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arXiv:2610.38338

Time evolution in the multi-component dynamical Keldysh models

M. N. Kiselev

cond-mat.mes-hallcond-mat.dis-nncond-mat.str-elquant-phSubmitted Oct 28, 2026• Announced Oct 29, 2026• 0 citations
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Abstract

We construct a time evolution operator for a dynamical quantum models with a multi-component random Gaussian field characterized by a long correlation time. We analyze three paradigmatic cases: non-Abelian $SU(2)$ symmetry class for the three-component Gaussian noise and Abelian symmetry classes $U(1)$ and $Z_2$ for the two- and one-component noises correspondingly. Starting with a limiting case of the infinite correlation time (Keldysh model) we shown that statistics of the two-component random field with a non-zero mean is described by the Rice distribution. We derived and solved the system of coupled linear differential equations for the exact diagonal and off-diagonal single-particle Green's functions. We showed how the finite time correlation in Keldysh models are accounted by a continuous fractions method.

AI Summary

The paper studies disorder-averaged single-particle dynamics in multi-component dynamical Keldysh models: a two-level system coupled to classical Gaussian fields with correlation $D(t)=W^2e^{-\gamma|t|}$. It considers one-, two-, and three-component fields, emphasizing the infinite-correlation-time limit $\gamma=0$ and extensions to finite correlation time. Using a Wei–Norman factorization, it constructs the evolution operator for each static field realization and then averages over disorder. For zero-mean fields, radial averaging uses Rayleigh and Maxwell–Boltzmann distributions in two and three components, respectively, recovering known Green’s functions with densities of states vanishing as $|\epsilon|$ and $\epsilon^2$ near zero energy. The main new calculation treats a two-component field with nonzero mean $\eta$. Its magnitude follows a Rice distribution, yielding integral expressions for diagonal and off-diagonal retarded Green’s functions involving modified Bessel functions $I_0$ and $I_1$. In units $W=1$, these satisfy the exact coupled equations $G'+(\epsilon-1/\epsilon)G-\eta F=1$ and $F'+\epsilon F-\eta G=0$. Finite correlation time is incorporated through series over higher-component zero-mean Green’s functions represented by continued fractions. The paper discusses how finite $\gamma$ fills the pseudogap and can change the spectrum from two peaks to one. The explicit nonzero-mean treatment is restricted to two components. Nonzero-mean three-component models, two-particle Green’s functions, larger symmetry groups, and finite-correlation-time Ward identities are left for further analysis.

No sign of AI-generated textIncrementalOverclaimsPossible errorUnclear writing

Central claim may not hold: Both finite-correlation-time series contain an extra factor of 1/2, violating Green's-function normalization and the diagonal zero-mean limit. Even after normalization is repaired, the proposed finite-correlation-time mixture has an incorrect high-frequency moment for nonzero mean and therefore is not an exact solution of the stated noise model. The three-component calculation uses an angular measure with total weight 2, and Eq. (MB) independently lacks the Dawson-term factor needed for the correct large-energy asymptote. The printed off-diagonal Fourier transform has the wrong energy parity; its Bessel-function argument must use the absolute value of energy.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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7 pages, 1 figure

