On the spectrum of the Heisenberg Hamiltonian

The quantum, antiferromagnetic, spin-1/2 Heisenberg Hamiltonian on thed-dimensional cubic lattice ^d is considered for any dimensiond. First the anisotropic case is considered for small transversal coupling and a convergent expansion is given for a family of eigenprojections which is complete in all finite-volume truncations. Then the general case is considered, for which an upper bound to the ground-state energy is given which is optimal for strong enough anisotropy. This bound is expressed through a functional involving the statistical expectation value at finite temperature of a certain correlation function of an Ising model defined on the lattice ^d itself.     

On the Gibbs states for one-dimensional lattice Boson systems with a long-range interaction

We consider an infinite chain of interacting quantum (anharmonic) oscillators. The pair potential for the oscillators at lattice distanced is proportional to {d ²[log(d+1)]F(d)}^–1 where _rZ [rF(r)]^–1 < . We prove that for any value of the inverse temperature> 0 there exists a limiting Gibbs state which is translationally invariant and ergodic. Furthermore, it is analytic in a natural sense. This shows the absence of phase transitions in the systems under consideration for any value of the thermodynamic parameters.     

Markov and stability properties of equilibrium states for nearest-neighbor interactions

Consider models on the lattice ^d with finite spin space per lattice point and nearest-neighbor interaction. Under the condition that the transfer matrix is invertible we use a transfer-matrix formalism to show that each Gibbs state is determined by its restriction to any pair of adjacent (hyper)planes. Thus we prove that (also in multiphase regions) translationally invariant states have a global Markov property. The transfer-matrix formalism permits us to view the correlation functions of a classicald-dimensional system as obtained by a linear functional on a noncommutative (quantum) system in (d – 1)-dimensions. More precisely, for reflection positive classical states and an invertible transfer matrix the linear functional is a state. For such states there is a decomposition theory available implying statements on the ergodic decompositions of the classical state ind dimensions. In this way we show stability properties of _ev ^d -ergodic states and the absence of certain types of breaking of translational invariance.     

A note on γ traces for the Wilson action in the continuum limit

In d=4 and d=2 dimensions we calculate averages of certain products of matrices with respect to closed lattice paths of length L. The approach to the asymptotic behaviour for large L is considered and found to be quite different in d=4 and d=2 dimensions.Institute für Theoretische Physik der Universität Hamburg, F.R.G.     

Existence of Baryons,Baryon Spectrum and Mass Splitting in Strong Coupling Lattice QCD

We consider a functional integral formulation for one-flavor lattice Quantum Chromodynamics in d=2,3 space dimensions and imaginary time, and work in the regime of the small hopping parameter , and zero plaquette coupling. Following the standard construction, this model exhibits positivity which is used to obtain the underlying physical Hilbert space . Then, using a Feynman-Kac formalism, we write the correlation functions for the model as functional integrals over the space of Grassmannian (fermionic) fields for one quark specie and the SU(3) gauge fields. We determine the energy-momentum spectrum associated with gauge invariant local baryon (anti-baryon) fields which are composites of three quark (anti-quark) fields. With the associated correlation functions, we establish a Feynman-Kac formula, and a spectral representation for the Fourier transform of the two-point functions. This representation allows us to show that baryons and anti-baryons arise as tightly bound, bound states of three (anti-)quarks. Labelling the components of the baryon fields by s=3/2,1/2,-1/2,-3/2, we show that the baryon and anti-baryon mass spectrum only depends on |s|, and the associated masses are given by M_s= –3ln+r_s(), where r_s() is real analytic in , for each d=2,3. The mass splitting is M_3/2–M_1/2=18⁶, for d=2 and, if any, is at least of (⁷), for d=3. In the subspace _o generated by an odd number of fermions, the baryon and anti-baryon energy-momentum dispersion curves are isolated up to near the baryon-meson threshold –5ln (upper gap property), identical and are determined up to (⁵). The symmetries of coordinate reflections, spatial lattice rotations, parity and charge conjugation are established for the correlation functions, and are shown to be implemented on by unitary (anti-unitary, for time reversal) operators.     

Localization for random potentials supported on a subspace

We consider the lattice Schrödinger operator acting onl ² ( ^d ) with random potential (independent, identically distributed random variables), supported on a subspace of dimension 1 v <d. We use the multiscale analyses scheme to prove that this operator exhibits exponential localization at the edges of the spectrum for any disorder or outside the interval [-2d, 2d] for sufficiently high disorder.     

