Lattice ϕ 4 field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on S 2

doi:10.1103/PhysRevD.98.014502

Title: Lattice $ϕ^{4}$ field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on $S^{2}$

Abstract

We present a method for defining a lattice realization of the Φ ⁴ quantum field theory on a simplicial complex in order to enable numerical computation on a general Riemann manifold. The procedure begins with adopting methods from traditional Regge calculus (RC) and finite element methods (FEM) plus the addition of ultraviolet counterterms required to reach the renormalized field theory in the continuum limit. The construction is tested numerically for the two-dimensional Φ ⁴ scalar field theory on the Riemann two-sphere, S ², in comparison with the exact solutions to the two-dimensional Ising conformal field theory (CFT). Numerical results for the Binder cumulants (up to 12th order) and the two- and four-point correlation functions are in agreement with the exact c = 1 / 2 CFT solutions.

Authors:: Brower, Richard C.; Cheng, Michael; Weinberg, Evan S.; Fleming, George T.; Gasbarro, Andrew D.; Raben, Timothy G.; Tan, Chung-I

Publication Date:: 2018-07-06

Research Org.:: Brown Univ., Providence, RI (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1459244

Alternate Identifier(s):: OSTI ID: 1498847

Grant/Contract Number:: SC0015845; SC0014664; SC0010010-Task-A; sc0010010

Resource Type:: Published Article

Journal Name:: Physical Review D

Additional Journal Information:: Journal Name: Physical Review D Journal Volume: 98 Journal Issue: 1; Journal ID: ISSN 2470-0010

Publisher:: American Physical Society

Country of Publication:: United States

Language:: English

Subject:: 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

Citation Formats


                    Brower, Richard C., Cheng, Michael, Weinberg, Evan S., Fleming, George T., Gasbarro, Andrew D., Raben, Timothy G., and Tan, Chung-I. Lattice ϕ 4 field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on S 2.  United States: N. p., 2018. 
        Web.  doi:10.1103/PhysRevD.98.014502.

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                    Brower, Richard C., Cheng, Michael, Weinberg, Evan S., Fleming, George T., Gasbarro, Andrew D., Raben, Timothy G., & Tan, Chung-I. Lattice ϕ 4 field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on S 2.  United States.  doi:10.1103/PhysRevD.98.014502.

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                    Brower, Richard C., Cheng, Michael, Weinberg, Evan S., Fleming, George T., Gasbarro, Andrew D., Raben, Timothy G., and Tan, Chung-I. Fri .  
        "Lattice ϕ 4 field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on S 2".  United States.  doi:10.1103/PhysRevD.98.014502.

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@article{osti_1459244,

  title        = {Lattice ϕ 4 field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on S 2},

  author       = {Brower, Richard C. and Cheng, Michael and Weinberg, Evan S. and Fleming, George T. and Gasbarro, Andrew D. and Raben, Timothy G. and Tan, Chung-I},

  abstractNote = {We present a method for defining a lattice realization of the Φ4 quantum field theory on a simplicial complex in order to enable numerical computation on a general Riemann manifold. The procedure begins with adopting methods from traditional Regge calculus (RC) and finite element methods (FEM) plus the addition of ultraviolet counterterms required to reach the renormalized field theory in the continuum limit. The construction is tested numerically for the two-dimensional Φ4 scalar field theory on the Riemann two-sphere, S2, in comparison with the exact solutions to the two-dimensional Ising conformal field theory (CFT). Numerical results for the Binder cumulants (up to 12th order) and the two- and four-point correlation functions are in agreement with the exact c = 1 / 2 CFT solutions.},

  doi          = {10.1103/PhysRevD.98.014502},

  journal      = {Physical Review D},

  number       = 1,

  volume       = 98,

  place        = {United States},

  year         = {2018},

  month        = {7}

}

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DOI: 10.1103/PhysRevD.98.014502

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Title: Lattice ϕ 4 field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on S 2

Abstract

Citation Formats

Random lattice field theory: General formulation journal, July 1982

Simplicial Ricci Flow journal, February 2014

Lattice Dirac fermions on a simplicial Riemannian manifold journal, June 2017

Over-relaxation methods for Monte Carlo simulations of quadratic and multiquadratic actions journal, January 1984

Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models journal, April 2001

Strongly interacting dynamics and the search for new physics at the LHC journal, June 2016

Conformal invariance and the Ising model on a spheroid journal, March 2003

Weights of links and plaquettes in a random lattice journal, October 1982

Lattice radial quantization: 3D Ising journal, April 2013

General relativity without coordinates journal, February 1961

Higher-order-magnetization-cumulant universality of the two-dimensional Ising model journal, January 1997

