Entanglement Entropy in Many-Fermion System

Physical Review X

The logarithmic violations of the area law, i.e. an "area law" with logarithmic correction of the form S ∼ L d−1 log L, for entanglement entropy are found in both 1D gapless system and for high dimensional free fermions. The purpose of this work is to show that both violations are of the same origin, and in the presence of Fermi liquid interactions such behavior persists for 2D fermion systems. In this paper we first consider the entanglement entropy of a toy model, namely a set of decoupled 1D chains of free spinless fermions, to relate both violations in an intuitive way. We then use multi-dimensional bosonization to re-derive the formula by Gioev and Klich [Phys. Rev. Lett. 96, 100503 (2006)] for free fermions through a low-energy effective Hamiltonian, and explicitly show the logarithmic corrections to the area law in both cases share the same origin: the discontinuity at the Fermi surface (points). In the presence of Fermi liquid (forward scattering) interactions, the bosonized theory remains quadratic in terms of the original local degrees of freedom, and after regularizing the theory with a mass term we are able to calculate the entanglement entropy perturbatively up to second order in powers of the coupling parameter for a special geometry via the replica trick. We show that these interactions do not change the leading scaling behavior for the entanglement entropy of a Fermi liquid. At higher orders, we argue that this should remain true through a scaling analysis. arXiv:1110.3004v2 [cond-mat.stat-mech]

Journal of Statistical Mechanics: Theory and Experiment, 2011

We introduce a systematic framework to calculate the bipartite entanglement entropy of a compact spatial subsystem in a one-dimensional quantum gas which can be mapped into a noninteracting fermion system. We show that when working with a finite number of particles N , the Rényi entanglement entropies grow as ln N , with a prefactor that is given by the central charge. We apply this novel technique to the ground state and to excited states of periodic systems. We also consider systems with boundaries. We derive universal formulas for the leading behavior and for subleading corrections to the scaling. The universality of the results allows us to make predictions for the finite-size scaling forms of the corrections to the scaling.

arXiv (Cornell University), 2023

We provide a prescription to construct Rényi and von Neumann entropy of a system of interacting fermions from a knowledge of its correlation functions. We show that Rényi entanglement entropy of interacting fermions in arbitrary dimensions can be represented by a Schwinger Keldysh free energy on replicated manifolds with a current between the replicas. The current is local in real space and is present only in the subsystem which is not integrated out. This allows us to construct a diagrammatic representation of entanglement entropy in terms of connected correlators in the standard field theory with no replicas. This construction is agnostic to how the correlators are calculated, and one can use calculated, simulated or measured values of the correlators in this formula. Using this diagrammatic representation, one can decompose entanglement into contributions which depend on the one-particle correlator, two particle correlator and so on. We provide analytic formula for the one-particle contribution and a diagrammatic construction for higher order contributions. We show how this construction can be extended for von-Neumann entropy through analytic continuation. For a practical implementation of a quantum state, where one usually has information only about fewparticle correlators, this provides an approximate way of calculating entanglement commensurate with the limited knowledge about the underlying quantum state.

Physical Review B

We formulate a new "Wigner characteristics" based method to calculate entanglement entropies of subsystems of Fermions using Keldysh field theory. This bypasses the requirements of working with complicated manifolds for calculating Rényi entropies for many body systems. We provide an exact analytic formula for Rényi and von-Neumann entanglement entropies of non-interacting open quantum systems, which are initialised in arbitrary Fock states. We use this formalism to look at entanglement entropies of momentum Fock states of one-dimensional Fermions. We show that the entanglement entropy of a Fock state can scale either logarithmically or linearly with subsystem size, depending on whether the number of discontinuities in the momentum distribution is smaller or larger than the subsystem size. This classification of states in terms number of blocks of occupied momenta allows us to analytically estimate the number of critical and non-critical Fock states for a particular subsystem size. We also use this formalism to describe entanglement dynamics of an open quantum system starting with a single domain wall at the center of the system. Using entanglement entropy and mutual information, we understand the dynamics in terms of coherent motion of the domain wall wavefronts, creation and annihilation of domain walls and incoherent exchange of particles with the bath.