A rigorous bound on the critical exponent for the number of lattice trees,animals, and polygons

The number ofn-site lattice trees (up to translation) is believed to behave asymptotically asCn ^{–0 n} , where is a critical exponent dependent only on the dimensiond of the lattice. We present a rigorous proof that (d–1)/d for anyd2. The method also applies to lattice animals, site animals, and two-dimensional self-avoiding polygons. We also prove that v whend=2, wherev is the exponent for the radius of gyration.     

1/d corrections for interacting spinless fermions: One-particle properties

One-particle properties of the spinless fermion model with repulsion at half filling are calculated within an approach correct to first order in the inverse of the lattice dimensiond. Continuity of the limitd requires a scaling of the nearest-neighbour hopping proportional to and of the nearest-neighbour interaction proportional to 1/d. Due to this scaling the Hartree approximation becomes exact in infinite dimensions. We show that 1/d corrections comprise the Fock diagram and the local correlation diagram in the self-consistent Dyson equation. This approach is applied to simple-cubic systems in dimensiond=1, 2 and 3. Ground state properties and the charge-density wave phase diagram are calculated. AtT=0 the inclusion of 1/d terms gives only small corrections to the leading Hartree contribution ind=2, 3. ForT>0, however, the 1/d corrections are important. They lead to a non-negligible reduction of the critical temperature. Ind=1 the 1/d corrections are very large, but they do not succeed in removing the spurious phase transition atT>0. The 1/d approach provides a good and tractable approximation ind=3 and probably ind=2, which allows also further systematic improvement.     

Correction to scaling for directed branched polymers (lattice animals)

Correction to scaling effects for the directed branched polymer (lattice animal) problem are calculated both from the field theoretic model due to Day and Lubensky (dynamics at the Yang-Lee edge singularity) and enumeration data. The universal correction to scaling exponent for the number of distinct animal configurations is estimated by numerical methods and field theoretic renormalization group results (=7–d expansion) extrapolated to exact results ford=2 andd=3; estimates of nonuniversal amplitudes are included.     

On singularities in the disordered phase of a driven diffusive system

Using renormalization group techniques, we investigate the large distance behavior of a driven, interacting lattice gas in the disordered phase. Unlike the equilibrium Ising model, its behavior, ford>2, is controlled by aline of fixed points, each of which is interpreted as a dynamical system violating the fluctuation-dissipation theorem (FDT). As a consequence, correlation functions at large distances typically decay according to a power law instead of an exponential. Ford2, the renormalization group flows towards an FDT-satisfying fixed point, which corresponds to the high-temperature, strong-drive limit. In the steady state of such a model (a driven, free lattice gas), correlations are known to be exactly zero. Nevertheless, our correlations are still dominated by power laws, since the FDT-breaking operators aredangerously irrelevant (marginal ind=2). Thus, for anyd, the long wavelength properties cannot be obtained by taking either the non-interacting or theT limit, unlike for the equilibrium Ising model.     

Statistical mechanical methods in particle structure analysis of lattice field theories

We illustrate on simple examples a new method to analyze the particle structure of lattice field theories. We prove that the two-point function in Ising and rotator models has an Ornstein-Zernike correction at high temperature. We extend this to Ising models at low temperatures if the lattice dimensiond3. We prove that the energy-energy correlation function at high temperatures (for Ising orN=2 rotators) decays according to mean field theory (i.e. with the square of the Ornstein-Zernike correction) ifd4. We also study some surface models mimicking the strong-coupling expansion of the glueball correlation function. In the latter model, besides Ornstein-Zernike decay, we establish the presence of two nearly degenerate bound states.     

A new construction of thermodynamic mean-field theories of itinerant fermions: application to the Falicov-Kimball model

We present a general scheme for a construction of mean-field-like theories of itinerant lattice fermions based on solutions to the exactd= grand canonical potential. The general construction is explicitly demonstrated on the exactly solvabled= Falicov-Kimball model. A mean-field zero-temperature behaviour of this sample model is studied quantitatively.Work supported by the A. von Humboldt Foundation.     

Infinite hierarchy of exponents in a two-component random resistor network

We have studied the voltage distribution for a two-component random mixture of conductances _a and _b. A scaling theory is developed for the moments of the distribution, which predicts, for small values ofh=_a/_b, an infinite number of crossover exponents, one for each moment, for Euclidean dimensiond >2, and only one crossover exponent ford=2. Monte Carlo results on the square lattice confirm this prediction.     

