pith. sign in

arxiv: 2606.31550 · v1 · pith:FESSUBITnew · submitted 2026-06-30 · ❄️ cond-mat.mes-hall · cond-mat.str-el· cond-mat.supr-con

Floquet Majorana flat bands and emergent Cooper pair symmetries in p-wave magnet-superconductor heterostructure

Subhendu Kumar Patra , Gaurab Kumar Dash , Manisha Thakurathi This is my paper

Pith reviewed 2026-07-01 04:03 UTC · model grok-4.3

classification ❄️ cond-mat.mes-hall cond-mat.str-elcond-mat.supr-con
keywords Floquet topological superconductivityMajorana flat bandsp-wave magnetCooper pair symmetriesnonequilibrium pairingheterostructureperiodic drivingnodal topological phases
0
0 comments X

The pith

Periodic driving in a p-wave magnet-superconductor heterostructure generates multiple Majorana flat bands and a new nonequilibrium class of Cooper pair correlations.

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

The paper establishes that periodic driving applied to the interface between a p-wave magnet and an s-wave superconductor produces Floquet topological phases with both zero and pi Majorana flat bands. This driving multiplies nodal points to support higher winding numbers and increases the number of flat bands, while the Floquet degree of freedom doubles the allowed Cooper pair correlations by adding a distinct class confined to odd-Floquet sectors. A sympathetic reader would care because these effects open nonequilibrium pairing channels unavailable in static systems and suggest more robust topological modes for potential applications. The static analysis first identifies seven nodal topological phases marked by Majorana zero-energy flat bands and quantized conductance peaks, along with specific triplet and orbital-singlet pairing symmetries induced by inter-orbital hopping.

Core claim

Periodic driving fundamentally reshapes the topological and superconducting landscape by generating multiple nodal points that support higher winding numbers and multiple Majorana flat bands, while the emergent Floquet degree of freedom doubles the number of symmetry-allowed Cooper-pair correlations. The first class of correlations is hosted by the even-Floquet sectors and has a direct counterpart in the static limit. In contrast, the second is a distinct Floquet-generated class that confines to the odd-Floquet sectors, representing a fundamentally nonequilibrium pairing channel that cannot exist in static systems.

What carries the argument

The Floquet degree of freedom from periodic driving, which splits Cooper-pair correlations into even- and odd-Floquet sectors while multiplying nodal points and winding numbers.

If this is right

  • Seven distinct nodal topological phases appear in the static case, each with Majorana zero-energy flat bands and quantized zero-bias conductance peaks.
  • Driven phases host both zero and pi Majorana flat bands whose transport signatures follow the Floquet sum rule.
  • The effective p-wave character produces spin-triplet correlations of even-frequency odd-parity and odd-frequency even-parity types.
  • Inter-orbital hopping adds an orbital-singlet pairing term that is simultaneously odd-parity and odd-frequency.
  • The topological modes remain robust against strong disorder.

Where Pith is reading between the lines

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

  • The odd-Floquet pairing channel might appear in time-resolved tunneling spectra or in the frequency dependence of conductance under drive.
  • Higher winding numbers from multiple nodal points could allow systematic engineering of flat-band multiplicity in other driven heterostructures.
  • If the nonequilibrium class proves stable, periodic driving offers a route to expand the space of symmetry-allowed pairings beyond equilibrium constraints.
  • Disorder robustness suggests these modes could be tested in fabricated interfaces with controlled impurities.

Load-bearing premise

The chosen tight-binding Hamiltonian for the p-wave magnet-superconductor interface plus the high-frequency Floquet approximation accurately describe the system without significant heating or higher-order corrections.

What would settle it

An experiment on a periodically driven p-wave magnet-superconductor device that detects neither pi Majorana flat bands nor pairing correlations unique to the odd-Floquet sectors would falsify the claim of fundamentally new nonequilibrium channels.

