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Title: Topological charge in 1+1 dimensional lattice {phi}{sup 4} theory

Abstract

We investigate the topological charge in 1+1 dimensional {phi}{sup 4} theory on a lattice with antiperiodic 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 {phi}{sup 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.

Authors:
; ; ;  [1]
  1. Theory Group, Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata 700064 (India)
Publication Date:
OSTI Identifier:
20713852
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. D, Particles Fields; Journal Volume: 72; Journal Issue: 9; Other Information: DOI: 10.1103/PhysRevD.72.094504; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BOUNDARY CONDITIONS; COMPARATIVE EVALUATIONS; LATTICE FIELD THEORY; ONE-DIMENSIONAL CALCULATIONS; ORDER PARAMETERS; PERIODICITY; PHASE TRANSFORMATIONS; PHI4-FIELD THEORY; RENORMALIZATION; SYMMETRY BREAKING; TOPOLOGY

Citation Formats

De, Asit K., Harindranath, A., Maiti, Jyotirmoy, and Sinha, Tilak. Topological charge in 1+1 dimensional lattice {phi}{sup 4} theory. United States: N. p., 2005. Web. doi:10.1103/PhysRevD.72.094504.
De, Asit K., Harindranath, A., Maiti, Jyotirmoy, & Sinha, Tilak. Topological charge in 1+1 dimensional lattice {phi}{sup 4} theory. United States. doi:10.1103/PhysRevD.72.094504.
De, Asit K., Harindranath, A., Maiti, Jyotirmoy, and Sinha, Tilak. Tue . "Topological charge in 1+1 dimensional lattice {phi}{sup 4} theory". United States. doi:10.1103/PhysRevD.72.094504.
@article{osti_20713852,
title = {Topological charge in 1+1 dimensional lattice {phi}{sup 4} theory},
author = {De, Asit K. and Harindranath, A. and Maiti, Jyotirmoy and Sinha, Tilak},
abstractNote = {We investigate the topological charge in 1+1 dimensional {phi}{sup 4} theory on a lattice with antiperiodic 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 {phi}{sup 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.},
doi = {10.1103/PhysRevD.72.094504},
journal = {Physical Review. D, Particles Fields},
number = 9,
volume = 72,
place = {United States},
year = {Tue Nov 01 00:00:00 EST 2005},
month = {Tue Nov 01 00:00:00 EST 2005}
}
  • In this work we perform a detailed numerical analysis of (1+1) dimensional lattice {phi}{sup 4} theory. We explore the phase diagram of the theory with two different parametrizations. We find that symmetry breaking occurs only with a negative mass-squared term in the Hamiltonian. The renormalized mass m{sub 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{sub R}, Z and the order parameter are extracted using a finite-size scaling analysis of the data formore » several volumes. The scaling behavior of Z has the interesting consequence that <{phi}{sub R}> does not scale in 1+1 dimensions. We also calculate the renormalized coupling constant {lambda}{sub R} in the broken symmetry phase. The ratio {lambda}{sub R}/m{sub R}{sup 2} does not scale and appears to reach a value independent of the bare parameters in the critical region in the infinite volume limit.« less
  • We compute, in the (1+1)-dimensional [lambda][Phi][sup 4] model on the lattice, the soliton mass by means of two very different numerical methods. First, we make use of a creation operator'' formalism, measuring the decay of a certain correlation function. Second, we measure the shift of the vacuum energy between the symmetric and the antiperiodic systems. The obtained results are fully compatible. We compute the continuum limit of the mass from the perturbative renormalization group equations. Special attention is paid to ensure that we are working in the scaling region, where physical quantities remain unchanged along any renormalization group trajectory. Wemore » compare the continuum value of the soliton mass with its perturbative value up to one-loop calculation. Both quantities show a quite satisfactory agreement. The first is slightly bigger than the perturbative one; this may be due to the contributions of higher-order corrections.« less
  • We study spontaneous symmetry breaking in (1+1)-dimensional [phi][sup 4] theory using the light-front formulation of field theory. Since the physical vacuum is always the same as the perturbative vacuum in light-front field theory the fields must develop a vacuum expectation value through the zero-mode components of the field. We solve the nonlinear operator equation for the zero mode in the one-mode approximation. We find that spontaneous symmetry breaking occurs at [lambda][sub critical]=4[pi](3+ [radical]3 )[mu][sup 2], which is consistent with the value [lambda][sub critical]=54.27[mu][sup 2] obtained in the equal-time theory. We calculate the vacuum expectation value as a function of themore » coupling constant in the broken phase both numerically and analytically using the [delta] expansion. We find two equivalent broken phases. Finally we show that the energy levels of the system have the expected behavior for the broken phase.« less
  • We investigate (1+1)-dimensional [phi][sup 4] field theory in the symmetric and broken phases using discrete light-front quantization. We calculate the perturbative solution of the zero-mode constraint equation for both the symmetric and broken phases and show that standard renormalization of the theory yields finite results. We study the perturbative zero-mode contribution to two diagrams and show that the light-front formulation gives the same result as the equal-time formulation. In the broken phase of the theory, we obtain the nonperturbative solutions of the constraint equation and confirm our previous speculation that the critical coupling is logarithmically divergent. We discuss the renormalizationmore » of this divergence but are not able to find a satisfactory nonperturbative technique. Finally we investigate properties that are insensitive to this divergence, calculate the critical exponent of the theory, and find agreement with mean field theory as expected.« less
  • We discuss spontaneous symmetry breaking of (1+1)-dimensional [phi][sup 4] theory in light-front field theory using a Tamm-Dancoff truncation. We show that, even though light-front field theory has a simple vacuum state which is an eigenstate of the full Hamiltonian, the field can develop a nonzero vacuum expectation value. This occurs because the zero mode of the field must satisfy an operator-valued constraint equation. In the context of (1+1)-dimensional [phi][sup 4] theory we present solutions to the constraint equation using a Tamm-Dancoff truncation to a finite number of particles and modes. We study the behavior of the zero mode as amore » function of coupling and Fock space truncation. The zero mode introduces new interactions into the Hamiltonian which breaks the [ital Z][sub 2] symmetry of the theory when the coupling is stronger than the critical coupling. We investigate the energy spectrum in the symmetric and broken phases, show that the theory does not break down in the vicinity of the critical coupling, and discuss the connection to perturbation theory. Finally, we study the spectrum of the field [phi] and show that, in the broken phase, the field is localized away from [phi]=0 as one would expect from equal-time calculations. We explicitly show that tunneling occurs.« less