Pith. sign in

REVIEW 1 major objections 4 minor 58 references

Path Integral Formalism for Quantum Open Systems

T0 review · 1 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A detailed coherent-state path-integral derivation shows that bosonic and fermionic open systems share one exponential-quadratic influence functional, with a generating functional method that recovers environment observables from system…

desk verdict Solid, self-contained derivation of open-system path integrals, with a correctable sign error in one intermediate imaginary-time result. read the letter →

arxiv 2506.08802 v1 pith:M4FD6UEG submitted 2025-06-10 quant-ph cond-mat.stat-mech

Ruofan Chen This is my paper
classification quant-phcond-mat.stat-mech MSC 81S4081V7082C10 PACS 03.65.Yz05.30.-d
keywords pathintegralquantumopensystemscoherentstatesinfluencefunctionalKeldyshcontourKadanoffgeneratingGreen'sfunctions
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to show that quantum open systems with a noninteracting environment and linear system–bath coupling admit a single, fully detailed path-integral description, whether the bath particles are bosons or fermions. Using coherent states, it derives the influence functional—the object that records everything the bath does to the system after the bath is integrated out—on three time contours: imaginary time for equilibrium, Keldysh for nonequilibrium dynamics, and Kadanoff for equilibrium initial states with real-time evolution. It then develops a generating functional technique that extracts environment information, such as coupling energy, heat and particle currents, and the dressed environment Green's functions, from system correlation functions alone. This matters because it turns the bath into a closed-form kernel and supplies the analytical starting point for numerical tensor-network evaluations of open-system dynamics.

What carries the argument

The machinery is the coherent-state Gaussian integral. Coherent states are eigenstates of annihilation operators (for bosons they are parameterized by complex numbers; for fermions by Grassmann variables), and the identity resolution built from them turns the trace over the bath into a Gaussian integral. With a Trotter-Suzuki splitting of the short-time propagator, the bath integral becomes $\int D[\bar\varphi\varphi]\, e^{-\bar\varphi S\varphi + \text{linear sources}}$, whose closed form is $\propto e^{\text{source} \cdot S^{-1}\cdot \text{source}}$. The load-bearing identity is that the inverse of the discretized contour matrix $S$ equals, up to standard factors, the free-environment contour Green's function, so $S^{-1}$ need not be computed: it is the familiar $D_0$ or $G_0$. This is what converts the environment into an exponential-quadratic influence functional and what makes the generating functional method work.

What would settle it

Take a small open system, for example a two-level system coupled to one or two bosonic modes, and compute its equilibrium partition function and two-time correlation function exactly by numerical diagonalization of the full Hamiltonian over a grid in the coupling strength and inverse temperature; compare these to the paper's influence-functional expressions (2.25) and (3.10). If the closed-form results match the exact numbers at finite coupling and temperature, the central derivation is supported; a mismatch would show the influence functional or the contour assignments are wrong. The same check applies to the fermion expressions (5.22) and (5.67) with a single-level system hybridized to one or two fermionic modes.

Watch

Extended reading notes

Core claim

The paper's central claim is that integrating out the environment in a bosonic open system with linear coupling, and in the corresponding fermionic hybridization model, yields an influence functional of the exponential-quadratic form $I[s]=\exp\!\big(-\int_{\mathcal{C}}\!dt'\!\int_{\mathcal{C}}\!dt''\, s(t')\Lambda(t',t'')s(t'')\big)$ for bosons and $I[\bar a,a]=\exp\!\big(-\int_{\mathcal{C}}\!dt'\!\int_{\mathcal{C}}\!dt''\, \bar a(t')\Delta(t',t'')a(t'')\big)$ for fermions, where the kernels $\Lambda$ and $\Delta$ are built from the free-environment contour Green's function and the spectral function. This form holds identically on the imaginary-time axis, the Keldysh contour, and the Kadanoff contour, with the choice of contour only changing the meaning of the kernel. The generating functional part claims that derivatives of the partition function with respect to inserted source terms return environment observables: coupling energy, heat and particle currents, and the full environment Green's function expressed through a T-matrix relation in terms of the bare Green's function and the system correlation function.

