# Lattice ${\varphi}^{4}$ field theory on Riemann manifolds: Numerical tests for the 2D Ising CFT on ${\mathbb{S}}^{2}$

## 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:

- Publication Date:

- 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.
```

```
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:https://doi.org/10.1103/PhysRevD.98.014502
```

```
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:https://doi.org/10.1103/PhysRevD.98.014502.
```

```
@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}

}

DOI: https://doi.org/10.1103/PhysRevD.98.014502

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