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arXiv:2501.17368 · Physical review. B./Physical review. B

Singularity and universality from von Neumann to Rényi entanglement entropy and disorder operator in Motzkin chains

Jianyu Wang, Zenan Liu, Zheng Yan, Congjun Wu

Published Jul 25, 2025• 3 citations
arXivHTMLPDF

Abstract

The Rényi entanglement entropy is widely used in studying quantum entanglement properties in strongly correlated systems, whose analytic continuation as the Rényi index $n \to 1$ is often believed to yield the von Neumann entanglement entropy. However, earlier findings indicate that this process exhibits a singularity for the colored Motzkin spin chain problem, leading to different scaling behaviors of $\sim \sqrt{l}$ and $\sim \log{l}$ for the von Neumann and Rényi entropies, respectively. Our analytical and numerical calculations confirm this transition, which can be explained by the exponentially increasing density of states in the entanglement spectrum that we extract numerically. Disorder operators are further employed under various symmetries to study such a system. Both analytical and numerical results demonstrate that the scaling of the disorder operators also follows $\log{l}$ as the leading behavior, matching that of the Rényi entropy. We propose that the coefficient of the term $\log{l}$ is a universal constant shared by both the Rényi entropies and disorder operators. This universal constant could potentially help capture the underlying constraint physics of Motzkin walks.

