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Time evolution in the multi-component dynamical Keldysh models

This paper constructs a time evolution operator for multi-component dynamical Keldysh models with long-correlated random Gaussian fields, analyzing non-Abelian $SU(2)$ and Abelian $U(1)$ and $Z_2$ symmetry classes to derive exact Green's functions, identify Rice distribution statistics for non-zero mean fields, and account for finite time correlations using a continued fractions method.

Original authors: M. N. Kiselev

Published 2026-10-30

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CC BY 4.0

Original authors: M. N. Kiselev

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum physics, particles do not move through empty space; they navigate a landscape filled with invisible, shifting obstacles. Imagine an electron trying to travel through a material where the electric potential is not smooth and predictable, but instead jitters randomly due to impurities or thermal noise. For decades, physicists have relied on a specific mathematical framework, known as the Keldysh model, to describe how these particles behave when the noise is so slow and persistent that it acts like a fixed, albeit random, wall rather than a fleeting bump. This framework has been essential for understanding everything from the flow of electricity in heavily doped semiconductors to the mysterious energy gaps found in high-temperature superconductors. However, real-world systems are rarely so simple. They often involve multiple types of noise acting simultaneously, and these noises can have a distinct average direction or a specific duration of persistence. Understanding how a quantum particle evolves when subjected to this complex, multi-layered chaos has remained a difficult challenge, leaving gaps in our ability to predict the behavior of advanced quantum devices.

A researcher has now constructed a precise mathematical map for this chaotic evolution, focusing on systems where the random noise comes in one, two, or three distinct components. The scientist began by tackling the most extreme version of the problem: a scenario where the noise is so slow that it never changes during the observation, a state known as infinite correlation time. In this limit, they successfully derived the exact rules for how a quantum system evolves over time when bombarded by random fields that point in different directions. They found that for a system with a single component of noise, the behavior is straightforward. But when the noise has two components, the statistics of the system's energy follow a specific pattern known as the Rice distribution, a result that had been suspected but not rigorously derived in this context. When the noise has three components, the mathematics becomes significantly more intricate, requiring the researcher to solve a complex set of linked equations to determine the particle's probability of being in a certain state.

The true breakthrough of this work lies in moving beyond the idealized, frozen-noise scenario to address the messy reality of finite time. In nature, noise fluctuates; it has a characteristic lifespan before it changes. The researcher demonstrated how to account for this finite duration by using a sophisticated mathematical technique involving continuous fractions. This method allowed them to bridge the gap between the slow, frozen noise and the rapid, random noise that characterizes everyday thermal fluctuations. They applied this approach to a system with a non-zero average noise, which corresponds to a situation where the random field has a slight bias in one direction, such as a weak magnetic field pushing a particle in a specific way. By solving the resulting equations, they discovered that the average noise and the speed of the fluctuations compete with each other. The average noise tends to smooth out the energy landscape from the top, while the speed of the fluctuations fills in the gaps from the bottom.

The outcome of this competition is a dramatic change in the material's properties. The researcher found that as the noise becomes faster, a deep dip in the energy spectrum, known as a pseudogap, begins to fill in. Eventually, at a critical speed of fluctuation, this dip disappears entirely, transforming the spectrum from a valley with two peaks into a single central peak. This transition marks a fundamental shift in how the system conducts energy and responds to external forces. The work provides a complete, exact solution for these dynamical models, offering a new toolset for scientists studying coupled quantum dots, which are tiny traps for electrons used in quantum computing, as well as for those investigating the complex instabilities found in superconducting and magnetic materials. By clarifying how time and randomness interact to shape quantum behavior, this study lays the groundwork for designing more stable and efficient quantum technologies.

Technical Summary: Time Evolution in Multi-Component Dynamical Keldysh Models

Problem Statement
The paper addresses the time evolution of dynamical quantum systems subject to classical random Gaussian fields with long correlation times. While the original Keldysh model successfully described electron motion in smooth, long-range spatial disorder, its time-domain extension (dynamical Keldysh models) requires handling time-dependent potentials defined by "colored" noise (finite correlation time $\tau_{noise} = 1/\gamma$ ). The specific challenge tackled here is the construction of an exact time-evolution operator (TEO) for multi-component systems (one, two, and three components) and the subsequent derivation of exact single-particle Green's functions (GFs). This includes analyzing systems with both zero and non-zero mean fields, a regime where off-diagonal correlations emerge, and extending the analysis from the infinite correlation time limit (static disorder) to finite correlation times.

