Statistical Geometry and Lattices

Statistical geometry furnishes the tool that allows the transfer of results from a lattice with finite lattice parameter to the continuum. Since lattice simulations are simpler than continuum ones, this suggests that larger scale simulations for the continuum might be more effectively carried out on a lattice with finite lattice parameter followed by the indicated transfer. We also show that a statistical geometry, peculiar to hard particles on a lattice, can be developed. Among other things, this opens the possibility that a scaled particle theory on a lattice might be derived.     

Bound states in the continuum in a single-level Fano-Anderson model

Bound states in the continuum (BIC) are shown to exist in a single-level Fano-Anderson model with a colored interaction between the discrete state and a structured tight-binding continuum, which may describe mesoscopic electron or photon transport in a semi-infinite one-dimensional lattice. The existence of BIC is explained in the lattice realization as a boundary effect induced by lattice truncation.     

Fermion propagators on a four-dimensional random-block lattice

Fermion propagators, composite boson propagators and the fermion condensate are calculated numerically on the four-dimensional random-block lattice, respectively. The ensemble-averaged fermion propagator agrees with the continuum propagator for distances greater than three average lattice spacings. The results on the fermion condensate show that the chiral symmetry of the doubled modes is broken in the continuum limit. The Goldstone boson arising from the broken symmetry is revealed by examining the composite pseudo-scalar propagator. The doubled fermion and the Goldstone boson both acquire masses of the order of inverse lattice spacing and thus decouple from the theory in the continuum limit.     

Lattice model with power-law spatial dispersion for fractional elasticity

A lattice model with a spatial dispersion corresponding to a power-law type is suggested. This model serves as a microscopic model for elastic continuum with power-law non-locality. We prove that the continuous limit maps of the equations for the lattice with the power-law spatial dispersion into the continuum equations with fractional generalizations of the Laplacian operators. The suggested continuum equations, which are obtained from the lattice model, are fractional generalizations of the integral and gradient elasticity models. These equations of fractional elasticity are solved for two special static cases: fractional integral elasticity and fractional gradient elasticity.     

An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice

This paper presents a derivation of a second-order isotropic continuum from a 2D lattice. The derived continuum is isotropic and dynamically consistent in the sense that it is unconditionally stable and prohibits the infinite speed of energy propagation. The Lagrangian density of the continuum is obtained from the Lagrange function of the underlying lattice. This density is used to obtain the expressions for standard and higher-order stresses in direct correspondence with the equations of the continuum motion. The derived continuum is characterized by two additional parameters relative to the classical elastic continuum. These are the characteristic lengthscale and a dimensionless continualization parameter, which characterizes indirectly the timescale of the derived continuum. The margins for the latter parameter are found from the stability analysis. It is envisaged that the continualization parameter could be measured employing a high-frequency pulse propagating along the surface of the continuum. Excitation and propagation of such pulse is studied theoretically in this paper.     

Universality in Ising model gauge-field analogues

The lattice dependence of a class of gauge-invariant Ising models is investigated. Any lattice dependence would indicate that the lattice could not be regarded as irrelevent and that it would be incorrect to define gauge models on a lattice as a basis for investigating the continuum limit. The models investigated lie within the class of multispin Ising models which show a wide variety of lattice-dependent behaviour and so these models should provide a significant test of the importance of the gauge-invariance constraint. Two and three dimensional models are investigated and lattice independence is confirmed. This indicates that imposing gauge symmetries on lattice models can restrict the possible behaviour in such a way that lattice independent continuum limits can be defined.     

Änderung der Laue-Bragg-Reflexe und des Volumens eines Kristalls durch statistisch verteilte Punktfehlstellen

It is proved that from the change in length of a sample due to randomly distributed point defects, and from the corresponding change of the Laue-Bragg maxima in x-ray experiments one measures the same average lattice constant. Furthermore it is shown that a homogeneous density of point defects in a continuum gives rise to a homogeneous deformation of the crystal. The problem is also treated by lattice theory. If the lattice is harmonic and if the displacements vary slowly this corresponds to the continuum theory. The double forces of continuum theory can be determined from the coupling parameters of lattice theory.     

