Pith. sign in

REVIEW 4 major objections 3 minor 1 cited by

Exact many-body wavefunction of the Kondo model with time-dependent interaction strength

T0 review · 4 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read The paper claims an exact many-body wavefunction for the Kondo model with time-dependent coupling $J(t)=\lambda t+p(t)\pm\sqrt{(\lambda t+p(t))^2+4/3}$ on a ring.

desk verdict Real first for time-dependent Kondo via qKZ, but the exact-solution claim rests on an undefined infinite sum. read the letter →

arxiv 2509.05640 v2 pith:GWYLP7ZA submitted 2025-09-06 cond-mat.str-el

Parameshwar R. Pasnoori , Emil. A. Yuzbashyan This is my paper · ORCID
classification cond-mat.str-el
keywords Kondomodeltime-dependentintegrabilityquantumKnizhnik-Zamolodchikovequationsoff-shellBetheansatznonstationarySchrödingerequationperiodicboundaryconditionsdrivenimpuritynonequilibriumdynamics
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

The paper claims to have found the first exact many-body solution of the time-dependent Kondo model, in which a magnetic impurity exchanges spin with conduction electrons through a coupling $J(t)$ that varies in time. It shows that for couplings of the form $J(t)=\lambda t+p(t)\pm\sqrt{(\lambda t+p(t))^2+4/3}$, with $p(t)$ an arbitrary periodic function, the nonstationary Schrödinger equation with periodic boundary conditions can be solved exactly, and it writes down the full $N$-particle wavefunction. A sympathetic reader would care because continuously driven interacting quantum systems are usually intractable; an exact wavefunction gives direct access to real-time observables such as impurity polarization, spin correlations, and entanglement entropy in a strongly correlated system far from equilibrium. The construction works by reducing the periodic boundary conditions to a system of quantum Knizhnik–Zamolodchikov difference equations, which are solved by an off-shell Bethe ansatz.

What carries the argument

The load-bearing object is the off-shell Bethe ansatz solution of the quantum Knizhnik–Zamolodchikov (qKZ) difference equations. The qKZ equations are a system of finite-difference equations for a vector-valued amplitude $\phi(y_0,\dots,y_N)$ with step $\kappa$, whose transport operators are ordered products of XXX $R$-matrices; here they arise from the periodic boundary conditions imposed on the Kondo wavefunction. The paper generalizes the existing off-shell construction, developed for fixed step $\kappa=2$ and crossing parameter $c=1$, to arbitrary ratio $\kappa/c$: the amplitude is written as a sum over products of creation operators $B(\{y_i\},u_j)$ acting on a fully polarized reference state, with each $u_j=\tilde u_j-l_j$ summed over integer shifts $l_j$, and the weight function $w(\{y_i\},\{u_j\})$ is chosen so that the unwanted terms, produced when the transfer matrix is commuted past the $B$-operators, cancel pairwise. This cancellation, together with the Gamma-function solutions of the resulting functional equations, is what makes the wavefunction exact.

What would settle it

Evaluate the truncated version of the sum in Eq. (99) for a small number of electrons (e.g., $N=1$ or $N=2$) and a concrete spin sector, substitute it into the periodic-boundary difference equation (66) and into the time-dependent Schrödinger equation (14), and check whether the residual goes to zero as more integer shifts $l_j$ are included. A direct numerical check for a coupling $J(t)$ outside the family (2) of whether the transport-operator consistency condition (68) fails would also test the claim that Eq. (2) is the most general solvable form.

Watch

Extended reading notes

Core claim

The central claim is that the nonstationary Schrödinger equation for the Kondo Hamiltonian on a ring is exactly solvable for a one-parameter family of time-dependent exchange couplings, $J(t)=\lambda t+p(t)\pm\sqrt{(\lambda t+p(t))^2+4/3}$, and that this family is the most general one for which the exact solution exists. The solution is not a plane-wave Bethe state: away from the impurity the amplitudes are arbitrary functions of the light-cone coordinates $z_j=x_j-t$, and all sector amplitudes are generated from a single reference amplitude by particle–particle and particle–impurity $S$-matrices that obey the Yang–Baxter algebra. Periodic boundary conditions convert the constraint on the reference amplitude into matrix difference equations, which coincide with the quantum Knizhnik–Zamolodchikov equations when the coupling has the above form. Solving those equations by the off-shell Bethe ansatz yields the explicit $N$-particle wavefunction, Eq. (99), as a sum over shifted lattice rapidities and Gamma-function kernels.

Load-bearing premise

The argument assumes that the infinite sum over integer shifts $l_j$ in the off-shell Bethe ansatz solution defines a legitimate single-valued function for arbitrary complex parameters $\tilde u_j$; convergence and analytic continuation of this sum are never discussed, and if it diverges or is multivalued the explicit wavefunction in Eq. (99) is not well-defined.

Editorial extensions

If this is right

  • For any particle number $N$ and any initial spin sector, the exact time-evolving state is explicitly known, so observables such as the impurity polarization $\langle S^z_{\rm imp}\rangle$, the electron spin-density profile, and the impurity–electron spin correlator can be computed from closed-form expressions rather than by simulation.
  • In the adiabatic limit $\lambda\to0$ the system tracks the instantaneous ground state, forming a singlet between the impurity and an electron at $x=0$, with $\langle S^z_{\rm imp}\rangle=0$ and entanglement entropy $\ln 2$; in the diabatic limit $\lambda\to\infty$ the state freezes, with $\langle S^z_{\rm imp}\rangle=\tfrac12$ and zero entanglement entropy.
  • The overall phase factor $e^{i\sum_j\gamma(z_j)}$ is common to all amplitudes and cancels from equal-time correlation functions, so physical predictions do not require constructing the phase function $\gamma$ explicitly.
  • The same strategy—linear dispersion plus integrability of the static model—is expected to yield exact time-dependent solutions of other one-dimensional integrable field theories, in particular the Gross–Neveu and Thirring models.

