Detection of a Rényi Index Dependent Transition in Entanglement Entropy Scaling

TL;DR

This paper demonstrates that entanglement scaling in a symmetry-constrained many-body state can depend on the Rényi index $\alpha$, challenging the common practice of using $S_2$ as a proxy for the von Neumann entropy $S_1$. By constructing a number-conserving state with symmetry-resolved blocks, the authors show $S_{\alpha>1}$ can scale as $\ln\ell$ while $S_1$ scales as $\ell^{1/2}\ln\ell$ (and $S_{\alpha<1}$ can be even more extensive), illustrating a genuine Rényi-index transition. They introduce a symmetry-aware lower bound $\bar{S}_α=\sum_{q_A} P_{q_A}S_α(q_A)+H_1(\{P_{q_A}\})$ that satisfies $S_1\ge\bar{S}_α\ge S_α$ for $α>1$ and can be computed from accessible data $S_2$ and the charge distribution $P_{q_A}$, enabling experimental/QLMC diagnostics of anomalous scaling. The framework provides a practical diagnostic for detecting entanglement-scaling transitions and highlights the importance of symmetry sectors in interpreting entanglement measures, with potential relevance to MBQC resources and nontrivial symmetry-protected structures.

Abstract

The scaling of entanglement with subsystem size encodes key information about phases and criticality, but the von Neumann entropy is costly to access in experiments and simulations, often requiring full state tomography. The second Rényi entropy is readily measured using two-copy protocols and is often used as a proxy for the von Neumann entanglement entropy, where it is assumed to track its asymptotic scaling. Sugino and Korepiny (Int. J. Mod. Phys. B 32, 1850306 (2018)) revealed that in the ground state of some highly constrained spin models, the scaling of the von Neumann and \ren entropies can differ, varying from power law to logarithmic scaling as a function of the \ren index. Here, we construct a number-conserving many-body state that demonstrates a Rényi-index-dependent change in the leading entanglement scaling, generalizing previous results to the case of interacting fermions. We introduce a symmetry-aware lower bound on the von Neumann entropy built from charge-resolved Rényi entropies that can provide a protocol for diagnosing anomalous entanglement scaling from experimentally accessible data.

Figures (5)

• Figure 1: Particle number distribution $P_n$ and symmetry-resolved entanglement entropy $S_\alpha(n)$ for the example discussed in Section \ref{['sec:EntropyScalingAnalysis']} for a system with $L=800$ sites and $\mu(N)\asymp \sqrt{N}$.
• Figure 2: The entanglement spectrum displays diverse entanglement entropy scaling with a partition size of $\ell$. Panel (a) shows the Rényi entanglement for $\alpha = 1/2,1,2$ vs. $\sqrt{\ell}$. The asymptotic scaling can be seen more clearly in the lower panels: (b) $S_{\alpha>1} \asymp \ln \ell$, (c) $S_{1} \asymp \sqrt{\ell}$, and (d) $S_{\alpha<1} \asymp \ell$. Here $\mu(N)\asymp \sqrt{N}$. The values $\mathcal{N}_\alpha$ are normalization constants chosen to allow a common $y$-axis with their values shown in the lower panels.
• Figure 3: Particle number distribution $P_n$ and symmetry-resolved entanglement entropy $S_\alpha(n)$ analysis using the quantum state $\vert\Psi\rangle$ introduced in Eq. (\ref{['Eq:StateExample']}). Here, the state $\vert\Psi\rangle$ describes $N=400$ particles with $\mu(N)\asymp \sqrt{N}$.
• Figure 4: Different measures of entanglement entropy applied to the state $\vert\Psi\rangle$ defined in Eq. (\ref{['Eq:StateExample']}) for $\mu(N)\asymp \sqrt{N}$. Asymptotically, $S_{\alpha = 2} \asymp \ln \ell$ (left), $S_1 \asymp \sqrt{\ell}\ln\ell$ (middle), and $S_{\alpha=1/2<1} \asymp \ell$ (right). The values $\mathcal{N}_\alpha$ are normalization constants chosen to allow a common $y$-axis.
• Figure 5: The von Neumann entanglement entropy $S_1$ shows a distinct scaling from that of the Rényi entanglement entropy $S_2$, while $\bar{S}_2$ exhibits compatible scaling with $S_1$ for the many-body state considered in section \ref{['Sec:ManyBodyStateExample']}.