Studying the Perturbed Wess-Zumino-Novikov-Witten SU(2)k Theory Using the Truncated Conformal Spectrum Approach

TL;DR

The paper analyzes the (1+1)-D $SU(2)_k$ Wess-Zumino-Novikov-Witten theory perturbed by the trace of the adjoint primary using the Truncated Conformal Spectrum Approach (TCSA) complemented by renormalization-group improvements. By combining semiclassical insights at large $k$ with nonperturbative TCSA+RG results, it shows a sign- and parity-dependent infrared structure: for even $k$ the low-energy theory is effectively the massive $O(3)$ (or SU(2)$_{1}$-protected) sector, while for odd $k$ the system flows toward a massless $SU(2)_1$ fixed point; the $k=2$ case reduces to three free Majorana fermions. The work develops a robust TCSA framework for arbitrary $k$ and adjoint perturbations, implements vacuum and excited-state renormalization terms, and demonstrates strong agreement with semiclassical predictions through finite-volume spectra and mass gaps. These results illuminate non-integrable perturbations of WZNW models and have potential implications for strongly correlated electron systems and related spin-ladder and cold-atom setups.

Abstract

We study the $SU(2)_k$ Wess-Zumino-Novikov-Witten (WZNW) theory perturbed by the trace of the primary field in the adjoint representation, a theory governing the low-energy behaviour of a class of strongly correlated electronic systems. While the model is non-integrable, its dynamics can be investigated using the numerical technique of the truncated conformal spectrum approach combined with numerical and analytical renormalization groups (TCSA+RG). The numerical results so obtained provide support for a semiclassical analysis valid at $k\gg 1$. Namely, we find that the low energy behavior is sensitive to the sign of the coupling constant, $λ$. Moreover for $λ>0$ this behavior depends on whether $k$ is even or odd. With $k$ even, we find definitive evidence that the model at low energies is equivalent to the massive $O(3)$ sigma model. For $k$ odd, the numerical evidence is more equivocal, but we find indications that the low energy effective theory is critical.

Figures (9)

• Figure 1: The potential surface for $k=4$. The black lines show the elementary cell of the periodic potential, blue dots show the two minima relevant for $\lambda<0$ that are connected via a saddle point. The red dot is a maximum of the potential, which becomes the only minimum (per elementary cell) for $\lambda>0$.
• Figure 2: $k=3$ (top) and $4$ (bottom), $\lambda>0$ ground state energy data obtained by TCSA+NRG for cutoff levels $N=5,6,7,8$ and $N=4,5,6,7$ (dashed lines); the results with the counter term (Eqn. \ref{['eq:vac_counterterm']}) (red lines); and the data involving the counter term and the RG improvement (Eqn. \ref{['eq:GW_energy_levels']}) (green lines). The insets show the same results blown up on the $1<L<14$ interval. Note that in the direction of increasing cut-off $N$ the subtracted (red), and the subtracted and renormalized (green) levels move less as the cut-off grows, which is a further confirmation of the validity of the renormalized TCSA.
• Figure 3: Finite volume gap in the perturbed $SU(2)_{4}$ model for $\lambda>0$ at cut-offs $N=2$, $3$, $4$ and $5$. The plot shows the raw TCSA data (without NRG, blue circles), and those after the RC (dashed lines) and the CT (magenta squares) corrections were applied. The gap can be estimated to be $\Delta=0.36\pm0.03$, and the error is approximated by the difference between the gap estimates for $N=3$ and 6.
• Figure 4: Finite volume spectra in the perturbed $SU(2)_{k}$ models with $k=3,4,5$ for negative coupling constant at cut-offs $n_{max}=7$, $6$, $5$, respectively. We show raw TCSA data since renormalization here had an effect that is not visible on these figures. Colors represent energy levels in the integer (black) and half integer (blue) sectors. The red arrows show the gaps corresponding to one-particle states. For $k=5$, two of these are already higher than the two-particle threshold (shown as the thick dashed line), and due to non-integrability they are expected to correspond to resonances.
• Figure 5: TCSA mass gap for negative coupling constant as a function of $k^{-1/2}$. We show data coming from $k=2,$$3$, $4$ and $5$ models. We also put error bars on the data points which we calculated by subtracting the gap estimates with and without applying the RC improvement, but they are so small that they are practically invisible in the plot.