Methodology
The author employs the Wei-Norman representation to construct the TEO for Hamiltonians possessing dynamical symmetries. The Hamiltonian is defined as a linear combination of compact group generators (Pauli matrices for $SU(2)$ ) with time-dependent coefficients representing the random Gaussian field.

Exact TEO Construction: Instead of averaging the TEO directly, the author first solves the non-linear differential equations governing the Wei-Norman parameters ( $f_\pm, f_z$ ) for a specific realization of the disorder. This yields an exact analytical form for the TEO $\hat{U}_\Delta(t,0)$ .
Disorder Averaging: The TEO is then averaged over the statistical distribution of the random field.
- For zero-mean fields, the author utilizes the Rayleigh distribution for the two-component case and the Maxwell-Boltzmann distribution for the three-component case.
- For non-zero mean fields, the fluctuating field is decomposed into a mean ( $\eta$ ) and a zero-mean fluctuation ( $\tilde{\Delta}$ ), leading to integrals involving modified Bessel functions ( $I_0, I_1$ ), which correspond to the Rice distribution statistics.
Finite Correlation Time: To account for finite but large correlation times, the author utilizes a continuous fractions method. They express the Green's functions of the finite-correlation-time model as infinite series expansions involving Green's functions of zero-mean, finite-correlation-time spherical models.

Key Contributions and Results

Exact Time-Evolution Operators: The paper derives explicit forms for the TEO in three paradigmatic cases:
- $SU(2)$ (3-component): Non-Abelian symmetry.
- $U(1)$ (2-component): Abelian symmetry.
- $Z_2$ (1-component): Abelian symmetry.
  The solutions are expressed in terms of trigonometric functions of the field magnitude and angles, valid for infinite correlation time.
Green's Functions for Zero-Mean Fields:
- For the 2-component ( $U(1)$ ) model, the averaged retarded Green's function is derived, showing a pseudogap in the density of states (DoS) that evolves linearly at low energies.
- For the 3-component ( $SU(2)$ ) model, the DoS exhibits a pseudogap characterized by an $\epsilon^2$ dependence at low energies.
- These results for zero-mean fields are shown to coincide with known results derived via Feynman diagrammatics and Dyson equations, validating the TEO approach.
Green's Functions for Non-Zero Mean Fields:
- The author derives exact expressions for both diagonal and off-diagonal Green's functions in the presence of a non-zero mean field ( $\eta$ ).
- The statistics of the field magnitude are identified as the Rice distribution.
- The diagonal ( $G_2$ ) and off-diagonal ( $F_2$ ) Green's functions are expressed as integrals involving Bessel functions.
- Crucially, the author derives a system of coupled linear differential equations (Eq. 31) that these Green's functions satisfy, along with their boundary conditions at infinite energy.
Finite Correlation Time Effects:
- The author demonstrates how to transition from the Keldysh limit (infinite correlation time) to the white-noise limit using a continuous fraction expansion.
- They analyze the interplay between finite correlation time ( $\gamma$ ) and non-zero mean ( $\eta$ ) on the DoS.
- Result: Finite $\gamma$ fills the pseudogap from the bottom (creating non-zero DoS at zero energy), while finite $\eta$ reduces the peak height from the top. The combination leads to a phase transition where the pseudogap vanishes, transforming the DoS from a central dip to a central peak. A critical line $\gamma(\eta)$ defines the boundary between these two phases.

Significance and Claims
The paper claims to provide an exact solution for the time evolution and Green's functions of dynamical models with classical random Gaussian disorder on spheres $S^{d-1}$ .

Validation: For zero-mean fields, the results recover established diagrammatic findings, confirming the validity of the TEO-based approach.
Novelty: The derivation of exact diagonal and off-diagonal Green's functions for multi-component models with non-zero mean is presented as a new result. This includes the formulation of the coupled differential equations governing these functions.
Methodological Utility: The application of the Wei-Norman parametrization is highlighted as a powerful tool for analyzing dynamical models with symmetries beyond $SU(2)$ (e.g., $SU(3)$ , $SU(1,1)$ ) and on various manifolds.
Physical Insight: The work elucidates how finite correlation times and non-zero mean fields modify the pseudogap phenomena, offering a framework to understand transitions in density of states relevant to coupled quantum dots, Landau-Zener transitions, and non-equilibrium instabilities.

The author maintains a modest tone regarding future implications, noting that while the method opens avenues for higher symmetry groups and two-particle Green's functions (e.g., singlet-triplet cases), these remain unsolved problems. The primary contribution is the rigorous construction of the exact time evolution and the resulting single-particle Green's functions for the specified multi-component dynamical Keldysh models.

Technical Summary: Time Evolution in Multi-Component Dynamical Keldysh Models

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