arXiv: Statistical Mechanics, 2018

We propose a new method of calculating entanglement entropy of a many-body interacting Bosonic system (open or closed) in a field theoretic approach without replica methods. The Wigner function and Renyi entropy of a Bosonic system undergoing arbitrary non-equilibrium dynamics can be obtained from its Wigner characteristic function, which we identify with the Schwinger Keldysh partition function in presence of quantum sources turned on at the time of measurement. For non-interacting many body systems, starting from arbitrary density matrices, we provide exact analytic formulae for Wigner function and entanglement entropy in terms of the single particle Green's functions. For interacting systems, we relate the Wigner characteristic to the connected multi-particle correlators of the system. We use this formalism to study the evolution of an open quantum system from a Fock state with negative Wigner function and zero entropy, to a thermal state with positive Wigner function and fin...

Physical Review A, 2013

We study interacting dipolar atomic bosons in a triple-well potential within a ring geometry. This system is shown to be equivalent to a three-site Bose-Hubbard model. We analyze the ground state of dipolar bosons by varying the effective on-site interaction. This analysis is performed both numerically and analytically by using suitable coherent-state representations of the ground state. The latter exhibits a variety of forms ranging from the su(3) coherent state in the delocalization regime to a macroscopic cat-like state with fully localized populations, passing for a coexistence regime where the ground state displays a mixed character. We characterize the quantum correlations of the ground state from the bi-partition perspective. We calculate both numerically and analytically (within the previous coherent-state representation) the single-site entanglement entropy which, among various interesting properties, exhibits a maximum value in correspondence to the transition from the cat-like to the coexistence regime. In the latter case, we show that the ground-state mixed form corresponds, semiclassically, to an energy exhibiting two almost-degenerate minima.

Physical Review Letters, 2011

We introduce a systematic framework to calculate the bipartite entanglement entropy of a spatial subsystem in a one-dimensional quantum gas which can be mapped into a noninteracting fermion system. To show the wide range of applicability of the proposed formalism, we use it for the calculation of the entanglement in the eigenstates of periodic systems, in a gas confined by boundaries or external potentials, in junctions of quantum wires and in a time-dependent parabolic potential.

Two bound, entangled fermions form a composite boson, which can be treated as an elementary boson as long as the Pauli principle does not affect the behavior of many such composite bosons. The departure from ideal bosonic behavior is quantified by the normalization ratio of N-composite-boson states. We derive the two-fermion states that extremize the normalization ratio for a fixed single-fermion purity P, and establish general tight bounds for this indicator. For very small purities, P<1/N^2, the upper and lower bounds converge, which allows us to quantify accurately the departure from perfectly bosonic behavior, for any state of many composite bosons.

2009

We review some of the recent progress on the study of entropy of entanglement in many-body quantum systems. Emphasis is placed on the scaling properties of entropy for one-dimensional multi-partite models at quantum phase transitions and, more generally, on the concept of area law. We also briefly describe the relation between entanglement and the presence of impurities, the idea of particle entanglement, the evolution of entanglement along renormalization group trajectories, the dynamical evolution of entanglement and the fate of entanglement along a quantum computation.

2006

The entanglement entropy of a distinguished region of a quantum many-body system reflects the entanglement present in its pure ground state. In this work, we establish scaling laws for this entanglement for critical quasifree fermionic and bosonic lattice systems, without resorting to numerical means. We consider the geometrical setting of D-dimensional half-spaces which allows us to exploit a connection to the one-dimensional case. Intriguingly, we find a difference in the scaling properties depending on whether the system is bosonic-where an area-law is first proven to hold-or fermionic, extending previous findings for cubic regions. For bosonic systems with nearest neighbor interaction we prove the conjectured area-law by computing the logarithmic negativity analytically. We identify a length scale associated with entanglement, different from the correlation length. For fermions we determine the logarithmic correction to the area-law, which depends on the topology of the Fermi surface. We find that Lifshitz quantum phase transitions are accompanied with a non-analyticity in the prefactor of the leading order term.