I compute the leading contribution to the ground state Renyi entropy $S_α$ for a region of linear size $L$ in a Fermi liquid. The result contains a universal boundary law violating term simply related the more familiar entanglement entropy. I also obtain a universal crossover function that smoothly interpolates between the zero temperature result and the ordinary thermal Renyi entropy of a Fermi liquid. Formulas for the entanglement entropy of more complicated regions, including non-convex and disconnected regions, are obtained from the conformal field theory formulation of Fermi surface dynamics. These results permit an evaluation of the quantum mutual information between different regions in a Fermi liquid. I also study the number fluctuations in a Fermi liquid. Taken together, these results give a reasonably complete characterization of the low energy quantum information content of Fermi liquids.
We study the Renyi mutual information of classical systems characterized by a transfer matrix. We first establish a general relationship between the Renyi mutual information of such classical mixtures of configuration states, and the Renyi entropy of a corresponding Rokhsar-Kivelson--type quantum superposition. We then focus on chiral and nonchiral kagome-ice systems, classical spin liquids on the kagome lattice, which respectively have critical and short-range correlations. Through a mapping of the chiral kagome ice to the quantum Liftshitz critical field theory, we predict a universal sublea…
The mutual information is a measure of classical and quantum correlations of great interest in quantum information. It is also relevant in quantum many-body physics, by virtue of satisfying an area law for thermal states and bounding all correlation functions. However, calculating it exactly or approximately is often challenging in practice. Here, we consider alternative definitions based on Rényi divergences. Their main advantage over their von Neumann counterpart is that they can be expressed as a variational problem whose cost function can be efficiently evaluated for families of states lik…
We implement a Wang-Landau sampling technique in quantum Monte Carlo (QMC) for the purpose of calculating the Renyi entanglement entropies and associated mutual information. The algorithm converges an estimate for an analogue to the density of states for Stochastic Series Expansion QMC allowing a direct calculation of Renyi entropies without explicit thermodynamic integration. We benchmark results for the mutual information on two-dimensional (2D) isotropic and anisotropic Heisenberg models, 2D transverse field Ising model, and 3D Heisenberg model, confirming a critical scaling of the mutual i…
We study the Shannon mutual information in one-dimensional critical spin chains, following a recent conjecture (Phys. Rev. Lett. 111, 017201 (2013)), as well as Rényi generalizations of it. We combine conformal field theory arguments with numerical computations in lattice discretizations with central charge $c=1$ and $c=1/2$. For a periodic system of length $L$ cut into two parts of length $\ell$ and $L-\ell$, all our results agree with the general shape-dependence $I_n(\ell,L)=(b_n/4)\ln \left(\frac{L}π\sin \frac{π\ell}{L}\right)$, where $b_n$ is a universal coefficient. For the free boson CF…
We present a new algorithm for calculating the Renyi entanglement entropy of interacting fermions using the continuous-time quantum Monte Carlo method. The algorithm only samples interaction correction of the entanglement entropy, which by design ensures efficient calculation of weakly interacting systems. Combined with Monte Carlo reweighting, the algorithm also performs well for systems with strong interactions. We demonstrate the potential of this method by studying the quantum entanglement signatures of the charge-density-wave transition of interacting fermions on a square lattice.
