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A Zoo of Fermionic Relative Rényi Negativities in the Sachdev-Ye-Kitaev Models

This paper introduces a family of relative Rényi negativities as entanglement witnesses for strongly correlated fermionic systems, utilizing a path-integral representation to evaluate them in Sachdev-Ye-Kitaev models via large- $N$ expansion and revealing first-order entanglement transitions invisible to simple observables.

Original authors: Linjing Jiang, Pengfei Zhang

Published 2026-10-30

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CC BY 4.0

Original authors: Linjing Jiang, Pengfei Zhang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, particles can become so deeply linked that they cease to be independent entities, sharing a single existence across space. This phenomenon, known as entanglement, is the engine behind many of the most promising technologies of the future, from unhackable communication networks to computers that solve problems in seconds that would take today's supercomputers millennia. For systems that are perfectly isolated and in a pure state, scientists have long had a reliable way to measure this connection: a calculation called entanglement entropy. It works like a thermometer for quantum links, telling researchers how much information is shared between two parts of a system. However, the real world is rarely so pristine. Most quantum systems interact with their environment, becoming "mixed" states where the clean signal of entanglement is drowned out by the noise of heat and thermal jitters. In these messy, realistic conditions, the old tools fail. The standard measures either become impossible to calculate or give false alarms, suggesting a connection exists where there is none.

A team of researchers at Fudan University and the Hefei National Laboratory has now developed a new set of tools designed specifically to cut through this noise. They focused on a class of theoretical models known as the Sachdev–Ye–Kitaev models, which describe collections of particles that interact in a chaotic, all-to-all fashion. These models are famous for being difficult to solve but are also believed to hold the key to understanding how black holes process information. The researchers introduced a family of new measurements they call "relative Rényi negativities." Unlike previous attempts to gauge entanglement in mixed states, these new metrics are designed to be strict witnesses: if the number is zero, there is no entanglement; if it is positive, entanglement is definitely present. By treating the quantum system as a vast network of interacting particles and using a sophisticated mathematical technique called a path integral to trace their histories, the team was able to calculate these values for systems too large to be simulated by brute force.

The results revealed a landscape of quantum behavior that was previously invisible. The researchers found that not all of their new measurements behaved the same way. Some acted as reliable indicators, steadily decreasing as the system got hotter, much like a thermometer dropping as a room cools. These specific measurements appear to be the best candidates for replacing the older, more difficult-to-calculate standards. However, other members of this new family behaved strangely. As the temperature changed, these specific measurements did not just rise or fall smoothly; they suddenly jumped from one value to another. This sudden shift, known as a first-order transition, signaled a fundamental reorganization of how the particles were connected. Crucially, these dramatic changes were completely hidden from the view of standard thermodynamic observables. A scientist looking only at the system's energy or heat capacity would see nothing unusual, yet the entanglement structure had fundamentally flipped.

This discovery suggests that the quantum world contains hidden layers of complexity that only reveal themselves through the right kind of probe. The researchers identified that these sudden jumps occur because the mathematical description of the system can settle into different stable configurations, or "saddles," depending on the temperature. At certain critical points, the system swaps from one configuration to another, causing the entanglement measure to leap. This behavior was observed in both the standard chaotic model and a more complex version where two separate systems are coupled together, mimicking the geometry of a wormhole connecting two black holes. In the coupled system, the new measurements detected two distinct transitions as the temperature dropped, revealing a sequence of changes in how the two sides of the system were linked. One transition corresponded to the known thermal shift of the system, but the second was a new phenomenon intrinsic to the entanglement itself, occurring at a different temperature scale.

The study confirms that these relative Rényi negativities are not just abstract mathematical curiosities but practical tools for diagnosing the health of quantum connections in noisy environments. By showing that these measures can detect phase transitions that standard physics misses, the work opens a new window into the behavior of strongly correlated fermionic systems. The researchers propose that these measurements might serve as a new kind of free energy, a thermodynamic potential that governs the stability of entangled states. While the work was conducted within the realm of theoretical models and large-scale numerical simulations, the findings suggest that future experiments on quantum simulators could potentially observe these hidden transitions. The ability to see these invisible shifts in entanglement could be vital for building robust quantum technologies, ensuring that the delicate links holding quantum information together are not silently broken by the heat of the environment.

Technical Summary: A Zoo of Fermionic Relative Rényi Negativities in the Sachdev–Ye–Kitaev Models

Problem Statement
Characterizing quantum entanglement in mixed states of strongly correlated fermionic systems remains a significant challenge. While the von Neumann entanglement entropy is well-defined for pure states, it conflates thermal and entanglement contributions in mixed states. The logarithmic negativity, $N_A = \ln \|\rho^{T_A}\|_1$ , is a standard entanglement monotone for mixed states, but its evaluation in interacting systems is computationally prohibitive, typically restricted to small system sizes via exact diagonalization.

