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

Abstract

Quantum entanglement is a fundamental resource that gives rise to correlations with no classical counterpart in strongly correlated fermionic systems. However, its characterization in mixed quantum states remains challenging. Established entanglement diagnostics, such as logarithmic negativity, are computationally demanding in interacting systems, and their evaluation by exact diagonalization is restricted to small system sizes. Meanwhile, conventional Rényi proxies, including twisted and untwisted Rényi negativities, can be positive even for separable states. In this work, we introduce a family of \textit{relative Rényi negativities} that serve as entanglement witnesses. By relating fermionic partial time reversal to the size superoperator, we establish a path-integral representation for these Rényi negativities, enabling their evaluation in thermal states of the Sachdev-Ye-Kitaev (SYK) model and its generalizations via a large-$N$ expansion. Applying this method, we investigate entanglement in both the original SYK model and the Maldacena-Qi coupled SYK model. We identify a special class of relative Rényi negativities which serve as plausible proxies for logarithmic negativity and uncover first-order entanglement transitions that are invisible to simple observables. These results highlight the utility of relative Rényi negativities as probes of mixed-state entanglement in strongly correlated fermionic systems and suggest the existence of a new class of transitions where these negativities serve as free energies.

The paper asks how to construct computable mixed-state entanglement witnesses for interacting fermions, since conventional Rényi negativity ratios can be positive for separable states. It defines moments $\mathcal M_A^{\bm f}$ from ordered products of the partial-time-reversed density matrix $\rho^{T_A}$ and its adjoint, and relative negativities $\mathcal P_A^{\bm f}=-\ln(|\mathcal M_A^{\bm f}|/\mathcal M_A^{\bm f_t})$, using the alternating product as reference. Hölder’s inequality establishes $\mathcal P_A^{\bm f}\geq0$. These quantities vanish for fermionically separable states built from parity-conserving local density matrices, so positive values witness entanglement; they are also symmetric under exchanging the subsystems. Writing partial time reversal as $T_A=e^{i\pi S_A/2}$, with $S_A$ the subsystem operator-size superoperator, yields a fixed-integer-replica path integral. Disorder averaging and numerical solutions of large-$N$ bilocal saddle equations then evaluate the witnesses in SYK and Maldacena–Qi coupled SYK models. In single SYK, sequences differing from the alternating reference by one sign show decreasing entanglement indicators with increasing temperature at replica orders $2,4,6$. Other sequences can be nonmonotonic; competing saddles produce first-order transitions as the subsystem fraction changes. In coupled SYK, the second- and fourth-order witnesses exhibit two successive transitions associated separately with reference and numerator partition functions. The minimal-deviation quantities are proposed as proxies for logarithmic negativity, not established entanglement monotones. The saddle structure is primarily characterized numerically; analytical large-$q$/low-energy explanations, gravitational interpretations, and extensions to spatially local models and nonequilibrium settings remain open.