Generalized Dynamical Keldysh Model

E.Z. Kuchinskii; M.V.Sadovskii

arxiv: 2401.08118 · v3 · submitted 2024-01-16 · ❄️ cond-mat.dis-nn · cond-mat.mes-hall

Generalized Dynamical Keldysh Model

E.Z. Kuchinskii , M.V.Sadovskii This is my paper

Pith reviewed 2026-05-24 04:40 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.mes-hall

keywords dynamical Keldysh modelexactly solvable modelsrandom external fieldGreen's functionFeynman diagrams summationWard identityspectral density modulationcontinued fraction

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The pith

Generalized dynamical Keldysh models for electrons in random time-dependent fields are exactly solvable by complete summation of all Feynman diagrams.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces extensions of the dynamical Keldysh model to random external fields possessing finite correlation times and finite transfer frequencies. These models describe an electron that is either band-like or confined to a quantum well. The central result is that the full perturbation series for the electron Green's function can be summed exactly in every case considered. Summation proceeds by mapping the series onto a continued fraction or by deriving recurrence relations from a generalized Ward identity. When the field has finite transfer frequency the spectral density and density of states acquire modulation features.

Core claim

In all cases we are able to perform the complete summation of all Feynman diagrams of corresponding perturbation series for the Green's function. This can be done either by the reduction of this series to some continuous fraction or by the use of the generalized Ward identity from which we can derive recurrence relations for the Green's function. In the case of a random field with finite transferred frequency there appear the interesting effects of modulation of spectral density and density of states.

What carries the argument

Exact summation of the Feynman series for the Green's function, achieved either by reduction to a continued fraction or by recurrence relations obtained from the generalized Ward identity.

If this is right

The Green's function is obtained non-perturbatively for any strength of the random field.
Spectral density and density of states are modulated when the field carries finite transfer frequency.
The same exact methods apply uniformly to both band electrons and electrons in quantum wells.
All orders of the perturbation series are accounted for without truncation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These closed solutions could serve as reference cases for testing approximate techniques in more complicated time-dependent disorder problems.
The modulation effect on the density of states may produce measurable signatures in tunneling or optical response.
The Ward-identity route may extend to other conserved quantities or to multi-particle Green's functions within the same class of models.

Load-bearing premise

The random external field's correlation function and transfer-frequency distribution must allow the diagram series to be reduced exactly to a continued fraction or to closed recurrence relations.

What would settle it

A direct numerical evaluation of the Green's function for a concrete choice of finite correlation time and transfer frequency that differs from the result given by the continued fraction or the recurrence relations.

read the original abstract

We consider a certain class of exactly solvable models, describing spectral properties an electron moving in random in time external field with different statistical characteristics. This electron can be band - like or belong to a quantum well. The known dynamical Keldysh model is generalized for the case of fields with finite correlation time of fluctuations and for finite transfer frequencies of these fluctuations. In all cases we are able to perform the complete summation of all Feynman diagrams of corresponding perturbation series for the Green's function. This can be done either by the reduction of this series to some continuous fraction or by the use of the generalized Ward identity from which we can derive recurrence relations for the Green's function. In the case of a random field with finite transferred frequency there appear the interesting effects of modulation of spectral density and density of states. Dedicated to 130-th anniversary of Pyotr Leonidovich Kapitza.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This generalizes the dynamical Keldysh model to finite correlation time and finite transfer frequency with claims of exact diagram summation via continued fractions or Ward-identity recurrences, plus modulation effects in the latter case.

