Investigations in two dimensional field theories on Euclidean lattice

Collaborators:
Asit De
Jyotirmoy Maiti
Tilak Sinha

Euclidean lattice formulation is another convenient method for the norperturbative study of quantum field theories. Following is a summary of our investigations in two dimensional phi⁴ theory.

We performed a detailed numerical analysis of (1+1) dimensional lattice phi⁴ theory [1] . We explored the phase diagram of the theory with two different parameterizations. We found 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 were 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 were extracted using a finite size scaling analysis of the data for several volumes. The scaling behavior of Z has the interesting consequence that R> does not scale in 1+1 dimensions. We also calculated the renormalized coupling constant lambda_R in the broken symmetry phase. The ratio lambda_R/m_R² does not scale and appears to reach a value independent of the bare parameters in the critical region in the infinite volume limit.

We further investigated [2] the topological charge in 1+1 dimensional phi⁴ theory on a lattice with Anti Periodic Boundary Condition (APBC) in the spatial direction. We propose a simple order parameter for the lattice theory with APBC and we demonstrate its effectiveness. Our study suggests that kink condensation is a possible mechanism for the order-disorder phase transition in the 1+1 dimensional phisup>4 theory. With renormalizations performed on the lattice with Periodic Boundary Condition (PBC), the topological charge in the renormalized theory is given as the ratio of the order parameters in the lattices with APBC and PBC. We present a comparison of topological charges in the bare and the renormalized theory and demonstrate invariance of the charge of the renormalized theory in the broken symmetry phase.

[1] Investigations in 1+1 dimensional lattice phi⁴ theory, Asit K. De, A. Harindranath, Jyotirmoy Maiti, Tilak Sinha, Phys. Rev. D 72, 094503 (2005).
[2] Topological charge in in 1+1 dimensional lattice phi⁴ theory, Asit K. De, A. Harindranath, Jyotirmoy Maiti, Tilak Sinha, Phys. Rev. D 72, 094504 (2005).