Hamiltonian Truncation Study of the Phi^4 Theory in Two Dimensions II. The Z_2-Broken Phase and the Chang Duality

TL;DR

The paper investigates the two‑dimensional $\phi^4$ theory in the $\mathbb{Z}_2$ broken phase using an enhanced Hamiltonian truncation approach with refined zero‑mode treatment. It verifies Chang duality by comparing spectra obtained from the direct symmetric description and a dual broken description on finite volumes, showing consistent results even at strong coupling. The study splits the problem into perturbative and non‑perturbative sectors, confirming perturbative masses and vacuum energies with standard calculations and matching the semiclassical kink mass predictions for finite‑volume splittings. This work demonstrates the viability of exact diagonalization methods for non‑integrable QFTs, providing a path toward higher‑dimensional applications and direct kink sector analyses in future research.

Abstract

The Fock-space Hamiltonian truncation method is developed further, paying particular attention to the treatment of the scalar field zero mode. This is applied to the two-dimensional Phi^4 theory in the phase where the Z_2-symmetry is spontaneously broken, complementing our earlier study of the Z_2-invariant phase and of the critical point. We also check numerically the weak/strong duality of this theory discussed long ago by Chang.

Figures (11)

• Figure 1: Left panel: equation $F(X)=f(x)$ has two solutions for $x>x_*$. Right panel: the two solution branches $X_{1,2}(x)$. We are mostly interested in the lower branch $X_1(x)$ which becomes weakly coupled as $x\to\infty$.
• Figure 2: The ground state energy (left) and the spectrum of excitations (right) for the direct and the dual theory as a function of $g$ for $m=1$, $L=5$. The excitation plot shows the energies of the $\mathbb{Z}_2$ odd and $\mathbb{Z}_2$ even energy levels. See the text for the details.
• Figure 3: The ground state energy density and the low-energy excitation spectrum as a function of $g$ for $L=12$; see the text. Results extracted from Coser:2014lla are shown by crosses (whose size does not reflect the uncertainty), see section \ref{['sec:translation']}.
• Figure 4: Same as in figure \ref{['fig:vsG_L20']} but for $L=20$.
• Figure 5: Results for $g=0.1$ plotted as a function of $L$.