DOI
10.1103/wffk-7ycs

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The Rényi entanglement entropy is widely used in studying quantum entanglement properties in strongly correlated systems, whose analytic continuation as the Rényi index n→1n \to 1n→1 is often believed to yield the von Neumann entanglement entropy. However, earlier findings indicate that this process exhibits a singularity for the colored Motzkin spin chain problem, leading to different scaling behaviors of ∼l\sim \sqrt{l}∼l​ and ∼log⁡l\sim \log{l}∼logl for the von Neumann and Rényi entropies, respectively. Our analytical and numerical calculations confirm this transition, which can be explained by the exponentially increasing density of states in the entanglement spectrum that we extract numerically. Disorder operators are further employed under various symmetries to study such a system. Both analytical and numerical results demonstrate that the scaling of the disorder operators also follows log⁡l\log{l}logl as the leading behavior, matching that of the Rényi entropy. We propose that the coefficient of the term log⁡l\log{l}logl is a universal constant shared by both the Rényi entropies and disorder operators. This universal constant could potentially help capture the underlying constraint physics of Motzkin walks.
At a quantum critical point, bipartite entanglement entropies have universal quantities which are subleading to the ubiquitous area law. For Renyi entropies, these terms are known to be similar to the von Neumann entropy, while being much more amenable to numerical and even experimental measurement. We show here that when calculating universal properties of Renyi entropies, it is important to account for unusual corrections to scaling that arise from relevant local operators present at the conical singularity in the multisheeted Riemann surface. These corrections grow in importance with increasing Renyi index. We present studies of Renyi correlation functions in the 1+11+11+1 transverse-field Ising model (TFIM) using conformal field theory, mapping to free fermions, and series expansions, and the logarithmic entropy singularity at a corner in 2+12+12+1 for both free bosonic field theory and the TFIM, using numerical linked cluster expansions. In all numerical studies, accurate results are only obtained when unusual corrections to scaling are taken into account. In the worst case, an analysis ignoring these corrections can get qualitatively incorrect answers, such as predicting a decrease in critical exponents with the Renyi index, when they are actually increasing. We discuss a two-step extrapolation procedure that can be used to account for the unusual corrections to scaling.
Entanglement is one of the most intriguing features of quantum theory and a main resource in quantum information science. Ground states of quantum many-body systems with local interactions typically obey an “area law” which means that the entanglement entropy is proportional to the boundary length. It is exceptional when the system is gapless, and the area law had been believed to be violated by at most a logarithm over two decades. Recent discovery of Motzkin and Fredkin spin chain models is striking, since these models provide significant violation of the entanglement beyond the belief, growing as a square root of the volume in spite of local interactions. In this paper, we first analytically compute the Rényi entropy of the Motzkin and Fredkin models by careful treatment of asymptotic analysis. The Rényi entropy is an important quantity, since the whole spectrum of an entangled subsystem is reconstructed once the Rényi entropy is known as a function of its parameter. We find nonanalytic behavior of the Rényi entropy with respect to the parameter, which is a novel phase transition never seen in any other spin chain studied so far. Interestingly, similar behavior is seen in the Rényi entropy of Rokhsar–Kivelson states in two dimensions.
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 Rényi entropies can differ, varying from power law to logarithmic scaling as a function of the Rényi 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.
Extracting operationally meaningful quantities such as von Neumann entropy and mutual information is central to characterizing many-body quantum systems, yet experiments and numerics often provide direct access only to integer Rényi entropies. Whereas conventional extrapolation relies on a prescribed fitting ansatz, in PRL 137, 100202 we introduced an alternative approach based on stabilized analytic continuation (SAC), which avoids such an ansatz and is inherently robust to noise. Here, we further develop this framework in the noiseless setting and benchmark its performance across several nontrivial many-body problems. We first show that, besides the intrinsic ambiguity of reconstructing the von Neumann entropy from finitely many Rényi samples, analytic continuation can also fail because of genuine nonanalyticities in the Rényi function arising from zeros of Trρz\text{Tr}ρ^zTrρz. We derive universal zero-free domains for Trρz\text{Tr}ρ^zTrρz and systematically sharpen these bounds using additional spectral information about ρρρ. We then benchmark the performance of SAC in three settings: (iii) extracting topological entanglement entropy from DMRG Rényi data for the toric code and Kagome Heisenberg models; (iiiiii) reconstructing the mutual information between disjoint intervals in a (1+1)d(1+1)d(1+1)d compact-boson CFT; and (iiiiiiiii) recovering finite-time ballistic growth of the von Neumann entropy from integer Rényi entropies that cross over toward subballistic growth in diffusive quantum dynamics. In the latter setting, analytic continuation is expected to fail in the long time limit. We illustrate this mechanism with a physically motivated two-sector model, where the asymptotic separation between von Neumann and higher-Rényi growth is accompanied by a zero of Trρz\text{Tr}ρ^zTrρz pinching the real axis at z=1z=1z=1, thereby producing a first-order Rényi phase transition.
The R\'enyi-Shannon entropy allows extraction of some universal information about many-body wave functions. For a critical spin chain with central charge c=1c=1c=1, we show that it exhibits a phase transition at some value nc{n}_{c}nc​ of the R\'enyi parameter nnn which depends on the Luttinger parameter RRR. A replica-free formulation establishes a connection to boundary entropies in conformal field theory and reveals that the transition is triggered by a vertex operator which becomes relevant at the boundary. Our numerical results (XXZ and ${J}_{1}\ensuremath{-}{J}_{2}$ spin chains) match the continuum limit prediction, confirming its universal character. The replica approach used in previous works turns out to be correct only for n<ncn<{n}_{c}n<nc​. From the point of view of two-dimensional Rokhsar-Kivelson states, this transition reveals a singularity in the entanglement spectra.
Abstract We investigate the dynamics of the Rényi Operator Space Entanglement (OSE) entropies Sn across several one-dimensional integrable and chaotic models. As a paradigmatic integrable system, we first consider the so-called rule 54 chain. Our numerical results reveal that the Rényi OSE entropies of diagonal operators with nonzero trace saturate at long times, in contrast with the behavior of von Neumann entropy. Oppositely, the Rényi entropies of traceless operators exhibit logarithmic growth with time, with the prefactor of this growth depending in a nontrivial manner on n. Notably, at long times, the complete operator entanglement spectrum of an operator can be reconstructed from the spectrum of its traceless part. We observe a similar pattern in the XXZ chain, suggesting Universal behavior. Additionally, we consider dynamics in nonintegrable deformations of the XXZ chain. Finite-time corrections do not allow to access the long-time behavior of the von Neumann entropy. On the other hand, for n > 1 the growth of the entropies is milder, and it is compatible with a sublinear growth, at least for operators associated with global conserved quantities. Finally, we show that in finite-size integrable systems, Sn exhibit strong revivals, which are washed out when integrability is broken.