On deforming and breaking integrability

Abstract

In this paper we study nearest-neighbour deformations of integrable models. After expanding in the deformation parameter, we identify four possible types of deformations. First there are deformations that simply break or preserve integrability. Then we find two different subtle cases. The first case is where the deformation is only integrable if all orders of the deformation parameter are taken into account. An example of these are the long-range deformations that appear in holographic models. The second case is when the deformation is perturbatively integrable to some order in the deformation parameter but can not be extended to an integrable model. In this paper we work this out for the XXZ spin chain and discuss the level statistics of each of these cases. We find numerical evidence that the onset of chaos occurs differently in each of these models. For the perturbatively integrable models, we find that the deformation strength at which chaos appears demonstrates a volume-scaling intermediate between strong and weak integrability breaking models.

Figures (6)

• Figure 1: Level-spacing distributions for integrable, $\epsilon=0.0$, and chaotic, $\epsilon=0.5$, regimes in $L=21$ chain in momentum-one sector with even magnetisation which has dimension 49,929 for (Left) $H_{\text{QInt}}$ with $J_z=0.68$, $\alpha=0.82$, $\beta=1.43$, $\gamma=1$. (Right) $H_{\text{dXYZ}}$ with $J_x=1.63, J_y= 0.73, J_z =1.18$.
• Figure 2: Brody parameter $\omega_B$ as a function of deformation strength for $L=21$ in the momentum-one, even-magnetisation sector (dim. 49,929) for (Left) $H_{\text{dXYZ}}$ with with $J_x= 1.63, J_y= 0.73, J_z =1.18$ and $H_{\text{QInt}}$ with $J_z=0.68$, $\alpha=0.82$, $\beta=1.43$, $\gamma=1$. Dashed lines are for a rescaled, effective coupling. (Right) $H_{\text{QInt}}$ for $\sigma_{xy}=1$, $\alpha=0.82$, $\beta=1.43$, $\gamma=1$ and different values of $J_z$.
• Figure 3: Brody parameter $\omega_B$ as a function of deformation strength $\epsilon$ for $L\in [16,21]$ in the momentum-one, even magnetisation sector. (Left) $H_{\text{dXYZ}}$ with with $J_x= 1.63, J_y= 0.73, J_z =1.18$ (Right) $H_{\text{QInt}}$ with $J_z=0.68$, $\alpha=0.82$, $\beta=1.43$, $\gamma=1$.
• Figure 4: (Left) Scaling of critical coupling $\epsilon_c$ as a function of $L$ in the momentum-one, even-magnetisation sector of the dXYZ model, with $J_x= 1.63, J_y= 0.73, J_z =1.18$, and QInt model $\alpha=0.82$, $\beta=1.43$, $\gamma=1$ for different values of $J_z$. Curves are best-fit power-law scaling. (Right) Values of power-law scaling exponent $b$ for QInt model for values of $J_z$. Dashed line is a linear fit to the last four points.
• Figure 5: Information entropy for states in the $L=18$ in the momentum-one, even-magnetisation sector (dim$=7252$) of the (Top left) deformed XYZ model with $\Delta_{xy}=0.45$, $\sigma_{xy}=J_z=1.18$, and coupling $\epsilon$ (Top right) QInt model with $\sigma_{xy}=1$, $J_z=0.68$, $\gamma=1$, $\alpha=0.82$, $\beta=1.4$ (Bottom left) XYZ model with $\Delta_{xy}=0.45$, $\sigma_{xy}=0.59+1.18 \epsilon$, $J_z=1.18$. (Bottom right) R.M.S. of differences to a fitted quadratic curve for the three models.