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arxiv: 2507.10856 · v1 · submitted 2025-07-14 · 🪐 quant-ph · cond-mat.quant-gas· cond-mat.stat-mech· cond-mat.supr-con· math-ph· math.MP

Higher spin Richardson-Gaudin model with time-dependent coupling: Exact dynamics

Suvendu Barik , Lieuwe Bakker , Vladimir Gritsev , Ji\v{r}\'i Min\'a\v{r} , Emil A. Yuzbashyan This is my paper

Pith reviewed 2026-05-19 04:03 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.quant-gascond-mat.stat-mechcond-mat.supr-conmath-phmath.MP
keywords Richardson-Gaudin modeltime-dependent couplinghigher spinexact dynamicsasymptotic wavefunctionnon-thermal steady statemean-field theory
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0 comments X

The pith

A spin-s Richardson-Gaudin model with 1/t coupling has an exact asymptotic wavefunction that must be derived separately for each spin size.

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

The paper determines the exact long-time many-body wavefunction for this integrable model when the interaction strength falls as 1/t and the system begins in its ground state right after t=0. A reader would care because the result produces a non-thermal steady state that does not match the generalized Gibbs ensemble familiar from the spin-1/2 case. The work also establishes that mean-field theory becomes exact for any finite product of spin operators on distinct sites, with direct implications for experiments in cavity QED and trapped ions.

Core claim

We determine the exact asymptotic many-body wavefunction of a spin-s Richardson-Gaudin model with a coupling inversely proportional to time, for time evolution starting from the ground state at t = 0^+ and for arbitrary s. Contrary to common belief, the resulting wavefunction cannot be derived from the spin-1/2 case by merging spins, but instead requires independent treatment for each spin size. The steady state is non-thermal and does not conform to a natural Generalized Gibbs Ensemble. Mean-field theory is exact for any product of a finite number of spin operators on different sites.

What carries the argument

The integrability structure of the Richardson-Gaudin model, which permits an exact asymptotic solution precisely when the coupling takes the inverse-time form and evolution starts from the ground state at t=0^+.

If this is right

  • The asymptotic wavefunction must be constructed independently for each spin value s rather than by combining spin-1/2 results.
  • The steady state reached at late times is non-thermal and lies outside the natural generalized Gibbs ensemble.
  • Mean-field theory exactly reproduces the dynamics of any finite product of spin operators acting on different sites.
  • The protocol can be implemented and measured in cavity QED and trapped-ion experiments.

Where Pith is reading between the lines

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

  • Time-dependent protocols in other integrable models may similarly produce non-equilibrium states that evade standard ensemble descriptions.
  • The exact mean-field property for local spin products suggests that correlation functions in these systems can be computed without solving the full many-body problem.
  • Realizing the inverse-time ramp in quantum simulators could test whether the non-thermal character persists under weak integrability-breaking perturbations.

Load-bearing premise

The specific inverse-time form of the coupling together with the initial ground-state condition at t=0^+ permits an exact asymptotic solution via the model's integrability structure.

What would settle it

A numerical time-evolution calculation for small lattices with s greater than 1/2 whose late-time wavefunction deviates from the analytic form derived here would falsify the claim.

Figures

Figures reproduced from arXiv: 2507.10856 by Emil A. Yuzbashyan, Ji\v{r}\'i Min\'a\v{r}, Lieuwe Bakker, Suvendu Barik, Vladimir Gritsev.

Figure 1
Figure 1. Figure 1: display a sharp discontinuity at ∆ = 0 for α = 1—a feature absent when α ̸= 1. We note that α = 1 corre￾sponds to the integrable model [32, 33, 38, 39]. Thus, the observed non-analytic behavior in this finite-size system is a signature of integrability and persists for any ν > 0. To see this explicitly, consider two scenarios: (A) Evolve the system and then take the limit ∆ → 0. (B) Set ∆ = 0 first and the… view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. A sample contour [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Simulation of the saddle point coordinates [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Overlap between the asymptotic wavefunction [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Probabilities [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Overlap between the asymptotic wavefunction [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of the expectation values [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Difference between numerically computed expecta [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Difference between [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Pictorial representation of combining spin-1 [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Probability distribution of a numerically simulated wavefunction (green bars), evaluated at times [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Numerical simulation of local magnetization per [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Examples of Trotter-Suzuki (TS) implementation. (a) TS approximation of Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p020_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Ratios [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. Overlap between the asymptotic wavefunction [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. Overlap between the asymptotic wavefunction [PITH_FULL_IMAGE:figures/full_fig_p025_17.png] view at source ↗
read the original abstract

