Truncated Conformal Space Approach for 2D Landau-Ginzburg Theories

TL;DR

This work extends the truncated conformal space approach (TCSA) to two-dimensional Landau-Ginzburg theories by employing a compactified boson as the computational basis, enabling a nonperturbative study of both broken and unbroken phases. It validates the method against the free massive boson and against two-loop perturbation theory in the unbroken Φ^4 phase, and confirms semiclassical kink-bound-state predictions in the broken phase, including Φ^4 and Φ^6 in two-well potentials. Finite-momentum sectors prove crucial for distinguishing bound states (notably the second neutral bound state B2) and for testing the consistency of S-matrix and Bethe-Ansatz descriptions in finite volume. The results establish TCSA as a robust tool for non-integrable 1+1D QFTs and suggest avenues toward connecting LG theories with minimal models via renormalization effects.

Abstract

We study the spectrum of Landau-Ginzburg theories in 1+1 dimensions using the truncated conformal space approach employing a compactified boson. We study these theories both in their broken and unbroken phases. We first demonstrate that we can reproduce the expected spectrum of a $Φ^2$ theory (i.e. a free massive boson) in this framework. We then turn to $Φ^4$ in its unbroken phase and compare our numerical results with the predictions of two-loop perturbation theory, finding excellent agreement. We then analyze the broken phase of $Φ^4$ where kink excitations together with their bound states are present. We confirm the semiclassical predictions for this model on the number of stable kink-antikink bound states. We also test the semiclassics in the double well phase of $Φ^6$ Landau-Ginzburg theory, again finding agreement.

Figures (13)

• Figure 1: Potential $U(\phi)$ of a quantum field theory with kink excitations.
• Figure 2: Residue equation for the matrix element on the kink states.
• Figure 3: In panel \ref{['subfig:phi2_sp']} we show the energy levels (with the ground state energy subtracted) for the $\Phi^2$ perturbation. Here $\beta=0.1$, $g_2=0.0016$, and $N_{\text{tr}}=6$. The numerical mass coincides with the expected value of $m_0=\sqrt{g_2} = 0.04$. The dashed black line represents the prediction for the energy of two particles moving with momentum $\pm 2\pi/R$, that is $E=2\sqrt{(2\pi/R)^2+m_0^2}$. We see that it provides a good match to the numerical data. In panel \ref{['subfig:phi2_mg']} we show the energy (points) of the first two levels as a function of $m_0=\sqrt{g_2}$ for a fixed $m_0 R=4$. Their values are compared with the analytically expected values, $E=m_0$ and $E=2m_0$, plotted as solid lines.
• Figure 4: We show here for a $\Phi^2$ theory with $g_2=0.0016$, and $N_{\text{tr}}=6$ how the first two excited states (\ref{['subfig:phi2_beta_1']}: the single particle state of nominal mass $m_0$ and \ref{['subfig:phi2_beta_2']}: the two-particle state with nominal energy $2m_0$) in the zero momentum sector depend on the compactification radius, $2\pi/\beta$. We plot these energies as a function of $\beta$ at different fixed values of $m_0R$. We see that the spectrum displays the greatest sensitivity to non-zero values of $\beta$ when $m_0R$ is small.
• Figure 5: Two spectra for $\Phi^4$ LG theories with \ref{['subfig:phi4pos_sp_set1']}$g_2=0.01$, $g_4=2\times 10^{-6}$ and \ref{['subfig:phi4pos_sp_set9']}$g_2=0.01$, $g_4=8\times 10^{-5}$. In both cases $\beta=0.07$, $N_{\text{tr}}=6$ and the ground state energy has been subtracted. All energies are normalized with respect to $m_0$ and are plotted as function of $m_0R$. The first level (blue) is a one-particle state of a zero momentum particle, while the second level (red) is a state with two such particles.