Load-bearing premise

The derivation assumes the environment has no internal interactions and the system couples to it linearly, so the environment path integral is a Gaussian integral that can be done in closed form; if those assumptions fail, the influence functional is no longer a simple exponential quadratic in the system variable.

Editorial extensions

If this is right

  • For any open system whose bath is noninteracting and linearly coupled, the environment can be removed analytically, leaving an effective system-only path integral whose only bath memory is the kernel $\Lambda$ or $\Delta$.
  • The same formula works on the imaginary-time, Keldysh, and Kadanoff contours, so equilibrium thermodynamics and out-of-equilibrium dynamics are treated by one derivation rather than parallel ones.
  • Environment observables—coupling energy, heat current, particle current, and the dressed environment Green's function—can be obtained from system correlation functions through the generating functional, without explicitly simulating the bath.
  • The fermion derivation shows that Grassmann sign factors are encoded partly in the Green's function and partly, on the Keldysh contour, in an explicit sign factor $P_{t't''}$; the final influence functional is still the same exponential-quadratic form.
  • The closed-form influence functionals give numerical tensor-network algorithms a compact analytical input, since the bath is fully characterized by the spectral function.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same Gaussian-integral strategy should extend to other contours, such as those used for out-of-time-ordered correlation functions, as long as the system–bath coupling remains linear.
  • The T-matrix relations for environment Green's functions imply a Dyson-type integral equation connecting system and bath correlations; inverting it could let an experimenter reconstruct the bath spectral function from measured system correlation functions, a step the paper does not take.
  • For multiple baths at different temperatures, the additivity of the influence functional suggests heat and particle currents through each bath are governed separately by the same system correlation function, which could be exploited in thermoelectric transport calculations.
  • Treating bath self-interactions as a perturbation around the Gaussian influence functional is the natural next test of how much of this structure survives beyond the noninteracting-bath assumption.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. This paper presents a systematic derivation of the coherent-state path integral formalism for quantum open systems, covering both bosonic (Caldeira-Leggett) and fermionic (Toulouse/Anderson-type) environments. The author derives influence functionals on the imaginary-time axis, the Keldysh contour, and the Kadanoff contour, and develops a generating-functional technique for extracting environment information—coupling energies, heat/particle currents, and dressed environment Green's functions—from system correlation functions. The presentation is largely self-contained, with explicit matrices, Gaussian integrals, and Green's function identities, and it is oriented toward tensor-network methods such as TEMPO and GTEMPO.

Significance. If the derivations are correct, the paper provides a useful single reference for a class of path integral results that are currently scattered across the literature. Its strengths are the explicit evaluation of the inverse matrices and the consistent representation of influence functionals in terms of standard contour-ordered Green's functions, as well as the unified treatment of boson and fermion cases. The generating-functional relations for environment Green's functions and currents are of practical value for numerical tensor-network approaches. The paper does not claim new physics or numerical benchmarks, but as a foundational technical derivation it could be a useful resource. The central derivations are standard and reproduce known Feynman-Vernon and hybridization-function results, with no obvious circularity in the use of references.