We determine the exact asymptotic many-body wavefunction of a spin-$s$ Richardson-Gaudin model with a coupling inversely proportional to time, for time evolution starting from the ground state at $t = 0^+$ and for arbitrary $s$. Contrary to common belief, the resulting wavefunction cannot be derived from the spin-$1/2$ case by merging spins, but instead requires independent treatment for each spin size. The steady state is non-thermal and, in contrast to the spin-$1/2$ case, does not conform to a natural Generalized Gibbs Ensemble. We show that mean-field theory is exact for any product of a finite number of spin operators on different sites. We discuss how these findings can be probed in cavity QED and trapped ion experiments.

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

1 major / 2 minor

Summary. The manuscript claims to derive the exact asymptotic many-body wavefunction for a spin-s Richardson-Gaudin model with time-dependent coupling inversely proportional to time, starting from the ground state at t=0^+, for arbitrary s. It states that this cannot be obtained by merging spins from the s=1/2 case and requires independent treatment for each s. The resulting steady state is non-thermal and does not conform to a natural Generalized Gibbs Ensemble (unlike the s=1/2 case). Mean-field theory is shown to be exact for any product of a finite number of spin operators on different sites, with discussion of experimental probes in cavity QED and trapped-ion systems.

Significance. If the central claims hold, the work provides a valuable extension of exact solvability to higher-spin integrable models under time-dependent driving, a setting where closed-form solutions are rare. The demonstration of exact mean-field behavior for multi-site spin-operator products is a clear strength, enabling simplified yet accurate calculations of observables. The non-thermal steady state and its departure from GGE expectations for s>1/2 offer concrete insight into the boundaries of generalized ensembles in integrable systems. The paper's use of integrability for the asymptotic limit is a positive feature that could inform related studies of non-equilibrium dynamics.

major comments (1)
  1. [Asymptotic analysis section] Asymptotic analysis section: The central claim of an exact closed-form wavefunction for arbitrary s rests on the independent treatment for s>1/2. The manuscript should explicitly show that the time-dependent Gaudin algebra (or equivalent Bethe-equation limit) yields a unique asymptotic state without sector mixing or additional root-distribution assumptions that appear only for s>1/2; otherwise the exactness for general s remains unverified beyond the s=1/2 case.
minor comments (2)
  1. [Experimental discussion] The experimental-proposal paragraph would be strengthened by naming at least one concrete observable (e.g., a specific spin-correlation function) whose mean-field exactness could be measured in cavity-QED or ion-trap setups.
  2. [Notation throughout] Notation for the time-dependent coupling strength and the initial time t=0^+ should be used uniformly in all equations and figure captions to prevent ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comment on the asymptotic analysis. We address the point in detail below and will incorporate an explicit clarification in the revised version.

read point-by-point responses

  1. Referee: [Asymptotic analysis section] Asymptotic analysis section: The central claim of an exact closed-form wavefunction for arbitrary s rests on the independent treatment for s>1/2. The manuscript should explicitly show that the time-dependent Gaudin algebra (or equivalent Bethe-equation limit) yields a unique asymptotic state without sector mixing or additional root-distribution assumptions that appear only for s>1/2; otherwise the exactness for general s remains unverified beyond the s=1/2 case.

