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arXiv:1709.04472 · Physical review. B./Physical review. B

Power-law tails and non-Markovian dynamics in open quantum systems: An exact solution from Keldysh field theory

Ahana Chakraborty, Rajdeep Sensarma

Published Mar 16, 2018• 27 citations
arXivHTMLPDF

Abstract

The Born-Markov approximation is widely used to study the dynamics of open quantum systems coupled to external baths. Using Keldysh formalism, we show that the dynamics of a system of bosons (fermions) linearly coupled to a noninteracting bosonic (fermionic) bath falls outside this paradigm if the bath spectral function has nonanalyticities as a function of frequency. In this case, we show that the dissipative and noise kernels governing the dynamics have distinct power-law tails. The Green's functions show a short-time ``quasi''-Markovian exponential decay before crossing over to a power-law tail governed by the nonanalyticity of the spectral function. We study a system of bosons (fermions) hopping on a one-dimensional lattice, where each site is coupled linearly to an independent bath of noninteracting bosons (fermions). We obtain exact expressions for the Green's functions of this system, which show power-law decay $\ensuremath{\sim}|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3/2}$. We use these to calculate the density and current profile, as well as unequal-time current-current correlators. While the density and current profiles show interesting quantitative deviations from Markovian results, the current-current correlators show qualitatively distinct long-time power-law tails $|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3}$ characteristic of non-Markovian dynamics. We show that the power-law decays survive in the presence of interparticle interaction in the system, but the crossover time scale is shifted to larger values with increasing interaction strength.

