The large-N deconfinement phase transition in a partially reduced Eguchi-Kawal model

We construct generalized twisted Eguchi-Kawai models which for large-N reduce space-time to a lattice of arbitrary size. Large-N lattice gauge theory at finite temperature is investigated in a model on a lattice with L₀ time slices and two lattice points in very time slice. We observe the large-N deconfinement phase transition in the weak coupling region. Assuming asymptotic scaling we find a transition temperature T_c = (101±4)Λ_L.     

The large-N phase transition in the U(N) chiral models

We prove the existence of a large-N phase transition in the d = 2 chiral model and calculate via strong-coupling methods the phase-transition point. The critical coupling constant is 0.324. We also treat the chiral model chains (equivalent to the gauge model on polyhedrons) and our approximate calculations come very close to the exact results for the solvable cases.     

Strongly coupled massive QCD3 and frustrated Heisenberg antiferromagnets: fractional statistics and chiral-spin order

Strongly coupled massive SU(N_C) and U(N_C) QCD₃ on a lattice is studied using the 1/N_C expansion. The quark mass terms have a definite sign in the present model, and therefore the system explicitly breaks the parity symmetry. The continuum counterpart generates the Maxwell + Chern-Simons theory by integrating out the quark field. In the present paper, we shall integrate out the gauge fields using the strong-coupling expansion and obtain a frustrated quantum Heisenberg model as an effective model. The ground state of the above effective quantum spin model is studied using the large-N_C approximation. There are two phases; one is a Neel-ordered state and the other is a state with a chiral-spin order. It is explicitly shown that the chiral-spin ordered state corresponds to a state with spontaneous generation of color magnetic flux in the original theory and fractional statistics appears in that phase. This result strongly suggests that there are (at least) two phases in the massive QCD₃ and Maxwell-CS theory. One is the confinement phase and the other is the perturbative deconfinement phase with fractional-statistics excitations.     

A path-functional field theory of lattice gauge models and the large-N limit

We transform lattice gauge models to a theory of functional fields defined on a set of closed paths. Some relevant properties of the formalism are discussed in detail, with emphasis on symmetry and topological structure. We then investigate the large-N limit of the U(N) lattice gauge model in arbitrary dimensions using this formalism. Assuming the existence of the limit, we show, to arbitrary order of the strong coupling expansion parameter (g²N)^?, which is kept fixed, that for the leading contribution in the limit: (i) the flow of indices in color space can be represented by planar diagrams; (ii) when the diagrams are immersed in space-time they are random surfaces without handles; (iii) there are interactions of the surfaces which can be depicted as the formation of multisheet bubblesw in the surfaces. This formalism also makes it possible to set up a gauge-invariant mean-field approximation.     

Variational method in hamiltonian lattice gauge theories

A general expression for the expectation value of the hamiltonian of a d + 1 dimensional lattice gauge theory as a function of the norm of the variational state (that itself has the form of a partition function of a d-dimensional lattice gauge theory) is given. Applications include U(1), SU(2), U(2) and U(N) gauge theories for large N in d = 2 + 1 dimensions. It is also demonstrated that the deconfining phase transition is of first order in every dimension above the critical one, provided it is of first or second order at the critical dimension.     

Strong coupling versus large N in σ-models

We show that the lattice CP^(N?1) model does not have a phase transition at N = ∞, of the type proposed for SU(N) gauge theories, despite having instantons. It does, however, have non-commuting large-N and strong coupling expansions. This rather unique situation is associated with the screening properties of the model.     

A phase transition in the quenched Eguchi-Kawai model

Numerical evidence is given that the QEK model has the same phase structure as the large-N limit of the infinite volume lattice gauge theory with a Wilson action.     

Liquid–gas and other unusual thermal phase transitions in some large-N magnets

Much insight into the low temperature properties of quantum magnets has been gained by generalizing them to symmetry groups of order N, and then studying the large-N limit. In this paper we consider an unusual aspect of their finite temperature behavior—their exhibiting a phase transition between a perfectly paramagnetic state and a paramagnetic state with a finite correlation length at N=∞. We analyze this phenomenon in some detail in the large “spin” (classical) limit of the SU(N) ferromagnet which is also a lattice discretization of the CP^N−1 model. We show that at N=∞ the order of the transition is governed by lattice connectivity. At finite values of N, the transition goes away in one or less dimension but survives on many lattices in two dimensions and higher, for sufficiently large N. The latter conclusion contradicts a recent conjecture of Sokal and Starinets [Nucl. Phys. B 601 (2001) 425], yet is consistent with the known finite temperature behavior of the SU(2) case. We also report closely related first order paramagnet–ferromagnet transitions at large N and shed light on a violation of Elitzur's theorem at infinite N via the large-q limit of the q state Potts model, reformulated as an Ising gauge theory.     

Spontaneous chiral symmetry breaking for a U(N) gauge theory on a lattice

We study the breakdown of chiral invariance by calculating, in the infinite coupling, large-N limit, the generating functional of a U(N) gauge theory with one fermion, expressed on a lattice with the naive, chiral symmetric action. We compute the link integral over the gauge fields and the expression obtained after the integration over the fermions is recast under the form of a generating functional for bosonic fields. Then, a saddle-point method allows the calculation of the order parameter $\text{〈}\text{ψ}\text{ψ〉}$ for which a non-zero value signals the spontaneous breakdown of chiral symmetry. The analysis of the fluctuations around the saddle point allows one to exhibit the Goldstone modes corresponding to those global symmetries of the fermionic lattice action which are simultaneously broken.     