    Authors: We agree that an explicit verification of uniqueness strengthens the central claim. The time-dependent Gaudin algebra is formulated directly in the general spin-s representation, with the defining commutation relations and Lax operators independent of the specific value of s. In the asymptotic limit t→∞ the coupling vanishes as 1/t, and the resulting Bethe equations reduce to a set of algebraic conditions whose solution is uniquely fixed by the initial ground-state quantum numbers. Because the driving preserves the relevant conserved quantities (total spin projection and the Gaudin invariants), no sector mixing occurs and no s-dependent root-distribution assumptions are introduced. This structure is verified explicitly for s=1 and s=3/2 in the main text and appendix by direct substitution into the limiting equations. To make the argument fully transparent for arbitrary s we will add a short subsection in the asymptotic analysis that isolates the uniqueness proof from the algebra alone, without reference to the s=1/2 reduction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on direct integrability analysis for the specific time-dependent Hamiltonian.

full rationale

The paper determines the exact asymptotic wavefunction for arbitrary spin s using the integrability structure of the Richardson-Gaudin model with inverse-time coupling, starting from the ground state at t=0^+. The abstract explicitly states that the result cannot be obtained by merging spins from the s=1/2 case and requires independent treatment, indicating a direct algebraic construction rather than reduction to prior fitted results or self-definitional mappings. Mean-field exactness for finite products of spin operators follows as a consequence of the derived wavefunction. No load-bearing step reduces by construction to its own inputs, self-citations, or ansatzes imported without independent verification; the central claim remains self-contained against the model's standard Bethe/Gaudin algebraic framework.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; the central claim rests on the integrability of the Richardson-Gaudin Hamiltonian under the specific 1/t time dependence and the validity of the asymptotic limit from the ground state at t=0^+. No explicit free parameters, new entities, or ad-hoc axioms are stated in the provided text.

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Reference graph

Works this paper leans on

109 extracted references · 109 canonical work pages

  1. [1]

    To obtain a nontrivial and well-defined thermodynamic limit, the coupling constant g(t) = 1 νt must be scaled to offset the N2 scaling of the interaction

    Primary considerations In the thermodynamic limit, the Zeeman term in the Hamiltonian (6) scales linearly with the system size N, while the interaction term P j,k ˆs+ j ˆs− k scales as N2. To obtain a nontrivial and well-defined thermodynamic limit, the coupling constant g(t) = 1 νt must be scaled to offset the N2 scaling of the interaction. This motivate...

    work page

  2. [2]

    (51) Using this, we define an n-local operator ˆB for some fixed positive integer n as ˆB = nY i=1 (ˆspi i ˆsqi i ) , p i, qi ∈ {0, +, −, z}

    Multi-site operators Consider the following notation for an arbitrary (bi- linear) single-site operator: ˆsp i ˆsq i , p, q ∈ {0, +, −, z}, ˆs0 = I3×3. (51) Using this, we define an n-local operator ˆB for some fixed positive integer n as ˆB = nY i=1 (ˆspi i ˆsqi i ) , p i, qi ∈ {0, +, −, z}. (52) It turns out that the specific choice of sites i used to c...

    work page

  3. [3]

    Single-site observables 2 4 6 8 10 site index ( i) −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 ⟨ˆsz i ⟩(1) N = 2 N = 3 N = 4 N = 5 N = 6 N = 7 N = 8 N = 9 N = 10 FIG. 7. Comparison of the expectation values ⟨ˆsz i ⟩(1)com- puted numerically at t = 10 3 and evaluated in the asymp- totic state (24) as functions of N for J z = 0. The dashed lines describe the asym...

    work page

  4. [4]

    (57), we obtain N∆N+ ⟨Ψ(N++∆N+)∞ (t)| ˆB |Ψ(N+)∞ (t)⟩ = nY i=1 ⟨ˆsp i ˆsq i ⟩mf

    Agreement with mean-field predictions Returning to the general n-local operator ˆB in (52) and applying the result of Eq. (57), we obtain N∆N+ ⟨Ψ(N++∆N+)∞ (t)| ˆB |Ψ(N+)∞ (t)⟩ = nY i=1 ⟨ˆsp i ˆsq i ⟩mf . (65) This result is valid up to order O(n/N) and becomes exact in the thermodynamic limit. Similar procedures for calculating expectation values fail for...

    work page

  5. [5]

    emergent GGE

    Variance and higher moments of ˆsz j for large spins In investigating spin-1 operators, we also address gen- eral spin-s single-site observables to understand the s ≫ 1/2 regime in the thermodynamic limit. We first con- sider the form of ωj in (35d) in this limit. Recalling that ν = N/η [Eq. (45)], we readily determine ωj from (33) in the η → 0+ (stiff-ra...

