We show the equivalence of the 2D Ising model to standard free Euclidean lattice fermions of the Wilson Majorana type. The equality of the loop representations for the partition functions of both systems is established exactly for finite lattices with well-defined boundary conditions. The honeycomb lattice is particularly simple in this context and therefore discussed first and only then followed by the more familiar square lattice case.
We obtain correlation functions for lattice gauge theories with action Boltzmann factor given by bond variables described by transverse Ising model and apply them to obtain area decay of the Wilson loop observable in a range of the coupling parameter larger than that obtained from mean field theory considerations.
This paper rests to a large extend on a paper I wrote some time ago on 'Duality in generalized Ising models and phase transitions without local order parameter'. It deals with Ising models with interactions containing products of more than two spins. In contrast to the old paper I will first give examples before I come to the general statements. Of particular interest is a gauge invariant Ising model in four dimensions. It has important properties in common with models for quantum chromodynamics as developed by Ken Wilson. One phase yields an area law for the Wilson-loop yielding an interactio…
We construct a map between the quantum field theory of free Weyl or Majorana fermions and the probability distribution of a classical statistical ensemble for Ising spins or discrete bits. More precisely, a Grassmann functional integral based on a real Grassmann algebra specifies the time evolution of the real wave function $q_τ(t)$ for the Ising states $τ$. The time dependent probability distribution of a generalized Ising model obtains as $p_τ(t)=q^2_τ(t)$. The functional integral employs a lattice regularization for single Weyl or Majorana spinors. We further introduce the complex structure…
An Euclidean representation of bosonized Majorana fermions, prior to imposing constraints, is derived in three space-time dimensons. The difference with the standard three dimensional Ising system is epmhasized. The mild sign problem, does not preclude Monte Carlo simulations in intermediate volumes. Implementing constraints is briefly outlined.
To investigate the properties of the large $N$ limit of $\mathcal{N} = 1$ SUSY Yang-Mills theory, we have started a study for a reduced matrix model with an adjoint Majorana fermion. The gauge action is based on the Wilson action and the adjoint-fermion one is the Wilson-Dirac action on a reduced lattice with twisted gauge boundary condition. We employ the RHMC algorithm in which the absolute value of the Pfaffian is incorporated. The sign of the Pfaffian is included with the re-weighting method and separately measured as an observable. In this talk, we show the configuration generation status…
The partition function (quantum transition amplitude) of the gauge system with gauge group $Z_2$ coupled with Majorana fermions is calculated on the regular 3D cubic lattice.
It is known that the normal three-dimensional (3D) Ising model on a cubic lattice is dual to the Wegner's 3D $Z_2$ lattice gauge theory. Here we find an unusual $Z_2$ lattice gauge theory which is dual to the 3D Ising model with not only nearest-neighbor (nn) coupling, but also next-nearest-neighbor (nnn) coupling. Our gauge theory has on each edge four $Z_2$ variables that have product $+1$, each located on a vector perpendicular to the edge. The nn coupling in the Ising model maps to the plaquette term in the gauge theory where the four variables multiplied have their vectors pointing inward…
Recently, two solutions have been proposed to the long standing problem of $\mathcal{CP}$-symmetry on the lattice, which is particularly evident when considering the construction of chiral gauge theories. The first, based on a lattice modification of $\mathcal{CP}$ was presented by Igarashi and Pawlowski; the second by myself using the renormalisation group and Ginsparg-Wilson relation. In this work, I combine the two approaches and show that they are each part of a more general framework related to an underlying renormalisation group. I continue by formulating Weyl and Majorana fermions on th…
In the mirror fermion approach with Ginsparg-Wilson fermions, it has been argued that the mirror fermions do not decouple: in the 345 model with Dirac- and Majorana-Yukawa couplings to XY-spin field, the two-point vertex function of the (external) gauge field in the mirror sector shows a singular non-local behavior in the PMS phase. We re-examine why the attempt seems a "Mission: Impossible" in the 345 model. We point out that the effective operators to break the fermion number symmetries ('t Hooft operators plus others) in the mirror sector do not have sufficiently strong couplings even in th…
In the framework of effective string theory (EST), the asymptotic behavior of a large Wilson loop in confining gauge theories can be expressed via Laplace determinant with Dirichlet boundary condition on the Wilson contour. For a general polygonal region, Laplace determinant can be computed using the conformal anomaly and Schwarz-Christoffel transformation. One can construct ratios of polygonal Wilson loops whose large-size limit can be expressed via computable Laplace determinants and is independent of the (confining) gauge group. These ratios are computed for hexagon polygons both in EST and…