A phase cell approach to Yang-Mills theory

For the abelian Yang-Mills theory, a one-to-one correspondence is established between continuum gauge potentials and compatible lattice configurations on an infinite sequence of finer and finer lattices. The compatibility is given by a block spin transformation determining the configuration on a lattice in terms of the configuration on any finer lattice. Thus the configuration on any single lattice is not an approximation to the continuum field, but rather a subset of the variables describing the field.It is proven that the Wilson actions on the lattices monotonically increase to the continuum action as one passes to finer and finer lattices. Configurations that minimize the continuum action, subject to having the variables fixed on some lattice, are studied.This work was supported in part by the National Science Foundation under Grant No. PHY-85-02074     

Some predictions for an improved fermion action on the lattice

We propose an improved fermion action on the lattice by adding a next nearest neightbor interaction term to Wilson action. The proposed action is expected to approach the continuum limit more rapidly. Using the improved action, the predictions for the critical value of the hopping parameter at weak and strong coupling are given. The relationship between quark masses on the lattice and in the continuum is also discussed.     

On the thermodynamics of an ideal fermi gas on the lattice at finite density

We study the ideal gas of fermions on a lattice at finite density for both naive and Wilson fermions. Comparing the thermodynamical quantities thus calculated with the known results in the continuum theory, we are led to propose a modification of the naive form of the lattice action, which is same for both the naive and the Wilson fermions. The thermodynamical quantities, calculated by using this form, are shown to have the correct continuum limit.     

Universality of the continuum limit of lattice quantum theories

The lattice approximation of the naïve continuum action in quantum mechanics or in field theory is not uniquely determined. We investigate to what extent corrections to the lattice action, which vanish in the naïve continuum limit, affect the continuum limit when taking quantum fluctuations into account. In the quantum mechanical case, modifications of the lattice action may induce non-trivial corrections to the potential of the system and thereby change the structure of the theory completely. We verify this statement analytically as well as numerically by performing a Monte Carlo simulation. In the field theoretical case we argue that the lattice corrections considered do not affect the physics of the continuum limit, at least not for asymptotically free gauge field theories. In four dimensions, one might encounter finite renormalization of CP violating terms.     

Monte Carlo simulation of non-compact QCD with stochastic gauge fixing

A non-compact lattice model of quantum chromodynamics is studied numerically. Whereas in Wilson's lattice theory the basic variables are the elements of a compact Lie group, the present lattice model resembles the continuum theory in that the basic variables A are elements of the corresponding Lie algebra, a non-compact space. The lattice gauge invariance of Wilson's theory is lost. As in the continuum, the action is a quartic polynomial in A, and a stochastic gauge fixing mechanism - which is covariant in the continuum and avoids Faddeev-Popov ghosts and the Gribov ambiguity — is also transcribed to the lattice. It is shown that the model is self-compactifying, in the sense that the probability distribution is concentrated around a compact region of the hyperplane div A = 0 which is bounded by the Gribov horizon. The model is simulated numerically by a Monte Carlo method based on the random walk process. Measurements of Wilson loops, Polyakov loops and correlations of Polyakov loops are reported and analyzed. No evidence of confinement is found for the values of the parameters studied, even in the strong coupling regime.     

Numerical simulations of two-dimensional QED

We describe the computer simulation of two-dimensional QED on a 64 × 64 Euclidean space-time lattice using the Susskind lattice fermion action. The order parameter for chiral symmetry breaking and the low-lying meson masses are calculated for both the model with two continuum flavours, which arises naturally in this formulation, and the model with one continuum flavour obtained by including a nonsymmetric mass term and setting one fermion mass equal to the cut-off. Results are compared with those obtained using the quenched approximation, and with analytic predictions.     

QCD sum rules,the spontaneous breakdown of chiral symmetry and short-distance behaviour in lattice gauge theories

We analyze the behaviour that correlation functions ought to have on the lattice in order to reproduce QCD sum rules in the continuum limit. We formulate a set of relations between lattice correlation functions of meson operators at small time separation and the quark condensates responsible for spontaneous breakdown of chiral symmetery. We suggest that the degree to which such relations are satisfied will provide a set of consistency checks on the ability of lattice Monte Carlo simulations to reproduce the correct spontaneous chiral symmetry breaking of the continuum theory.     