    work page

  6. [6]

    The single-site magnetization in the asymptotic state has a non-Gaussian distribution, as previ- ously discussed in Sec. IV C 5. For higher spin val- ues, the distribution becomes increasingly peaked, approaching a delta function. Numerical results for a 5-site spin −1 RG Hamiltonian are compared against the thermodynamic and GGE predictions in Fig. 12

    work page

  7. [7]

    We illustrate the expected results in Fig

    For s > 1/2, the probability distribution of a given configuration of local magnetizations (76) no longer conforms to the natural generalized Gibbs ensemble at late times, unlike for the s = 1 /2 case. We illustrate the expected results in Fig. 11

    work page

  8. [8]

    For non-integrable couplings of the form g(t) ∝ 1/tα, α ̸= 1, the local magnetization exhibit a de- pendence on ε, as we demonstrate in Fig

    These local magnetizations for the given 1 /t cou- pling used in (6) are special in that they remain invariant under variations of ε in the Zeeman term. For non-integrable couplings of the form g(t) ∝ 1/tα, α ̸= 1, the local magnetization exhibit a de- pendence on ε, as we demonstrate in Fig. 13

    work page

  9. [9]

    qubits” for the spin-1/2 systems and “qudits

    The transition probabilities of the model exhibit discontinuities when εj = εk, j ̸= k. In contrast, we suggest that such discontinuities do not arise for non-integrable couplings. The integrable point at α = 1 distinctly exhibits this discontinuous behav- ior. VI. TOW ARDS EXPERIMENT AL REALIZA TIONS In this section we sketch possible pathways towards an...

    work page

  10. [10]

    π 2 cosh 2π ν # 1 2 eirτ τ 1 2 + 2i ν h J 1 2 + 2i ν (τ) |2⟩ − iJ− 1 2 + 2i ν (τ) |1⟩ i , (A2b) |ψ(2,1) 1 (t)⟩ =

    Spin 1 For J z = −2 magnetization sector, the solution |ψ(N,s) J z ⟩ is trivial: |ψ(2,1) −2 (t)⟩ = exp [2i(ε1 + ε2)t] |−1, −1⟩ . (A2a) For J z = ±1, the solutions are: |ψ(2,1) −1 (t)⟩ = " π 2 cosh 2π ν # 1 2 eirτ τ 1 2 + 2i ν h J 1 2 + 2i ν (τ) |2⟩ − iJ− 1 2 + 2i ν (τ) |1⟩ i , (A2b) |ψ(2,1) 1 (t)⟩ = " π 2 cosh 2π ν # 1 2 e−irτ τ 1 2 + 4i ν h J 1 2 + 2i ν ...

    work page

  11. [11]

    Spin 3/2 Given the already lengthy presentation of the J z = 0 solution to two-site s = 1 problem, we only present the asymptotic solutions for J z = ±1: lim t→∞ |ψ(2,3/2) −1 (t)⟩ = N (ν)C1(t, ν, ε)× e−γ0,1 e− 6π ν e−4iε1t |1/2, −3/2⟩ + e−(γ2,0−γ0,1)e− 9π ν e−2i(ε1+ε2)t |−1/2, −1/2⟩ + e− 12π ν e−4iε2t |−3/2, 1/2⟩ , C1(t, ν, ε) = t 2i ν e2i(ε1+ε2)te 9π ν ,...

    work page

  12. [12]

    Fan and A

    B. Fan and A. M. Garc´ ıa Garc´ ıa, Quenched dynamics and pattern formation in clean and disordered Bogoliubov-de Gennes superconductors, SciPost Phys. 17, 049 (2024)

    work page 2024

  13. [13]

    L. D. Landau, A theory of energy transfer. II, Phys. Z. Sowjetunion 2, 63 (1932)

    work page 1932

  14. [14]

    Zener and R

    C. Zener and R. H. Fowler, Non-adiabatic crossing of energy levels, Proc. R. Soc. A 137, 696 (1997)

    work page 1997

  15. [15]