major comments (1)
  1. [Sec. 2.1, Eq. (2.30)] The sign in the exponent of Eq. (2.30) is incorrect. Starting from Eq. (2.25) with Λ(τ′,τ′′)=V²D0(τ′,τ′′), and using Eq. (2.23), the double integral over the full square satisfies ∫∫ s(τ′)D0(τ′−τ′′)s(τ′′) dτ′ dτ′′ = −∫_{τ′>τ′′} s(τ′)α(τ′−τ′′)s(τ′′) dτ′ dτ′′, where α is defined in Eq. (2.31). Therefore I[s] = exp[+∫_{τ′>τ′′} s(τ′)α(τ′−τ′′)s(τ′′)], not exp[−∫_{τ′>τ′′} s(τ′)α(τ′−τ′′)s(τ′′)] as printed. A static check confirms this: for constant s the shifted-oscillator partition function gives I = e^{+βV²s²/ω0}, whereas Eq. (2.30) gives the inverse. This error does not appear to propagate into the final influence functionals (2.25), (2.66), (2.88), (5.22), (5.46), and (5.64), but Eq. (2.30) is a claimed intermediate result and must be corrected before the derivation can be used as a reliable reference.
minor comments (4)
  1. [Sec. 3.1, after Eq. (3.9)] The paper explicitly states that non-time-ordered correlation functions require a reconstruction whose details are not discussed. Since the section is entitled "System Correlation Functions" and the abstract advertises a detailed derivation, this omission should be either filled or explicitly acknowledged in the abstract and conclusion so that the scope of the claim is clear.
  2. [Sec. 5.2, Eqs. (5.30), (5.40), (5.53)] There are several typographical errors that should be fixed: in Eq. (5.30) the first factor in the exponent is missing the bar on a(t′); in Eq. (5.40) the Green's function is written as G0(ε;t′,t′) but should be G0(ε;t′,t′′); and the rendered text of Eq. (5.53) contains a garbled matrix element that should read ⟨a−_{N−1}|e^{iĤSδt}|a−_N⟩.
  3. [Throughout] The term "Kadanoffcontour" appears without a space in the title and in several section headings; the text also contains minor typos such as "succeedst" and "suceed." A careful proofreading pass is recommended.
  4. [Sec. 5.2, Eq. (5.47)] The extra sign factor P_{t′t′′} in the Keldysh hybridization function is important but is only introduced in one sentence; a short explicit definition of P_{t′t′′} for the three contour branches would improve readability and reduce the risk of sign errors in applications.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the path-integral derivation is self-contained and reproduces standard influence functionals from coherent-state Gaussian integrals.

full rationale

The paper's central derivation starts from coherent-state identities (1.9)-(1.12), Trotter-Suzuki decomposition (2.10), and Gaussian integration (2.18)/(5.17), and obtains the influence functionals (2.25), (2.66), (2.88), (5.22), (5.46), and (5.64) by direct manipulation of the free-environment Green's functions. The generating functional method (Sections 4 and 6) differentiates source-modified influence functionals to derive T-matrix relations such as (4.18), (4.34), (6.17), and (6.21); these are derived relations, not inputs. Self-cited works (Refs. [29,30,41,42,50,51]) are cited for context, notation, or numerical application (e.g., GTEMPO) and are not used as the premise of any derivation. No parameters are fitted to the target results, and no 'prediction' is equivalent by construction to an input. The reader-reported sign issue in Eq. (2.30) would be a mathematical error, not a circularity, and it does not affect the key influence functionals. Thus no circularity is found.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data; model parameters such as ω_k, V_k, ε_k, and spectral functions are inputs from the Hamiltonian. The derivations rely on standard mathematical assumptions (Trotter-Suzuki, coherent-state completeness, Gaussian integration) and on the physical domain assumption of linear coupling to a noninteracting bath. No new particles, forces, or entities are introduced.

assumptions (4)
  • standard math First-order Trotter-Suzuki decomposition e^{-δτ Ĥ} ≈ e^{-δτ V ŝ b†} e^{-δτ Ĥ_S} e^{-δτ Ĥ_E} e^{-δτ V ŝ b} is valid in the continuum limit.
    Invoked in Eq. (2.10) for bosons and Eq. (5.9) for fermions; standard for path integral discretization.
  • domain assumption The environment is noninteracting and linearly coupled to the system, so the coherent-state path integral is Gaussian and can be integrated exactly.
    Hamiltonians (2.2) and (5.1); without this, the influence functional would not reduce to the quadratic form in Eqs. (2.25) and (5.22).
  • domain assumption Initial density matrix for Keldysh dynamics is a product state of the system and a thermal environment.
    Assumed in Eq. (2.36); needed for the factorized expression Z(t_f) = Z_E^{(0)} Σ_s K[s] I[s] in Eq. (2.46).
  • standard math Coherent-state closure relations and Gaussian integral formulas for bosons (Eqs. (1.9), (1.12)) and fermions (Eqs. (1.23), (1.26)).
    Used throughout Sections 2 and 5 to evaluate traces and to integrate out the environment degrees of freedom.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Path Integral Formalism for Quantum Open Systems." pith.science (2026). https://pith.science/paper/M4FD6UEG

