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.
Many-body entanglement unveils additional aspects of quantum matter and offers insights into strongly correlated physics. While ground-state entanglement has received much attention in the past decade, the study of mixed-state quantum entanglement using negativity in interacting fermionic systems remains largely unexplored. We demonstrate that the partially transposed density matrix of interacting fermions, similar to their reduced density matrix, can be expressed as a weighted sum of Gaussian states describing free fermions, enabling the calculation of rank-n Rényi negativity within the determinant quantum Monte Carlo framework. We calculate the rank-two Rényi negativity for the half-filled Hubbard model and the spinless t-V model. Our calculation reveals that the area law coefficient of the Rényi negativity for the spinless t-V model has a logarithmic finite-size scaling at the finite-temperature transition point. Our work contributes to the calculation of entanglement and sets the stage for future studies on quantum entanglement in various fermionic many-body mixed states.
Abstract We investigate several entanglement-related quantities at finite-temperature criticality in the three-dimensional quantum spherical model, both as a function of temperature T and of the quantum parameter g , which measures the strength of quantum fluctuations. While the von Neumann and the Rényi entropies exhibit a volume-law for any g and T , the mutual information obeys an area law. The prefactors of the volume-law and of the area-law are regular across the transition, reflecting that universal singular terms vanish at the transition. This implies that the mutual information is dominated by nonuniversal contributions. This hampers its use as a witness of criticality, at least in the spherical model. We also study the logarithmic negativity. For any value of g , T , the negativity exhibits an area-law. The negativity vanishes deep in the paramagnetic phase, it is larger at small temperature, and it decreases upon increasing the temperature. For any g , it exhibits the so-called sudden death, i.e. it is exactly zero for large enough T . The vanishing of the negativity defines a ‘death line’, which we characterise analytically at small g . Importantly, for any finite T the area-law prefactor is regular across the transition, whereas it develops a cusp-like singularity in the limit . Finally, we consider the single-particle entanglement and negativity spectra. The vast majority of the levels are regular across the transition. Only the larger ones exhibit singularities. These are related to the presence of a zero mode, which reflects the symmetry breaking. This implies the presence of sub-leading singular terms in the entanglement entropies. Interestingly, since the larger levels do not contribute to the negativity, sub-leading singular corrections are expected to be suppressed for the negativity.
Entanglement entropy has been a powerful tool for analyzing phases and criticality in pure ground states via quantum Monte Carlo (QMC). However, mixed-state entanglement, relevant to systems with dissipation, finite temperature, and disjoint regions, remains less explored due to the lack of efficient numerical methods. In this work, we present a practical and easy-to-implement QMC method within the reweight-annealing framework, enabling efficient computation of the entanglement R\'enyi negativity by tracking its variation along given parameter paths. This method is scalable, parallelizable, and well suited for high-dimensional and large-scale simulations. Applying it to diverse scenarios, including one- and two-dimensional systems, ground and thermal states, and bipartite and tripartite partitions, not only is the information of the underlying conformal field theory achieved, but the role of entanglement in quantum and thermal phase transitions is revealed.
We investigate several entanglement-related quantities at finite-temperature criticality in the three-dimensional quantum spherical model, both as a function of temperature
T and of the quantum parameter
g, which measures the strength of quantum fluctuations. While the von Neumann and the R\\'enyi entropies exhibit the volume-law for any
g and
T, the mutual information obeys an area law. The prefactors of the volume-law and of the area-law are regular across the transition, reflecting that universal singular terms vanish at the transition. This implies that the mutual information is dominated by nonuniversal contributions. This hampers its use as a witness of criticality, at least in the spherical model. We also study the logarithmic negativity. For any value of
g,T, the negativity exhibits an area-law. The negativity vanishes deep in the paramagnetic phase, it is larger at small temperature, and it decreases upon increasing the temperature. For any
g, it exhibits the so-called sudden death, i.e., it is exactly zero for large enough
T. The vanishing of the negativity defines a "death line", which we characterise analytically at small
g. Importantly, for any finite
T the area-law prefactor is regular across the transition, whereas it develops a cusp-like singularity in the limit
Tto0. Finally, we consider the single-particle entanglement and negativity spectra. The vast majority of the levels are regular across the transition. Only the larger ones exhibit singularities. These are related to the presence of a zero mode, which reflects the symmetry breaking. This implies the presence of sub-leading singular terms in the entanglement entropies. Interestingly, since the larger levels do not contribute to the negativity, sub-leading singular corrections are expected to be suppressed for the negativity.
We show that the bipartite logarithmic entanglement negativity (EN) of quantum spin models obeys an area law at all nonzero temperatures. We develop numerical linked cluster (NLC) expansions for the "area-law" logarithmic entanglement negativity as a function of temperature and other parameters. For one-dimensional models the results of NLC are compared with exact diagonalization on finite systems and are found to agree very well. The NLC results are also obtained for two dimensional XXZ and transverse field Ising models. In all cases, we find a sudden onset (or sudden death) of negativity at a finite temperature above which the negativity is zero. We use perturbation theory to develop a physical picture for this sudden onset (or sudden death). The onset of EN or its magnitude are insensitive to classical finite-temperature phase transitions, supporting the argument for absence of any role of quantum mechanics at such transitions. On approach to a quantum critical point at T=0, negativity shows critical scaling in size and temperature.