    J. B. Delos, Theory of electronic transitions in slow atomic collisions, Rev. Mod. Phys. 53, 287 (1981)

    work page 1981

  16. [16]

    Nakatsukasa, K

    T. Nakatsukasa, K. Matsuyanagi, M. Matsuo, and K. Ya- bana, Time-dependent density-functional description of nuclear dynamics, Rev. Mod. Phys. 88, 045004 (2016)

    work page 2016

  17. [17]

    S.-Y. Bai, C. Chen, H. Wu, and J.-H. An, Quantum con- trol in open and periodically driven systems, Adv. Phys.: X 6, 1870559 (2021)

    work page 2021

  18. [18]

    P. W. Claeys, M. Pandey, D. Sels, and A. Polkovnikov, Floquet-engineering counterdiabatic protocols in quan- tum many-body systems, Phys. Rev. Lett. 123, 090602 (2019)

    work page 2019

  19. [19]

    Mizuta and K

    K. Mizuta and K. Fujii, Optimal hamiltonian simulation for time-periodic systems, Quantum 7, 962 (2023)

    work page 2023

  20. [20]

    Wilczek, Quantum Time Crystals, Phys

    F. Wilczek, Quantum Time Crystals, Phys. Rev. Lett. 109, 160401 (2012)

    work page 2012

  21. [21]

    N. Y. Yao and C. Nayak, Time crystals in periodically driven systems, Phys. Today 71, 40 (2018)

    work page 2018

  22. [22]

    Khemani, R

    V. Khemani, R. Moessner, and S. L. Sondhi, A Brief His- tory of Time Crystals, arXiv 10.48550/arXiv.1910.10745 (2019)

    work page doi:10.48550/arxiv.1910.10745 1910

  23. [23]

    Goldman and J

    N. Goldman and J. Dalibard, Periodically driven quan- tum systems: Effective hamiltonians and engineered gauge fields, Phys. Rev. X 4, 031027 (2014)

    work page 2014

  24. [24]

    Geier, N

    S. Geier, N. Thaicharoen, C. Hainaut, T. Franz, A. Salzinger, A. Tebben, D. Grimshandl, G. Z¨ urn, and M. Weidem¨ uller, Floquet hamiltonian engineering of an isolated many-body spin system, Science 374, 1149 (2021)

    work page 2021

  25. [25]

    Gritsev and A

    V. Gritsev and A. Polkovnikov, Integrable Floquet dy- namics, SciPost Phys. 2, 021 (2017)

    work page 2017

  26. [26]

    R. Fan, Y. Gu, A. Vishwanath, and X. Wen, Emergent Spatial Structure and Entanglement Localization in Flo- quet Conformal Field Theory, Phys. Rev. X 10, 031036 (2020)

    work page 2020

  27. [27]

    d’Alessandro, Introduction to quantum control and dy- namics (Chapman and hall/CRC, 2021)

    D. d’Alessandro, Introduction to quantum control and dy- namics (Chapman and hall/CRC, 2021)

    work page 2021

  28. [28]

    X. Yuan, S. Endo, Q. Zhao, Y. Li, and S. C. Benjamin, Theory of variational quantum simulation, Quantum 3, 191 (2019)

    work page 2019

  29. [29]

    Cerezo, A

    M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, et al. , Variational quantum algorithms, Nat. Rev. Phys. 3, 625 (2021)

    work page 2021

  30. [30]

    M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information (Cambridge University Press, 2010)

    work page 2010

  31. [31]

    I. M. Georgescu, S. Ashhab, and F. Nori, Quantum sim- ulation, Rev. Mod. Phys. 86, 153 (2014)

    work page 2014

  32. [32]

    Fauseweh, Quantum many-body simulations on digi- tal quantum computers: State-of-the-art and future chal- lenges, Nat

    B. Fauseweh, Quantum many-body simulations on digi- tal quantum computers: State-of-the-art and future chal- lenges, Nat. Commun. 15, 2123 (2024)

    work page 2024

  33. [33]

    Zhang, T

    X.-M. Zhang, T. Li, and X. Yuan, Quantum state prepa- ration with optimal circuit depth: Implementations and applications, Phys. Rev. Lett. 129, 230504 (2022)

    work page 2022

  34. [34]

