Entanglement Rényi Negativity across the Finite-Temperature Transition in the O(3) Universality Class

Abstract

The fate of quantum entanglement at finite-temperature phase transitions remains an open question, particularly for continuous symmetry breaking where zero-temperature Goldstone modes generate long-range correlations. Using large-scale quantum Monte Carlo simulations, we investigate the third Rényi negativity across the O(3) transition in the three-dimensional Heisenberg antiferromagnet. The first such study for a thermal critical point with continuous symmetry. We uncover two fundamental results. First, the negativity exhibits a pure area law at the critical point, with the subleading constant term vanishing within statistical uncertainty. This demonstrates that thermal fluctuations completely destroy the long-range entanglement present at zero temperature. The divergent classical correlation length leaves no imprint on quantum entanglement itself. Second, despite this absence of singular behavior in the negativity, its temperature derivative follows the exact scaling of the specific heat, yielding critical exponents -α/ν=0.190(1) and 1/ν=1.350(5) in precise agreement with the O(3) universality class. Our work establishes that while quantum entanglement is blind to thermal criticality, its thermodynamic derivatives encode the full universal scaling, revealing an unexpected connection between entanglement and classical phase transitions. Furthermore, this also provides further evidence that Rényi negativity can still effectively shield classical correlations in systems with continuous symmetry.

Figures (5)

• Figure 1: (a) Schematic of the partitioned subsystems A and B on a three-dimensional cubic lattice, sharing an interface $\partial A \cap \partial B$. (b) The space-time manifold in path integral used to compute the Rényi negativity $R_{3}$. The horizontal axis denotes the spatial spin direction and the vertical axis corresponds to imaginary time. Spins of the same color are connected along the imaginary time direction.
• Figure 2: The Rényi negativity $R_3$ was computed for the three-dimensional Heisenberg model and fitted to the area law $R_3 = aL^2 + \gamma$. (a) $R_3$ as a function of temperature $T$.(b) Temperature dependence of the fitted coefficient $a$.
• Figure 3: The Rényi negativity $R_3$ was computed for the three-dimensional Heisenberg model. (a) $R_3$ follows the area law $R_3 = aL^2 + \gamma$ at the rescaled temperature $3T_c$. (b) The temperature dependence of the fitted subleading coefficient $\gamma$ near $3T_c$, showing that it is consistent with zero within the statistical uncertainty.
• Figure 4: Finite-size scaling collapse of data for system sizes $L=10,12,14$ using Eq.(\ref{['eq:data_collaps']}). The horizontal axis is the scaled variable $tL^{1/\nu}$, where $t=(T-3T_{c})/3T_{c}$ is the reduced temperature. The vertical axis represents the singular part of the interface density of the derivative, scaled as $(1/\partial A) [(dR_{3}/dT)-(dR_{3}/dT)\vert_{T=3T_{c}}]L^{-\alpha/\nu}$, which captures the critical behavior near $3T_{c}$. Inset: The behavior of the derivative of the area-law coefficient near the rescaled temperature $3T_{c}$.
• Figure S1: The horizontal axis denotes the spatial spin direction, while the vertical axis corresponds to imaginary time. Spins of the same color are connected along the imaginary time direction. (a) Manifold representing $Z_{3}$, where three replicas are connected along the imaginary time direction for the entire system $A \cup B$. (b) The twisted manifold corresponding to $Z^{T_{B}}_{3}$, in which the three replicas are connected within subsystem $A$, while in subsystem $B$, the connections are no longer cyclic due to the partial transpose operation $T_{B}$.