YM2:Continuum expectations,lattice convergence,and lassos

The two dimensional Yang-Mills theory (YM₂) is analyzed in both the continuum and the lattice. In the complete axial gauge the continuum theory may be defined in terms of a Lie algebra valued white noise, and parallel translation may be defined by stochastic differential equations. This machinery is used to compute the expectations of gauge invariant functions of the parallel translation operators along a collection of curvesC. The expectation values are expressed as finite dimensional integrals with densities that are products of the heat kernel on the structure group. The time parameters of the heat kernels are determined by the areas enclosed by the collectionC, and the arguments are determined by the crossing topologies of the curves inC. The expectations for the Wilson lattice models have a similar structure, and from this it follows that in the limit of small lattice spacing the lattice expectations converge to the continuum expectations. It is also shown that the lasso variables advocated by L. Gross [36] exist and are sufficient to generate all the measurable functions on the YM₂-measure space.     

Chiral and conformal anomalies in lattice gauge theory

The continuum limit of the chiral and conformal (Weyl) Ward-Takahashi identities in the lattice Wilson action is studied. The Wilson term works for the chiral anomaly, but it gives rise to-15 times the conventional conformal anomaly for a smallr-parameter and a very sensitiver-dependence of the Λ-parameter. This shows that the strong symmetry breaking by the Wilson term by itself does not necessarily generate correct anomalies. In the lattice regularization the functional Jacobian factors becomec-numbers and do not contribute to anomalies, corresponding to the cut-off of short distance components; the naive continuum limit of lattice WT identities can thus behave differently from continuum ones. To reconstruct conventional identities from lattice relations, the lattice composite operators should be rewritten in terms of relevant continuum operators. In general, this identification of relevant operators is facilitated either by the procedure corresponding to Zimmermann's normal product algorithm or simply by the use of auxiliary regulators such as the dimensional regulator.     

A statistical mechanics view of quantum chromodynamics: Lattice gauge theory

Recent developments in lattice gauge theory are discussed from a statistical mechanics viewpoint. The basic physics problems of quantum chromodynamics (QCD) are reviewed for an audience of critical phenomena theorists. The idea of local gauge symmetry and color, the connection between statistical mechanics and field theory, asymptotic freedom and the continuum limit of lattice gauge theories, and the order parameters (confinement and chiral symmetry) of QCD are reviewed. Then recent developments in the field are discussed. These include the proof of confinement in the lattice theory, numerical evidence for confinement in the continuum limit of lattice gauge theory, and perturbative improvement programs for lattice actions. Next, we turn to the new challenges facing the subject. These include the need for a better understanding of the lattice Dirac equation and recent progress in the development of numerical methods for fermions (the pseudofermion stochastic algorithm and the microcanonical, molecular dynamics equation of motion approach). Finally, some of the applications of lattice gauge theory to QCD spectrum calculations and the thermodynamics of. QCD will be discussed and a few remarks concerning future directions of the field will be made.Supported in part by the NSF under grant No. PHY82-01948     

Emergent spin

Quantum mechanics and relativity in the continuum imply the well known spin–statistics connection. However for particles hopping on a lattice, there is no such constraint. If a lattice model yields a relativistic field theory in a continuum limit, this constraint must “emerge” for physical excitations. We discuss a few models where a spin-less fermion hopping on a lattice gives excitations which satisfy the continuum Dirac equation. This includes such well known systems such as graphene and staggered fermions.     

Supersymmetric lattice theories

We attempt to construct supersymmetric lattice theories using the staggered lattice fermions of Kogut and Susskind. Although we are able to construct lattice field theories with many of the properties of standard supersymmetric models, all of our interacting models violate Lorentz invariance in the continuum limit.     

Locally resonant acoustic metamaterials with 2D anisotropic effective mass density

A two-dimensional (2D) lattice model with anisotropic resonant microstructures is found to provide an anisotropic band gap structure. A 2D continuum with anisotropic effective mass density is introduced to represent this lattice system. Two methods are proposed to derive the equivalent continuum. In the first method, the effective mass density of the equivalent continuum is obtained by matching the dispersion relations for harmonic waves propagating in the principal directions. The second approach employs an approximate estimation of the effective mass density by volume-averaging an effective mass that represents the resonant microstructure. For both equivalent continuum models, the effective mass density is frequency-dependent and may become negative in certain frequency ranges. Subsequently, the effective mass density of the equivalent continuum assumes the form of a second-order tensor. Thus, it suffices to determine the effective mass density tensor with respect to the principle directions. It is shown numerically that the local-resonance effect is accurately described by the equivalent continuum model. In addition, the effect of anisotropic mass density on wave propagation is numerically illustrated and discussed.