    Albash and D

    T. Albash and D. A. Lidar, Adiabatic quantum compu- tation, Rev. Mod. Phys. 90, 015002 (2018)

    work page 2018

  35. [35]

    Monroe, W

    C. Monroe, W. Campbell, L.-M. Duan, Z.-X. Gong, A. Gorshkov, P. Hess, R. Islam, K. Kim, N. Linke, G. Pagano, P. Richerme, C. Senko, and N. Yao, Pro- grammable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93, 025001 (2021)

    work page 2021

  36. [36]

    Senese, J

    R. Senese, J. Robertson, and F. Essler, Out-of- equilibrium full counting statistics in Gaussian theories of quantum magnets, SciPost Phys. 17, 139 (2024)

    work page 2024

  37. [37]

    Puzzuoli, S

    D. Puzzuoli, S. F. Lin, M. Malekakhlagh, E. Pritchett, B. Rosand, and C. J. Wood, Algorithms for perturbative analysis and simulation of quantum dynamics, J. Com- put. Phys. 489, 112262 (2023)

    work page 2023

  38. [38]

    Oka and S

    T. Oka and S. Kitamura, Floquet Engineering of Quan- tum Materials, Annu. Rev. Condens. Matter Phys. 10, 387 (2019)

    work page 2019

  39. [39]

    Scopa, G

    S. Scopa, G. T. Landi, and D. Karevski, Lindblad- Floquet description of finite-time quantum heat engines, Phys. Rev. A 97, 062121 (2018)

    work page 2018

  40. [40]

    Scopa, G

    S. Scopa, G. T. Landi, A. Hammoumi, and D. Karevski, Exact solution of time-dependent Lindblad equations with closed algebras, Phys. Rev. A 99, 022105 (2019)

    work page 2019

  41. [41]

    L. R. Bakker, V. I. Yashin, D. V. Kurlov, A. K. Fe- dorov, and V. Gritsev, Lie-algebraic approach to one- dimensional translationally invariant free-fermionic dis- sipative systems, Phys. Rev. A 102, 052220 (2020)

    work page 2020

  42. [42]

    Patra and E

    A. Patra and E. A. Yuzbashyan, Quantum integrability in the multistate Landau–Zener problem, J. Phys. A: Math. Theor. 48, 245303 (2015)

    work page 2015

  43. [43]

    N. A. Sinitsyn, E. A. Yuzbashyan, V. Y. Chernyak, A. Patra, and C. Sun, Integrable Time-Dependent Quan- tum Hamiltonians, Phys. Rev. Lett. 120, 190402 (2018)

    work page 2018

  44. [44]

    E. A. Yuzbashyan, Integrable time-dependent Hamiltoni- ans, solvable Landau–Zener models and Gaudin magnets, Ann. Phys. 392, 323 (2018)

    work page 2018

  45. [45]

    C. Sun, V. Y. Chernyak, A. Piryatinski, and N. A. Sinit- syn, Cooperative light emission in the presence of strong inhomogeneous broadening, Phys. Rev. Lett.123, 123605 (2019)

    work page 2019

  46. [46]

    V. Y. Chernyak, N. A. Sinitsyn, and C. Sun, Multitime Landau–Zener model: classification of solvable Hamilto- nians, J. Phys. A: Math. Theor. 53, 185203 (2020)

    work page 2020

  47. [47]

    V. Y. Chernyak and N. A. Sinitsyn, Integrability in the multistate Landau-Zener model with time-quadratic commuting operators, J. Phys. A: Math. Theor. 54, 115204 (2021)

    work page 2021

  48. [48]

    R. K. Malla, V. Y. Chernyak, C. Sun, and N. A. Sinitsyn, Coherent reaction between molecular and atomic bose- einstein condensates: Integrable model, Phys. Rev. Lett. 129, 033201 (2022)

    work page 2022

  49. [49]

    Zabalo, A.-K

    A. Zabalo, A.-K. Wu, J. H. Pixley, and E. A. Yuzbashyan, Nonlocality as the source of purely quantum dynamics of BCS superconductors, Phys. Rev. B 106, 104513 (2022)

    work page 2022

  50. [50]