@misc{pith2026250608802,
  author       = {Pith},
  title        = {Pith review of: Path Integral Formalism for Quantum Open Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/M4FD6UEG}},
  note         = {Machine review of arXiv:2506.08802}
}
read the original abstract

This article provides a detailed derivation of the path integral formalism for both boson and fermion quantum open systems using coherent states. The formalism on the imaginary-time axis, Keldysh contour, and Kadanoff contour are given. The corresponding generating functional technique, which can be used to retrieve the environment information from the system correlation function, is also discussed.

Figures

Figures reproduced from arXiv: 2506.08802 by the authors.

Figure 1
Figure 1. The Keldysh contour C = C1 ∪ C2, where the upper part is the forward (1st) branch and lower part is the backward (2nd) branch. Now we define the contour time interval dt for which dt = δt if it is on the forward branch and dt = −δt on the backward branch. Denote s = (s + 0 , s + 1 , . . . , s + N = s − N , . . . , s − 1 , s − 0 ), the influence functional (2.48) can be written as I[s] = 1 Z (0) E Z D[φφ¯ ]e − P2N j,… view at source ↗
Figure 2
Figure 2. The L-shaped Kadanoff-Baym contour C = C1 ∪ C2 ∪ C3, where the arrows indicate the ordering of the contour. Splitting tf = Nδt, β = Mδτ with N → ∞, M → ∞, the partition function can be written as Z(β, tf) = X s Z D[φφ¯ ]e −φφ¯ ⟨s ∼ Mφ ∼ M|e −δτHˆ |s ∼ M−1φ ∼ M−1 ⟩· · · ⟨s ∼ 1 φ ∼ 1 |e −δτHˆ |s ∼ 0 φ ∼ 0 ⟩ × ⟨s − 0 φ − 0 |e iHˆ δt |s − 1 φ − 1 ⟩· · · ⟨s − N−1φ − N−1 |e iHˆ δt |s − Nφ − N ⟩ × ⟨s + Nφ + N |e −iHˆ δt |s… view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

58 extracted references · 29 canonical work pages

  1. [1]

    R. P. Feynman, Space-time approach to non-relativistic quantum mechanics, Reviews of Modern Physics 20 (2) (1948) 367–387.doi:10.1103/revmodphys.20.367. URLhttps://doi.org/10.1103/revmodphys.20.367

    doi:10.1103/revmodphys.20.367 1948

  2. [2]

    R. P. Feynman, F. L. Vernon, The theory of a general quantum system interacting with a linear dissipative system, Annals of Physics 24 (1963) 118–173.doi:10.1016/0003-4916(63)90068-X. URLhttps://doi.org/10.1016/0003-4916(63)90068-X

    doi:10.1016/0003-4916(63)90068-x 1963

  3. [3]

    R. P. Feynman, A. R. Hibbs, Quantum Mechanics and Path Integrals, Mc Graw-Hill, New York, 1965

    work page 1965

  4. [4]

    R. P. Feynman, Statistical Mechanics, Benjamin, Reading, Mass., 1972

    work page 1972

  5. [5]

    L. S. Schulman, Techniques and Applications of Path Integration, Wiley, 1981

    work page 1981

  6. [6]

    Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, 2006

    H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific, 2006

    work page 2006

  7. [7]

    Weiss, Quantum Dissipative Systems, World Scientific, Singapore, 1993.doi:10.1142/8334

    U. Weiss, Quantum Dissipative Systems, World Scientific, Singapore, 1993.doi:10.1142/8334. URLhttps://doi.org/10.1142/8334

    doi:10.1142/8334 1993

  8. [8]

    A. O. Caldeira, A. J. Leggett, Influence of dissipation on quantum tunneling in macroscopic systems, Physical Review Letters 46 (4) (1981) 211–214.doi:10.1103/physrevlett.46.211. URLhttps://doi.org/10.1103/physrevlett.46.211 35

    doi:10.1103/physrevlett.46.211 1981

Show all 58 references
  1. [9]

