Strong-coupling field theory. I. Variational approach to φ^{4} theory

1979

We develop and apply a Hamiltonian variational approach to the study of quantum electrodynamics formulated on a spatial lattice in both 2+1 and 3+1 dimensions. Two lattice versions of QED are considered: a non-compact version which reproduces the physics of continuum QED, and a compact version constructed in correspondence with lattice formulations of non-Abelian theories. Our focus is on photon dynamics with charged particles treated in the static limit. We are especially interested in the nonperturbative aspects of the solutions in the weak-coupling region in order to clarify and establish aspects of confinement. In particular we find, in accord with Polyakov, that the compact QED leads to linear confinement for any nonvanishing coupling, no matter how small, in 2+1 dimensions, but that a phase transition to an unconfined phase for sufficiently weak couplings occurs in 3+1 dimensions. We identify and describe the causes of confinement.

Nuclear Physics B, 1983

It is shown that at weak coupling physical quantities in hamiltonian U(1) lattice gauge (or global symmetric) theories of arbitrary dimension are provided as expectation values in a d-1 dimensional lagrangian Z(2) gauge (or spin) theory with calculable long-range interactions.

Physical Review D, 1976

We investigate X$ theory in the random-phase approximation as a prototypic model for understanding the formation of bound states (here 2-meson bound states) and the nature of the effective interaction between such states. We give the random-phase approximation a functional-derivative interpretation which allows us to determine the effective expansion parameter of the random-phase approximation t'o be (A,z'/N)Dz(q'), where DJ, (q') is the propagator of the collective mode. The field theory in this approximation has both mass and coupling-constant renormalization, and we derive expressions for unrenormalized and renormalized 2-, 4-, and 6-point functions for the original scalar field, and determine the 4-point function for bound-state-bound-state scattering as well as the effective coupling between bound states. We show the relation between the randomphase approximation and the O(N) model for large N and prove that without single-field symmetry breaking there is a range of renormalized coupling constant where there is no ghost.

2012

Physical Review D, 2005

In this work we perform a detailed numerical analysis of (1+1) dimensional lattice φ 4 theory. We explore the phase diagram of the theory with two different parameterizations. We find that symmetry breaking occurs only with a negative mass-squared term in the Hamiltonian. The renormalized mass m R and the field renormalization constant Z are calculated from both coordinate space and momentum space propagators in the broken symmetry phase. The critical coupling for the phase transition and the critical exponents associated with m R , Z and the order parameter are extracted using a finite size scaling analysis of the data for several volumes. The scaling behavior of Z has the interesting consequence that φ R does not scale in 1 + 1 dimensions. We also calculate the renormalized coupling constant λ R in the broken symmetry phase. The ratio λ R /m 2 R does not scale and appears to reach a value independent of the bare parameters in the critical region in the infinite volume limit.

Nuclear Physics B - Proceedings Supplements, 2002

Eprint Arxiv Cond Mat 9503085, 1995

We propose a perturbative-variational approach to interacting fermion systems on 1D and 2D lattices at half-filling. We address relevant issues such as the existence of Long Range Order, quantum phase transitions and the evaluation of ground state energy. In 1D our method is capable of unveiling the existence of a critical point in the coupling constant at (t/U) c = 0.7483 as in fact occurs in the exact solution at a value of 0.5. In our approach this phase transition is described as an example of Bifurcation Phenomena in the variational computation of the ground state energy. In 2D the van Hove singularity plays an essential role in changing the asymptotic behaviour of the system for large values of t/U. In particular, the staggered magnetization for large t/U does not display the Hartree-Fock law (t/U)e −2π √ t/U but instead we find the law (t/U)e − π 2 3 t/U. Moreover, the system does not exhibit bifurcation phenomena and thus we do not find a critical point separating a CDW state from a fermion "liquid" state.

Physical Review B

Some of the exciting phenomena uncovered in strongly correlated systems in recent years -for instance quantum topological order, deconfined quantum criticality and emergent gauge symmetries -appear in systems where the Hilbert space is effectively projected at low energies in a way that imposes local constraints on the original degrees of freedom. Cases in point include spin liquids, valence bond systems, dimer and vertex models. In this work, we use a slave boson description coupled to a large-S path integral formulation to devise a generalised route to obtain effective field theories for such systems. We demonstrate the validity and capability of our approach by studying quantum dimer models and by comparing our results with the existing literature. Field theoretic approaches to date are limited to bipartite lattices, they depend on a gauge-symmetric understanding of the constraint, and lack generic quantitative predictive power for the coefficients of the terms that appear in the Lagrangians of these systems. Our method overcomes all these shortcomings and we show how the results up to quadratic order compare with the known height description of the square lattice quantum dimer model, as well as with the numerical estimate of the speed of light of the photon excitations on the diamond lattice. Finally, instanton considerations allow us to infer properties of the finite temperature behaviour in two dimensions.

Physical Review D, 1995

Using the monomer--dimer representation of the lattice Schwinger model, with $N_f =1$ Wilson fermions in the strong--coupling regime ($\beta=0$), we evaluate its partition function, $Z$, exactly on finite lattices. By studying the zeroes of $Z(k)$ in the complex plane $(Re(k),Im(k))$ for a large number of small lattices, we find the zeroes closest to the real axis for infinite stripes in temporal direction and spatial extent $S=2$ and 3. We find evidence for the existence of a critical value for the hopping parameter in the thermodynamic limit $S\rightarrow \infty$ on the real axis at about $k_c \simeq 0.39$. By looking at the behaviour of quantities, such as the chiral condensate, the chiral susceptibility and the third derivative of $Z$ with respect to $1/2k$, close to the critical point $k_c$, we find some indications for a continuous phase transition.

Journal of Statistical Physics, 1986

Monte-Carlo calculations are performed for the model Hamiltonian ~= Yl [(r/2) #2(i) + (u/4) #4(i)] + Y<v> (C/2)[#(i)-q~(j)]2 for various values of the parameters r, u, C in the crossover region from the Ising limit (r ~-oo, u ~ +or) to the displacive limit (r = 0). The variable ~b(i) is a scalar continuous spin variable which can lie in the range-oo < ~6(i)< +0% for each lattice site (i). ~b(i) is a priori selected proportional to the single-site probability in our Monte Carlo algorithm. The critical line is obtained in very good agreement with other previous approaches. A decrease of apparent critical exponents, deduced from a finite-size scaling analysis, is attributed to a crossover toward mean-field values at the displacive limit. The relation of this model to the coarse-grained Landau-Ginzburg Wilson free-energy functional of Ising models is discussed in detail, and, by matching local moments (r (~4(i)) to corresponding averages of subsystem blocks of Ising systems with linear dimensions l= 5 to l= 15, an explicit construction of this coarse-grained free energy is attempted; self-consistency checks applied to this matching procedure show qualitatively reasonable behavior, but quantitative difficulties remain, indicating that higher-order terms are needed for a quantitatively satisfactory description.