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.     

Unitary fermions and Lscher's formula on a crystal

We consider the low-energy particle-particle scattering properties in a periodic simple cubic crystal. In particular, we investigate the relation between the two-body scattering length and the energy shift experienced by the lowest-lying unbound state when this is placed in a periodic finite box. We introduce a continuum model for s-wave contact interactions that respects the symmetry of the Brillouin zone in its regularisation and renormalisation procedures, and corresponds to the nae continuum limit of the Hubbard model. The energy shifts are found to be identical to those obtained in the usual spherically symmetric renormalisation scheme upon resolving an important subtlety regarding the cutoff procedure. We then particularize to the Hubbard model, and find that for large finite lattices the results are identical to those obtained in the continuum limit. The results reported here are valid in the weak,intermediate and unitary limits. These may be used to significantly ease the extraction of scattering information, and therefore effective interactions in condensed matter systems in realistic periodic potentials. This can achieved via exact diagonalisation or Monte Carlo methods, without the need to solve challenging, genuine multichannel collisional problems with very restricted symmetry simplifications.     

Der Grenzübergang zum starren Körper für das quantenmechanische Dreikörperproblem

It is a well known difficulty to introduce the concept of rigid body into quantum mechanics. The authors show how this problem may be solved rigorously for three particles by using a suitable set of coordinates. After subjecting the hamiltonian to a canonical transformation, the coupling term of rotational and vibrational motion vanishes in the limit of infinitesimal vibrational amplitudes, making them separable and thus leading back toWang's naïve quantum theory of the rigid body rotations.     

Phase transitions in quantum field theory. I. Phases of the Thirring model

In this series of papers we exhibit and analyse phase transitions in quantum field theory. In this paper we consider the Thirring model. We show that when the interaction becomes sufficiently attractive there is a transition to a vacuum that is ‘dead” in the sense there are no finite energy excitations. Nevertheless the corresponding continuum Green's functions exist. We make this demonstration precise by considering the model on a lattice and constructing the continuum limit explicitly on either side of the critical point. For this we extensively use the connection between the $\text{spin}\text{-}\text{1}\text{2}$x-y-z chain and the lattice model. We also show a new continuum theory with four fermion interactions exists in 1 + 1 dimensions. This theory corresponds to taking the continuum limit of the spin chain in absence of any external magnetic field. Its Hamiltonian differs from that of the Thirring model by addition of fermion number operator with an infinite coefficient and is not renormalizable in the conventional sense. It has more interesting critical properties and a different spectrum.     

On the effective action of center vortices in continuum Yang-Mills theory

The effective action of sufficiently smooth center vortices in continuum Yang-Mills theory is investigated taking into account some of their basic properties observed in lattice simulations. The obtained action is given by the action of an ideal vortex configuration, which is local on the vortex worldsheet, nonlocal quantum corrections due to virtual gluon exchanges, which can be reliably calculated using perturbation theory, and the contribution of Abelian magnetic monopoles. Smooth center vortices appear to have negative rigidity. This implies that, in agreement with lattice data, in the ground state of the theory center vortices are not regular surfaces. The text was submitted by the authors in English.     

Lattice spinor gravity

We propose a regularized lattice model for quantum gravity purely formulated in terms of fermions. The lattice action exhibits local Lorentz symmetry, and the continuum limit is invariant under general coordinate transformations. The metric arises as a composite field. Our lattice model involves no signature for space and time, describing simultaneously a Minkowski or euclidean theory. It is invariant both under Lorentz transformations and euclidean rotations. The difference between space and time arises from expectation values of composite fields. Our formulation includes local gauge symmetries beyond the generalized Lorentz symmetry. The lattice construction can be employed for formulating models with local gauge symmetries purely in terms of fermions.     

Geometry and symmetries in lattice spinor gravity

Lattice spinor gravity is a proposal for regularized quantum gravity based on fermionic degrees of freedom. In our lattice model the local Lorentz symmetry is generalized to complex transformation parameters. The difference between space and time is not put in a priori, and the euclidean and the Minkowski quantum field theory are unified in one functional integral. The metric and its signature arise as a result of the dynamics, corresponding to a given ground state or cosmological solution. Geometrical objects as the vierbein, spin connection or the metric are expectation values of collective fields built from an even number of fermions. The quantum effective action for the metric is invariant under general coordinate transformations in the continuum limit. The action of our model is found to be also invariant under gauge transformations. We observe a “geometrical entanglement” of gauge- and Lorentz-transformations due to geometrical objects transforming non-trivially under both types of symmetry transformations.     

Field theories on a lattice

Lattice field theories are described as a way to regularize continuum quantum field theories. They are obtained by replacing ordinary space time by a lattice, space time derivatives by suitable differences and Minkowski by Euclidean space. The connection between a quantum field theory isd space dimension and classical statistical mechanics in (d+1) dimensions is brought outvia elementary examples. The problem of regaining the continuum limit and of handling nonabelian gauge theories are briefly discussed.     