Figures

Figures reproduced from arXiv: 2606.31550 by Gaurab Kumar Dash, Manisha Thakurathi, Subhendu Kumar Patra.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic setup of the two-dimensional het [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Illustrates the phase diagram for nodal points as [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Showcases the eigenspectrum of the static het [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Topological phase diagram with varying [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Real-space spatial probability distributions [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Depicts transport signatures and zero-bias conduc [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Depicts the Floquet quasi-energy ( [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Showcases Floquet topological phase diagrams un [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Differential conductance in square-wave periodic driv [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Disordered energy spectrum by superimposing 100 [PITH_FULL_IMAGE:figures/full_fig_p013_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Disordered quasi-energy spectrum by superimpos [PITH_FULL_IMAGE:figures/full_fig_p014_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Depicts the Floquet quasi-energy ( [PITH_FULL_IMAGE:figures/full_fig_p015_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Showcases Floquet topological phase diagrams under [PITH_FULL_IMAGE:figures/full_fig_p016_13.png] view at source ↗
read the original abstract

We investigate the emergence of topological superconductivity at the two-dimensional heterostructure interface between a $p$-wave magnet (pWM) and an $s$-wave superconductor. By analyzing nodal gap closings, we identify seven distinct nodal topological phases, each characterized by the presence of Majorana zero-energy flat bands and quantized zero-bias conductance peaks. We demonstrate that the effective $p$-wave nature of the system gives rise to spin-triplet pairing correlations with even-frequency, odd-parity and odd-frequency, even-parity symmetries. Notably, the introduction of inter-orbital hopping induces an exotic orbital-singlet term characterized by simultaneous odd-parity and odd-frequency. Furthermore, we explore the transition from static phases to Floquet topological regimes through periodic driving. These driven phases host both zero and $\pi$ Majorana flat bands, with transport signatures governed by the Floquet sum rule. Most significantly, we show that periodic driving fundamentally reshapes the topological and superconducting landscape by generating multiple nodal points that support higher winding numbers and multiple Majorana flat bands, while the emergent Floquet degree of freedom doubles the number of symmetry-allowed Cooper-pair correlations. The first class of correlations is hosted by the even-Floquet sectors and has a direct counterpart in the static limit. In contrast, the second is a distinct Floquet-generated class that confines to the odd-Floquet sectors, representing a fundamentally nonequilibrium pairing channel that cannot exist in static systems. Finally, we demonstrate the robustness of these topological modes against strong disorder, confirming their potential for stable fault-tolerant applications.

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

2 major / 2 minor

Summary. The manuscript studies a 2D p-wave magnet / s-wave superconductor heterostructure. It reports seven distinct nodal topological phases identified via gap closings, each hosting Majorana zero-energy flat bands and quantized zero-bias conductance. Static phases exhibit spin-triplet even-frequency odd-parity and odd-frequency even-parity pairings, plus an orbital-singlet odd-parity odd-frequency term induced by inter-orbital hopping. Periodic driving is shown to generate Floquet phases with zero- and π-Majorana flat bands, higher winding numbers, multiple flat bands, and an additional class of Cooper-pair correlations confined to odd-Floquet sectors that have no static counterpart; transport obeys a Floquet sum rule and the modes remain robust to strong disorder.

Significance. If the central claims hold, the work would establish that high-frequency driving can both multiply topological features (higher winding numbers, multiple flat bands) and open an entirely new nonequilibrium pairing channel forbidden in equilibrium, thereby enlarging the symmetry classification of superconducting correlations. The explicit mapping of seven static phases plus disorder robustness would also provide a concrete platform for fault-tolerant topological applications.

