Generalized lattice fermion actions and applications

We derive the general form of lattice fermion action consistent with the requirements of gauge invariance, translation invariance, reflection positivity and invariance under 90° rotations, and involving only bilinear, nearest neighbour couplings. The meaning of the parameters occuring in the action is discussed analyzing the spectrum, the symmetries and the axial Ward identities of the theory, and their renormalization is studied within the Migdal-Kadanoff approximation. In particular we give the relation between the dependence of the vacuum energy density on the CP phases appearing in the action and the mean topological density and susceptibility.     

Derivations commuting with abelian gauge actions on lattice systems

Let be an action of a compact abelian groupG on aC*-algebraA, and assume that the fixed-point subalgebraA is an AF-algebra. We show that if is a closed *-derivation onA commuting with , and the restriction of toA generates a one-parameter group of *-automorphisms, then itself is a generator. In particular, the result applies if is an infinite product action ofG on a UHF algebra. Furthermore, if in this situation ₁ and ₂ are two derivations both satisfying the hypotheses on , and ₁ and ₂ have the same restriction toA , then there exists a one-parameter subgroup of the action with generator ₀ such thatD(₁)D(₂)D(₀) is a joint core for the three derivations, and ₂=₁+₀ on this core.     

Phase diagrams of lattice systems with residual entropy

We introduce an equivalence relation on the family of ground states and generalize the Peierls and Pirogov-Sinai theory of phase transitions to systems with residual entropy. The idea consists in the replacement of the periodic ground states by equivalence classes together with an entropy factor. We apply these results to discuss the phase diagrams of diluted spin-1/2 systems.On leave of absence from the Central Research Institute for Physics, Budapest, Hungary.     

Translation invariant equilibrium states of ferromagnetic Abelian lattice systems

The structure of the set of all translation invariant equilibrium states is determined for all temperatures, for which the free energy is differentiable. Models with several phase transitions are discussed rigorously.     

Model band structure calculations for heavy fermion systems

We present calculations of the quasiparticle band structure for simple heavy fermion systems, based on a mean-field approximation of the Anderson Hamiltonian. The aim of this investigation is to study the influence of the parameters of the Anderson Hamiltonian, position of thef-level and hybridization, on the quasiparticle bands and the form of the Fermi surface. We also calculate the static susceptibility.     

Entanglement and criticality in translationally invariant harmonic lattice systems with finite-range interactions

We discuss the relation between entanglement and criticality in translationally invariant harmonic lattice systems with nonrandom, finite-range interactions. We show that the criticality of the system as well as validity or breakdown of the entanglement area law are solely determined by the analytic properties of the spectral function of the oscillator system, which can easily be computed. In particular, for finite-range couplings we find a one-to-one correspondence between an area-law scaling of the bipartite entanglement and a finite correlation length. This relation is strict in the one-dimensional case and there is strong evidence for the multidimensional case. We also discuss generalizations to couplings with infinite range. Finally, to illustrate our results, a specific 1D example with nearest and next-nearest-neighbor coupling is analyzed.     

Phase transition in lattice surface systems with gonihedric action

We prove the existence of an ordered low-temperature phase in a model of soft-self-avoiding closed random surfaces on a cubic lattice by a suitable extension of Peierls contour method. The statistical weight of each surface configuration depends only on the mean extrinsic curvature and on an interaction term arising when two surfaces touch each other along some contour. The model was introduced by F.J. Wegner and G.K. Savvidy as a lattice version of the gonihedric string, which is an action for triangulated random surfaces.     

Homology theory of lattice fermion doubling

I give a general discussion of the phenomenon of spectrum degeneracy in transcribing continuum field equations to the lattice, using concepts of homology theory. This leads to a topological understanding of the problems in transcribing the Dirac equation and a unified treatment of the many lattice fermion schemes in the literature. The connection between spectrum degeneracy and chiral symmetry is explained geometrically without appealing to quantum effects such as anomalies.     

