Oddities in the Entanglement Scaling of the Quantum Six-Vertex Model

Abstract

We investigate the entanglement properties of the Quantum Six-Vertex Model on a cylinder, focusing on the Shannon-Renyi entropy in the limit of Renyi order $n = \infty$. This entropy, calculated from the ground state amplitudes of the equivalent XXZ spin-1/2 chain, allows us to determine the Renyi entanglement entropy of the corresponding Rokhsar-Kivelson wavefunctions, which describe the ground states of certain conformal quantum critical points. Our analysis reveals a novel logarithmic correction to the expected entanglement scaling when the system size is odd. This anomaly arises from the geometric frustration of spin configurations imposed by periodic boundary conditions on odd-sized chains. We demonstrate that the scaling prefactor of this logarithmic term is directly related to the compactification radius of the low-energy bosonic field theory description, or equivalently, the Luttinger parameter. Thus, this correction provides a direct probe of the underlying Conformal Field Theory (CFT) describing the critical point. Our findings highlight the crucial role of system size parity in determining the entanglement properties of this model and offer insights into the interplay between geometry, frustration, and criticality.

Figures (3)

• Figure 1: (a) The 2D quantum system lives on a cylinder of circumference $L$ and height $2h$; it is bipartite into two subsystems each of height $h$. (b) With degrees of freedom on the links and local constraints around each vertex, the boundary configurations $i$ enumerate the Schmidt eigenvalues and eigenvectors of the decomposition of the RK wavefunction; this is possible because by fixing $i$, we also constraint the possible configurations of $A$ and $B$, which allows writing the Schmidt eigenvectors again in an RK form. (c) Allowed vertex configurations of the 6VM with their weights; (d) The different maximal states (Néel states) when $L$ is even or odd. In the even case (left) there are only two possibilities independent of system size; in the odd case (right) the total spin is $S^z = \pm 1/2$ and there is always a pair of parallel spins. It can be in $L$ possible positions, leading to an extensive degeneracy in the maximal state when $L$ is odd.
• Figure 2: (a) The min-entropy $S_{\infty}$ at $\Delta = 1/2$ for $L = 5, 7, \dots, 23$ obtained from exact diagonalization. The solid line is the analytical expression \ref{['eq:minent_delta_half']} derived from razumov2001spin. The full expression \ref{['eq:minent_delta_half_theor']} can be found in Appendix \ref{['app:computation_of_minent_at_delta_half']}. Inset. The plot of min-entropy minus the linear term shows a clear $\log$ behavior. (b) The entropy difference $S_{\infty}^{\text{(o)}} - S_{\infty}^{\text{(e)}}$ at $\Delta = 0$ for $L = 5, 7, \dots, 51$, computed numerically from the Slater determinant \ref{['eq:slater_vandermonde_det']}. The fit against the function $a L + b \log L + c$ (with $L$ odd) shows $a \sim 10^{-4}$ (negligible), $b = 0.2566+-0.0006$ (very close to $1/4$) and $c = 0.136+-0.001$. Inset. The entropies $S_{\infty}^{\text{(o)}}$ and $S_{\infty}^{\text{(e)}}$ are shown separately in the same range of $L$. (c) The log term coefficient $b$ of $S_{\infty}$ for $\Delta \in (-1, 1]$ (20 points). $S_{\infty}$ has been computed for $L = 7, 9, \dots, 23$ with exact diagonalization and then fitted against $a L + b \log L + c$. The fit errors on $b$ are of order e-3--e-4. As a comparison, the log coefficient obtained in the same manner from the iMPS Ansatz \ref{['eq:IMPS']} (same $L$s) and the theoretical curve $\Delta = - \cos(2 \pi b)$ have also been plotted.
• Figure 3: The log term coefficient $b_n$, as a function of $n$, in the scaling of the Rényi entropy $S_n$ for some specific values of $\Delta$. The entropies have been obtained numerically with exact diagonalization and have been fitted against the function $S_n = a_n L + b_n \log L + c_n$. For the odd case the sizes $L = 11, 13, \dots, 23$ were considered, while for the even case $L = 10, 12, \dots, 20$. The enlarged region shows only the odd case near the von Neumann entropy limit $n \to 1$.