComment: Latex,10 pages,DFUPG 70/9

Multiplet Effects in the Quasiparticle Band Structure of the f1−f2f^1-f^2 Anderson Model

In this paper, we examine the mean field electronic structure of the $f^1-f^2$ Anderson lattice model in a slave boson approximation, which should be useful in understanding the physics of correlated metals with more than one f electron per site such as uranium-based heavy fermion superconductors. We find that the multiplet structure of the $f^2$ ion acts to quench the crystal field splitting in the quasiparticle electronic structure. This is consistent with experimental observations in such metals as $UPt_3$.Comment: 9 pages, revtex, 3 uuencoded postscript figures attached at en

Absence of a metallic phase in random-bond Ising models in two dimensions: applications to disordered superconductors and paired quantum Hall states

When the two-dimensional random-bond Ising model is represented as a noninteracting fermion problem, it has the same symmetries as an ensemble of random matrices known as class D. A nonlinear sigma model analysis of the latter in two dimensions has previously led to the prediction of a metallic phase, in which the fermion eigenstates at zero energy are extended. In this paper we argue that such behavior cannot occur in the random-bond Ising model, by showing that the Ising spin correlations in the metallic phase violate the bound on such correlations that results from the reality of the Ising couplings. Some types of disorder in spinless or spin-polarized p-wave superconductors and paired fractional quantum Hall states allow a mapping onto an Ising model with real but correlated bonds, and hence a metallic phase is not possible there either. It is further argued that vortex disorder, which is generic in the fractional quantum Hall applications, destroys the ordered or weak-pairing phase, in which nonabelian statistics is obtained in the pure case.Comment: 13 pages; largely independent of cond-mat/0007254; V. 2: as publishe

Interacting Open Wilson Lines in Noncommutative Field Theories

In noncommutative field theories, it was known that one-loop effective action describes propagation of non-interacting open Wilson lines, obeying the flying dipole's relation. We show that two-loop effective action describes cubic interaction among `closed string' states created by open Wilson lines. Taking d-dimensional noncommutative [\Phi^3] theory as the simplest setup, we compute nonplanar contribution at low-energy and large noncommutativity limit. We find that the contribution is expressible in a remarkably simple cubic interaction involving scalar open Wilson lines only and nothing else. We show that the interaction is purely geometrical and noncommutative in nature, depending only on sizes of each open Wilson line.Comment: v1: 27 pages, Latex, 7 .eps figures v2: minor wording change + reference adde

The thermal decomposition of studtite: analysis of the amorphous phase

Studtite is known to exist at the back-end of the nuclear fuel cycle as an intermediate phase formed in the reprocessing of spent nuclear fuel. In the thermal decomposition of studtite, an amorphous phase is obtained at calcination temperatures between 200 and 500 °C. This amorphous compound, referred to elsewhere in the literature as U2O7, has been characterised by analytical spectroscopic methods. The local structure of the amorphous compound has been found to contain uranyl bonding by X-ray absorption near edge (XANES), Fourier transform infrared and Raman spectroscopy. Changes in bond distances in the uranyl group are discussed with respect to studtite calcination temperature. The reaction of the amorphous compound with water to form metaschoepite is also discussed and compared with the structure of schoepite and metaschoepite by X-ray diffraction. A novel schematic reaction mechanism for the thermal decomposition of studtite is proposed

Modular Invariants in the Fractional Quantum Hall Effect

We investigate the modular properties of the characters which appear in the partition functions of nonabelian fractional quantum Hall states. We first give the annulus partition function for nonabelian FQH states formed by spinon and holon (spinon-holon state). The degrees of freedom of spin are described by the affine SU(2) Kac-Moody algebra at level $k$. The partition function and the Hilbert space of the edge excitations decomposed differently according to whether $k$ is even or odd. We then investigate the full modular properties of the extended characters for nonabelian fractional quantum Hall states. We explicitly verify the modular invariance of the annulus grand partition functions for spinon-holon states, the Pfaffian state and the 331 states. This enables one to extend the relation between the modular behavior and the topological order to nonabelian cases. For the Haldane-Rezayi state, we find that the extended characters do not form a representation of the modular group, thus the modular invariance is broken.Comment: Latex,21 pages.version to appear in Nucl.Phys.

The two-dimensional random-bond Ising model, free fermions and the network model

We develop a recently-proposed mapping of the two-dimensional Ising model with random exchange (RBIM), via the transfer matrix, to a network model for a disordered system of non-interacting fermions. The RBIM transforms in this way to a localisation problem belonging to one of a set of non-standard symmetry classes, known as class D; the transition between paramagnet and ferromagnet is equivalent to a delocalisation transition between an insulator and a quantum Hall conductor. We establish the mapping as an exact and efficient tool for numerical analysis: using it, the computational effort required to study a system of width $M$ is proportional to $M^{3}$, and not exponential in $M$ as with conventional algorithms. We show how the approach may be used to calculate for the RBIM: the free energy; typical correlation lengths in quasi-one dimension for both the spin and the disorder operators; even powers of spin-spin correlation functions and their disorder-averages. We examine in detail the square-lattice, nearest-neighbour $\pm J$ RBIM, in which bonds are independently antiferromagnetic with probability $p$, and ferromagnetic with probability $1-p$. Studying temperatures $T\geq 0.4J$, we obtain precise coordinates in the $p-T$ plane for points on the phase boundary between ferromagnet and paramagnet, and for the multicritical (Nishimori) point. We demonstrate scaling flow towards the pure Ising fixed point at small $p$, and determine critical exponents at the multicritical point.Comment: 20 pages, 25 figures, figures correcte