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We construct a time evolution operator for a dynamical quantum models with a multi-component random Gaussian field characterized by a long correlation time. We analyze three paradigmatic cases: non-Abelian SU(2)SU(2)SU(2) symmetry class for the three-component Gaussian noise and Abelian symmetry classes U(1)U(1)U(1) and Z2Z_2Z2​ for the two- and one-component noises correspondingly. Starting with a limiting case of the infinite correlation time (Keldysh model) we shown that statistics of the two-component random field with a non-zero mean is described by the Rice distribution. We derived and solved the system of coupled linear differential equations for the exact diagonal and off-diagonal single-particle Green's functions. We showed how the finite time correlation in Keldysh models are accounted by a continuous fractions method.
The paper studies disorder-averaged single-particle dynamics in multi-component dynamical Keldysh models: a two-level system coupled to classical Gaussian fields with correlation D(t)=W2e−γ∣t∣D(t)=W^2e^{-\gamma|t|}D(t)=W2e−γ∣t∣. It considers one-, two-, and three-component fields, emphasizing the infinite-correlation-time limit γ=0\gamma=0γ=0 and extensions to finite correlation time.
Using a Wei–Norman factorization, it constructs the evolution operator for each static field realization and then averages over disorder. For zero-mean fields, radial averaging uses Rayleigh and Maxwell–Boltzmann distributions in two and three components, respectively, recovering known Green’s functions with densities of states vanishing as ∣ϵ∣|\epsilon|∣ϵ∣ and ϵ2\epsilon^2ϵ2 near zero energy.
The main new calculation treats a two-component field with nonzero mean η\etaη. Its magnitude follows a Rice distribution, yielding integral expressions for diagonal and off-diagonal retarded Green’s functions involving modified Bessel functions I0I_0I0​ and I1I_1I1​. In units W=1W=1W=1, these satisfy the exact coupled equations G′+(ϵ−1/ϵ)G−ηF=1G'+(\epsilon-1/\epsilon)G-\eta F=1G′+(ϵ−1/ϵ)G−ηF=1 and F′+ϵF−ηG=0F'+\epsilon F-\eta G=0F′+ϵF−ηG=0. Finite correlation time is incorporated through series over higher-component zero-mean Green’s functions represented by continued fractions. The paper discusses how finite γ\gammaγ fills the pseudogap and can change the spectrum from two peaks to one.
The explicit nonzero-mean treatment is restricted to two components. Nonzero-mean three-component models, two-particle Green’s functions, larger symmetry groups, and finite-correlation-time Ward identities are left for further analysis.
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Equations for correlation functions in nonequilibrium systems are derived using a generalization of the diagrammatic Keldysh method. The four field operators of the correlation function are ordered on a time contour that consists of four axes extending from -\ensuremath{\infty} to +\ensuremath{\infty} and from +\ensuremath{\infty} to -\ensuremath{\infty} (doubled Keldysh contour). For those correlation functions that may be considered as correlation functions of one-particle Green-functions' fluctuations, a closed system of equations is obtained in the case of weak interaction of the particles with the scattering system. Its structure may be related to the Langevin approach: on the left-hand side a linear operator acts on each of two pairs of the correlation-function arguments, each of these two operators being just the operator of the linearized quantum kinetic equation. The expression for the right-hand side of this equation is derived. It has the meaning of the correlation function of extraneous Langevin sources, and the equation may be called the quantum Boltzmann-Langevin equation. In the quasi-classical limit it reduces to the usual Boltzmann-Langevin equation for the particles' distribution-function fluctuations. Possible applications of the method are discussed.
In this work we derive the replica field theory for monitored quantum many-body systems evolving under the quantum jumps protocol, corresponding to a non-Hermitian evolution interspersed with random quantum jumps whose distribution is state-dependent. We show that the density matrix of RRR replicas evolves according to a master equation where the non-Hermitian term is replica-diagonal while coupling among replicas are due to quantum jumps. We write down the associated Keldysh action and study its behavior for the specific case of the Ising Chain with monitoring of particle density and tunable anisotropy, interpolating between free fermions with strong U(1) symmetry and the Ising chain with Z2_22​ symmetry. We derive the effective field theory in terms of slowly varying fields and obtain the replica-diagonal saddle point, which we show to describe the average state. We then go beyond saddle point and derive the effective field theory describing the replica off-diagonal sector, which takes the form of a Non-Linear Sigma Model. The symmetry class is either DIII or D, depending on the parameters of the Ising chain, except at a special symmetric point, where we recover the results for free fermions. We discuss the implications of these findings for the entangling phase observed numerically for the monitored Ising chain.
A new class of random quantum-dynamical systems in continuous space is introduced. Each member of the class is characterized by a Hamiltonian which is the sum of two parts. While one part is deterministic, time independent, and quadratic, the Weyl-Wigner symbol of the other part is a homogeneous Gaussian random field which is $\ensuremath{\delta}$ correlated in time and arbitrary, but smooth in position and momentum. Exact expressions for the time evolution of both averaged states and observables are obtained. If the deterministic part is that of a particle subject to a constant magnetic field, spatial variance of the averaged state grows diffusively for long times independent of the initial state.
We show that operator dynamics in U(1) symmetric systems are constrained by two independent conserved charges and construct a non-Abelian operator size basis that respects both, enabling a symmetry-resolved characterization of operator growth. The non-Abelian operator size depends on the operator's nonlocal structure and is organized by an SU(2) algebra. Operators associated with large total angular momentum are relatively simple, while those with small angular momentum are more complex. Operator growth is thus characterized by a reduction in angular momentum and can be probed using out-of-time-ordered correlators. Using the charge-conserving Brownian Sachdev-Ye-Kitaev model, we derive an exact classical master equation that governs the size distribution, the distribution of an operator expanded in this basis, for arbitrary system sizes. The resulting dynamics reveal that the size distribution follows a chi-squared form, with the two conserved charges jointly determining the overall time scale and the shape of the distribution. In particular, single-particle operators retain a divergent peak at large angular momentum throughout the time evolution.