A Cheap Alternative to the Lattice?

TL;DR

This paper introduces a nonperturbative Hamiltonian approach using the Truncated Conformal Space Approach (TCSA) to study quantum field theories in non-integer dimensions, demonstrating its effectiveness through analysis of phi^4 theory in 2.5 dimensions.

Contribution

The paper develops and applies a TCSA-based method for accurate, controlled calculations in quantum field theories in non-integer dimensions, including detailed analysis of phi^4 theory in 2.5 dimensions.

Findings

Successfully observed different phases of the theory across coupling strengths.

Measured approximate masses and critical exponents.

Found non-unitarity in theories with non-integer dimensions, with limited low-energy effects.

Abstract

We show how to perform accurate, nonperturbative and controlled calculations in quantum field theory in d dimensions. We use the Truncated Conformal Space Approach (TCSA), a Hamiltonian method which exploits the conformal structure of the UV fixed point. The theory is regulated in the IR by putting it on a sphere of a large finite radius. The QFT Hamiltonian is expressed as a matrix in the Hilbert space of CFT states. After restricting ourselves to energies below a certain UV cutoff, an approximation to the spectrum is obtained by numerical diagonalization of the resulting finite-dimensional matrix. The cutoff dependence of the results can be computed and efficiently reduced via a renormalization procedure. We work out the details of the method for the phi^4 theory in d dimensions with d not necessarily integer. A numerical analysis is then performed for the specific case d = 2.5, a value chosen in the range where UV divergences are absent. By going from weak to intermediate to strong coupling, we are able to observe the symmetry-preserving, symmetry-breaking, and conformal phases of the theory, and perform rough measurements of masses and critical exponents. As a byproduct of our investigations we find that both the free and the interacting theories in non integral d are not unitary, which however does not seem to cause much effect at low energies.