major comments (2)
  1. The assertion that odd-Floquet-sector Cooper-pair correlations constitute a 'fundamentally nonequilibrium pairing channel that cannot exist in static systems' rests on the high-frequency Magnus (or equivalent) expansion cleanly separating even- and odd-Floquet sectors. No explicit bound on the magnitude of omitted commutators, no comparison of odd-sector amplitudes at reduced driving frequencies, and no check that higher-order terms do not generate effective static-like pairings are provided; this directly undermines the load-bearing claim of a new symmetry class.
  2. The seven nodal topological phases are identified solely by nodal gap closings together with the appearance of Majorana flat bands and quantized conductance. Without explicit evaluation of the winding numbers (or other topological invariants) across the full parameter space of inter-orbital hopping and driving amplitude/frequency, it remains possible that some reported phases are not topologically distinct or that additional phases exist.
minor comments (2)
  1. The abstract and main text refer to 'Floquet sum rule' for transport without defining the precise relation between zero- and π-mode contributions; a short derivation or reference would clarify the statement.
  2. Notation for the inter-orbital hopping term and the driving vector potential should be introduced once with explicit matrix structure before being used in the symmetry classification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comments point by point below, providing our responses and indicating the revisions we plan to make.

read point-by-point responses

  1. Referee: The assertion that odd-Floquet-sector Cooper-pair correlations constitute a 'fundamentally nonequilibrium pairing channel that cannot exist in static systems' rests on the high-frequency Magnus (or equivalent) expansion cleanly separating even- and odd-Floquet sectors. No explicit bound on the magnitude of omitted commutators, no comparison of odd-sector amplitudes at reduced driving frequencies, and no check that higher-order terms do not generate effective static-like pairings are provided; this directly undermines the load-bearing claim of a new symmetry class.

    Authors: We agree that additional evidence is needed to rigorously support the claim of a new nonequilibrium pairing class. While the Floquet formalism naturally introduces the odd sectors absent in static cases, the high-frequency approximation requires validation. In the revised manuscript, we will provide explicit bounds on the commutator terms in the Magnus expansion, compare the odd-sector pairing amplitudes at lower driving frequencies to show their persistence as a distinct feature, and confirm that higher-order terms do not produce effective static pairings in the odd sector. This will be included in a new appendix or section. revision: yes

  2. Referee: The seven nodal topological phases are identified solely by nodal gap closings together with the appearance of Majorana flat bands and quantized conductance. Without explicit evaluation of the winding numbers (or other topological invariants) across the full parameter space of inter-orbital hopping and driving amplitude/frequency, it remains possible that some reported phases are not topologically distinct or that additional phases exist.

    Authors: We acknowledge that relying primarily on gap closings leaves room for further verification. The phases are distinguished by their unique combinations of flat band multiplicities and conductance features, which are topological indicators. To strengthen this, we will explicitly calculate the winding numbers over the full parameter space of inter-orbital hopping, driving amplitude, and frequency in the revised version. This will map the topological invariants, confirm the seven phases are distinct, and check for any additional phases. revision: yes

Circularity Check

0 steps flagged

No circularity: claims follow from explicit model + standard Floquet analysis

full rationale

The paper constructs an explicit tight-binding Hamiltonian for the p-wave magnet–s-wave superconductor interface, applies the high-frequency Magnus expansion to obtain effective Floquet Hamiltonians, locates topological transitions by tracking nodal gap closings and computing winding numbers, and extracts even/odd-frequency pairing symmetries from the resulting Bogoliubov–de Gennes spectrum or Green’s functions. The separation into even- and odd-Floquet sectors and the appearance of odd-sector correlations are direct algebraic consequences of the time-periodic drive term; they are not obtained by fitting parameters to the target observables, by redefining static quantities, or by invoking self-citations as uniqueness theorems. The derivation therefore remains self-contained within the chosen microscopic model and conventional Floquet machinery.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on an effective tight-binding model whose parameters (inter-orbital hopping, driving amplitude/frequency) are not specified in the abstract, plus the standard applicability of Floquet theory to the driven heterostructure.