    A. O. Caldeira, A. J. Leggett, Quantum tunnelling in a dissipative system, Annals of Physics 149 (2) (1983) 374–456.doi:10.1016/0003-4916(83)90202-6. URLhttps://doi.org/10.1016/0003-4916(83)90202-6

    1983 doi

  2. [10]

    A. O. Caldeira, A. J. Leggett, Path integral approach to quantum brownian motion, Physica A 121 (3) (1983) 587–616.doi:10.1016/0378-4371(83)90013-4. URLhttps://doi.org/10.1016/0378-4371(83)90013-4

    1983 doi

  3. [11]

    A. J. Bray, M. A. Moore, Influence of dissipation on quantum coherence, Physical Review Letters 49 (21) (1982) 1545–1549.doi:10.1103/physrevlett.49.1545. URLhttp://dx.doi.org/10.1103/PhysRevLett.49.1545

    1982 doi

  4. [12]

    A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, W. Zwerger, Dynamics of the dissipative two-state system, Reviews of Modern Physics 59 (1) (1987) 1–85.doi:10.1103/revmodphys.59.1. URLhttps://doi.org/10.1103/revmodphys.59.1

    1987 doi

  5. [13]

    D. E. Makarov, N. Makri, Path integrals for dissipative systems by tensor multiplication. condensed phase quantum dynamics for arbitrarily long time, Chemical Physics Letters 221 (5-6) (1994) 482.doi:10.1016/ 0009-2614(94)00275-4. URLhttps://doi.org/10.1016/0009-2614(94)00275-4

    1994 doi

  6. [14]

    Makri, Numerical path integral techniques for long time dynamics of quantum dissipative systems, Journal of Mathematical Physics 36 (5) (1995) 2430.doi:10.1063/1.531046

    N. Makri, Numerical path integral techniques for long time dynamics of quantum dissipative systems, Journal of Mathematical Physics 36 (5) (1995) 2430.doi:10.1063/1.531046. URLhttps://doi.org/10.1063/1.531046

    1995 doi

  7. [15]

    D. E. Makarov, N. Makri, Control of dissipative tunneling dynamics by continuous wave electromagnetic fields: Localization and large-amplitude coherent motion, Physical Review E 52 (6) (1995) 5863.doi:10.1103/ physreve.52.5863. URLhttps://doi.org/10.1103/physreve.52.5863

    1995 doi

  8. [16]

    D. E. Makarov, N. Makri, Stochastic resonance and nonlinear response in double-quantum-well structures, Phys- ical Review B 52 (4) (1995) R2257.doi:10.1103/physrevb.52.r2257. URLhttps://doi.org/10.1103/physrevb.52.r2257

    1995 doi

  9. [17]

    Thorwart, P

    M. Thorwart, P. Reimann, P. Hänggi, Iterative algorithm versus analytic solutions of the parametrically driven dis- sipative quantum harmonic oscillator, Physical Review E 62 (4) (2000) 5808–5817.doi:10.1103/physreve. 62.5808. URLhttp://dx.doi.org/10.1103/PhysRevE.62.5808

    2000 doi

  10. [18]

    J. Shao, N. Makri, Iterative path integral formulation of equilibrium correlation functions for quantum dissipative systems, The Journal of Chemical Physics 116 (2) (2002) 507–514.doi:10.1063/1.1423936. URLhttps://doi.org/10.1063/1.1423936

    2002 doi

  11. [19]

    K. Dong, N. Makri, Quantum stochastic resonance in the strong-field limit, Physical Review A 70 (4) (2004) 042101.doi:10.1103/physreva.70.042101. URLhttp://dx.doi.org/10.1103/PhysRevA.70.042101

    2004 doi

  12. [20]

    Nalbach, M

    P. Nalbach, M. Thorwart, Landau-zener transitions in a dissipative environment: Numerically exact results, Physical Review Letters 103 (22) (2009) 220401.doi:10.1103/physrevlett.103.220401. URLhttps://doi.org/10.1103/physrevlett.103.220401 36

    2009 doi

  13. [21]

