On the theory of superconductivity in Kondo lattice systems

An attempt is made to explain the occurrence of superconductivity in Kondo lattice systems with special reference to CeCu₂Si₂. Starting point is the Fermi liquid approach. It is generalized from a Kondo impurity to the Kondo lattice by means of the Korringa-Kohn-Rostocker method. From it a hybridization model is derived and discussed in detail. Two electron-phonon mechanisms are investigated which appear in Kondo lattices. One results from the additional phase shifts caused by the Kondo ions while the other is responsible for the so-called Kondo volume collapse. It is shown that the latter is sufficiently strong in order to explain why CeCu₂Si₂ is a superconductor while LaCu₂Si₂ is not. An estimate for the superconducting transition temperatureT _c produces the right order of magnitude.     

Density functional theory for general hard-core lattice gases

We put forward a general procedure to obtain an approximate free-energy density functional for any hard-core lattice gas, regardless of the shape of the particles, the underlying lattice, or the dimension of the system. The procedure is conceptually very simple and recovers effortlessly previous results for some particular systems. Also, the obtained density functionals belong to the class of fundamental measure functionals and, therefore, are always consistent through dimensional reduction. We discuss possible extensions of this method to account for attractive lattice models.     

More evidence for universality in the SU(2) lattice gauge theory

The O⁺ glueball mass in the fundamental-adjoint SU(2) la ttice gauge theory is extracted from the Monte Carlo data on the correlation functions at time distances t = 1 and t = 2 on a 8⁴ lattice. The ratio $\text{m}{}_{\mathrm{<mtext>g</mtext>}}\text{σ}$ is constant in the β_F ?β_A region explored giving more evidence in favour of the universality hypothesis.     

Random-walk theory and the decay of pair correlations in Ornstein-Zernike lattice systems

A random-walk formalism is applied to some general Ornstein-Zernike lattice systems to obtain information as to the asymptotic form of the total correlation function. Calculations in terms of the Percus-Yevick approximation are then presented for certain lattice gases with interactions extending over a few orders of neighbours, illustrating the circumstances in which the decay of the total correlation function may be monotonic or oscillatory.     

A general method for constructing vector integrable lattice systems

A method for generating vector-value integrable analogies of integrable lattice systems or integrable differential-difference equations is presented. The basic ingredient of the method is to insert permutation matrices. We formulate the zero-curvature representations and Hamiltonian structures of the resulting vector lattice systems. The effectiveness of the method is illustrated using some examples such as the Volterra lattice, the Belov–Chaltikian lattice, the Ablowitz–Ladik lattice and the Heisenberg ferromagnet lattice.     

Progress in lattice gauge theory

Two topics of lattice gauge theory are reviewed. They include string tension and β-function calculations by strong coupling Hamiltonian methods for SU(3) gauge fields in 3 + 1 dimensions, and a 1/N-expansion for discrete gauge and spin systems in all dimensions. The SU(3) calculations give solid evidence for the coexistence of quark confinement and asymptotic freedom in the renormalized continuum limit of the lattice theory. The crossover between weak and strong coupling behavior in the theory is seen to be a weak coupling but non-perturbative effect. Quantitative relationships between perturbative and non-perturbative renormalization schemes are obtained for the O(N) nonlinear sigma models in 1 + 1 dimensions as well as the range theory in 3 + 1 dimensions. Analysis of the strong coupling expansion of the β-function for gauge fields suggests that it has cuts in the complex 1/g²-plane. A toy model of such a cut structure which naturally explains the abruptness of the theory's crossover from weak to strong coupling is presented. The relation of these cuts to other approaches to gauge field dynamics is discussed briefly.The dynamics underlying first order phase transitions in a wide class of lattice gauge theories is exposed by considering a class of models-P(N) gauge theories - which are soluble in the N → ∞ limit and have non-trivial phase diagrams. The first order character of the phase transitions in Potts spin systems for N #62; 4 in 1 + 1 dimensions is explained in simple terms which generalizes to P(N) gauge systems in higher dimensions. The phase diagram of Ising lattice gauge theory coupled to matter fields is obtained in a $\text{1}\text{N}$ expansion. A one-plaquette model (1 time-0 space dimensions) with a first-order phase transitions in the N → ∞ limit is discussed.     

First order phase transitions in lattice and continuous systems: Extension of Pirogov-Sinai theory

We generalize the notion of ground states in the Pirogov-Sinai theory of first order phase transitions at low temperatures, applicable to lattice systems with a finite number of periodic ground states to that of restricted ensembles with equal free energies. A restricted ensemble is a Gibbs ensemble, i.e. equilibrium probability measure, on a restricted set of configurations in the phase space of the system. When a restricted ensemble contains only one configuration it coincides with a ground state. In the more general case the entropy is also important.An example of a system we can treat by our methods is theq-state Potts model where we prove that forq sufficiently large there exists a temperature at which the system coexists inq+1 phases;q-ordered phases are small modifications of theq perfectly ordered ground states and one disordered phase which is a modification of the restricted ensemble consisting of all perfectly disordered (neighboring sites must have different spins) configurations. The free energy thus consists entirely of energy in the firstq-restricted ensembles and of entropy in the last one.Our main motivation for this work is to develop a rigorous theory for phase transitions in continuum fluids in which there is no symmetry between the phases, e.g. the liquid-vapour phase transition. The present work goes a certain way in that direction.Supported in part by NSF Grant N^r DMR81-14726-02     

A theory for the Anderson lattice

Zeitschrift für Physik B Condensed Matter - A perturbational analysis of the Anderson lattice problem in the limit of infinite local Coulomb repulsion is given. It furnishes a clear conceptual...     

