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arxiv: 1907.11695 · v1 · pith:246EEQKPnew · submitted 2019-07-26 · ❄️ cond-mat.stat-mech · physics.chem-ph

Time-dependent coupled cluster theory on the Keldysh contour for non-equilibrium systems

Alec F. White , Garnet Kin-Lic Chan This is my paper

Pith reviewed 2026-05-24 15:13 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech physics.chem-ph
keywords coupled cluster theoryKeldysh contournon-equilibrium dynamicsfinite temperatureHubbard modelwarm dense mattertime-dependent methods
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The pith

Keldysh contour extends finite-temperature coupled cluster theory to non-equilibrium driven systems.

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

The paper adapts coupled cluster methods for quantum systems at finite temperature to handle cases where external drives push the system out of equilibrium. It places the theory on the Keldysh contour to manage time-dependent processes and implements the singles and doubles level for practical calculations. Tests on a Hubbard model with a Peierls phase and a model of silicon under an XUV pulse demonstrate the approach. If the method holds, it opens calculations of real-time dynamics in driven thermal materials that were previously limited to equilibrium or ground-state treatments.

Core claim

Leveraging the Keldysh formalism extends finite temperature coupled cluster theory to thermal systems driven out of equilibrium, yielding Keldysh coupled cluster theory with implemented equations for singles and doubles calculations of finite temperature dynamics, as shown on a Hubbard model with Peierls phase and an ab initio model of warm-dense silicon under an ultrafast XUV pulse.

What carries the argument

Keldysh coupled cluster theory on the contour, which encodes time-dependent non-equilibrium evolution while retaining the coupled cluster expansion for thermal systems.

If this is right

  • Finite-temperature dynamics become accessible for driven lattice models such as the Hubbard chain under a Peierls phase.
  • Ab initio calculations of warm-dense matter response to ultrafast pulses, such as XUV irradiation of silicon, can be performed at the CCSD level.
  • The Keldysh contour implementation supplies the working equations needed to propagate the cluster amplitudes in time for non-equilibrium thermal states.

Where Pith is reading between the lines

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

  • The same contour-based extension could be applied to other truncations or to open systems coupled to baths.
  • Direct benchmarks against time-dependent exact methods on small clusters would quantify the error introduced by the doubles truncation under drive.
  • The framework suggests a route to embed Keldysh CC into larger embedding schemes for extended materials.

Load-bearing premise

The singles and doubles truncation of coupled cluster remains accurate enough for the non-equilibrium dynamics on the Keldysh contour.

What would settle it

Exact diagonalization of the driven Hubbard model with Peierls phase would show whether Keldysh CCSD matches the true dynamics at the same parameters.

Figures

Figures reproduced from arXiv: 1907.11695 by Alec F. White, Garnet Kin-Lic Chan.

Figure 1
Figure 1. Figure 1: FIG. 1. A cartoon depiction of the Keldysh contour in the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Relative error in the dipole moment as a function of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Population difference as a function of time for the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The number of electrons per unit cell for the Si system [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Difference in the population of the valence band (solid [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Hugenholtz skeletons contributing to (a) second or [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Hugenholtz skeletons representing the iteration in (a) [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Absolute difference in particle number relative to [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Absolute difference in particle number relative to [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Absolute difference in particle number relative to [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Absolute difference in particle number relative to [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Absolute difference in particle number relative to [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
read the original abstract

We leverage the Keldysh formalism to extend our implementation of finite temperature coupled cluster theory [\textit{J. Chem. Theory Comput.} 2018, \textit{14}, 5690-5700] to thermal systems that have been driven out of equilibrium. The resulting Keldysh coupled cluster theory is discussed in detail. We describe the implementation of the equations necessary to perform Keldysh coupled cluster singles and doubles calculations of finite temperature dynamics, and we apply the method to some simple systems including a Hubbard model with a Peierls phase and an {\it ab initio} model of warm-dense silicon subject to an ultrafact XUV pulse.

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.

Referee Report

0 major / 1 minor

Summary. The manuscript extends the authors' prior finite-temperature coupled cluster theory to non-equilibrium thermal systems by leveraging the Keldysh formalism. It discusses the resulting Keldysh coupled cluster theory in detail, describes the implementation of the equations required for Keldysh CCSD calculations of finite-temperature dynamics, and applies the method to a Hubbard model with a Peierls phase and an ab initio model of warm-dense silicon subject to an ultrafast XUV pulse.

Significance. If the algebraic construction of the time-dependent amplitudes on the contour is formally correct and the truncation is applied consistently, this provides a new computational framework for simulating driven finite-temperature quantum many-body systems. The work directly extends a previously published equilibrium method and includes concrete applications to model systems, which is a strength for a methods paper in this area.

minor comments (1)
  1. The abstract contains a typographical error ('ultrafact XUV pulse' should read 'ultrafast XUV pulse').

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The report provides a concise summary of the work but does not raise any specific major comments requiring point-by-point rebuttal.