    Strathearn, P

    A. Strathearn, P. Kirton, D. Kilda, J. Keeling, B. W. Lovett, Efficient non-markovian quantum dynamics using time-evolving matrix product operators, Nature Communications 9 (1) (2018) 3322.doi:10.1038/ s41467-018-05617-3. URLhttps://doi.org/10.1038/s41467-018-05617-3

    2018 doi

  14. [22]

    Strathearn, Modelling Non-Markovian Quantum Systems Using Tensor Networks, Springer International Publishing, Cham, Switzerland, 2020.doi:10.1007/978-3-030-54975-6

    A. Strathearn, Modelling Non-Markovian Quantum Systems Using Tensor Networks, Springer International Publishing, Cham, Switzerland, 2020.doi:10.1007/978-3-030-54975-6. URLhttp://doi.org/10.1007/978-3-030-54975-6

    2020 doi

  15. [23]

    M. R. Jørgensen, F. A. Pollock, Exploiting the causal tensor network structure of quantum processes to ef- ficiently simulate non-markovian path integrals, Physical Review Letters 123 (24) (2019) 240602.doi: 10.1103/physrevlett.123.240602. URLhttp://dx.doi.org/10.1103/PhysRevL...

    2019 doi

  16. [24]

    G. E. Fux, E. P. Butler, P. R. Eastham, B. W. Lovett, J. Keeling, Efficient exploration of hamiltonian parameter space for optimal control of non-markovian open quantum systems, Physical Review Letters 126 (20) (2021) 200401.doi:10.1103/physrevlett.126.200401. URLhttp://dx.doi...

    2021 doi

  17. [25]

    Popovic, M

    M. Popovic, M. T. Mitchison, A. Strathearn, B. W. Lovett, J. Goold, P. R. Eastham, Quantum heat statistics with time-evolving matrix product operators, PRX Quantum 2 (2) (2021) 020338.doi:10.1103/prxquantum.2. 020338. URLhttp://dx.doi.org/10.1103/PRXQuantum.2.020338

    2021 doi

  18. [26]

    Gribben, A

    D. Gribben, A. Strathearn, G. E. Fux, P. Kirton, B. W. Lovett, Using the environment to understand non- markovian open quantum systems, Quantum 6 (2021) 847.doi:10.22331/q-2022-10-25-847. URLhttps://doi.org/10.22331/q-2022-10-25-847

    2021 doi

  19. [27]

    Gribben, D

    D. Gribben, D. M. Rouse, J. Iles-Smith, A. Strathearn, H. Maguire, P. Kirton, A. Nazir, E. M. Gauger, B. W. Lovett, Exact dynamics of nonadditive environments in non-markovian open quantum systems, PRX Quantum 3 (1) (2022) 010321.doi:10.1103/prxquantum.3.010321. URLhttp://dx.d...

    2022 doi

  20. [28]

    Otterpohl, P

    F. Otterpohl, P. Nalbach, M. Thorwart, Hidden phase of the spin-boson model, Physical Review Letters 129 (12) (2022) 120406.doi:10.1103/physrevlett.129.120406. URLhttp://dx.doi.org/10.1103/PhysRevLett.129.120406

    2022 doi

  21. [29]

    R. Chen, X. Xu, Non-markovian effects in stochastic resonance in a two level system, The European Physical Journal Plus 138 (2023) 194.doi:10.1140/epjp/s13360-023-03835-3. URLhttps://doi.org/10.1140/epjp/s13360-023-03835-3

    2023 doi

  22. [30]

    Chen, Heat current in non-markovian open systems, New Journal of Physics 25 (2023) 033035.doi:10

    R. Chen, Heat current in non-markovian open systems, New Journal of Physics 25 (2023) 033035.doi:10. 1088/1367-2630/acc60a. URLhttps://doi.org/10.1088/1367-2630/acc60a

    2023 doi

  23. [31]

    G. E. Fux, D. Kilda, B. W. Lovett, J. Keeling, Thermalization of a spin chain strongly coupled to its environment, Physical Review Research 5 (2023) 033078.doi:10.1103/PhysRevResearch.5.033078. URLhttps://doi.org/10.1103/PhysRevResearch.5.033078 37