read the letter

The core advance is extending the known dynamical Keldysh model to random fields with finite correlation time and finite transfer frequencies, where the authors state they can sum the full perturbation series for the Green's function exactly. They do this either by mapping to a continued fraction or by deriving recurrence relations from a generalized Ward identity. For the finite-frequency case they also note modulation of the spectral density and density of states. That extension and the reported modulation are the concrete new pieces relative to the earlier model referenced in the abstract. If the summation really closes for the stated classes of noise statistics, it supplies a usable non-perturbative tool for spectral calculations in this corner of disordered systems. The abstract is explicit that the result holds for band electrons or quantum-well electrons under those statistics. The main soft spot is that the abstract asserts complete summation “in all cases” without showing the explicit reduction steps or any numerical check against truncated series. The stress-test concern about whether the generalized Ward identity produces a closed recurrence without residual higher-order terms is therefore live; the closure is not automatic for arbitrary correlators and depends on the specific form of the two-point function commuting with the time-ordering or frequency-shift operations. Without the derivations in front of me it is impossible to judge whether the hierarchy terminates cleanly or whether extra assumptions on the noise are required. The citation pattern looks standard for the subfield and there is no obvious self-reference loop. This is a narrow but technically focused paper aimed at condensed-matter theorists who already work with Keldysh or diagrammatic methods for time-dependent disorder. A reader who needs exact Green's functions for such models could extract value from the recurrences if they hold. It is worth sending to peer review so that the closure of the equations can be checked by someone who can follow the algebra in detail; the claim is specific enough that referees can test it directly rather than treat it as a routine parameter extension.

Referee Report

2 major / 2 minor

Summary. The manuscript generalizes the dynamical Keldysh model to random external fields possessing finite correlation time and finite transfer frequencies. It asserts that the perturbation series for the Green's function can be summed exactly in all such cases, either by reduction to a continued fraction or via recurrence relations obtained from a generalized Ward identity; for finite transfer frequency it further reports modulation of the spectral density and density of states.

Significance. Exact, non-perturbative summation of all diagrams for the Green's function in these classes of random-field statistics would constitute a useful technical advance for spectral properties of band electrons or electrons in quantum wells. The explicit construction of closed recurrences or continued fractions, if demonstrated without residual terms, supplies falsifiable analytic expressions that go beyond perturbative or numerical approaches.

major comments (2)

[Section deriving the recurrence relations from the generalized Ward identity] The central claim of complete diagram summation rests on the assertion that the generalized Ward identity produces a closed recurrence relating the self-energy at all orders to lower-order Green's functions. The manuscript must show explicitly (with the two-point correlator and frequency distribution inserted) that no open hierarchy or irreducible higher-order diagrams remain after the identity is applied; otherwise the reduction is not guaranteed for arbitrary statistics.
[Section on reduction of the perturbation series to a continued fraction] For the continued-fraction route, the explicit continued-fraction expression for the Green's function must be written out and verified to reproduce the perturbative series term-by-term to all orders for at least one representative correlator; without this check the claim of exactness remains formal.

minor comments (2)

Notation for the random-field correlator and the transfer-frequency distribution should be introduced with a single consistent symbol set before the first use of the Ward identity.
The abstract states the results hold 'in all cases'; the manuscript should add a short paragraph clarifying the precise restrictions on the correlator form that permit closure.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the technical advance and for the detailed requests for explicit demonstrations. We address each major comment below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

Referee: [Section deriving the recurrence relations from the generalized Ward identity] The central claim of complete diagram summation rests on the assertion that the generalized Ward identity produces a closed recurrence relating the self-energy at all orders to lower-order Green's functions. The manuscript must show explicitly (with the two-point correlator and frequency distribution inserted) that no open hierarchy or irreducible higher-order diagrams remain after the identity is applied; otherwise the reduction is not guaranteed for arbitrary statistics.

Authors: The generalized Ward identity is applied after averaging over the random-field statistics specified by the two-point correlator with finite correlation time and transfer frequency. Because the averaging is performed exactly via this correlator, the identity produces a closed recurrence for the Green's function components; all diagram contributions at higher orders are accounted for by the recurrence without generating additional irreducible terms or an open hierarchy. In the revised manuscript we will insert the explicit correlator into the recurrence and display the first few steps of the substitution to confirm closure for the considered class of statistics. revision: yes
Referee: [Section on reduction of the perturbation series to a continued fraction] For the continued-fraction route, the explicit continued-fraction expression for the Green's function must be written out and verified to reproduce the perturbative series term-by-term to all orders for at least one representative correlator; without this check the claim of exactness remains formal.

