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 finite entropy. The evolution of the Renyi entropy is non-monotonic in time for both Markovian and non-Markovian dynamics. The entropy is also found to be anti-correlated with negativity of the Wigner function of a
2 -mode open quantum system.
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 R\'enyi entropies for many-body systems. We provide an exact analytic formula for R\'enyi and von Neumann entanglement entropies of noninteracting open quantum systems, which are initialized 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. 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.
Using a Wigner-function-based approach, we study the Renyi entropy of a subsystem
A of a system of bosons interacting with a local repulsive potential. The full system is assumed to be in thermal equilibrium at a temperature
T and density $\ensuremath{\rho}$. For a
U(N)-symmetric model, we show that the Renyi entropy of the system in the large-
N limit can be understood in terms of an effective noninteracting system with a spatially varying mean field potential, which has to be determined self-consistently. The Renyi entropy is the sum of two terms: (a) the Renyi entropy of this effective system and (b) the difference in thermal free energy between the effective system and the original translation-invariant system, scaled by
T. We determine the self-consistent equation for this effective potential within a saddle-point approximation. We use this formalism to look at one- and two-dimensional Bose gases on a lattice. In both cases, the potential profile is that of a square well, taking one value in subsystem
A and a different value outside it. The potential varies in space near the boundary of subsystem
A on the scale of density-density correlation length. The effect of interaction on the entanglement entropy density is determined by the ratio of the potential barrier to the temperature and peaks at an intermediate temperature, while the high- and low-temperature regimes are dominated by the noninteracting answer.
Schwinger Keldysh field theory is a widely used paradigm to study nonequilibrium dynamics of quantum many-body systems starting from a thermal state. We extend this formalism to describe nonequilibrium dynamics of quantum systems starting from arbitrary initial many-body density matrices. We show how this can be done for both bosons and fermions, and for both closed and open quantum systems, using additional sources coupled to bilinears of the fields at the initial time, calculating Green's functions in a theory with these sources, and then taking appropriate set of derivatives of these Green's functions with respect to initial sources to obtain physical observables. The set of derivatives depends on the initial density matrix. The physical correlators in a dynamics with arbitrary initial conditions do not satisfy Wick's theorem, even for noninteracting systems. However, our formalism constructs intermediate ``$n\ensuremath{-}\mathrm{particle}$ Green's functions'' that obey Wick's theorem and provide a prescription to obtain physical correlation functions from them. This allows us to obtain analytic answers for all physical many-body correlation functions of a noninteracting system even when it is initialized to an arbitrary density matrix. We use these exact expressions to obtain an estimate of the violation of Wick's theorem and relate it to presence of connected multiparticle initial correlations in the system. We illustrate this new formalism by calculating density and current profiles in many-body fermionic and bosonic open quantum systems initialized to nontrivial density matrices. We have also shown how this formalism can be extended to interacting many-body systems.
We investigate the time-dependent R\'enyi entanglement entropy after a quantum quench starting from the Mott-insulating and charge-density-wave states in a one-dimensional free boson system. The second R\'enyi entanglement entropy is found to be the negative of the logarithm of the permanent of a matrix consisting of time-dependent single-particle correlation functions. From this relation and a permanent inequality, we obtain rigorous conditions for satisfying the volume-law entanglement growth. We also succeed in calculating the time evolution of the R\'enyi entanglement entropy in extremely large systems by brute-force computations of the permanent. We discuss possible applications of our findings to the real-time dynamics of noninteracting bosonic systems.
Abstract We developed a perturbative calculation for entropy dynamics, which considers a sudden coupling between a system and a bath. The theory we developed can work in a general environment without Markovian approximation. A perturbative formula is given for bosonic environments and fermionic environments, respectively. We find the Rényi entropy response is only related to the spectral functions of the system and the environment, together with a statistical kernel distribution function. We find at2growth/decay in the short time limit and a lineartgrowth/decay in a longer time scale for the second Rényi entropy response. A non-monotonic behavior of Rényi entropy for fermionic systems is found to be quite general when the environmental temperature is the lower one. A Fourier’s law in heat transport is obtained when two systems’ temperatures are close to each other. A consistency check is made for Sachdev-Ye-Kitaev model coupling to free fermions, a Page curve alike dynamics is found in a process dual to black hole evaporation. An oscillation of Rényi entropy is found for an environment with a gapped spectrum.