A Cheap Alternative to the Lattice?

TL;DR

This work extends the Truncated Conformal Space Approach (TCSA) from $d=2$ to general $d$, including fractional dimensions, by regulating the theory on a sphere $S^{d-1}_R$ and constructing a finite Hamiltonian from the UV CFT data. Applying TCSA to the Landau–Ginzburg flow generated by perturbing the free scalar with $: abla^2 abla:$- and $: abla^4 abla:$-type operators (specifically $: ext{φ}^2:$ and $: ext{φ}^4:$) in $d=2.5$, the authors observe symmetry-preserving, symmetry-breaking, and IR conformal phases, and extract rough masses and critical exponents. They show that, for fractional $d$, the free and interacting theories are not unitary due to negative-norm states, which can yield complex energy levels at high energies, though low-energy spectra can remain real with proper renormalization. A central methodological contribution is a detailed renormalization program that computes $ riangle H$ via OPEs and runs couplings with RG-like equations, markedly improving convergence and reducing cutoff dependence. The results suggest TCSA as a computationally cheap alternative to the lattice for certain nonperturbative QFT flows, while highlighting challenges from exponential state growth and non-unitarity in fractional dimensions.

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.

Figures (22)

• Figure 1: The graph corresponding to the only scalar operator which can be obtained by contracting indices in (\ref{['eq:optyp']}).
• Figure 2: The number of scalar $P$-even states in the extended Hilbert space of free massless scalar theory in $d=3$ on the cylinder. Blue squares: all states (physical + null). Red dots: null states. Black dotted curve: just physical states. The proportion of null states grows quickly: at $\Delta=18$, which is the maximal cutoff we will be working with, about a quarter of all states are null. In future studies one should perhaps separate the null states to speed up the numerics.
• Figure 3: The ground state energy of the $\phi^2$ flow in $d=3$ as a function of $R$ (we set $m=1$). Solid black curve: theory prediction (\ref{['eq:E0']}). Dotted black: theory limit at large $R$, Eq. (\ref{['eq:E0lim']}). Blue curves marked 'raw': raw TCSA results, i.e. before applying any correction. Red curves marked 'ren.': renormalized TCSA results, see section \ref{['sec:RG-phi2']}. Dashed and solid TCSA curves correspond to cutoff $\Delta_{\rm max} = 12(18)$.
• Figure 4: A few lowest massive excitations from the raw TCSA spectra at $\Delta_{\max}=18$ (blue dots connected with a line to guide the eye) vs exact spectrum (magenta lines). Left(right): $\mathbb{Z}_2$-even (odd) sector. The gray region indicates the sliding UV cutoff (\ref{['eq:sliding']}).
• Figure 5: Same as the previous figure, but for the renormalized TCSA spectra.