Truncated Conformal Space Approach for Perturbed Wess-Zumino-Witten $SU(2)_k$ Models

TL;DR

This work extends the truncated conformal space approach (TCSA) to perturbations of SU(2)$_k$ WZW models, providing a practical framework for non-perturbative, one-dimensional quantum field theories. By constructing the SU(2)$_k$ conformal basis and computing perturbation matrix elements with current-algebra structure constants, the authors test the method on three cases: SU(2)$_1$ perturbed by the spin-1/2 field (sine-Gordon at $\beta^2=2\pi$), SU(2)$_1$ with a marginal current-current perturbation, and SU(2)$_2$ perturbed by the spin-1 field (three massive Majorana fermions). The results show precise replication of known spectra and finite-size effects, reveal regulator-dependent universal terms in the marginal case, and demonstrate that an NRG extension enables accurate treatment of large Hilbert spaces and marginal perturbations. The study lays groundwork for applying TCSA to spin-chain physics and integrable perturbations of higher-level WZW models, with promising avenues like the dimerized-frustrated $J_1$-$J_2$-$\delta$ Heisenberg model and possible integrable perturbations for $k>1$.

Abstract

We outline the application of the truncated conformal space approach (TCSA) to perturbations of $SU(2)_k$ Wess-Zumino-Witten theories. As examples of this methodology, we consider two distinct perturbations of $SU(2)_1$ and one of $SU(2)_2$. $SU(2)_1$ is first perturbed by its spin-1/2 field, a model which is equivalent to the sine-Gordon model at a particular value of its coupling $β$. The sine-Gordon spectrum is correctly reproduced as well as the corresponding finite size corrections. We next study $SU(2)_1$ with a marginal current-current perturbation. The TCSA results can be matched to perturbation theory within an appropriate treatment of the UV divergences. We find however that these results do not match field theoretic computations on the same model performed with a Lorentz invariant regulator. Finally, we consider $SU(2)_2$ perturbed by its spin-1 field, which is equivalent to three decoupled massive Majorana fermions. In this case as well the TCSA reproduces accurately the known spectrum.

Figures (12)

• Figure 1: Plot of the TCSA data for the lowest six excited states in the $S_z=0$ sector with the ground state energy subtracted. The lowest two excited states correspond to $B_1$ (the $S_z=0$ state of the triplet) and $B_2$. The next four excited states are two particle states. The data are computed with a truncation level $N_{tr}=9$. The fundamental triplet of particles can be seen to have mass M=1.0016.
• Figure 2: Single particle state $B_2$: Analytic prediction for its mass with finite size effects (continuous line) compared to the TCSA data.
• Figure 3: Ground state energy: analytic prediction (continuous line) compared to the TCSA results. Black squares: TCSA with $N_{tr}=9$; orange circles: TCSA with $N_{tr}=10$.
• Figure 4: Two-particle states: comparison between the analytic results and the TCSA data. Left panel: a two particle state involving two triplet particles with total $S_z=0$ and $(n_1,n_2)=(-1,1)$. Right panel: a two-breather $B_2$ state with $(n_1,n_2)=(-1,1)$. Continuous line: analytical results derived from Eqn. (\ref{['quant']}). Dots: TCSA data.
• Figure 5: Plots of the ground state energy of $SU(2)_1+\bar{J}_L\cdot\bar{J}_R$ as a function of the marginal coupling $g$. Solid lines give the perturbative computation for both $g>0$ relevant (blue) and $g<0$ irrelevant (red). The black line shows the second order perturbative correction in $g$. The data points represent the corresponding numerical data from the TCSA.