[Paper Review] Non-perturbative Methods in Modal Field Theory

Nathan Salwen|ArXiv.org|Dec 19, 2002

Numerical methods for differential equations3 citations

TL;DR

This paper develops non-perturbative computational methods in modal field theory, focusing on differential renormalization in spherical field theory and efficient many-body techniques for strongly correlated systems. It introduces the quasi-sparse eigenvector (QSE) method and stochastic error correction to compute the ground state energy of the 8×8 Hubbard model with U/t=4 at 25/64 filling, achieving a result of −9.74±0.05, within 1% of the expected value.

ABSTRACT

Several issues in the modal approach to quantum field theory are discussed. Within the formalism of spherical field theory, differential renormalization is presented and shown to result in a finite number of renormalization parameters. Computations of the massless Thirring model in 1+1 dimensions are presented using this approach. Diagonalization techniques in periodic field theory are demonstrated. Issues of very large Hilbert spaces are considered and several approaches are presented. The quasi sparse eigenvector (QSE) approach takes advantage of the relatively small number of basis states that typically contribute significantly to any particular eigenvector. Stochastic correction methods use Monte Carlo calculations to calculate higher order corrections to the quasi sparse result. The quasi sparse eigenvector method and stochastic error correction are applied to the Hubbard model. With U/t=4, the shift in the ground energy below the U=0 value is found to within 1% for the 8x8 Hubbard model with 25/64 filling.

Motivation & Objective

• To develop non-perturbative techniques for quantum field theories that avoid the divergences and asymptotic series of perturbation theory.
• To address the challenge of large Hilbert spaces in lattice field theories by introducing efficient eigenvector approximation methods.
• To apply differential renormalization in spherical field theory to ensure translational invariance and finiteness with a finite number of counterterms.
• To compute the ground state energy of the 8×8 Hubbard model with U/t=4 at 25/64 filling using QSE and stochastic correction, achieving high precision.
• To test the efficacy of the power method and modified stochastic Lanczos methods for strongly correlated fermionic systems.

Proposed method

• Applies differential renormalization in spherical field theory to remove ultraviolet divergences using a finite set of local counterterms.
• Uses the quasi-sparse eigenvector (QSE) method to exploit the fact that only a small number of basis states significantly contribute to any eigenvector.
• Employs stochastic correction methods with Monte Carlo sampling to compute higher-order corrections to QSE results.
• Implements the intermediate state method in path integral Monte Carlo to extend the reach of calculations before the sign problem dominates.
• Uses the power method and modified stochastic Lanczos method to extract ground state energy from matrix elements of the Hamiltonian.
• Applies the Brillouin-Wigner perturbation theory and power method to analyze convergence and accuracy of energy estimates.

Experimental results

Research questions

• RQ1 Can differential renormalization in spherical field theory yield a finite, translationally invariant quantum field theory with only a finite number of counterterms?
• RQ2 How accurately can the ground state energy of the 8×8 Hubbard model with U/t=4 and 25/64 filling be computed using QSE and stochastic correction?
• RQ3 To what extent do the QSE and stochastic correction methods reduce computational cost while maintaining accuracy in large Hilbert spaces?
• RQ4 How do the power method and stochastic Lanczos method compare in precision and stability when applied to the Hubbard model?
• RQ5 Can the intermediate state method in Monte Carlo path sampling extend the effective range of matrix element calculations before the sign problem becomes prohibitive?

Key findings

• Differential renormalization in spherical field theory successfully removes ultraviolet divergences with a finite set of local counterterms, ensuring translational invariance.
• For the 8×8 Hubbard model at U/t=4 and 25/64 filling, the ground state energy is computed as −9.74±0.05, within 1% of the expected value.
• The power method yields the most precise result among tested methods, with a best-fit asymptotic ratio of H^n / H^{n-1} approaching −39.74.
• The 6×6 Hubbard model yields a ground state energy of −4.887 with a distance cutoff of 6, improving to −4.94 with a cutoff of 8.
• The stochastic intermediate state method improves reach in path sampling but requires careful sampling to avoid bias from poorly sampled intermediate states.
• First-order corrections in the QSE method show apparent convergence but remain far from the correct ground state energy, indicating the need for higher-order corrections.