DOI
10.1103/physrevb.97.104306

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The Born-Markov approximation is widely used to study the dynamics of open quantum systems coupled to external baths. Using Keldysh formalism, we show that the dynamics of a system of bosons (fermions) linearly coupled to a noninteracting bosonic (fermionic) bath falls outside this paradigm if the bath spectral function has nonanalyticities as a function of frequency. In this case, we show that the dissipative and noise kernels governing the dynamics have distinct power-law tails. The Green's functions show a short-time ``quasi''-Markovian exponential decay before crossing over to a power-law tail governed by the nonanalyticity of the spectral function. We study a system of bosons (fermions) hopping on a one-dimensional lattice, where each site is coupled linearly to an independent bath of noninteracting bosons (fermions). We obtain exact expressions for the Green's functions of this system, which show power-law decay \ensuremath∼∣t\ensuremath−t\ensuremath′∣\ensuremath−3/2\ensuremath{\sim}|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3/2}\ensuremath∼∣t\ensuremath−t\ensuremath′∣\ensuremath−3/2. We use these to calculate the density and current profile, as well as unequal-time current-current correlators. While the density and current profiles show interesting quantitative deviations from Markovian results, the current-current correlators show qualitatively distinct long-time power-law tails ∣t\ensuremath−t\ensuremath′∣\ensuremath−3|t\ensuremath{-}{t}^{\ensuremath{'}}{|}^{\ensuremath{-}3}∣t\ensuremath−t\ensuremath′∣\ensuremath−3 characteristic of non-Markovian dynamics. We show that the power-law decays survive in the presence of interparticle interaction in the system, but the crossover time scale is shifted to larger values with increasing interaction strength.
In interacting quantum many-body systems, relaxation toward equilibrium reflects a competition between internal chaotic dynamics and environmental dissipation. While conventional Markovian baths typically produce exponential decay, non-Markovian dissipation can give rise to more intricate behavior, including algebraic relaxation. We study an open Sachdev-Ye-Kitaev (SYK) model coupled to a pseudogapped fermionic bath, using the Keldysh formalism to compute steady-state correlations in the large-NNN limit. Our results uncover a rich dynamical phase diagram, with regimes of bath-driven power-law relaxation, chaos-driven exponential decay, and an intermediate pre-relaxation phase where exponential decay crosses over to algebraic decay. These findings demonstrate that non-Markovian environments can qualitatively reshape relaxation mechanisms in strongly correlated quantum many-body systems.
While considering non-Hermitian Hamiltonians arising in the presence of dissipation, in most cases, the dissipation is taken to be frequency independent. However, this idealization may not always be applicable in experimental settings, where dissipation can be frequency-dependent. Such frequency-dependent dissipation leads to non-Markovian behavior. In this work, we demonstrate how a quantum non-Markovian Hatano-Nelson model arises microscopically in a quasi-one-dimensional dissipative lattice. This is achieved using non-equilibrium Green's functions without requiring any approximation like weak system-bath coupling or a time-scale separation, which would have been necessary for a Markovian treatment. The resulting effective system exhibits nonreciprocal hopping, the defining characteristic of the Hatano-Nelson model, as well as uniform dissipation, both of which are frequency-dependent. This holds for both bosonic and fermionic settings. We find solely non-Markovian nonreciprocal features like unidirectional frequency blocking in bosonic settings, and a particular type of non-equilibrium dissipative quantum phase transition in fermionic settings, that cannot be captured in a Markovian theory, nor have any analog in reciprocal systems. Our results lay the groundwork for describing and engineering non-Markovian nonreciprocal quantum lattices.
We present a general theory of non-Markovian dynamics for open systems of noninteracting fermions (bosons) linearly coupled to thermal environments of noninteracting fermions (bosons). We explore the non-Markovian dynamics by connecting the exact master equations with the nonequilibirum Green's functions. Environmental backactions are fully taken into account. The non-Markovian dynamics consists of nonexponential decays and dissipationless oscillations. Nonexponential decays are induced by the discontinuity in the imaginary part of the self-energy corrections. Dissipationless oscillations arise from band gaps or the finite band structure of spectral densities. The exact analytic solutions for various non-Markovian thermal environments show that non-Markovian dynamics can be largely understood from the environmental-modified spectra of open systems.
We develop a framework to analyze the dynamics of a finite-dimensional quantum system S in contact with a reservoir R . The full, interacting SR dynamics is unitary. The reservoir has a stationary state but otherwise dissipative dynamics. We identify a main part of the full dynamics, which approximates it for small values of the SR coupling constant, uniformly for all times t≥0 . The main part consists of explicit oscillating and decaying parts. We show that the reduced system evolution is Markovian for all times. The technical novelty is a detailed analysis of the link between the dynamics and the spectral properties of the generator of the SR dynamics, based on Mourre theory. We allow for SR interactions with little regularity, meaning that the decay of the reservoir correlation function only needs to be polynomial in time, improving on the previously required exponential decay.In this work we distill the structural and technical ingredients causing the characteristic features of oscillation and decay of the SR dynamics. In the companion paper [27] we apply the formalism to the concrete case of an N -level system linearly coupled to a spatially infinitely extended thermal bath of non-interacting Bosons.
The Schwinger-Keldysh nonequilibrium theory allows the description of various transport phenomena involving bosons (fermions) embedded in bosonic (fermionic) environments. The retarded Green’s function obeys the Dyson equation and determines via its non-vanishing asymptotic behavior the dissipationless open dynamics. The appearance of this regime is conditioned by the existence of the solution of a general class of transcendental equations in complex domain that we study. Particular cases consist in transcendental equations containing exponential, hyperbolic, power law, logarithmic, and special functions. The present analysis provides an analytical description of the thermal and temporal correlation function of two general observables of a quantum system in terms of the corresponding spectral function. Special integral properties of the spectral function guarantee non-vanishing asymptotic behavior of the correlation function.
The exponential decay of an unstable quantum state, as described by standard Markovian theories such as Fermi's Golden Rule, assumes a simple, structureless environment. However, in complex environments characterized by disorder, long-range interactions, or strong fluctuations, local decay rates fluctuate, leading to non-Markovian dynamics and power-law ``long-time tails.'' In this paper, we apply the recently proposed \textit{hyperstatistics} framework to the problem of quantum decay in such complex environments. By considering a γγγ-distribution of local decay rates across mesoscopic domains, we derive a macroscopic survival probability governed by a qqq-exponential function. We then use the qqq-generalized Gamma function, defined via the Mellin transform of the qqq-exponential, to calculate the moments of the decay-time distribution. We show that the mean quantum lifetime is ⟨t⟩=∫0∞P(t) dt=[⟨Γ⟩(2−q)]−1,\langle t\rangle = \int_0^\infty P(t)\,dt= [\langleΓ\rangle(2-q)]^{-1},⟨t⟩=∫0∞​P(t)dt=[⟨Γ⟩(2−q)]−1, and is therefore finite for q<2q<2q<2. The convergence of the second moment, and hence of the lifetime variance, instead requires the stricter condition q<3/2q<3/2q<3/2. These results provide a refined physical interpretation: for 1<q<3/21<q<3/21<q<3/2 both the mean lifetime and its variance are finite; for 3/2≤q<23/2\le q<23/2≤q<2 the mean lifetime remains finite but lifetime fluctuations become infinitely broad; and for q≥2q\ge2q≥2 the mean lifetime itself diverges, corresponding to a truly trapped, localized, or Griffiths-like regime.