Chaos in Systems with Quantum Group Symmetry

TL;DR

This work demonstrates that quantum group global symmetry can persist in non-integrable, non-unitary systems by constructing a $U_q(sl_2)$-invariant XXZ-type spin chain and adding a carefully chosen next-to-nearest deformation that breaks integrability. Through exact diagonalization and symmetry-based projections, the authors show that the resulting spectrum exhibits chaotic eigenvalue statistics, with real spectra at some parameter ranges and complex-conjugate pairs when PT-symmetry is broken. The real-spectrum regime shows level statistics transitioning from Poisson (integrable) toward GOE (chaotic), while complex spectra align with GinUE-like behavior as the fraction of complex eigenvalues grows. In the infrared, the model flows to the same conformal field theory as the undeformed integrable model, indicating the perturbation is RG-irrelevant to the IR, yet chaos persists in the spectrum. Overall, the paper provides evidence that quantum group symmetries can coexist with non-integrable, PT-symmetric chaos, suggesting new avenues for realizing QG symmetries in non-integrable settings and potentially in higher-dimensional quantum field theories.

Abstract

Quantum groups have a long and fruitful history of applications in integrable systems. Can quantum group symmetries exist in the absence of integrability? We provide an explicit example of a system with quantum group global symmetry which is chaotic. The example is a spin chain with next-to-nearest interaction term. We show the chaotic behavior of the system by studying the Eigenvalue statistics. The spin chain is non-unitary but PT-symmetric and, in addition to chaos, exhibits an interesting transition after which the eigenvalues become complex.

Figures (7)

• Figure 1: Level spacing distribution $P(s)$ for real eigenvalues from the $q$-parity even sector with $\{\ell=1,S^z=1\}$ with fixed parameters $L=16,q=e^{i\pi/10.4}$ and different $\lambda$. The blue and red solid lines are respectively Poissonian and Wigner distributions \ref{['eq:PW_GOE']} for the GOE ensemble.
• Figure 2: Spacing ratio distribution $P(\Tilde{r})$ for real eigenvalues from the $q$-parity even sector with $\{\ell=1,S^z=1\}$ with fixed parameters $L=16,q=e^{i\pi/10.4}$ and different $\lambda$. The blue and red solid lines are respectively Poissonian and Wigner distributions approximating ratio distributions \ref{['eq:PW_GOE_ratio']} for the GOE ensemble.
• Figure 3: Value of average CLSR, $\braket{r}$, as a function of $\lambda$ for different values of parameter $q$. Solid horizontal lines are the values of $\braket{r}$ for different ensembles taken from Table \ref{['tbl:average_r']}. For the values of $q$ close to 1 all the eigenvalues in the maximally chaotic regime are real and the values of $\braket{r}$ approach GOE ensemble. For $q$ far away from 1 almost all the eigenvalues become complex in chaotic regime and $\braket{r}$ reaches the values comparable with GinUE ensemble.
• Figure 4: Complex level spacing ratio distributions $P(z)$ for a fixed value of $q=e^{i\pi/4.8}$, fixed length $L=16$, and different values of $\lambda$. Eigenvalues are from $q$-parity even sector with $\{\ell=1,S^z=1\}$.
• Figure 5: Plot of conformal dimensions $\Delta_{e,m}(\mu)$ for several lightest bulk XXZ operators labeled by quantum numbers $(e,m)$, which are also singlets under the lattice $U(1)_{\text{lat}}$. $\Delta$ is defined as $\Delta = \kappa+\bar{\kappa}$, see \ref{['eq:compBos_dim']}. Operators labeled by $(0,\pm 1)$ are relevant for all values of $\mu$, whereas operators $(0,\pm 2)$ are instead irrelevant with approaching marginality at $\mu \to \infty$.