    Barik, L

    S. Barik, L. Bakker, V. Gritsev, and E. A. Yuzbashyan, Knizhnik-Zamolodchikov equations and integrable hy- perbolic Landau-Zener models, SciPost Phys. 18, 212 (2025)

    work page 2025

  51. [51]

    Suzuki, R

    F. Suzuki, R. K. Malla, and N. A. Sinitsyn, Competing bosonic reactions: Insight from exactly solvable time- 27 dependent models, arXiv 10.48550/arXiv.2504.13027 (2025)

    work page doi:10.48550/arxiv.2504.13027 2025

  52. [52]

    P. W. Anderson, Random-Phase Approximation in the Theory of Superconductivity, Phys. Rev. 112, 1900 (1958)

    work page 1900

  53. [53]

    Coleman, Introduction to Many-Body Physics (Cam- bridge University Press, Cambridge, 2015)

    P. Coleman, Introduction to Many-Body Physics (Cam- bridge University Press, Cambridge, 2015)

    work page 2015

  54. [54]

    Richardson and N

    R. Richardson and N. Sherman, Exact eigenstates of the pairing-force Hamiltonian, Nucl. Phys. 52, 221 (1964)

    work page 1964

  55. [55]

    Richardson, Exact eigenstates of the pairing-force Hamiltonian

    R. Richardson, Exact eigenstates of the pairing-force Hamiltonian. II, J. Math. Phys. 6, 1034 (1965)

    work page 1965

  56. [56]

    Gaudin and J.-S

    M. Gaudin and J.-S. Caux, The Bethe wavefunction (Cambridge University Press, Cambridge, United King- dom, 2014)

    work page 2014

  57. [57]

    E. K. Sklyanin, Separation of variables in the Gaudin model, J. Sov. Math. 47, 2473 (1989)

    work page 1989

  58. [58]

    Child, Curve-crossing and the WKB approximation, Mol

    M. Child, Curve-crossing and the WKB approximation, Mol. Phys. 20, 171 (1971)

    work page 1971

  59. [59]

    Bandrauk, On long-range curve crossings, Mol

    A. Bandrauk, On long-range curve crossings, Mol. Phys. 24, 661 (1972)

    work page 1972

  60. [60]

    V. I. Osherov and V. G. Ushakov, Threshold anomaly in S scattering with Coulomb charge exchange, Phys. Rev. A 53, 3157 (1996)

    work page 1996

  61. [61]

    V. N. Ostrovsky, Nonstationary multistate Coulomb and multistate exponential models for nonadiabatic transi- tions, Phys. Rev. A 68, 012710 (2003)

    work page 2003

  62. [62]

    N. A. Sinitsyn, Exact results for models of multichan- nel quantum nonadiabatic transitions, Phys. Rev. A 90, 062509 (2014)

    work page 2014

  63. [63]

    E. E. Nikitin, The Theory of Nonadiabatic Transitions: Recent Development with Exponential Models, in Ad- vances in Quantum Chemistry , Vol. 5, edited by P.-O. L¨ owdin (Academic Press, 1970) pp. 135–184

    work page 1970

  64. [64]

    F. H. Mies, Effects of Anharmonicity on Vibrational En- ergy Transfer, J. Chem. Phys. 40, 523 (1964)

    work page 1964

  65. [65]

    V. A. Nascimento, L. L. Caliri, A. Schwettmann, J. P. Shaffer, and L. G. Marcassa, Electric field effects in the excitation of cold rydberg-atom pairs, Phys. Rev. Lett. 102, 213201 (2009)

    work page 2009

  66. [66]

    Baumgartner and H

    F. Baumgartner and H. Helm, Stark field control of nonadiabatic dynamics in triatomic hydrogen, Phys. Rev. Lett. 104, 103002 (2010)

    work page 2010

  67. [67]

    Feynman, J

    R. Feynman, J. Hollingsworth, M. Vennettilli, T. Budner, R. Zmiewski, D. P. Fahey, T. J. Carroll, and M. W. Noel, Quantum interference in the field ionization of rydberg atoms, Phys. Rev. A 92, 043412 (2015)

    work page 2015

  68. [68]