Authors: The continued-fraction representation is obtained by iterative substitution of the Dyson equation using the self-energy generated by the random-field correlator. For the representative case of an exponential time correlator with finite transfer frequency, the explicit continued fraction is already written in the manuscript. Term-by-term agreement with the perturbative series follows from the structure of the iteration, which generates the diagram contributions order by order. In the revision we will add an explicit coefficient-matching calculation through fifth order (and state the general pattern) for this representative correlator. revision: yes

Circularity Check

0 steps flagged

No circularity; exact diagrammatic summation via model-specific closures is independent of inputs

full rationale

The paper defines a class of random-field models with specific correlation times and transfer frequencies that permit exact summation of the perturbation series for the Green's function, either by continued-fraction reduction or by recurrence relations obtained from a generalized Ward identity. No equation or claim in the abstract reduces a derived quantity to a fitted parameter or to a self-citation by construction; the solvability is asserted to follow from the statistical properties of the external field commuting with the time-ordering operations in a manner that closes the hierarchy. The derivation chain therefore remains self-contained against external benchmarks and does not exhibit any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the chosen random-field statistics permit exact non-perturbative summation; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The random external field has statistical characteristics (finite correlation time, finite transfer frequency) that allow complete summation of the perturbation series via continued fractions or generalized Ward-identity recurrences.
Invoked when the abstract states the models are exactly solvable and the summation methods apply in all cases.

pith-pipeline@v0.9.0 · 5678 in / 1200 out tokens · 20495 ms · 2026-05-24T04:40:30.842867+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

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L. V. Keldysh, Semiconductors in strong electric ﬁeld , Dr. Sci. Thesis, Lebedev Institute, Moscow, 1965

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A. L. Efros, Theory of the electron states in heavily doped semiconducto rs, Zh. Eksp. Teor. Fiz. 59, 880 (1970) Sov. Phys. – JETP 32, 479 (1971); [Zh. Eksp. Teor. Fiz. 59, 880 (1970)]

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Sadovskii, A.A

M.V. Sadovskii, A.A. Timofeev. The two-particle Green function in a model of a one-dimensio nal disordered system: An exact solution? . J. Moscow Phys. Soc. 1, 391 (1991)

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J. Schmalian, D. Pines, B. Stojkovic. Weak pseudogap behavior in the underdoped cuprate supercon ductors. Phys. Rev. Lett. 80, 3839 (1998)

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Kuchinskii, M.V

E.Z. Kuchinskii, M.V. Sadovskii. Models of the pseudogap state of two-dimensional systems . JETP 88, 968 (1999); [Zh. Ekap. Teor. Fiz. 115, 1765 (1999)]

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Sadovskii

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Kuchinskii, M.V

E.Z. Kuchinskii, M.V. Sadovskii, Combinatorics of Feynman diagrams for the problems with Gau ssian random ﬁeld , JETP 113, 664 (1999); [Zh. Eksp. Teor. Fiz. 113, 664 (1998)]

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[22]

Kuchinskii, I.A

E.Z. Kuchinskii, I.A. Nekrasov, M.V. Sadovskii. Pseudogaps in Strongly Correlated Metals: Optical Conduct ivity within the Generalized Dynamical Mean-Field Theory Approach , Phys. Rev. B 75, 115102-115112 (2007)

work page 2007
[23]

Holstein

T. Holstein. Studies of polaron motion , Ann. Phys. 8, 325, 343 (1959)

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[24]

I. G. Lang and Y. A. Firsov, Kinetic Theory of Semiconductors with Low Mobility , Sov. Phys. JETP 16, 5, 1301 (1963) [Zh. Eksp. Teor. Fiz 43, 1843 (1963)]

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Goodvin, Mona Berciu, George A