Existing alternatives, such as untwisted and twisted Rényi negativities ( $N_A^{(2n)}$ and $\tilde{N}_A^{(2n)}$ ), offer computationally accessible proxies. However, these quantities suffer from a critical limitation: they can be positive even for separable states, meaning they do not strictly serve as entanglement witnesses. Furthermore, in fermionic systems, the partial transpose $\rho^{T_A}$ is generally non-Hermitian, complicating the analytic continuation required to recover the logarithmic negativity from Rényi ratios.

Methodology
The authors introduce a new family of observables, termed relative Rényi negativities, designed to serve as rigorous entanglement witnesses for fermionic mixed states.

Construction of Probes:
The authors exploit the rich structure of the fermionic partial transpose. By defining the partial time reversal operation $T_A$ via the size superoperator $S_A$ (where $T_A = e^{i\frac{\pi}{2}S_A}$ ), they establish a connection between the algebraic properties of the density matrix and operator size.
They construct generalized Rényi moments $M_A^f$ by taking the trace of products of the partially transposed density matrix $\rho^{T_A}$ and its Hermitian conjugate $(\rho^{T_A})^\dagger$ in various orderings defined by a sequence $f \in \{+, -\}^{2n}$ .
The relative Rényi negativity is defined as:
$P_A^f \equiv -\ln \left( \frac{|M_A^f|}{M_A^{f_t}} \right)$
where $f_t = (+, -, +, -, \dots)$ corresponds to the alternating sign sequence (the "twisted" case). By the generalized Hölder inequality, $P_A^f \geq 0$ , and $P_A^f = 0$ if and only if the state is separable. Thus, any non-zero value certifies entanglement.
Path-Integral Representation:
To evaluate these quantities in interacting models, the authors derive a path-integral representation for the moments $M_A^f$ at fixed integer replica numbers. The insertion of the superoperators $T_A$ and $T_A^{-1}$ is mapped to modified boundary conditions and two-point functions in the imaginary-time path integral. Specifically, the effect of the superoperators is incorporated into a modified free Green's function $G_{A}^{f,0}$ for the subsystem $A$ , while subsystem $B$ retains standard free propagation.
Large- $N$ Expansion in SYK Models:
The framework is applied to the Sachdev-Ye-Kitaev (SYK) model and the Maldacena-Qi coupled SYK model. Using the large- $N$ saddle-point approximation, the authors solve for the bilocal fields (Green's functions $G$ and self-energies $\Sigma$ ) on complex contours with $2n$ replicas. The disorder-averaged partition functions are computed by minimizing the effective action $S_{G\Sigma}$ .

Key Results

Validation as Entanglement Witnesses: The authors demonstrate that while conventional Rényi negativity ratios (e.g., $\tilde{R}_A^{(4)}$ ) can be positive for separable states, the relative Rényi negativities $P_A^f$ vanish strictly for separable states, making them valid entanglement witnesses.
Behavior in the Single SYK Model:
- For the "minimal-deviation" sequences (where $f$ differs from the alternating sequence $f_t$ by a single sign flip, e.g., $f=(+,+,+,-)$ ), the relative negativities exhibit monotonic decrease with increasing temperature. This behavior aligns with physical expectations for an entanglement measure, suggesting these specific members of the "zoo" are plausible proxies for logarithmic negativity.
- For the "untwisted" sequence ( $f=(+,+,+,+)$ ), the authors observe first-order entanglement transitions. At low temperatures, multiple saddle-point solutions coexist for the partition function. The physical solution is determined by the saddle with the lowest on-shell action. The exchange of dominance between these saddles leads to a discontinuous jump in the relative negativity, a transition invisible to simple thermodynamic observables.
Behavior in the Coupled SYK Model:
In the Maldacena-Qi model (two coupled SYK systems), the relative negativities reveal two successive first-order transitions as the temperature decreases (or equivalently, as the inverse temperature $\beta J$ $β J$ increases).
- The first transition (at lower $\beta J$ ) arises from the thermal phase transition of the reference partition function (related to the Hawking-Page transition).
- The second transition (at higher $\beta J$ ) is intrinsic to the relative negativity itself, originating from a change in the dominant saddle of the numerator partition function. This suggests a qualitative change in inter-replica correlations in the low-temperature phase, where replicas organize into nearly independent pairs.

Significance and Claims
The paper claims that relative Rényi negativities provide a robust and computationally tractable framework for probing mixed-state entanglement in strongly correlated fermionic systems. By relating fermionic partial time reversal to the size superoperator, the authors enable the evaluation of these quantities in thermal states via large- $N$ expansions, bypassing the need for exact diagonalization.

The primary significance lies in the discovery of entanglement transitions that are undetectable by conventional thermodynamic observables. The existence of competing replica saddles leads to first-order transitions in the relative negativities, suggesting that these quantities can serve as effective free energies for new classes of phase transitions. The authors posit that the "minimal-deviation" members of this family offer promising proxies for logarithmic negativity, while the full "zoo" of observables reveals a richer structure of entanglement and replica correlations in quantum many-body systems.

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