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arXiv:2609.30386

Rényi Phase Transitions and Analytic Continuation to the von Neumann Entropy

Ayush Raj, Akash Vijay, Hong-Chen Jiang, Laimei Nie

quant-phcond-mat.stat-mechSubmitted Sep 24, 2026• Announced Sep 28, 2026• 0 citations
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Abstract

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 $\text{Tr}ρ^z$. We derive universal zero-free domains for $\text{Tr}ρ^z$ and systematically sharpen these bounds using additional spectral information about $ρ$. We then benchmark the performance of SAC in three settings: ($i$) extracting topological entanglement entropy from DMRG Rényi data for the toric code and Kagome Heisenberg models; ($ii$) reconstructing the mutual information between disjoint intervals in a $(1+1)d$ compact-boson CFT; and ($iii$) 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 $\text{Tr}ρ^z$ pinching the real axis at $z=1$, thereby producing a first-order Rényi phase transition.

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30 pages, 13 figures

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Abstract An alternative method is presented for extracting the von Neumann entropy − Tr(ρ ln ρ) from Tr(ρn) for integer n in a quantum system with density matrix ρ. Instead of relying on direct analytic continuation in n, the method uses a generating function − Tr{ρ ln[(1 − zρ)/(1 − z)]} of an auxiliary complex variable z. The generating function has a Taylor series that is absolutely convergent within |z| < 1, and may be analytically continued in z to z = −∞ where it gives the von Neumann entropy. As an example, we use the method to calculate analytically the CFT entanglement entropy of two intervals in the small cross ratio limit, reproducing a result that Calabrese et al. obtained by direct analytic continuation in n. Further examples are provided by numerical calculations of the entanglement entropy of two intervals for general cross ratios, and of one interval at finite temperature and finite interval length.
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.
We prove rigorous bounds on the growth of $\ensuremath{\alpha}$-R\'enyi entropies ${S}_{\ensuremath{\alpha}}(t)$ (the von Neumann entropy being the special case $\ensuremath{\alpha}=1$) associated with any subsystem AAA of a general lattice quantum many-body system with finite on-site Hilbert space dimension. For completely nonlocal Hamiltonians, we show that the instantaneous growth rates $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ (with $\ensuremath{\alpha}\ensuremath{\ne}1$) can be exponentially larger than $|{S}_{1}^{\ensuremath{'}}(t)|$ as a function of the subsystem size ∣A∣|A|∣A∣. For DDD-dimensional systems with geometric locality, we prove bounds on $|{S}_{\ensuremath{\alpha}}^{\ensuremath{'}}(t)|$ that depend on the decay rate of interactions with distance. When $\ensuremath{\alpha}=1$, the bound is ∣A∣|A|∣A∣ independent for all power-law decaying interactions $V(r)\ensuremath{\sim}{r}^{\ensuremath{-}w}$ with w>2D+1w>2D+1w>2D+1. However, for $\ensuremath{\alpha}>1$, the bound is ∣A∣|A|∣A∣ independent only when the interactions are finite-range or decay faster than $V(r)\ensuremath{\sim}{e}^{\ensuremath{-}c\phantom{\rule{0.16em}{0ex}}{r}^{D}}$ for some ccc depending on the local Hilbert space dimension. Using similar arguments, we also prove bounds on kkk-local systems with or without geometric locality. A central theme of this work is that the value of $\ensuremath{\alpha}$ strongly influences the interplay between locality and instantaneous entanglement growth. In other words, the von Neumann entropy and the $\ensuremath{\alpha}$-R\'enyi entropies cannot be regarded as proxies for each other in studies of entanglement dynamics. We compare these bounds with analytic and numerical results on Hamiltonians with varying degrees of locality and find concrete examples that almost saturate the bound for nonlocal dynamics.
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.
Rényi coherent information, a computable proxy for the von Neumann coherent information, is widely used to study mixed-state phases of matter and decodability transitions in noisy quantum error-correcting codes. However, being a difference of two Rényi entropies, it need not be monotonic in the Rényi index, and lacks the operational meaning of its von Neumann counterpart. Here we address both issues for stabilizer codes. First, for Pauli noise generated by independent Bernoulli events, we prove that the Rényi-nnn coherent information is nondecreasing in n∈Z+n \in \mathbb{Z}^+n∈Z+. This follows from a general theorem: if independent random bits are mapped linearly to a fine label TTT and a coarse label CCC, then the Rényi entropy difference Hn(C)−Hn(T)H_n(C)-H_n(T)Hn​(C)−Hn​(T) is nondecreasing in nnn. For stabilizer codes, TTT is the joint syndrome--logical class and CCC is the syndrome, and the difference is the Rényi-nnn coherent information up to a constant. The same theorem covers classical linear codes and independent detector error models. Second, for arbitrary stochastic Pauli noise, we give the Rényi-nnn coherent information an operational meaning via postselection on matching syndromes between one data block and n−1n-1n−1 auxiliary blocks. We determine when this defines a quantum channel and show that saturation of the Rényi-nnn coherent information is equivalent to asymptotically perfect recovery of the postselected channel. Moreover, the Rényi-nnn coherent information also upper-bounds the ordinary coherent information achievable after any syndrome-conditioned recovery.
Concepts of information theory are increasingly used to characterize collective phenomena in condensed matter systems, such as the use of entanglement entropies to identify emergent topological order in interacting quantum many-body systems. Here, we employ classical variants of these concepts, in particular R\'enyi entropies and their associated mutual information, to identify topological order in classical systems. Like for their quantum counterparts, the presence of topological order can be identified in such classical systems via a universal, subleading contribution to the prevalent volume and boundary laws of the classical R\'enyi entropies. We demonstrate that an additional subleading O(1)O(1)O(1) contribution generically arises for all R\'enyi entropies S(n){S}^{(n)}S(n) with $n\ensuremath{\ge}2$ when driving the system towards a phase transition, e.g., into a conventionally ordered phase. This additional subleading term, which we dub connectivity contribution, tracks back to partial subsystem ordering and is proportional to the number of connected parts in a given bipartition. Notably, the Levin-Wen summation scheme, typically used to extract the topological contribution to the R\'enyi entropies, does not fully eliminate this additional connectivity contribution in this classical context. This indicates that the distillation of topological order from R\'enyi entropies requires an additional level of scrutiny to distinguish topological from nontopological O(1)O(1)O(1) contributions. This is also the case for quantum systems, for which we discuss which entropies are sensitive to these connectivity contributions. We showcase these findings by extensive numerical simulations of a classical variant of the toric code model, for which we study the stability of topological order in the presence of a magnetic field and at finite temperatures from a R\'enyi entropy perspective.