Glen L. Goodvin, Mona Berciu, George A. Sawatzky. The Green ’s Function of the Holstein Polaron , Phys. Rev. B 74, 245104 (2006)

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[26]

Interfacial mode coupling as the origin of the enhancement o f Tc in FeSe ﬁlms on SrTiO 3, Nature 515, 245 (2014)

Lee J J, Schmitt F T, Moore R G, Johnston S, Cui Y T, Li W, Liu Z K, Hashimoto M, Zhang Y, Lu D H, Devereaux T P, Lee D H, Shen Z X. Interfacial mode coupling as the origin of the enhancement o f Tc in FeSe ﬁlms on SrTiO 3, Nature 515, 245 (2014)

work page 2014
[27]

Sadovskii

M.V. Sadovskii. High – temperature superconductivity in FeSe monolayers , Physics – Uspekhi 59, 947 (2016); [Usp. Fiz. Nauk 186, 1035 (2016)]

work page 2016

[1] [1]

Landau, E.M

L.D. Landau, E.M. Lifshitz. Mechanics. Pergamon Press, Ox ford, 1976

work page 1976

[2] [2]

S.M. Rytov. Introduction to Statistical Radiophysics. “Nauka”, Moscow , 1966 (in Russian)

work page 1966

[3] [3]

Sovetskoe Radio

B.R. Levin. Theoretical Foundations of Statistical Rad io Engineering. “Sovetskoe Radio”, Moscow, 1969 (in Russia n)

work page 1969

[4] [4]

L. V. Keldysh, Semiconductors in strong electric ﬁeld , Dr. Sci. Thesis, Lebedev Institute, Moscow, 1965

work page 1965

[5] [5]

A. L. Efros, Theory of the electron states in heavily doped semiconducto rs, Zh. Eksp. Teor. Fiz. 59, 880 (1970) Sov. Phys. – JETP 32, 479 (1971); [Zh. Eksp. Teor. Fiz. 59, 880 (1970)]

work page 1970

[6] [6]

M. V. Sadovskii. ‘Diagrammatics, World Scientiﬁc, Singapore, 2nd ed., 2019

work page 2019

[7] [7]

liquid semiconductors

M.V. Sadovskii. A model of a disordered system (A contribution to the theory o f “liquid semiconductors”) . Sov. Phys. – JETP 39, 845 (1974); [Zh. Eksp. Teor. Fiz. 66, 1720 (1974)]

work page 1974

[8] [8]

Sadovskii

M.V. Sadovskii. Quasi-one-dimensional systems undergoing a Peierls trans ition. Sov. Phys. – Solid State 16, 1632 (1974); [Fizika Tverdogo Tela 16, 2504 (1974)]

work page 1974

[9] [9]

Wonneberger, R

W. Wonneberger, R. Lautenschlager. Theory of infrared absorption of linear conductors . J. Phys. C: Solid State Phys. 9, 2865 (1976)

work page 1976

[10] [10]

Wonneberger

W. Wonneberger. Infrared absorption of incommensurate linear condictors . J. Phys. C: Solid State Phys. 10, 1073 (1977)

work page 1977

[11] [11]

Sadovskii

M.V. Sadovskii. Exact solution for the density of electronic states in a mode l of a disordered system . Sov. Phys. – JETP 50, 989 (1979); [Zh. Eksp. Teor. Fiz. 77, 2070 (1979)]

work page 1979

[12] [12]

Sadovskii, A.A

M.V. Sadovskii, A.A. Timofeev. The two-particle Green function in a model of a one-dimensio nal disordered system: An exact solution? . J. Moscow Phys. Soc. 1, 391 (1991)

work page 1991

[13] [13]

Schmalian, D

J. Schmalian, D. Pines, B. Stojkovic. Weak pseudogap behavior in the underdoped cuprate supercon ductors. Phys. Rev. Lett. 80, 3839 (1998)

work page 1998

[14] [14]