Phase transitions in anisotropic lattice spin systems

A general method for proving the existence of phase transitions is presented and applied to six nearest neighbor models, both classical and quantum mechanical, on the two dimensional square lattice. Included are some two dimensional Heisenberg models. All models are anisotropic in the sense that the groundstate is only finitely degenerate. Using our method which combines a Peierls argument with reflection positivity, i.e. chessboard estimates, and the principle of exponential localization we show that five of them have long range order at sufficiently low temperature. A possible exception is the quantum mechanical, anisotropic Heisenberg ferromagnet for which reflection positivity isnot proved, but for which the rest of the proof is valid.Work partially supported by U.S. National Science Foundation grant no. MPS 75-11864Work partially supported by U.S. National Science Foundation grant no. MCS 75-21684 A01     

New classical solutions with fermion in conformal invariant field theories

New instanton type solutions for coupled non-linear equations of scalar and fermion are given. Invariance properties of the solutions under the six-dimensional conformal group are studied. Quantum significances are discussed, and the equations of motion for quantum fluctuations turn out to be the eigenvalue equations for the Casimir operators of the O(5) group.     

Shell structure and orbit bifurcations in finite fermion systems

We first give an overview of the shell-correction method which was developed by V.M. Strutinsky as a practicable and efficient approximation to the general self-consistent theory of finite fermion systems suggested by A.B. Migdal and collaborators. Then we present in more detail a semiclassical theory of shell effects, also developed by Strutinsky following original ideas of M.C. Gutzwiller. We emphasize, in particular, the influence of orbit bifurcations on shell structure. We first give a short overview of semiclassical trace formulae, which connect the shell oscillations of a quantum system with a sum over periodic orbits of the corresponding classical system, in what is usually called the “periodic orbit theory”. We then present a case study in which the gross features of a typical double-humped nuclear fission barrier, including the effects of mass asymmetry, can be obtained in terms of the shortest periodic orbits of a cavity model with realistic deformations relevant for nuclear fission. Next we investigate shell structures in a spheroidal cavity model which is integrable and allows for far-going analytical computation. We show, in particular, how period-doubling bifurcations are closely connected to the existence of the so-called “superdeformed” energy minimum which corresponds to the fission isomer of actinide nuclei. Finally, we present a general class of radial power-law potentials which approximate well the shape of a Woods-Saxon potential in the bound region, give analytical trace formulae for it and discuss various limits (including the harmonic oscillator and the spherical box potentials).     

Integrable fermion systems with fourth-order nonlinearity

One integrates the infinite-dimensional fermionic system governed by the superconductivity-like Hamiltonian. The theory of orthogonal polynomials is applied.     

Phase transitions and renormalized structure of lattice gauge theories with fermions

Using recursion relations of the type proposed by Migdal and Kadanoff we discuss the fixed points relevant to the renormalization of lattice gauge theories in four dimensions. The role of the topological excitation for the U(1) case is evaluated.     

Phase structure of U(1) models with mixed actions

We study PQED using an action which is the admixture of two terms with different periodicities. The Villain form of this action is rewritten as a Coulomb gas of two kinds of magnetic loops and one electric loop. Novel features of such models are the existence of partial confinement phases and a two-photon phase. Phase diagrams are obtained by naive condensate considerations and by Monte Carlo calculations.     

Phase structure of self-gravitating systems

The equilibrium properties of classical self-gravitating systems in the grand canonical ensemble are studied by using the correspondence with an euclidean field theory with infrared and ultraviolet cutoffs. It is shown that the system develops a first order phase transition between a low and a high density regime. In addition, due to the long range of the gravitational potential, the system is close to criticality within each phase, with the exponents of mean field theory. The coexistence of a sharp first order transition and critical behavior can explain both the presence of voids in large regions of the universe as well as the self-similar density correlations in terms of self-gravity alone.     

On the structure of gauge invariant classical observables in lattice gauge theories

For a lattice gauge theory a necessary and sufficient condition on the gauge group is stated, which assures that the linear span of products of Wilson loop observables is dense in the space of continuous, gauge invariant functions on the configuration space. Some groups which fulfill this condition are exhibited, among themU(N) andSU(N),N=1, 2, 3, ... Finally we prove that generically it is fulfilled for all connected, compact Lie groups.