Circularity Check

0 steps flagged

Minor self-citation to prior equilibrium CC method; new Keldysh derivation is independent

full rationale

The paper extends the authors' own 2018 finite-temperature CC implementation to the Keldysh contour for driven systems. This is a standard self-citation for continuity of code/method, but the central load-bearing content is the algebraic construction of the time-dependent CCSD amplitudes on the contour, which is presented as a formal extension without any reduction of predictions to fitted parameters, self-definitional loops, or load-bearing uniqueness theorems imported from the same authors. No equations are shown to collapse to inputs by construction, and the truncation accuracy is treated as an empirical question rather than a formal premise.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5640 in / 1084 out tokens · 18850 ms · 2026-05-24T15:13:06.695848+00:00 · methodology

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Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · 2 internal anchors

  1. [1]

    The quantum many-body problem for a realistic material

    work page

  2. [2]

    The statistical mixture of many-body states im- plied by a finite temperature

    work page

  3. [3]

    The generically non-thermal, time-dependent dis- tribution of states generated by the perturbation

    work page

  4. [4]

    For simplicity in this study we will neglect the descrip- tion of electron-nuclear coupling

    The coupling between electronic and nuclear de- grees of freedom. For simplicity in this study we will neglect the descrip- tion of electron-nuclear coupling. However, points 1.-

    work page

  5. [5]

    already pose major challenges for an electronic treat- ment. In ab initio real time dynamics, there have been applications of real-time density functional theory (DFT) to both materials as well as molecules 6–15, while time- dependent ab initio wavefunction methods such as time- dependent configuration interaction 16–18 and coupled cluster methods have bee...

    work page

  6. [6]

    In Section II A we describe the working equations of the FT-CC method presented in Ref. 49

    work page

  7. [7]

    In Section II B we describe some important aspects of the Keldysh formalism

    work page

  8. [8]

    In Section II C we generalize the imaginary time FT-CC to out of equilibrium systems using the Keldysh formalism

    work page

  9. [9]

    Time-dependent coupled cluster theory on the Keldysh contour for non-equilibrium systems

    In the final Sections (II D-II F) we discuss several important properties and extensions of the theory. arXiv:1907.11695v1 [cond-mat.stat-mech] 26 Jul 2019 2 A. Finite temperature coupled cluster In Ref. 49 we presented a finite temperature coupled cluster theory which provides an ansatz for the correla- tion contribution to the grand potential Ω such that ...

    work page internal anchor Pith review Pith/arXiv arXiv 1907

  10. [10]

    Practical improvements necessary to decrease the computational cost and memory requirements

    work page

  11. [11]

    Appendix A: Introduction to the Keldysh formalism We imagine a system in thermal equilibrium at time t = 0

    Modifications of the theory to lessen the impact of particle-number symmetry breaking and other unphysical effects at longer times ACKNOWLEDGMENTS This work is supported by the US Department of En- ergy, Office of Science, via grant number SC0018140. Appendix A: Introduction to the Keldysh formalism We imagine a system in thermal equilibrium at time t = 0. At...

    work page

  12. [12]

    (E1) The corresponding grand potential is denoted by Ω, and and we will can do perturbation theory in powers of f: Ω = Ω[0] + Ω[1] + Ω[2] +

    Preliminaries We will consider a 1-particle Hamiltonian of the form H =h0 +f(t). (E1) The corresponding grand potential is denoted by Ω, and and we will can do perturbation theory in powers of f: Ω = Ω[0] + Ω[1] + Ω[2] +... (E2) The value of some observable at time t is given by ⟨O⟩ (t1) = ∂ ∂α Ω[α] ⏐⏐⏐⏐ α=0 (E3) where H[α] =h0 +f(t) +αδC(t−t1)O (E4) as d...

    work page

  13. [13]

    CCS is complete to all orders in perturbation the- ory and is therefore exact

    work page

  14. [14]

    LCCS is complete only to 3rd order in perturbation theory

    work page

  15. [15]

    The 4th order terms missing in LCCS are those arising from the 3rd skeleton in Figure 6(c)

    work page

  16. [16]

    Second order perturbation theory The free energy at second order can be expressed as a contour integral Ω[2] = i β ∫ C dt ∑ ia fiasa i (t)[1] (E5) 14 a) b) c) FIG. 6. Hugenholtz skeletons contributing to (a) second or- der, (b) 3rd order, and (c) 4th order in perturbation theory. = + = + + a) b) FIG. 7. Hugenholtz skeletons representing the iteration in (...

    work page

  17. [17]

    Third order perturbation theory We can perform the same analysis for the 3rd order terms. We must use the second order amplitudes sa i (t)[2] =−i ∫ C(t) C(0) dt′ei∆a i (t′−t)  ∑ b fab(α)sb i(t′)[1]− ∑ j fji(α)sa j (t′)[1]   (E15) =− ∫ C(t) C(0) dt′ei∆a i (t′−t) [ ∑ b (1−na)fab(α) ∫ C(t′) C(0) dt′′ei∆b i(t′′−t′)fbi(α)ni(1−nb) − ∑ j nifji(α) ∫ C(t′) C(0...

    work page

  18. [18]

    Some features of 4th order perturbation theory 4th order perturbation theory becomes quite tedious, but it is sufficient for our purposes to consider the pair of 4th order diagrams that arise from the quadratic term in the CCS equations. These diagrams correspond to the skeleton listed 3rd in Figure 6(c): Ω[4]∗(α) = i β ∫ C dt ∑ ia fia(α)sa i (t)[3∗] = i β ...

    work page

  19. [19]

    Numerical results From this discussion and the discussion given in Sec- tion II F, we conclude several things about particle num- ber conservation in the 1-particle problem:

    work page

  20. [20]

    nth order perturbation theory should conserve par- ticle number

    work page

  21. [21]

    For 2nd and 3rd order perturbation theory, each diagram individually conserves the particle number

    work page

  22. [22]

    But this is not true of 4th order perturbation theory where the (∗) terms (and therefore the remaining terms also) do not conserve particle number

    work page

  23. [23]

    LCCS does not conserve particle number

    work page

  24. [24]

    Ab initio quantum dynamics using coupled-cluster

    CCS does conserve particle number by virtue of being exact: it is complete to all orders in pertur- bation theory. We will now verify numerically these statements. Be- cause there is error in the particle number due to the nu- merical integration, we will say that a given theory does not conserve particle number if the quantity changes in time and this ch...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0953-8984/21/6/064224 2000