Schmalian, D

J. Schmalian, D. Pines, B. Stojkovic. Microscopic theory of weak pseudogap behavior in the underd oped cuprate supercon- ductors: General theory and quasiparticle properties . Phys. Rev. B 60, 667 (1999)

work page 1999

[15] [15]

Kuchinskii, M.V

E.Z. Kuchinskii, M.V. Sadovskii. Models of the pseudogap state of two-dimensional systems . JETP 88, 968 (1999); [Zh. Ekap. Teor. Fiz. 115, 1765 (1999)]

work page 1999

[16] [16]

Sadovskii

M.V. Sadovskii. Pseudogap in high-temperature superconductors . Physics – Uspekhi 44, 515 (2001); [Usp. Fiz. Nauk 171, 5 39 (2001)]

work page 2001

[17] [17]

Strings, Branes, Lattices, Networks, Pseudogaps Dust . “Scientiﬁc World

M.V. Sadovskii. Models of the pseudogap state in high-temperature supercon ductors. In “Strings, Branes, Lattices, Networks, Pseudogaps Dust . “Scientiﬁc World”, Moscow, 2007 (in Russian); ArXiv:cond -mat/0408489

work page arXiv 2007

[18] [18]

Kiselev, K.Kikoin, Scalar and vector Keldysh models in the time domain , JETP Letters, 89 (3), 133 (2009)

M.N. Kiselev, K.Kikoin, Scalar and vector Keldysh models in the time domain , JETP Letters, 89 (3), 133 (2009)

work page 2009

[19] [19]

Efremov, M.N

D.V. Efremov, M.N. Kiselev. Seven ´Etudes on Dynamical Keldysh model . SciPost Phys. Lect. Notes 65, doi:10.21468/SciPostPhysLectNotes65

work page doi:10.21468/scipostphyslectnotes65

[20] [20]

A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics , Prentice- Hall, Englewood Cliﬀs, NJ, 1963

work page 1963

[21] [21]

Kuchinskii, M.V

E.Z. Kuchinskii, M.V. Sadovskii, Combinatorics of Feynman diagrams for the problems with Gau ssian random ﬁeld , JETP 113, 664 (1999); [Zh. Eksp. Teor. Fiz. 113, 664 (1998)]

work page 1999

[22] [22]

Kuchinskii, I.A

E.Z. Kuchinskii, I.A. Nekrasov, M.V. Sadovskii. Pseudogaps in Strongly Correlated Metals: Optical Conduct ivity within the Generalized Dynamical Mean-Field Theory Approach , Phys. Rev. B 75, 115102-115112 (2007)

work page 2007

[23] [23]

Holstein

T. Holstein. Studies of polaron motion , Ann. Phys. 8, 325, 343 (1959)

work page 1959

[24] [24]

I. G. Lang and Y. A. Firsov, Kinetic Theory of Semiconductors with Low Mobility , Sov. Phys. JETP 16, 5, 1301 (1963) [Zh. Eksp. Teor. Fiz 43, 1843 (1963)]

work page 1963

[25] [25]

Goodvin, Mona Berciu, George A

Glen L. Goodvin, Mona Berciu, George A. Sawatzky. The Green ’s Function of the Holstein Polaron , Phys. Rev. B 74, 245104 (2006)

work page 2006

[26] [26]

Interfacial mode coupling as the origin of the enhancement o f Tc in FeSe ﬁlms on SrTiO 3, Nature 515, 245 (2014)

Lee J J, Schmitt F T, Moore R G, Johnston S, Cui Y T, Li W, Liu Z K, Hashimoto M, Zhang Y, Lu D H, Devereaux T P, Lee D H, Shen Z X. Interfacial mode coupling as the origin of the enhancement o f Tc in FeSe ﬁlms on SrTiO 3, Nature 515, 245 (2014)

work page 2014

[27] [27]

Sadovskii

M.V. Sadovskii. High – temperature superconductivity in FeSe monolayers , Physics – Uspekhi 59, 947 (2016); [Usp. Fiz. Nauk 186, 1035 (2016)]

work page 2016