free parameters (2)
  • inter-orbital hopping strength
    Controls the appearance of the exotic orbital-singlet term; value chosen within the model.
  • driving amplitude and frequency
    Determine the Floquet phases, nodal points, and which sectors host the new pairing class.
axioms (2)
  • domain assumption The heterostructure is described by a tight-binding Hamiltonian combining p-wave magnetism and s-wave superconductivity.
    Foundation for identifying the seven nodal phases via gap closings.
  • domain assumption Floquet theory in the high-frequency limit applies without significant heating or multi-photon processes.
    Required to obtain the driven phases, sum rule, and odd-Floquet pairing class.

pith-pipeline@v0.9.1-grok · 5844 in / 1590 out tokens · 98948 ms · 2026-07-01T04:03:17.973452+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

42 extracted references · 6 canonical work pages · 2 internal anchors

  1. [1]

    Even-Floquet Correlations The first class of correlations corresponds to the even- Floquet pairing (F n,m =F −m,−n). In this sector, the pairing symmetries mirror those of the static case, where the total wave function satisfies the standard Berezin- skii constraint across frequency, spin, orbital, and parity (momentum) degrees of freedom. These are categ...

  2. [2]

    Odd-Floquet Correlations and Frequency Inversion The second class involves an odd-Floquet pairing (F n,m =−F −m,−n). Here, we observe a systematic in- version of the frequency index, while the parities of the spatial momentum, orbital, and spin degrees of freedom remain preserved relative to the static limit. The result- ing correlations are categorized i...

  3. [3]

    Qi and S.-C

    X.-L. Qi and S.-C. Zhang, Topological insulators and su- perconductors, Rev. Mod. Phys.83, 1057 (2011)

    2011

  4. [4]

    Tanaka, M

    Y. Tanaka, M. Sato, and N. Nagaosa, Symmetry and topology in superconductors –odd-frequency pairing and edge states–, Journal of the Physical Society of Japan81, 011013 (2012)

    2012

  5. [5]

    DeGottardi, M

    W. DeGottardi, M. Thakurathi, S. Vishveshwara, and D. Sen, Majorana fermions in superconducting wires: Ef- fects of long-range hopping, broken time-reversal symme- try, and potential landscapes, Phys. Rev. B88, 165111 (2013)

    2013

  6. [6]

    Klinovaja, P

    J. Klinovaja, P. Stano, A. Yazdani, and D. Loss, Topo- logical superconductivity and majorana fermions in rkky systems, Phys. Rev. Lett.111, 186805 (2013)

    2013

  7. [7]

    Sato and Y

    M. Sato and Y. Ando, Topological superconductors: a re- view, Reports on Progress in Physics80, 076501 (2017)

    2017

  8. [8]

    M. Sato, Y. Tanaka, K. Yada, and T. Yokoyama, Topol- ogy of andreev bound states with flat dispersion, Phys. Rev. B83, 224511 (2011)

    2011

  9. [9]

    Thakurathi, P

    M. Thakurathi, P. Simon, I. Mandal, J. Klinovaja, and D. Loss, Majorana kramers pairs in rashba double nanowires with interactions and disorder, Phys. Rev. B 97, 045415 (2018)

    2018

  10. [10]

    C. L. M. Wong, J. Liu, K. T. Law, and P. A. Lee, Majo- rana flat bands and unidirectional majorana edge states in gapless topological superconductors, Phys. Rev. B88, 060504(R) (2013)

    2013

  11. [11]

    Thakurathi, O

    M. Thakurathi, O. Deb, and D. Sen, Majorana modes and transport across junctions of superconductors and normal metals, Journal of Physics: Condensed Matter 27, 275702 (2015)

    2015

  12. [12]

    A. B. Hellenes, T. Jungwirth, R. Jaeschke-Ubiergo, A. Chakraborty, J. Sinova, and L. ˇSmejkal, P-wave mag- nets (2024), arXiv:2309.01607 [cond-mat.mes-hall]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  13. [13]

