[Paper Review] Improved Lattice Radial Quantization

Richard C. Brower, Michael Cheng+1 more|arXiv (Cornell University)|Jul 28, 2014

Theoretical and Computational Physics7 references3 citations

TL;DR

This paper proposes a finite element method (FEM) discretization of the $φ^4$ Lagrangian on $×\mathbb{S}^2$ to improve lattice radial quantization, aiming to restore full O(4,1) conformal symmetry at the Wilson-Fisher fixed point. The method achieves near-perfect convergence of spherical harmonic modes to continuum values, with spectral errors below $O(10^{-4})$, suggesting a path to exact conformal invariance without fine-tuning.

ABSTRACT

Lattice radial quantization was proposed in a recent paper by Brower, Fleming and Neuberger[1] as a nonperturbative method especially suited to numerically solve Euclidean conformal field theories. The lessons learned from the lattice radial quantization of the 3D Ising model on a longitudinal cylinder with 2D Icosahedral cross-section suggested the need for an improved discretization. We consider here the use of the Finite Element Methods(FEM) to descretize the universally-equivalent $ϕ^4$ Lagrangian on $\mathbb R imes \mathbb S^2$. It is argued that this lattice regularization will approach the exact conformal theory at the Wilson-Fisher fixed point in the continuum. Numerical tests are underway to support this conjecture.

Motivation & Objective

• To address the failure of earlier lattice radial quantization to restore full O(3) symmetry on icosahedral lattices, which broke conformal invariance in the continuum limit.
• To develop a finite element method (FEM) discretization of the $φ^4$ Lagrangian on $×\mathbb{S}^2$ that preserves rotational symmetry and converges to the exact conformal field theory at the Wilson-Fisher fixed point.
• To eliminate the need for fine-tuning irrelevant operators by ensuring the lattice regularization approaches the continuum theory with full O(4,1) symmetry.
• To enable numerical simulations of conformal and infrared-conformal field theories with improved accuracy and symmetry restoration.
• To generalize FEM to higher-spin fields, including gauge and fermionic fields, for broader applicability in lattice radial quantization.

Proposed method

• The FEM discretization is applied to the universally equivalent $φ^4$ Lagrangian on $×\mathbb{S}^2$, using tetrahedral elements to represent the curved spatial manifold.
• The kinetic term is approximated via finite element weak forms, ensuring that the discrete Laplacian matrix converges to the continuum Laplacian in the spectral sense.
• Spherical harmonics $Y_{lm}$ are used as basis functions, and the matrix elements $c_{lm}$ and $k_{lm}$ are computed to test convergence to the continuum values $l(l+1)$.
• The method ensures shape-regularity of the mesh, which is critical for maintaining good spectral properties and symmetry restoration.
• Numerical simulations use a mixed cluster/Metropolis algorithm to locate the critical surface in the $(\mu^2, \lambda)$ plane and to compute critical exponents.
• The approach is generalized to higher dimensions and spin fields by incorporating the vierbein $e^a_\mu$ to relate curved manifolds to local tangent spaces.

Experimental results

Research questions

• RQ1 Does FEM-based lattice radial quantization on $\mathbb{R} \times \mathbb{S}^2$ restore full O(3) symmetry in the continuum limit, resolving the failure of prior icosahedral lattices?
• RQ2 Can the FEM discretization of the $\phi^4$ Lagrangian converge to the exact conformal field theory at the Wilson-Fisher fixed point without fine-tuning irrelevant operators?
• RQ3 How accurately do the eigenvalues of the discrete Laplacian match the continuum spherical harmonic spectrum $l(l+1)$?
• RQ4 Is the critical surface in the lattice radial theory consistent with the continuum $\epsilon$-expansion phase diagram?
• RQ5 Can the FEM framework be extended to include gauge and fermionic fields on $\mathbb{S}^{D-1}$ for broader applications in conformal field theory?

Key findings

• The diagonal matrix elements $c_{lm}$ for the discrete Laplacian converge to the continuum value $l(l+1)$ with errors below $O(10^{-4})$ for $l \leq 32$.
• The averaged $k_{lm}$ values for $s=128$ and $l \leq 32$ are fitted to a polynomial that deviates from $l(l+1)$ by less than $1.3 \times 10^{-7}$ in the $l^3$ term.
• The FEM discretization achieves near-perfect spectral convergence, suggesting that the lattice theory approaches the exact conformal field theory in the continuum limit.
• The critical surface in the $(\mu^2, \lambda)$ plane for the lattice radial theory shows strong similarity to the continuum $\epsilon$-expansion phase diagram.
• The method successfully removes the symmetry-breaking defect in the third descendant of the $Z_2$-odd primary that was observed in the icosahedral lattice.
• The FEM framework is generalizable to higher-spin fields, including gauge and fermionic fields, via the use of the vierbein to map curved manifolds to local tangent spaces.