Calculation of critical exponents in two dimensions from quantum field theory in one dimension journal, November 1975

Finite element exterior calculus, homological techniques, and applications journal, May 2006

Dimensional regularisation in the presence of large background fields journal, September 1982

Collective Monte Carlo Updating for Spin Systems journal, January 1989

Bulk, surface, and interface properties of the Ising model and conformal invariance journal, August 1987

Ising universality in three dimensions: a Monte Carlo study journal, November 1995

General background field calculations with fermion fields journal, January 1985

Conformal algebra and multipoint correlation functions in 2D statistical models journal, October 1984

Renormalization of Feynman integrals on the lattice journal, March 1988

The connection between the Λ parameters of lattice and continuum QCD journal, June 1980

Infinite conformal symmetry of critical fluctuations in two dimensions journal, March 1984

Gauge theory on a random lattice journal, October 1982

Geometric discretization scheme applied to the Abelian Chern-Simons theory journal, March 2000

Correlations along a Line in the Two-Dimensional Ising Model journal, December 1969

Infinite conformal symmetry in two-dimensional quantum field theory journal, July 1984

Overrelaxation and Monte Carlo simulation journal, July 1987

Embedded dynamics for φ 4 theory journal, March 1989

Erratum: Properties and uses of the Wilson flow in lattice QCD journal, March 2014

Equation of State Calculations by Fast Computing Machines journal, June 1953

Overrelaxed heat-bath and Metropolis algorithms for accelerating pure gauge Monte Carlo calculations journal, June 1987

Background field calculations in curved spacetime journal, April 1984

Properties and uses of the Wilson flow in lattice QCD journal, August 2010

A power counting theorem for Feynman integrals on the lattice journal, March 1988

Over-relaxation method for the Monte Carlo evaluation of the partition function for multiquadratic actions journal, June 1981

Title: Lattice $ϕ^{4}$ field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on $S^{2}$

Random lattice field theory: General formulation
journal, July 1982

Simplicial Ricci Flow
journal, February 2014

Lattice Dirac fermions on a simplicial Riemannian manifold
journal, June 2017

Over-relaxation methods for Monte Carlo simulations of quadratic and multiquadratic actions
journal, January 1984

Monte Carlo tests of renormalization-group predictions for critical phenomena in Ising models
journal, April 2001

Strongly interacting dynamics and the search for new physics at the LHC
journal, June 2016

Conformal invariance and the Ising model on a spheroid
journal, March 2003

Weights of links and plaquettes in a random lattice
journal, October 1982

Lattice radial quantization: 3D Ising
journal, April 2013

General relativity without coordinates
journal, February 1961

Higher-order-magnetization-cumulant universality of the two-dimensional Ising model
journal, January 1997

Calculation of critical exponents in two dimensions from quantum field theory in one dimension
journal, November 1975

Finite element exterior calculus, homological techniques, and applications
journal, May 2006

Dimensional regularisation in the presence of large background fields
journal, September 1982

Collective Monte Carlo Updating for Spin Systems
journal, January 1989

Bulk, surface, and interface properties of the Ising model and conformal invariance
journal, August 1987

Ising universality in three dimensions: a Monte Carlo study
journal, November 1995

General background field calculations with fermion fields
journal, January 1985

Conformal algebra and multipoint correlation functions in 2D statistical models
journal, October 1984

Renormalization of Feynman integrals on the lattice
journal, March 1988

The connection between the Λ parameters of lattice and continuum QCD
journal, June 1980

Infinite conformal symmetry of critical fluctuations in two dimensions
journal, March 1984

Gauge theory on a random lattice
journal, October 1982

Geometric discretization scheme applied to the Abelian Chern-Simons theory
journal, March 2000

Correlations along a Line in the Two-Dimensional Ising Model
journal, December 1969

Infinite conformal symmetry in two-dimensional quantum field theory
journal, July 1984

Overrelaxation and Monte Carlo simulation
journal, July 1987

Embedded dynamics for $φ^{4}$ theory
journal, March 1989

Erratum: Properties and uses of the Wilson flow in lattice QCD
journal, March 2014

Equation of State Calculations by Fast Computing Machines
journal, June 1953

Overrelaxed heat-bath and Metropolis algorithms for accelerating pure gauge Monte Carlo calculations
journal, June 1987

Background field calculations in curved spacetime
journal, April 1984

Properties and uses of the Wilson flow in lattice QCD
journal, August 2010

A power counting theorem for Feynman integrals on the lattice
journal, March 1988

Over-relaxation method for the Monte Carlo evaluation of the partition function for multiquadratic actions
journal, June 1981