Correct Continuum Limit Research Articles

A solvable (1+1)-dimensional gauge model and the corresponding lattice model

A (1+1)-dimensional gauge model is proposed and the spectrum with the energy eigenstates is solved exactly in Hamiltonian formalism. The exact solution of the corresponding lattice model is found. The results show that the lattice model has the correct continuum limit.

Explicit solutions of the finite-element Schrodinger equation

Energy eigenvalues and eigenfunctions are obtained for the finite-element Schrodinger equation in the cases of linear and harmonic potentials. For the harmonic oscillator, the eigenfunctions in momentum space are related to Gegenbauer polynomials and have a simpler form than those for the usual lattice Schrodinger equation obtained using finite differences. In the case of linear potential the eigenfunctions are related to Whittaker functions. For each potential it is demonstrated both theoretically and numerically that the eigenvalues and eigenfunctions have the correct continuum limit.

Universality and mass gaps for a series of lattice Hamiltonians with exact ground states

A series of lattice Hamiltonians with exact ground states and the same correct continuum limit are obtained by using the arbitrariness in defining lattice gauge theories. The rigorous upper bounds of the mass gaps in (2+1)-dimensional SU(2) theory are obtained by the variational method. Trial wave functions for excited states contain loop variables up to 130\ifmmode\times\else\texttimes\fi{}130 Wilson loops, whose coefficients we take as variational parameters. The results show the same scaling behavior am=2.27${g}^{2}$ extended to a very deep weak-coupling region (1/${g}^{2}$\ensuremath{\sim}7) and confirm the universality in lattice gauge theory.

Comment on "Exact ground state, mass gap, and string tension in lattice gauge theory"

It is shown that the lattice Hamiltonian recently proposed by Guo, Zheng,and Liu does not have the correct continuum limit.

Schwinger model on the random block lattice

The fermion propagator and the vacuum polarization tensor of the Schwinger model are studied on the random block lattice, in the free field limit and weak coupling perturbation theory, respectively. The ensemble averaged fermion propagator and the vacuum polarization tensor agree with their respective values in the continuum theory for distances greater than three average lattice spacings. In momentum space, the ensemble averaged fermion propagator also agrees with the continuum propagator. These results support the possibility of having a chiral fermion interacting with the dynamical gauge field, on the random block lattice, with the correct continuum limit.

Parity violating vacuum currents on the random lattice

The Dirac operator is studied both with and without background gauge fieldson a three-dimensional random lattice. It is shown that the correct continuum limit for free fermions is recovered in the zero field case and the doubling problem evaded at tree level, the doubles acquiring masses of order the cut-off and decoupling from long-distance physics. The vacuum current is calculated in the presence of a nontrivial U(1) field and shown to be in agreement with continuum calculations.

Continuum limit of sine-Gordon and phi4 chains by use of an effective potential.

The effective potential previously introduced by the variational method is used to calculate thermodynamic quantities of quantum one-dimensional nonlinear fields in the weak-coupling approximation. The correct continuum limit is recovered both for integrable (sine-Gordon) and noninte- grable (phi/sup 4/) models. Good agreement is found with exact Bethe-ansatz results for the specific heat.

Interacting fermions on a random lattice

We extend previous work on the properties of the Dirac lagrangian on two-dimensional random lattices to the case where interaction terms are included. Although for free fermions the chiral symmetry of the doubles is spontaneously broken by their interaction with the lattice and they decouple from long-distance physics, our results in this paper show that all is undone by quantum corrections in an interacting field theory and that the end result is very similar to what is found with Wilson fermions. Two field-theoretical models with interacting fermions are studied by perturbation expansion in the field theory coupling constant. These are a model with one fermion and one boson species interacting via a scalar Yukawa coupling and the massive Thirring model. It is shown that on the random lattice ultraviolet finite diagrams and finite parts of ultraviolet divergent diagrams have the correct continuum limit. Ultraviolet divergent parts can be removed by the same renormalisation procedure as in the continuum, but do not exhibit the same dependence on the lagrangian mass. In the case of the massive Thirring model this causes a fermion mass correction of order the cut-off scale, which breaks the chiral symmetry of the remaining light fermion; there is consequently a fine-tuning problem. In the context of the same model we discuss the effect of the Goldstone boson associated with the spontaneous breakdown of the chiral symmetry of the doubles on two-dimensional models with vector couplings.

Fermion propagators in a four-dimensional random lattice

Fermion propagators on a four-dimensional random lattice are calculated numerically. They show correct continuum limits. We also show that with the use of the point-splitting technique the composite boson propagators also approach the correct continuum limits.

Fermion propagator and composite-boson propagators on random lattices

The ensemble averages of various propagators of a free-fermion system on random lattices in D = 1 and D = 2 are calculated numerically. The results indicate a correct continuum limit. The cancellation mechanism of the unphysical states is investigated.