    Das and A

    S. Das and A. Soori, Orientation dependent anomalous hall and spin hall currents at junctions of altermagnets withp-wave magnets, Phys. Rev. B112, 235309 (2025)

    2025

  14. [14]

    Soori, Crossed andreev reflection in collinearp-wave magnet/triplet superconductor junctions, Phys

    A. Soori, Crossed andreev reflection in collinearp-wave magnet/triplet superconductor junctions, Phys. Rev. B 111, 165413 (2025)

    2025

  15. [15]

    Priessnitz, A

    J. Priessnitz, A. B. Hellenes, R. Comin, and L. ˇSmejkal, Ferroelectricp-wave magnets (2026), arXiv:2603.19107 [cond-mat.mtrl-sci]

    work page arXiv 2026

  16. [16]

    $P$-wave Orbital Magnetism

    Y. Li and P. Sukhachov,p-wave orbital magnetism (2026), arXiv:2604.18695 [cond-mat.mes-hall]

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  17. [17]

    Maeda, Y

    K. Maeda, Y. Fukaya, K. Yada, B. Lu, Y. Tanaka, and J. Cayao, Classification of pair symmetries in supercon- ductors with unconventional magnetism, Phys. Rev. B 111, 144508 (2025)

    2025

  18. [18]

    Fukaya, B

    Y. Fukaya, B. Lu, K. Yada, Y. Tanaka, and J. Cayao, Superconducting phenomena in systems with unconven- tional magnets, Journal of Physics: Condensed Matter 37, 313003 (2025)

    2025

  19. [19]

    Fukaya, K

    Y. Fukaya, K. Maeda, K. Yada, J. Cayao, Y. Tanaka, and B. Lu, Josephson effect and odd-frequency pairing in superconducting junctions with unconventional magnets, Phys. Rev. B111, 064502 (2025)

    2025

  20. [20]

    Nagae, L

    Y. Nagae, L. Katayama, and S. Ikegaya, Flat-band zero- energy states and anomalous proximity effects inp-wave magnet–superconductor hybrid systems, Phys. Rev. B 111, 174519 (2025)

    2025

  21. [21]

    Fukaya, K

    Y. Fukaya, K. Yada, and Y. Tanaka,p-wave superconduc- tivity and josephson current inp-wave unconventional magnet/s-wave superconductor hybrid systems (2026), arXiv:2512.18636 [cond-mat.supr-con]

    work page arXiv 2026

  22. [22]

    A. Pal, P. Dutta, and A. Saha, Emergent superconduct- ing phases in unconventionalp-wave magnets: Topo- logical superconductivity, bogoliubov fermi surfaces and superconducting diode effect (2026), arXiv:2603.03221 [cond-mat.supr-con]

    work page arXiv 2026

  23. [23]

    Sharma and M

    L. Sharma and M. Thakurathi, Tunable josephson diode effect in singlet superconductor-altermagnet-triplet su- perconductor junctions, Phys. Rev. B112, 104506 (2025)

    2025

  24. [24]

    Yang and Z

    Z. Yang and Z. P. Niu, Josephson diode effect in un- conventionalp-wave magnet/superconductor josephson junctions, Phys. Rev. B113, 094515 (2026)

    2026

  25. [25]

    Sharma, B

    L. Sharma, B. Ghimire, and M. Thakurathi,p-wave magnet driven field-free josephson diode effect (2026), arXiv:2602.16677 [cond-mat.supr-con]. 18

    work page arXiv 2026

  26. [26]

    Thakurathi, A

    M. Thakurathi, A. A. Patel, D. Sen, and A. Dutta, Flo- quet generation of majorana end modes and topological invariants, Phys. Rev. B88, 155133 (2013)

    2013

  27. [27]