Bogoliubov theory on the disordered lattice

Quantum fluctuations of Bose-Einstein condensates trapped in disordered lattices are studied by inhomogeneous Bogoliubov theory. Weak-disorder perturbation theory is applied to compute the elastic scattering rate as well as the renormalized speed of sound in lattices of arbitrary dimensionality. Furthermore, analytical results for the condensate depletion are presented, which are in good agreement with numerical data.     

Scalar lattice gauge theory

Scalar lattice gauge theories are models for scalar fields with local gauge symmetries. No fundamental gauge fields, or link variables in a lattice regularization, are introduced. The latter rather emerge as collective excitations composed from scalars. For suitable parameters scalar lattice gauge theories lead to confinement, with all continuum observables identical to usual lattice gauge theories. These models or their fermionic counterpart may be helpful for a realization of gauge theories by ultracold atoms. We conclude that the gauge bosons of the standard model of particle physics can arise as collective fields within models formulated for other “fundamental” degrees of freedom.     

On the space-time interpretation of classical canonical systems I: The general theory

We define a canonical system as a canonical manifoldM plus a canonical vectorfield onM. For such systems a unique kinematical interpretation is deduced from a set of Kinematical Axioms satisfied by the algebra of differentiable functions onM. This algebra is required to contain a subalgebra which is maximal commutative under the Poisson bracket.M is shown to be diffeomorphic to the cotangent bundle over its quotient manifold, which is defined by the given subalgebra. Canonical systems satisfying these axioms are then classified. If the phase space interpretation is adopted they are shown to describe the motion of masspoints in some configuration space under the influence of and interacting by arbitrary vector and scalar potentials.     

Mixing properties in lattice systems

We study mixing or spatial cluster properties and some of their consequences in classical lattice systems, in particular complete regularity and the weaker notion of strong mixing. Introducing the notion of reflection positivity as a generalization ofT-positivity of [1], we construct a generalized transfer matrixP and relate complete regularity to a spectral gap inP. It is shown that all reflection invariant Ising systems with n.n. and ferromagnetic n.n.n. interaction satisfy reflection positivity. For Ising ferromagnets with reflection positivity, exponential decay of the truncated 2-point function implies complete regularity. In particular, the 2-dimensional spin-1/2 Ising model is completely regular, except at the critical point. This complements a result of [2] that strong mixing fails at the critical point of this model and in this case verifies the suggestion of Jona-Lasinio [3] that critical behaviour should be linked with failure of strong mixing. We then show that strong mixing imposes severe restrictions on the possible form of limits of block spins. Strong mixing in each direction allows onlyindependent Gaussians as non-zero limit if the 2-point function exists; strong mixing in a single direction only will allow infinitely divisible distributions.     

Finite-element lattice fermions in perturbation theory

A lattice Thirring model is defined using a finite-element differencing scheme. The momentum-space propagator is calculated to second order in the coupling, and it is shown that the form of the ultraviolet divergence as the lattice spacing is taken to zero is identical to that found in continuum perturbation theory. This provides further evidence for the absence of fermion species-doubling present in most other lattice formulations.     

Loop states in lattice gauge theory

We reformulate d-dimensional SU(2) lattice gauge theory in terms of gauge invariant loop state variables by solving the SU(2) Gauss law as well as the corresponding Mandelstam constraints. The loop states satisfying the Gauss law and the Mandelstam constraints in d dimension are explicitly constructed in terms of the SU(2) harmonic oscillator prepotential operators. We show that these mutually independent (orthonormal) loop states carry certain non-negative integer Abelian fluxes over the lattice links and are characterized by 3(d−1) gauge invariant angular momentum quantum numbers per lattice site. Thus, they provide a complete orthonormal loop basis in the physical Hilbert space of the gauge theory. Further, we derive the loop Hamiltonian and show that it counts, creates and annihilates the Abelian fluxes over the plaquettes. The generalization to SU(N) gauge group is discussed.     

Adjoint sources in lattice gauge theory

Variance reduction techniques for the evaluation of Wilson loops in lattice gauge theory are analysed. The method is extended to Wilson loops in the adjoint representation. Variational methods are also applied to adjoint sources. The combination of these techniques allows the potential V(R) between two static adjoint sources to be determined in SU(2) gauge theory. One isolated static adjoint source is also studied and the energy and distribution of the gluon field of this “glue-lump” is obtained. This is relevant to the saturation of the adjoint potential V(R) at large R.     

Some calculations in lattice gauge theory

We look at the action proposed by Wilson on a lattice and calculate static constants like f_π and two-body decay amplitudes in a certain approximation. Results are good to factors of four to six. There is good agreement for some of the predicted meson masses.