The topological susceptibility in SU(2) lattice gauge theory: An exploratory study

The local topological charge density of the (Monte Carlo generated) SU(2) vacuum is measured, using an algorithm that has the correct continuum limit and is not contaminated by perturbative contributions or dislocations. The resulting susceptibility, χ t ∼ (130 MeV) 4 (using m ρ measurements to set physical units), is smaller than that obtained by “geometric” methods and is suppressed at small physical lattice volumes. Various systematic biases are quantified: finite volume effects, reaching equilibrium and uniqueness of the topology. The role of instantons is commented upon and an inequality between χ t and the gluon condensate is presented. It is found that a significant fraction of the non-perturbative action resides in extended topological objects.

Lattice supersymmetry for N=4 Yang-Mills model

The authors introduce a supersymmetric four-dimensional N=4 Yang-Mills model on the space lattice The construction involves the Dirac-Kahler formalism. They obtain the Hamiltonian on the space lattice with the correct classical continuum limit.

A Dirac-Kähler approach to the two dimensional Wess-Zumino N = 2model on the lattice

We apply the Dirac-Kähler formalism to the two dimensional Wess-Zumino N = 2 model. On the hamiltonian lattice the model exhibits invariance under a lattice subalgebra of supersymmetry. We also describe the euclidean space-time lattice version of this model with the action invariant under the supersymmetry transformations which have the correct continuum limit.

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.

N = 2 two-dimensional Wess-Zumino model on the lattice

A lattice version of the N = 2 SUSY two-dimensional Wess-Zumino model was constructed and studied. The correct continuum limit is checked in perturbation theory. The strong-coupling limit is defined and investigated. We find that the ground state of the model has zero energy and infinite degeneracy. The connection between this degeneracy and the properties of the Nicolai-Parisi-Sourlas transformation is discussed.

A lattice version of the Wess-Zumino model

A lattice Lagrangian of the Wess-Zumino model is constructed using perturbation theory up to two loops. It is shown that the renormalized vertex-functions have the correct continuum limit if nonsupersymmetric counterterms up to dimension four are added to the Lagrangian. The structure of these terms is analysed with the Wilson prescription for the fermions.

Supersymmetric models on the lattice

We construct the d=2 Wess-Zumino model and the d=4 N=2 supersymmetric gauge theory on the lattice, in the hamiltonian formalism. The models have the correct tree level continuum limit. Using the Kähler-Dirac formalism, we show that on the lattice a full cubic symmetry is preserved. A strong coupling calculation shows the possibility of spontaneous breaking of supersymmetry for the d=2 model.

Extrapolation to the continuum limit from the cluster expansion on the lattice

In this paper the problem of obtaining the continuum-limit theory (which has manifest Lorentz covariance) from the lattice approximation for a weak-coupling as well as for a strong-coupling regime is studied. We use the lattice cluster expansion of the kinetic term around the static ultralocal approximation and we study a theory without infinite renormalization problems,i.e. the anharmonic oscillator. We find that, in the weak-coupling case, the lattice approximation gives us the correct continuum limit by simply taking the limit for zero lattice spacing; in the strong-coupling regime one needs a suitable analytic continuation.

Spectral degeneracy of the lattice dirac equation as a function of lattice shape

We investigate the free Dirac equation in 2 + 1 dimensions on square, triangular, and hexagonal lattices. For each lattice the spectrum exhibits a degeneracy not present in the continuum limit. In the square and hexagonal cases there is a 4-fold degeneracy corresponding to 2 independent symmetries of the Hamiltonian; the degeneracy is eliminated by diagonalizing these symmetries and projecting onto the subspace characterized by a particular pair of eigenvalues. For the triangular case the degeneracy is 6-fold, but the naive Hamiltonian does not possess enough symmetry to eliminate the degeneracy. Certain ambiguities in the lattice Hamiltonian are pointed out and by the addition of terms which vanish in the continuum limit, it is cast in a form with sufficient symmetry to remove the degeneracy entirely, just as was done for the hexagonal and square lattices. It is found that after elimination of the spectral degeneracy the Hamiltonians for the hexagonal and triangular lattices are identical. The solutions to these theories are shown to have the correct continuum limit.

Mass and heat transfer in a diatomic gas

The transfer of mass and heat through a diatomic gas bounded by two plane parallel surfaces is investigated by a kinetic model which includes translational, rotational and vibrational degrees of freedom. A full range moment method is employed in the solution. Numerical solutions obtained for small temperature and pressure differences show the effect of inelastic collisions on the mass and heat fluxes. Approximate analytical expressions are derived for the mass and heat transfer rates, which are valid over the entire density range and yield the correct free-molecule and continuum limits. The results agree well with existing heat transfer and density measurements.