Free fermion propagators and lattice finite-size effects

Efficient methods are described for calculating the various “hadron” (fermion bilinear/trilinear) propagators for free lattice fermions. For lattice sizes typical of current Monte Carlo calculations, the large finite-size effects are investigated. Periodic and antiperiodic boundary conditions in the spatial directions appear to yield upper and lower bounds respectively for both the meson-like and baryon-like propagators.     

Improved lattice action for Wilson fermions

An improved lattice action for gauge theories with Wilson fermions is presented.     

Optimal lattice domain-wall fermions

I show that the conventional formulations of lattice domain-wall fermion with any finite N(s) (in the fifth dimension) do not preserve the chiral symmetry optimally and propose a new action which preserves the chiral symmetry optimally for any finite N(s).     

Majorana fermions in finite-size strips with in-plane magnetic fields

The European Physical Journal B - In this article we study the effects of disorder and long-range Coulomb interactions on triple-Weyl semimetals using renormalization group up to one-loop order. We...     

Susskind fermions on a euclidean lattice

A version of euclidean lattice QCD obtained by introducing the fermions using Susskind's method is described and certain properties discussed. A U(1) axial current having the correct axial anomaly in the continuum limit is identified. We find $\text{Λ}{}_{\mathrm{<mtext>min</mtext>}}\text{Λ}{}_{\mathrm{<mtext>L,Suss</mtext>}}\text{= 28.78}$ for SU(3) with 4 flavors.     

Correlated fermions on a checkerboard lattice

A model of strongly correlated spinless fermions on a checkerboard lattice is mapped onto a quantum fully packed loop model. We identify a large number of fluctuationless states specific to the fermionic case. We also show that for a class of fluctuating states the fermionic sign problem can be gauged away. This claim is supported by numerical evaluation of the low-lying states. Furthermore, we analyze excitations at the Rokhsar-Kivelson point of this model using the relation to the height model and the single-mode approximation.     

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.     

Analytic and clustering properties of thermodynamic functions and distribution functions for classical lattice and continuum systems

Our most complete results concern the Ising spin system with purely ferromagnetic interactions in a magnetic fieldH (or the corresponding lattice gas model with fugacityz=const. exp(–2mH) wherem is the magnetic moment of each spin). We show that, in the limit of an infinite lattice, (i) the free energy per site and the distribution functionsn _s (x ₁, ...,x _s ; ,z) are analytic in the two variables andH if the reciprocal temperature >0 and the complex numberH is not a limit point of zeros of the grand partition function , and (ii) the Ursell functionsu _s (x ₁, ...,x _s ; ,z) tend to 0 as _s Max _{i, j} |x _i –x _j | if >0 and ReH0; in particular, if the interaction potential vanishes for separations exceeding some fixed cutoff value , then |u _s |<C exp [(–2 m |ReH|+) _s /] where is any small positive number andC is independent of _s . One consequence of the result (i) is that a phase transition can occur as is varied at constantH only ifH is a limit point of zeros of (which can happen only if ReH=0); this supplements Lee and Yang's result that the same condition is necessary for a phase transition whenH is varied at constant .For a lattice or continuum gas with non-negative interaction potential (corresponding, in the lattice case, to an Ising antiferromagnet), similar results are shown to hold provided >0 and the complex fugacityz is less than the radius of convergence of the Mayerz expansion; for the continuum gas, however,n _s andu _s must be replaced by their values integrated over small volumes surrounding each of the pointsx ₂, ...,x _s .It is shown that the pressurep is analytic in both andz, if it is analytic inz at fixed over a suitable range of values of andz, and further that, except for continuum systems without hard cores,p,n _s andu _s have convergent Maclaurin expansions in for small enoughz.Supported by the U.S. Air Force Office of Scientific Research under grant no. AF 68-1416.     

The sigma-model with Wilson lattice fermions

The σ-model with Wilson fermions is considered in one-loop lattice perturbation theory and in the hopping-parameter expansion at large bare couplings. Chiral-symmetry restoration in the large cut-off limit of perturbation theory is only possible if asymmetric counterterms are added to the lattice action. In the hopping-parameter expansion at infinitely large bare Yukawa coupling, dynamical parity doubling of the fermion occurs.     

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.     

Λ-parameters for asymmetric lattice gauge theories with fermions

Dimensional quantities obtained from Monte Carlo simulations on the lattice depend on the lattice mass parameter, Λ_L. To make a connection with continuum physics, a relationship is needed between Λ_L and the Λ-parameters of the continuum theory. This has been done for the euclidean symmetric lattice by others. However, in order to incorporate finite temperature into Monte Carlo studies, or to study the transition from the euclidean formulation to the hamiltonian formulation of gauge theories, asymmetric lattices (a_s≠a_t) may be used. In this paper, the assymetric calculations are extended and the ratio $\text{Λ}{}_{\mathrm{<mtext>min</mtext>}}\text{Λ}{}_{\mathrm{<mtext>L</mtext>}}$, where Λ_min is the continuum mass parameter in the minimal subtraction scheme, is given to one loop for n_f flavors of Wilson and Susskind massless fermions on an asymmetric four-dimensional lattice for two different asymmetric lattice actions.