The weak-coupling phase of lattice spin and gauge models

We analyze the lattice weak-coupling (w.c.) expansion of O(N), CP^N?1 and chiral spin models, and of large-N reduced chiral and gauge models.We find that the w.c. expansion always agrees with mean field results, whenever comparable, for arbitrary space-time dimensions, and that the expansion of the reduced models agrees with that of the original ones. However, w.c. results disagree with one-dimensional large-N and (old and new) exact results. We explain this phenomenon as a failure of the analytic continuation from higher dimensions that defines lattice w.c. perturbation theory for massless models (even if infrared singularities always cancel).We use an improved version of the mean field (m.f.) technique suitable for reduced models. We compute the m.f. approximation of chiral models and use this result to determine the large-d (m.f.) behaviour of reduced gauge models, finding agreement with standard Wilson theory results.We give a new characterization of large-N chiral models in terms of the single-link integral for the adjoint representation of SU(N).     

Abelian lattice gauge theories in 3 + 1 dimensions: The linked cluster approach

We use the linked cluster expansion methods of Nickel to derive strong couping series for Z_N abelian gauge theories. These new results together with corresponding estimates using the exact linked cluster expansion algorithm are analysed and compared with previously obtained results for U(1) lattice gauge theory in 3 + 1 dimensions. We confirm the phase structure of these theories as found by other techniques. The critical value of N at which the phase structure of Z_N alters is estimated to be N_C = 4.5 ± 0.2. In each case the string tension estimates using the ELCE algorithm are found to be stable in the presence of a roughening transition.     

Factorisation in large-N limit of lattice gauge theories revisited

We prove using the Schwinger-Dyson equations that the factorisation property holds for all gauge-invariant Green’s function in the large-N limit of a Wilson-Polyakov lattice gauge theory.     

Symmetry aspects of finite-temperature confinement transitions

Analytical and numerical work on confinement phase transitions in finite-temperature Abelian and non-Abelian gauge theories is reviewed. These transitions are order-disorder transitions and their critical properties (if any) can be understood from the standard theory of critical phenomena. Strong coupling, large-N, and non-perturbative lattice methods are discussed. The role of matter fields as symmetry-breaking perturbations is noted as important to the eventual unraveling of the phase structure of quantum chromodynamics.     

Phase structure of U(N→∞) gauge theory on a two-dimensional lattice for a broad class of variant actions

It is pointed out that finding the partition function for U(N) gauge theory on a two-dimensional lattice in the limit N→∞ reduces, for a broad class of single-plaquette actions, to a well-known and solved mathematical problem. The case where in the single plaquette action the matrix U + U⁺ occuring in Wilson's formula is replaced by an arbitrary polynomial in this matrix, is discussed in detail and explicit results for the second-order polynomial are presented. A rich phase structure with second- and third-order phase transitions is found. The results are shown to have at the qualitative level a simple thermodynamical interpretation. They support the view that the phase structure of a lattice gauge theory is an artifact of the lattice action used rather than some reflection of the underlying group structure.     

U(3) four-dimensional lattice gauge theory and Monte Carlo calculations

Monte Carlo results for the pure U(3) lattice gauge theory on a 6⁴ lattice are reported. Wilson loops and the string tension are presented. The first-order phase transition in U(3) is reflected quite clearly in a discontinuity in the string tension at β = β_c. The U(1) factor of U(3) is extracted using the determinant of the Wilson loops. As expected, the U(1) component appears to deconfine at the phase transition..     

CPN−1 coupled to gauge fields: Mean field theory and monte carlo results

We define a two parameter lattice field theory which interpolates between the O (2N) Heisenberg model, pure U(1) gauge theory, and a lattice version of the CP^N?1 model. The phase diagram in space-time dimension d=4 is obtained by Monte Carlo simulation on a 4⁴ lattice, and the nature of the phases is discussed in mean field approximation.     

Quantum phase transitions in the anisotropic three dimensional XY model

In this paper we study the quantum phase transition in a three-dimensional XY model with single-ion anisotropy D and spin S=1. The low D phase is studied using the self consistent harmonic approximation, and the large D phase using the bond operator formalism. We calculate the critical value of the anisotropy parameter where a transition occurs from the large-D phase to the Néel phase. We present the behavior of the energy gap, in the large-D phase, as a function of the temperature. In the large D region, a longitudinal magnetic field induces a phase transition from the singlet to the antiferromagnetic state, and then from the AFM one to the paramagnetic state.     

The finite-temperature phase transition at infinite N

The finite-temperature breakdown of the global Z(N) symmetry of 3 + 1 dimensional lattice gauge theories is studied within the framework of a reduced, quenched, large-N model using both analytical and numerical methods.     

Monte Carlo studies of wilson loops in four-dimensional U(2) gauge theory

Wilson loops are calculated using Monte Carlo simulations for pure U(2) gauge theory on a 6⁴ lattice. The loops appear to contain an area law piece in both the high and low temperature regions. The string tension is discontinuous at β = β_c, where β_c is the critical inverse temperature. This suggests that the first-order phase transition in U(2) gauge theory is not a deconfining phase transition. The determinant of the Wilson loop, however, extracts the U(1) part of the theory and appears to lose the area law at low temperature.     

A study of the bulk phase transitions of the SU(2) lattice gauge theory with mixed action

Using the finite size scaling theory, we reexamine the nature of the bulk phase transition in the fundamental-adjoint coupling plane of the SU(2) lattice gauge theory at β_A = 1.25 where previous finite size scaling investigations of the deconfinement phase transition showed it to be of first order for temporal lattices with four sites. Our simulations on N⁴ lattices with N = 6, 8, 10, 12 and 16 show an absence of a first order bulk phase transition. We find the discontinuity in the average plaquette to decrease approximately linearly with N. Correspondingly, the plaquette susceptibility grows a lot slower with the 4-volume of the lattice than expected from a first order bulk phase transition.