    Mondal, R

    D. Mondal, R. Kumari, T. Nag, and A. Saha, Transport signatures of single and multiple floquet majorana modes in a one-dimensional rashba nanowire and shiba chain, Phys. Rev. B111, 235441 (2025)

    2025

  28. [28]

    K. Roy, G. Paul, D. Debnath, K. Bhattacharyya, and S. Basu, Floquet-engineered diode performance of a topo- logical josephson junction composed of two kitaev chains coupled via a quantum dot, Phys. Rev. B112, 125407 (2025)

    2025

  29. [29]

    Jiang, T

    L. Jiang, T. Kitagawa, J. Alicea, A. R. Akhmerov, D. Pekker, G. Refael, J. I. Cirac, E. Demler, M. D. Lukin, and P. Zoller, Majorana fermions in equilibrium and in driven cold-atom quantum wires, Phys. Rev. Lett.106, 220402 (2011)

    2011

  30. [30]

    Thakurathi, D

    M. Thakurathi, D. Loss, and J. Klinovaja, Floquet majorana fermions and parafermions in driven rashba nanowires, Phys. Rev. B95, 155407 (2017)

    2017

  31. [31]

    P.-H. Fu, S. Mondal, J.-F. Liu, Y. Tanaka, and J. Cayao, Floquet engineering spin triplet states in unconventional magnets, Phys. Rev. Lett.136, 066703 (2026)

    2026

  32. [32]

    Triola and A

    C. Triola and A. V. Balatsky, Odd-frequency supercon- ductivity in driven systems, Phys. Rev. B94, 094518 (2016)

    2016

  33. [33]

    G. E. Blonder, M. Tinkham, and T. M. Klapwijk, Tran- sition from metallic to tunneling regimes in supercon- ducting microconstrictions: Excess current, charge im- balance, and supercurrent conversion, Phys. Rev. B25, 4515 (1982)

    1982

  34. [34]

    V. L. Berezinskii, New model of the anisotropic phase of superfluidhe 3, JETP Letters20, 628 (1974)

    1974

  35. [35]

    Balatsky and E

    A. Balatsky and E. Abrahams, New class of singlet su- perconductors which break the time reversal and parity, Phys. Rev. B45, 13125(R) (1992)

    1992

  36. [36]

    Brekke, P

    B. Brekke, P. Sukhachov, H. G. Giil, A. Brataas, and J. Linder, Minimal models and transport properties of unconventionalp-wave magnets, Phys. Rev. Lett.133, 236703 (2024)

    2024

  37. [37]

    A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Lud- wig, Classification of topological insulators and super- conductors in three spatial dimensions, Phys. Rev. B78, 195125 (2008)

    2008

  38. [38]

    D. S. Fisher and P. A. Lee, Relation between conductiv- ity and transmission matrix, Phys. Rev. B23, 6851(R) (1981)

    1981

  39. [39]

    Datta,Electronic Transport in Mesoscopic Systems, Cambridge Studies in Semiconductor Physics and Mi- croelectronic Engineering (Cambridge University Press, 1995)

    S. Datta,Electronic Transport in Mesoscopic Systems, Cambridge Studies in Semiconductor Physics and Mi- croelectronic Engineering (Cambridge University Press, 1995)

    1995

  40. [40]

    Sambe, Steady states and quasienergies of a quantum- mechanical system in an oscillating field, Phys

    H. Sambe, Steady states and quasienergies of a quantum- mechanical system in an oscillating field, Phys. Rev. A 7, 2203 (1973)

    1973

  41. [41]

    Kundu and B

    A. Kundu and B. Seradjeh, Transport signatures of flo- quet majorana fermions in driven topological supercon- ductors, Phys. Rev. Lett.111, 136402 (2013)

    2013

  42. [42]

    H. Aoki, N. Tsuji, M. Eckstein, M. Kollar, T. Oka, and P. Werner, Nonequilibrium dynamical mean-field theory and its applications, Rev. Mod. Phys.86, 779 (2014)

    2014