    2023 doi

  24. [32]

    Link, H.-H

    V . Link, H.-H. Tu, W. T. Strunz, Open quantum system dynamics from infinite tensor network contraction, Physical Review Letters 132 (20) (2024) 200403.doi:10.1103/physrevlett.132.200403. URLhttp://dx.doi.org/10.1103/PhysRevLett.132.200403

    2024 doi

  25. [33]

    Cygorek, J

    M. Cygorek, J. Keeling, B. W. Lovett, E. M. Gauger, Sublinear scaling in non-markovian open quantum systems simulations, Physical Review X 14 (1) (2024) 011010.doi:10.1103/physrevx.14.011010. URLhttp://dx.doi.org/10.1103/PhysRevX.14.011010

    2024 doi

  26. [34]

    G. D. Mahan, Many-Particle Physics, Springer; 3nd edition, 2000

    2000

  27. [35]

    L. D. Landau, E. M. Lifshitz, Course of Theoretical Physics V olume 3: Quantum Mechanics, Elsevier, 1965

    1965

  28. [36]

    J. W. Negele, H. Orland, Quantum Many-Particle Systems, Westview Press, 1998

    1998

  29. [37]

    Georges, G

    A. Georges, G. Kotliar, W. Krauth, M. J. Rozenberg, Dynamical mean-field theory of strongly correlated fermion systems and the limit of infinite dimensions, Reviews of Modern Physics 68 (1) (1996) 13–125.doi:10.1103/ revmodphys.68.13. URLhttps://doi.org/10.1103/revmodphys.68.13

    1996 doi

  30. [38]

    E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, P. Werner, Continuous-time monte carlo methods for quantum impurity models, Reviews of Modern Physics 83 (2) (2011) 349–404.doi:10.1103/ revmodphys.83.349. URLhttps://doi.org/10.1103/revmodphys.83.349

    2011 doi

  31. [39]

    N. Ng, G. Park, A. J. Millis, G. K.-L. Chan, D. R. Reichman, Real-time evolution of anderson impurity models via tensor network influence functionals, Physical Review B 107 (12) (2023) 125103.doi:10.1103/physrevb. 107.125103. URLhttp://dx.doi.org/10.1103/PhysRevB.107.125103

    2023 doi

  32. [40]

    Thoenniss, M

    J. Thoenniss, M. Sonner, A. Lerose, D. A. Abanin, Efficient method for quantum impurity problems out of equilibrium, Physical Review B 107 (20) (2023) L201115.doi:10.1103/physrevb.107.l201115. URLhttp://dx.doi.org/10.1103/PhysRevB.107.L201115

    2023 doi

  33. [41]

    R. Chen, X. Xu, C. Guo, Grassmann time-evolving matrix product operators for quantum impurity models, Physical Review B 109 (4) (2024) 045140.doi:10.1103/physrevb.109.045140. URLhttp://dx.doi.org/10.1103/PhysRevB.109.045140

    2024 doi

  34. [42]

    X. Xu, C. Guo, R. Chen, Grassmann time-evolving matrix product operators: An efficient numerical approach for fermionic path integral simulations, The Journal of Chemical Physics 161 (15) (2024) 151001.doi:10. 1063/5.0226167. URLhttp://dx.doi.org/10.1063/5.0226167

    2024 doi

  35. [43]

    Y .-F. Chiu, A. Strathearn, J. Keeling, Numerical evaluation and robustness of the quantum mean-force gibbs state, Physical Review A 106 (1) (2022) 012204.doi:10.1103/physreva.106.012204. URLhttp://dx.doi.org/10.1103/PhysRevA.106.012204

    2022 doi

  36. [44]

    H. F. Trotter, On the product of semi-groups of operators, Proceedings of the American Mathematical Society 10 (4) (1959) 545–545.doi:10.1090/s0002-9939-1959-0108732-6. URLhttps://doi.org/10.1090/s0002-9939-1959-0108732-6 38

    1959 doi

  37. [45]