We perform quantum Monte Carlo calculations to determine how the Renyi entropies, $S_n$, of the interacting Fermi liquid depend on Renyi order, $n$, and scale as a function of system size, $L$. Using the swap operator and an accurate Slater-Jastrow wave function, we compute Renyi entropies for spinless fermions interacting via the Coulomb and modified Pöschl-Teller potentials across a range of correlation strengths. Our results show that interactions increase the Renyi entropies and increase the prefactor of their scaling laws. The relationships between Renyi entropies of different order $n$ a…
In this work we compute subleading oscillating terms in the Renyi entropy of Fermi gases and Fermi liquids corresponding to $2k_F$-like oscillations. Our theoretical tools are the one dimensional formulation of Fermi liquid entanglement familiar from discussions of the logarithmic violation of the area law and quantum Monte Carlo calculations. The main result is a formula for the oscillating term for any region geometry and a spherical Fermi surface. We compare this term to numerical calculations of entanglement using the correlation function method and find excellent agreement. We also compar…
We develop a quantum Monte Carlo procedure to compute the Renyi mutual information of an interacting quantum many-body system at non-zero temperature. Performing simulations on a spin-1/2 XXZ model, we observe that for a subregion of fixed size embedded in a system of size L, the mutual information converges at large L to a limiting function which displays non-monotonic temperature behavior corresponding to the onset of correlations. For a region of size L/2 embedded in a system of size L, the mutual information divided by L converges to a limiting function of temperature, with apparently nont…
We present a new method for calculating Renyi entanglement entropies for fermionic field-theories originating from microscopic Hamiltonians. The method builds on an operator identity which we discover for the first time. The identity leads to the representation of traces of operator products, and thus Renyi entropies of a subsystem, in terms of fermionic-displacement operators. This allows for a very transparent path-integral formulation, both in and out-of-equilibrium, having a simple boundary condition on the fermionic fields. The method is validated by reproducing well known expressions for…
The spin-1 classical Blume-Capel model on a square lattice is known to exhibit a finite-temperature phase transition described by the tricritical Ising CFT in 1+1 space-time dimensions. This phase transition can be accessed with classical Monte Carlo simulations, which, via a replica-trick calculation, can be used to study the shape-dependence of the classical Rényi entropies for a torus divided into two cylinders. From the second Rényi entropy, we calculate the Geometrical Mutual Information (GMI) introduced by Stéphan et. al. [Phys. Rev. Lett. 112, 127204 (2014)] and use it to extract a nume…
Momentum space entanglement entropy probes quantum correlations in interacting fermionic phases. It is very sensitive to interactions, obeying volume-law scaling in general, while vanishing in the Fermi gas. We show that the Rényi entropy in momentum space has a systematic expansion in terms of the phase space volume of the partition, which holds at all orders in perturbation theory. This permits, for example, the controlled computation of the entropy of thin shells near the Fermi wavevector in isotropic Fermi liquids and BCS superconductors. In the Fermi liquid, the thin shell entropy is a un…
We study the dynamics of (Rényi) mutual information, logarithmic negativity, and (Rényi) reflected entropy after exciting the ground state by a local operator. Together with recent results from Ref. [1], we are able to conjecture a close-knit structure between the three quantities that emerges in states excited above the vacuum, including both local and global quantum quenches. This structure intimately depends on the chaoticity of the theory i.e. there exist distinct sets of equivalences for integrable and chaotic theories. For rational conformal field theories (RCFT), we find all quantities …
We show that the metastable, symmetry-breaking ground states of quantum many-body Hamiltonians have vanishing quantum mutual information between macroscopically separated regions, and are thus the most classical ones among all possible quantum ground states. This statement is obvious only when the symmetry-breaking ground states are simple product states, e.g. at the factorization point. On the other hand, symmetry-breaking states are in general entangled along the entire ordered phase, and to show that they actually feature the least macroscopic correlations compared to their symmetric superp…
We study the Rényi mutual information $\tilde{I}_n$ of the ground state of different critical quantum chains. The Rényi mutual information definition that we use is based on the well established concept of the Rényi divergence. We calculate this quantity numerically for several distinct quantum chains having either discrete $Z(Q)$ symmetries (Q-state Potts model with $Q=2,3,4$ and $Z(Q)$ parafermionic models with $Q=5,6,7,8$ and also Ashkin-Teller model with different anisotropies) or the $U(1)$ continuous symmetries(Klein-Gordon field theory, XXZ and spin-1 Fateev-Zamolodchikov quantum chains…
Entanglement measures such as the entanglement entropy have become an indispensable tool to identify the fundamental character of ground states of interacting quantum many-body systems. For systems of interacting spin or bosonic degrees of freedom much recent progress has been made not only in the analytical description of their respective entanglement entropies but also in their numerical classification. Systems of interacting fermionic degrees of freedom however have proved to be more difficult to control, in particular with regard to the numerical understanding of their entanglement propert…
We study the Renyi entropy in the finite temperature crossover regime of a Hubbard chain using quantum Monte Carlo. The ground state entropy has characteristic features such as a logarithmic divergence with block size and $2\kF$ oscillations that are a hallmark of its Luttinger liquid nature. The interplay between the (extensive) thermal entropy and the ground state features is studied and we analyze the temperature induced decay of the amplitude of the oscillations as well as the scaling of the purity. Furthermore, we show how the spin and charge velocities can be extracted from the temperatu…
In a recent article T. Grover [Phys. Rev. Lett. 111, 130402 (2013)] introduced a simple method to compute Renyi entanglement entropies in the realm of the auxiliary field quantum Monte Carlo algorithm. Here, we further develop this approach and provide a stabilization scheme to compute higher order Renyi entropies and an extension to access the entanglement spectrum. The method is tested on systems of correlated topological insulators.
