Hamiltonian truncation study of the ${\phi }^4$ theory in two dimensions

We defend the Fock-space Hamiltonian truncation method, which allows us to calculate numerically the spectrum of strongly coupled quantum field theories, by putting them in a finite volume and imposing a UV cutoff. The accuracy of the method is improved via an analytic renormalization procedure inspired by the usual effective field theory. As an application, we study the two-dimensional ${\varphi }^4$ theory for a wide range of couplings. The theory exhibits a quantum phase transition between the symmetry-preserving and symmetry-breaking phases. We extract quantitative predictions for the spectrum and the critical coupling and make contact with previous results from the literature. Future directions to further improve the accuracy of the method and enlarge its scope of applications are outlined.

Figure 1

A test of the $M(E)$ asymptotics; see the text.

Figure 2

Exact and numerical ground state energy for the ${\varphi }^2$ perturbation; see the text.

Figure 3

Exact and numerical spectra of excitations for the ${\varphi }^2$ perturbation; see the text.

Figure 4

Numerical spectra as a function of $g$ for $m=1$, $L=10$; see the text.

Figure 5

The vacuum energy (left) and the first odd excitation (right) determined numerically for $L=6,8,10$. The blue dashed line in the right plot is the fit to determine the critical coupling; see Sec. 4b.

Figure 6

Comparison with the CFT spectrum; see the text.

Figure 7

The vacuum energy density and the excitation spectrum for $g=1$, as a function of $L$.

Figure 8

Same as in Fig. 7, but for $g=2.97$.

Figure 9

Variation with $E_{\text{max}}$ and the effect of renormalization corrections for $g=1$.

Figure 10

Same as in Fig. 9 but for $g=3$.

Figure 11

Finite-volume spectrum of the ${\varphi }^4$ theory on a circle of length $L=10\pi m^{-1}$ (plot taken from [4]). In our notation $\lambda /4!=g$, $\mu =m$. Black solid lines with error bars—the results of QSE with 250 states. Black dashed line—the results of a lattice Monte Carlo simulation. On their plot we overlay our results for the lowest ${\mathbb{Z}}_2$-odd state on a circle of a smaller length $L=10m^{-1}$ (red band). The central value and the width of the red band are the same as in the conservative method of determining ${\overline{g}}_c$ in Sec. 4b.

Figure 12

The lattice and the continuum RG flows should agree in the IR. See the text.

Figure 13

Comparison of perturbative and numerical predictions; see the text.