Yang-Mills theory in three dimensions as a quantum gravity theory

We perform the dual transformation of theYang-Mills theory in three dimensions using the Wilson action on the cubic lattice. The dual lattice is made of tetrahedra triangulating a 3-dimensional curved manifold but which is embedded into a flat 6-dimensional space [for the SU(2) gauge group]. In the continuum limit, the theory can be reformulated in terms of 6-component gauge-invariant scalar fields having the meaning of the external coordinates of the dual lattice sites. These 6-component fields induce a metric and a curvature of the 3-dimensional dual-color space. The Yang-Mills theory can also be rewritten as a quantum gravity theory with the Einstein-Hilbert action but with a purely imaginary Newton constant plus a homogeneous “ether” term. The theory can be formulated in a gauge-invariant and local form without explicit color degrees of freedom.     

Dilute neutron matter on the lattice at next-to-leading order in chiral effective field theory

We discuss lattice simulations of the ground state of dilute neutron matter at next-to-leading order in chiral effective field theory. In a previous paper the coefficients of the next-to-leading-order lattice action were determined by matching nucleon-nucleon scattering data for momenta up to the pion mass. Here the same lattice action is used to simulate the ground state of up to 12 neutrons in a periodic cube using Monte Carlo simulations. We explore the density range from 2% to 8% of normal nuclear density and analyze the ground-state energy as an expansion about the unitarity limit with corrections due to finite scattering length, effective range, and P -wave interactions.     

Systematics of collective correlation energies from self-consistent mean-field calculations

We study nucleon-nucleon scattering on the lattice at next-to-leading order in chiral effective field theory. We determine phase shifts and mixing angles from the properties of two-nucleon standing waves induced by a hard spherical wall in the center-of-mass frame. At fixed lattice spacing we test model independence of the low-energy effective theory by computing next-to-leading-order corrections for two different leading-order lattice actions. The first leading-order action includes instantaneous one-pion exchange and same-site contact interactions. The second leading-order action includes instantaneous one-pion exchange and Gaussian-smeared interactions. We find that in each case the results at next-to-leading order are accurate up to corrections expected at higher order.     

Simplicial gauge theory and quantum gauge theory simulation

We propose a general formulation of simplicial lattice gauge theory inspired by the finite element method. Numerical tests of convergence towards continuum results are performed for several SU(2) gauge fields. Additionally, we perform simplicial Monte Carlo quantum gauge field simulations involving measurements of the action as well as differently sized Wilson loops as functions of β.     

A rigorous construction of the chiral anomaly on a spin manifold

The quantum field for Dirac fermion is rigorously formulated on a compact spin manifold by using the functional integral defined as the continuum limit of a lattice approximation with a new action. Within this framework, the chiral anomaly for a fermion interacting with gauge as well as gravitational fields is calculated with mathematical rigor.     

Finite Size Corrections for the Ising Model on Higher Genus Triangular Lattices

We study the topology dependence of the finite size corrections to the Ising model partition function by considering the model on a triangular lattice embedded on a genus two surface. At criticality we observe a universal shape dependent correction, expressible in terms of Riemann theta functions, that reproduces the modular invariant partition function of the corresponding conformal field theory. The period matrix characterizing the moduli parameters of the limiting Riemann surface is obtained by a numerical study of the lattice continuum limit. The same results are reproduced using a discrete holomorphic structure.     

Center vortex properties in the Laplace-center gauge of SU(2) Yang–Mills theory

Resorting to the the Laplace center gauge (LCG) and to the Maximal-center gauge (MCG), respectively, confining vortices are defined by center projection in either case. Vortex properties are investigated in the continuum limit of SU(2) lattice gauge theory. The vortex (area) density and the density of vortex crossing points are investigated. In the case of MCG, both densities are physical quantities in the continuum limit. By contrast, in the LCG the piercing as well as the crossing points lie dense in the continuum limit. In both cases, an approximate treatment by means of a weakly interacting vortex gas is not appropriate.     

Duals for Non-Abelian Lattice Gauge Theories by Categorical Methods

We introduce duals for non-Abelian lattice gauge theories in dimension at least three by using a categorical approach to the notion of duality in lattice theories. We first discuss the general concepts for the case of a dual-triangular lattice (i.e., the dual lattice is triangular) and find that the commutative tetrahedron condition of category theory can directly be used to define a gauge-invariant action for the dual theory. We then consider the cubic lattice (where the dual is cubic again). The case of the gauge group SU(2) is discussed in detail. We will find that in this case gauge connections of the dual theory correspond to SU(2) spin networks, suggesting that the dual is a discrete version of a quantum field theory of quantum simplicial complexes (i.e. the dual theory lives already on a quantized level in its classical form). We conclude by showing that our notion of duality leads to a hierarchy of extended lattice gauge theories closely resembling the one of extended topological quantum field theories. The appearance of this hierarchy can be understood by the quantum von Neumann hierarchy introduced by one of the authors in previous work.