Thed=2tJ model on a triangular lattice at large doping

We determine the spectra and interactions of quasiparticles in thed=2tJ model on a triangular lattice forJ2t and near 2/3 electron filling. We find coexistence of magnons and quasielectrons atJ=2t and a transition from repulsion among quasielectrons atJ<2t to attraction atJ>2t. The mathematical methods developed here involve graded cosets and are also applicable, in modified form, to the Fermi liquid regime of thetJ model on a square lattice.     

Migdal-Kadanoff renormalization group for theO(n) model

We perform a Migdal-Kadanoff renormalization group calculation on anO(n) symmetric model on ad-dimensional hypercubic lattice,d=2, 3. We find that in two dimensions the critical fixed point disappears asn=n _KT1.96, which is in good agreement with the exact valuen _KT=2. In three dimensions the fixed point persists much longer, albeit not all the way up to infinity. Surface critical phenomena in a semiinfiniteO(n) model are also considered.     

Phase transitions of ann-vector model on an elastic anisotropic lattice

Wegner's model for magnetic phase transitions on elastic isotropic lattices is generalized to elastic anisotropy and the general spin-lattice coupling linear in lattice distortion. Cyclic boundary conditions are employed. The elimination of the elastic degrees of freedom gives rise to an effective spin Hamiltonian which consists of a shortrange and a long-range part. Renormalization group analysis yields the following result: If the lattice couples to a relevant operator with an exponenty >d/2, the long-range part contains a relevant contribution with exponent 2y—d leading away from any fixed pointH ^* associated with the short-range interactions. In particular,H ^* is unstable for positive specific heat exponent, and, in most cases, if a spin-anisotropy is involved.     

Exact solution of a vertex model ind dimensions

We consider a class of vertex models describing directed lines on a lattice in arbitraryd dimensions, and solve the model exactly for the Cartesian lattice and in the case that each loop of lines carries a fugacity - 1. Our analysis, which can be carried out for arbitrary lattices, is based on an equivalence of the vertex model with a dimer problem. The dimer problem is, in turn, solved using the method of Pfaffians. It is found that the system is frozen below a critical temperatureT _cwith the critical exponent = (3 –d)/2.     

Gas Phase of Asymmetric Nearest Neighbor Ising Model

We consider an asymmetric d-dimensional, d>1, Ising model with the pair interaction I in one direction different from the pair interaction J in all other directions. We show that for any inverse temperature the system is in the gas phase as soon as |J|<C ^–1 d ^–2(1–tanh( |I|)) with C>0 being a small numeric constant.     

Random Walks on a Fractal Solid

It is established that the trapping of a random walker undergoing unbiased, nearest-neighbor displacements on a triangular lattice of Euclidean dimension d=2 is more efficient (i.e., the mean walklength n before trapping of the random walker is shorter) than on a fractal set, the Sierpinski tower, which has a Hausdorff dimension D exactly equal to the Euclidean dimension of the regular lattice. We also explore whether the self similarity in the geometrical structure of the Sierpinski lattice translates into a self similarity in diffusional flows, and find that expressions for n having a common analytic form can be obtained for sites that are the first- and second-nearest-neighbors to a vertex trap.     

Order parameters forn-vector spin glasses ind dimensions

We report Monte Carlo simulations of the time-dependent behavior of Edwards-Anderson spin glasses with Gaussian nearest-neighbor exchange, for both spin dimensionalityn and space dimensionalityd from 2 up to 6. A (nearly) logarithmic decay of the Edwards-Anderson order parameter with time is observed for alln and alld, similar to earlier studies forn=1. But the Monte Carlo data forn>1 suggest stronger than those forn=1 that all order parameters considered vanish in thermal equilibrium for nonzero temperature, because the decay forn>1 is faster at the temperatures of interest. For Heisenberg spins (n=3) no significant dependence of the Edwards-Anderson order parameterq on the size of the lattice was observed ford=2,3 and 4, whereas ford=5 and 6,q was smaller for smaller systems (in contrast to thed=5 Ising case). These results are the first Monte Carlo indication of a change in the bulk behavior of Heisenberg spin glasses at dimensionalityd=4. Quenching the system to zero temperature and then applying a field we find that the order parameter , measuring the alignment with respect to the state at zero field, is destroyed by a sufficiently strong magnetic field, for all observedn andd.Sonderforschungsbereich 125 Aachen-Jülich-Köln, FRG