    M. Suzuki, Generalized trotter’s formula and systematic approximants of exponential operators and inner deriva- tions with applications to many-body problems, Communications in Mathematical Physics 51 (2) (1976) 183– 190.doi:10.1007/bf01609348. URLhttps://doi.org/10.1007/bf01609348

    1976 doi

  38. [46]

    L. V . Keldysh, Diagram technique for non-equilibrium processes, Soviet Physics JETP 20 (1965) 1018.doi: 10.1142/9789811279461_0007. URLhttps://doi.org/10.1142/9789811279461_0007

    1965 doi

  39. [47]

    E. M. Lifshitz, L. P. Pitaevskii, Course of Theoretical Physics V olume 10: Physical Kinetics, Elsevier, 1981

    1981

  40. [48]

    Kamenev, A

    A. Kamenev, A. Levchenko, Keldysh technique and non-linearσ-model: Basic principles and applications, Advances in Physics 58 (3) (2009) 197–319.doi:10.1080/00018730902850504. URLhttps://doi.org/10.1080/00018730902850504

    2009 doi

  41. [49]

    J.-S. Wang, B. K. Agarwalla, H. Li, J. Thingna, Nonequilibrium green’s function method for quantum thermal transport, Frontiers of Physics 9 (6) (2013) 673–697.doi:10.1007/s11467-013-0340-x. URLhttps://doi.org/10.1007/s11467-013-0340-x

    2013 doi

  42. [50]

    R. Chen, Path integral formalism of open quantum systems with non-diagonal system-bath coupling, Communi- cations in Theoretical Physics 76 (11) (2024) 115701.doi:10.1088/1572-9494/ad696b. URLhttp://dx.doi.org/10.1088/1572-9494/ad696b

    2024 doi

  43. [51]

    R. Chen, C. Guo, Solving equilibrium quantum impurity problems on the l-shaped kadanoff-baym contour, Physical Review B 110 (16) (2024) 165114.doi:10.1103/physrevb.110.165114. URLhttp://dx.doi.org/10.1103/PhysRevB.110.165114

    2024 doi

  44. [52]

    L. P. Kadanoff, G. Baym, Quantum Statistical Mechnics, W. A. Benjamin, New York, 1962

    1962

  45. [53]

    H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Reviews of Modern Physics 86 (2) (2014) 779–837.doi:10.1103/revmodphys.86.779. URLhttps://doi.org/10.1103/revmodphys.86.779

    2014 doi

  46. [54]

    P. W. Anderson, Localized magnetic states in metals, Physical Review 124 (1) (1961) 41–53.doi:10.1103/ physrev.124.41. URLhttps://doi.org/10.1103/physrev.124.41

    1961 doi

  47. [55]

    Kanamori, Electron correlation and ferromagnetism of transition metals, Progress of Theoretical Physics 30 (3) (1963) 275–289.doi:10.1143/ptp.30.275

    J. Kanamori, Electron correlation and ferromagnetism of transition metals, Progress of Theoretical Physics 30 (3) (1963) 275–289.doi:10.1143/ptp.30.275. URLhttp://dx.doi.org/10.1143/PTP.30.275

    1963 doi

  48. [56]

    Georges, L

    A. Georges, L. de’ Medici, J. Mravlje, Strong correlations from hund’s coupling, Annual Review of Condensed Matter Physics 4 (1) (2013) 137–178.doi:10.1146/annurev-conmatphys-020911-125045. URLhttps://doi.org/10.1146/annurev-conmatphys-020911-125045

    2013 doi

  49. [57]

    Bertrand, S

    C. Bertrand, S. Florens, O. Parcollet, X. Waintal, Reconstructing nonequilibrium regimes of quantum many- body systems from the analytical structure of perturbative expansions, Physical Review X 9 (4) (2019) 041008. doi:10.1103/physrevx.9.041008. URLhttp://dx.doi.org/10.1103/P...

    2019 doi

  50. [58]

    J. Tuziemski, Out-of-time-ordered correlation functions in open systems: a feynman-vernon influence functional approach, Physical Review A 100 (6) (2019) 062106.doi:10.1103/physreva.100.062106. URLhttp://dx.doi.org/10.1103/PhysRevA.100.062106 40

    2019 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.