    R. S. Tantawi, A. S. Sabbah, J. H. Macek, and S. Y. Ovchinnikov, Two-state model for top-of-barrier pro- cesses, Phys. Rev. A 62, 042710 (2000)

    work page 2000

  69. [69]

    Affleck, T

    I. Affleck, T. Kennedy, E. H. Lieb, and H. Tasaki, Rig- orous results on valence-bond ground states in antiferro- magnets, Phys. Rev. Lett. 59, 799 (1987)

    work page 1987

  70. [70]

    Hung and C.-D

    H.-H. Hung and C.-D. Gong, Numerical evidence of a spin-1/2 chain approaching a spin-1 chain, Phys. Rev. B 71, 054413 (2005)

    work page 2005

  71. [71]

    E. Y. Andrei and A. H. MacDonald, Graphene bilayers with a twist, Nat. Mater. 19, 1265 (2020)

    work page 2020

  72. [72]

    Zhang, G

    Y. Zhang, G. Shavit, H. Ma, Y. Han, C. W. Siu, A. Mukherjee, K. Watanabe, T. Taniguchi, D. Hsieh, C. Lewandowski, F. von Oppen, Y. Oreg, and S. Nadj-Perge, Twist-programmable superconductivity in spin–orbit-coupled bilayer graphene, Nature 641, 625–631 (2025)

    work page 2025

  73. [73]

    Dagotto, J

    E. Dagotto, J. Riera, and D. Scalapino, Superconductiv- ity in ladders and coupled planes, Phys. Rev. B 45, 5744 (1992)

    work page 1992

  74. [74]

    Takahashi, A

    H. Takahashi, A. Sugimoto, Y. Nambu, T. Yamauchi, Y. Hirata, T. Kawakami, M. Avdeev, K. Matsubayashi, F. Du, C. Kawashima, H. Soeda, S. Nakano, Y. Uwa- toko, Y. Ueda, T. J. Sato, and K. Ohgushi, Pressure- induced superconductivity in the iron-based ladder ma- terial BaFe2S3, Nat. Mater. 14, 1008 (2015)

    work page 2015

  75. [75]

    Jiang, Z.-Y

    H.-C. Jiang, Z.-Y. Weng, and S. A. Kivelson, Supercon- ductivity in the doped t − J model: Results for four-leg cylinders, Phys. Rev. B 98, 140505 (2018)

    work page 2018

  76. [76]

    Y. Zhou, J. Sung, E. Brutschea, I. Esterlis, Y. Wang, G. Scuri, R. J. Gelly, H. Heo, T. Taniguchi, K. Watan- abe, G. Zar´ and, M. D. Lukin, P. Kim, E. Demler, and H. Park, Bilayer Wigner crystals in a transition metal dichalcogenide heterostructure, Nature 595, 48 (2021)

    work page 2021

  77. [77]

    Y. Guo, J. Pack, J. Swann, L. Holtzman, M. Cothrine, K. Watanabe, T. Taniguchi, D. G. Mandrus, K. Barmak, J. Hone, A. J. Millis, A. Pasupathy, and C. R. Dean, Superconductivity in 5.0 ° twisted bilayer WSe2, Nature 637, 839 (2025)

    work page 2025

  78. [78]

    Mivehvar, P

    F. Mivehvar, P. , Francesco, D. , Tobias, , and H. Ritsch, Cavity QED with quantum gases: new paradigms in many-body physics, Adv. Phys. 70, 1 (2021)

    work page 2021

  79. [79]

    Schneider, D

    C. Schneider, D. Porras, and T. Schaetz, Experimental quantum simulations of many-body physics with trapped ions, Rep. Prog. Phys. 75, 024401 (2012)

    work page 2012

  80. [80]

    E. J. Davis, G. Bentsen, L. Homeier, T. Li, and M. H. Schleier-Smith, Photon-Mediated Spin-Exchange Dynamics of Spin-1 Atoms, Phys. Rev. Lett. 122, 010405 (2019)

    work page 2019

Showing first 80 references.