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Entanglement renormalization is a method for coarse-graining a quantum state in real space, with the multi-scale entanglement renormalization ansatz (MERA) as a notable example. We obtain an entanglement renormalization scheme for finite-temperature (Gibbs) states by applying MERA to their canonical purification, the thermofield double state. As an example, we find an analytically exact renormalization circuit for finite temperature two-dimensional toric code which maps it to a coarse-grained system with a renormalized higher temperature, thus explicitly demonstrating its lack of topological order. Furthermore, we apply this scheme to one-dimensional free boson models at a finite temperature and find that the thermofield double corresponding to the critical thermal state is described by a Lifshitz theory. We numerically demonstrate the relevance and irrelevance of various perturbations under real space renormalization.
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We discuss the behavior of the entanglement entropy of the ground state in various collective systems. Results for general quadratic two-mode boson models are given, yielding the relation between quantum phase transitions of the system (signaled by a divergence of the entanglement entropy) and the excitation energies. Such systems naturally arise when expanding collective spin Hamiltonians at leading order via the Holstein-Primakoff mapping. In a second step, we analyze several such models (the Dicke model, the two-level BCS model, the Lieb-Mattis model and the Lipkin-Meshkov-Glick model) and investigate the properties of the entanglement entropy in the whole parameter range. We show that when the system contains gapless excitations the entanglement entropy of the ground state diverges with increasing system size. We derive and classify the scaling behaviors that can be met.
Using the nonperturbative renormalization group, we study the existence of bound states in the symmetry-broken phase of the scalar $phi^4$ theory in all dimensions between two and four and as a function of the temperature. The accurate description of the momentum dependence of the two-point function, required to get the spectrum of the theory, is provided by means of the Blaizot--Mendez-Galain--Wschebor approximation scheme. We confirm the existence of a bound state in dimension three, with a mass within 1% of previous Monte-Carlo and numerical diagonalization values.
We study the time evolution of the logarithmic negativity after a global quantum quench. In a 1+1 dimensional conformal invariant field theory, we consider the negativity between two intervals which can be either adjacent or disjoint. We show that the negativity follows the quasi-particle interpretation for the spreading of entanglement. We check and generalise our findings with a systematic analysis of the negativity after a quantum quench in the harmonic chain, highlighting two peculiar lattice effects: the late birth and the sudden death of entanglement.
We reconsider the moments of the reduced density matrix of two disjoint intervals and of its partial transpose with respect to one interval for critical free fermionic lattice models. It is known that these matrices are sums of either two or four Gaussian matrices and hence their moments can be reconstructed as computable sums of products of Gaussian operators. We find that, in the scaling limit, each term in these sums is in one-to-one correspondence with the partition function of the corresponding conformal field theory on the underlying Riemann surface with a given spin structure. The analytical findings have been checked against numerical results for the Ising chain and for the XX spin chain at the critical point.
The study of the entanglement dynamics plays a fundamental role in understanding the behaviour of many-body quantum systems out of equilibrium. In the presence of a globally conserved charge, further insights are provided by the knowledge of the resolution of entanglement in the various symmetry sectors. Here, we carry on the program we initiated in Phys. Rev. B 103, L041104 (2021), for the study of the time evolution of the symmetry resolved entanglement in free fermion systems. We complete and extend our derivations also by defining and quantifying a symmetry resolved mutual information. The entanglement entropies display a time delay that depends on the charge sector that we characterise exactly. Both entanglement entropies and mutual information show effective equipartition in the scaling limit of large time and subsystem size. Furthermore, we argue that the behaviour of the charged entropies can be quantitatively understood in the framework of the quasiparticle picture for the spreading of entanglement, and hence we expect that a proper adaptation of our results should apply to a large class of integrable systems. We also find that the number entropy grows logarithmically with time before saturating to a value proportional to the logarithm of the subsystem size.
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