Properties Of Lattice Gauge Theories Research Articles

Finite-representation approximation of lattice gauge theories at the continuum limit with tensor networks

It has been established that Matrix Product States can be used to compute the ground state and single-particle excitations and their properties of lattice gauge theories at the continuum limit. However, by construction, in this formalism the Hilbert space of the gauge fields is truncated to a finite number of irreducible representations of the gauge group. We investigate quantitatively the influence of the truncation of the infinite number of representations in the Schwinger model, one-flavour QED$_2$, with a uniform electric background field. We compute the two-site reduced density matrix of the ground state and the weight of each of the representations. We find that this weight decays exponentially with the quadratic Casimir invariant of the representation which justifies the approach of truncating the Hilbert space of the gauge fields. Finally, we compute the single-particle spectrum of the model as a function of the electric background field.

Chiral condensate in the Schwinger model with matrix product operators

Tensor network (TN) methods, in particular the matrix product states (MPS) ansatz, have proven to be a useful tool in analyzing the properties of lattice gauge theories. They allow for a very good precision, much better than standard Monte Carlo (MC) techniques for the models that have been studied so far, due to the possibility of reaching much smaller lattice spacings. The real reason for the interest in the TN approach, however, is its ability, shown so far in several condensed matter models, to deal with theories which exhibit the notorious sign problem in MC simulations. This makes it prospective for dealing with the nonzero chemical potential in QCD and other lattice gauge theories, as well as with real-time simulations. In this paper, using matrix product operators, we extend our analysis of the Schwinger model at zero temperature to show the feasibility of this approach also at finite temperature. This is an important step on the way to deal with the sign problem of QCD. We analyze in detail the chiral symmetry breaking in the massless and massive cases and show that the method works very well and gives good control over a broad range of temperatures, essentially from zero to infinite temperature.

Cooling, physical scales and the vacuum structure of Y-M theories

We present a cooling method controlled by a physical cooling radius that defines a scale below which fluctuations are smoothed out while leaving physics unchanged at all larger scales. This method can be generally used as a gauge invariant low pass filter to extract the physics from noisy MC configurations. Here we apply this method to study topological properties of lattice gauge theories where it allows to retain instanton-anti-instanton pairs.

Cooling, physical scales and topology

We develop a cooling method controlled by a physical cooling radius that defines a scale below which fluctuations are smoothed out while leaving physics unchanged at all larger scales. We apply this method to study topological properties of lattice gauge theories, in particular the behaviour of instantons, dislocations and instanton-anti-instanton pairs. Monte Carlo results for the SU(2) topology are presented. We find that the method provides a means to prevent instantonanti-instanton annihilation under cooling. While the instanton sizes are largely independent from the smoothing scale, the density and pair separations are determined by the particular choice made for this quantity. We discuss the questions this raises for the “physicality” of these concepts.

Lattice topological charge and boundary conditions

We discuss some effects of the boundary conditions on the topological properties of lattice gauge theories.

Chiral symmetry breaking in strongly coupled lattice gauge theory

The chiral properties of lattice gauge theories at strong coupling are investigated in the action formalism with naive fermions. An exact calculation of 〈 ψ ψ〉 to order 1/ gl = 1/ Ng 2 is presented in the limit of large N for SU( N) and U( N) theories. For any dimension d>2 and for large enough coupling λ, the chiral symmetry spontaneously breaks.

Concept and properties of lattice gauge theory

A finite lattice in four dimensions and correlation functions defined by integrals are used to formulate quantum gauge theory. Former results on Schwinger-Dyson equations and Ward-Takahashi identities are extended and the much richer structure of quantities and relations, which arises necessarily on the lattice, is discussed. The mechanism of gauge fixing is analyzed and various consequences of this analysis are pointed out. The implications of the generalized fermion degeneracy regularization for the position-space propagator and in the relations for the various currents are shown. An explicit solution for open boundaries is presented and compared with that of the case of the otherwise-used periodic conditions. The analog of continuum methods for dynamical masses and particular decompositions of the fermion determinants are considered. The connection between degeneracy regularization and axial-vector anomaly and the situation for weak interactions are discussed. Further, a number of important details are clarified.