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arXiv:1810.08692 · Physical review. B./Physical review. B

Nonequilibrium field theory for dynamics starting from arbitrary athermal initial conditions

Ahana Chakraborty, Pranay Gorantla, Rajdeep Sensarma

Published Feb 19, 2019• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.99.054306

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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−particlen\ensuremath{-}\mathrm{particle}n\ensuremath−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 consider a finite quantum system under slow driving and weakly coupled to thermal reservoirs at different temperatures. We present a systematic derivation of the quantum master equation for the density matrix and the out-of-time-order correlators. We start from the microscopic Hamiltonian and we formulate the equations ruling the dynamics of these quantities by recourse to the Schwinger-Keldysh nonequilibrium Green's function formalism, performing a perturbative expansion in the coupling between the system and the reservoirs. We focus on the adiabatic dynamics, which corresponds to considering the linear response in the ratio between the relaxation time due to the system-reservoir coupling and the time scale associated to the driving. We calculate the particle and energy fluxes. We illustrate the formalism in the case of a qutrit coupled to bosonic reservoirs and of a pair of interacting quantum dots attached to fermionic reservoirs, also discussing the relevance of coherent effects.
Nonequilibrium dynamics of unentangled and entangled pure states in interacting quantum systems is crucial for harnessing quantum information and to understand quantum thermalization. We develop a general Schwinger-Keldysh (SK) field theory for nonequilibrium dynamics of pure states of fermions. We apply our formalism to study the time evolution of initial density inhomogeneity and multipoint correlations of pure states in the complex Sachdev-Ye-Kitaev (SYK) models. We demonstrate a remarkable universality in the dynamics of pure states in the SYK model. We show that dynamics of almost all pure states in a fixed particle number sector is solely determined by a set of universal large-NNN Kadanoff-Baym equations. Moreover, irrespective of the initial state the site- and disorder-averaged Green's function thermalizes instantaneously, whereas local and nonlocal Green's functions have finite thermalization rate. We provide understanding of our numerical and analytical large-NNN results through random-matrix theory (RMT) analysis. Furthermore, we show that the thermalization of an initial pure product state in the noninteracting SYK2{\mathrm{SYK}}_{2}SYK2​ model is independent of fermion filling and an initial density inhomogeneity decays with weak but long lived oscillations due to dephasing. In contrast, the interacting ${\mathrm{SYK}}_{q\ensuremath{\ge}4}$ model thermalizes slower than the noninteracting model and exhibits filling-dependent monotonic relaxation of initial inhomogeneity. For evolution of entangled pure states, we show that the initial entanglement is encoded in the nonlocal and/or multipoint quantum correlations that relax as the system thermalizes.
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A fundamental challenge in quantum many-body physics is to understand the universal properties of strongly correlated systems. In this work, we establish a universal and exact identity from the Dyson-Schwinger equations within the Keldysh-Schwinger field theory for systems with generic bilinear interactions. Our derivation demonstrates the emergence of composite particles as "elementary excitations", whose Green's functions definitively determine the original single-particle Green's function. This universal relation uniquely identifies the composite particles governing correlations and rigorously connects their spectra to the observable single-particle spectrum. Thus, our exact identity reveals a new pathway toward a paradigm for understanding many-body correlated systems.
We propose a new field theoretic method for calculating Renyi entropy of a subsystem of many interacting bosons without using replica methods. This method is applicable to dynamics of both open and closed quantum systems starting from arbitrary initial conditions. Our method identifies the Wigner characteristic of a reduced density matrix with the partition function of the whole system with a set of linear sources turned on only in the subsystem, and uses this to calculate the subsystem's Renyi entropy. We use this method to study the evolution of Renyi entropy in a noninteracting open quantum system starting from an initial Fock state. We find a relation between the initial state and final density matrix which determines whether the entropy shows nonmonotonic behavior in time. For non-Markovian dynamics, we show that the entropy approaches its steady-state value as a power law with exponents governed by nonanalyticities of the bath. We illustrate that this field-theoretic method can be used to study large bosonic open quantum systems.
This chapter examines the Schwinger-Keldysh technique, applying path integrals, and the functional method of 2PI effective actions—with the latter serving to construct powerful symmetry-preserving nonperturbative approximations for equilibrium as well as nonequilibrium descriptions of strongly interacting quantum many-body systems. It provides a rigorous mathematical basis for exploring the quantum mechanical basis of thermalization and decoherence. Mesoscopic systems with quantum processes, in particular, play a key role. The domain of nonequilibrium mesoscopic systems is flooded with open questions some of which are solved but still others have provisional or incomplete solutions. A correlation function encodes the correlation between the field operators at different times and is an inference to all thermodynamic quantities. The beauty of the Keldysh time path is the definition of the Green's function, which plays an analogous role in the nonequilibrium formalism similar to the role the causal Green's function plays in the equilibrium theory.