I study the mutual information between spatial subsystems in a variety of scale invariant quantum field theories. While it is derived from the bare entanglement entropy, the mutual information offers a more refined probe of the entanglement structure of quantum field theories because it remains finite in the continuum limit. I argue that the mutual information has certain universal singularities that are a manifestation of the idea of "entanglement per scale". Moreover, I propose a method, based on an ansatz for higher dimensional twist operators, to compute the entanglement entropy, Renyi ent…
A remarkable yet mysterious property of black holes is that their entropy is proportional to the horizon area. This area law inspired the holographic principle, which was later realized concretely in gauge/gravity duality. In this context, entanglement entropy is given by the area of a minimal surface in a dual spacetime. However, discussions of area laws have been constrained to entanglement entropy, whereas a full understanding of a quantum state requires Renyi entropies. Here we show that all Renyi entropies satisfy a similar area law in holographic theories and are given by the areas of du…
In this work we consider the time evolution of charged Renyi entanglement entropies after exciting the vacuum with local fermionic operators. In order to explore the information contained in charged Renyi entropies, we perform computations of their excess due to the operator excitation in 2d CFT, free fermionic field theories in various dimensions as well as holographically. In the analysis we focus on the dependence on the entanglement charge, the chemical potential and the spacetime dimension. We find that excesses of charged (Renyi) entanglement entropy can be interpreted in terms of charge…
We consider the (Renyi) mutual information, $I^{(n)}(A,B) = S^{(n)}_A+S^{(n)}_{B} - S^{(n)}_{A \cup B}$, of distant compact spatial regions A and B in the vacuum state of a free scalar field. The distance r between A and B is much greater than their sizes $R_{A,B}$. It is known that $I^{(n)}(A,B) \sim C^{(n)}_{AB} \left<0| ϕ(r)ϕ(0) |0\right>^2$ . We obtain the direct expression of $C^{(n)}_{AB}$ for arbitrary regions A and B. We perform the analytical continuation of $n$ and obtain the mutual information. The direct expression is useful for the numerical computation. By using the direct expres…
Universal scaling terms occurring in Renyi entanglement entropies have the potential to bring new understanding to quantum critical points in free and interacting systems. Quantitative comparisons between analytical continuum theories and numerical calculations on lattice models play a crucial role in advancing such studies. In this paper, we exactly calculate the universal two-cylinder shape dependence of entanglement entropies for free bosons on finite-size square lattices, and compare to approximate functions derived in the continuum using several different ansatzes. Although none of these …
The symmetry-resolved Rényi entanglement entropy is the Rényi entanglement entropy of each symmetry sector of a density matrix $ρ$. This experimentally relevant quantity is known to have rich theoretical connections to conformal field theory (CFT). For a family of critical free-fermion chains, we present a rigorous lattice-based derivation of its scaling properties using the theory of Toeplitz determinants. We consider a class of critical quantum chains with a microscopic U(1) symmetry; each chain has a low energy description given by $N$ massless Dirac fermions. For the density matrix, $ρ_A$,…
Recent proposals of measuring bipartite Renyi entropy experimentally involve techniques that hold exactly for non-interacting quantum particles. Here we consider the difference between such measurements and the actual Renyi entropy for ground state fermion ab initio molecular systems. To calculate various entanglement measures we extended several different techniques for use with variational Monte Carlo, Hartree-Fock and quantum chemistry methods. Our results show that in systems with strong electron correlations the Renyi entropy may not be accurately determined with the proposed measurements…
We study the spread of Rényi entropy between two halves of a Sachdev-Ye-Kitaev (SYK) chain of Majorana fermions, prepared in a thermofield double (TFD) state. The SYK chain model is a model of chaotic many-body systems, which describes a one-dimensional lattice of Majorana fermions, with spatially local random quartic interaction. We find that for integer Rényi index $n>1$, the Rényi entanglement entropy saturates at a parametrically smaller value than expected. This implies that the TFD state of the SYK chain does not rapidly thermalize, despite being maximally chaotic: instead, it rapidly ap…