Reading between the lines

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

  • The unresolved convergence of the $l_j$ sum suggests a natural follow-up: establish the parameter domain ($\lambda$, $c$, particle number) on which Eq. (99) defines a normalizable state, and check whether analytic continuation extends the solution beyond that domain.
  • Because the solution is explicit and controlled by the single sweep rate $\lambda$, it offers a clean benchmark for approximation schemes on driven impurity systems; a testable prediction the authors do not develop is the shape of the adiabatic-to-diabatic crossover, including whether any sharp feature appears at a critical $\lambda$.
  • The contrast with classical-Yang–Baxter-based models suggests a broader organizing principle: models built from the quantum Yang–Baxter equation may admit time-dependent couplings in which static external parameters become light-cone coordinates, pointing to analogues in other $R$-matrix models with linear dispersion.
  • The claim that Eq. (2) is the most general solvable form rests on the qKZ map being the only route to exactness; a skeptic could attempt a coupling outside Eq. (2) and check whether some other ansatz still solves the Schrödinger equation.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. This paper claims an exact solution of the non-stationary Schrödinger equation for the spin-1/2 Kondo impurity model with a time-dependent exchange coupling J(t) of the form λt+p(t)±√((λt+p(t))²+4/3), where p(t) is an arbitrary periodic function, under periodic boundary conditions. The solution strategy is to write the N-particle wavefunction in terms of sector amplitudes related by particle-particle and particle-impurity S-matrices, impose periodic boundary conditions to obtain matrix difference equations, map those difference equations to quantum Knizhnik–Zamolodchikov (qKZ) equations with the XXX R-matrix, and then solve the qKZ equations by an off-shell Bethe ansatz (Appendix A). The central object is Eq. (99), the explicit N-particle wavefunction, which contains an unrestricted infinite sum over integer shifts of the Bethe parameters. The paper also claims that Eq. (2) gives the most general functional form of J(t) for which the problem is exactly solvable, and discusses extensions to Gross–Neveu and Thirring models.

Significance. If correct, this is a substantial step: it extends time-dependent integrability beyond the previously known Gaudin-type models (rooted in the classical Yang–Baxter equation) to a genuinely quantum Yang–Baxter structure, providing a rare example of an exact time-dependent many-body wavefunction for an interacting impurity model. The mapping from periodic boundary conditions to qKZ equations is an original and nontrivial construction, and the use of an external benchmark (the known qKZ system) reduces the risk of circular reasoning. The concrete family of J(t) and the explicit wavefunction offer a new tool for studying coherent nonequilibrium dynamics in a strongly correlated system. However, the main result is currently presented as a formal expression whose mathematical status is not fully specified, and the N-particle verification is sketched rather than demonstrated. The paper has the potential to be a strong contribution, but it requires nontrivial revision to make the central claim rigorous.

major comments (4)
  1. [Appendix A.3, Eq. (A.23); Section 5, Eq. (99)] The infinite sum over integer shifts l_j (with u_j = euj − l_j) defines the central object of the paper, but its convergence and summation order are never specified. For a single large shift l_m with other parameters fixed, the summand contains N+1 Gamma-function ratios Γ(y_i−u_m+1−ic)/Γ(y_i−u_m+1) ∼ (u_m)^{−ic}, (M−1) factors τ(u_i−u_m) ∼ (u_i−u_m)^{2ic}, and one factor from the B-operator ∼ 1/u_m, so the series is at best conditionally convergent with a slowly oscillating phase; in several summation orders it may be order-dependent or divergent. No regularization, analytic continuation, or restriction to parameters where convergence holds is provided. Since Eqs. (56), (60)-(61), (65), and (99) are solved only through this amplitude, the exact-solution claim is unsupported unless the sum is properly defined. The authors should either prove convergence in a suitable sense, define the sum via analytic continuation (e.g., as a Jackson-type integral or a contour integral), or state explicitly the restricted parameter domain where the expression is well-defined.
  2. [Section 2.3 and Appendix A.3] There is an internal inconsistency in the spin projection. Section 5 states S^z = (N+1)/2 − 2M for the state built from M B-operators, whereas Appendix A.3, Eq. (A.23) correctly gives S^z = (N+1)/2 − M for a state obtained by applying M B-operators to the all-up reference state. Since Eq. (99) contains exactly M B-operators, the section 5 formula assigns the wavefunction to the wrong spin sector. This misprint should be corrected, and the authors should verify that the resulting spin sector is compatible with the physical sector they intend to describe.
  3. [Section 4 and Introduction] The N-particle solution is asserted rather than explicitly verified. The derivation of the exact wavefunction relies on the claim that the ansatz (A.39) satisfies the qKZ equation (A.1) through pairwise cancellation of the unwanted terms UW^m_A and UW^m_D, but the cancellation is only summarized by Eq. (A.47) with the sentence 'One can verify...'. Moreover, the generalization from Babujian's κ=2, c=1 case to arbitrary κ/c is not demonstrated; the functional equations (A.46) are stated, but the computation showing that the right-hand side of Eq. (A.48) matches the left-hand side of the qKZ equation is omitted. An explicit verification of the cancellations for arbitrary κ/c, or a detailed reference that covers the general case, is needed to make the solution self-consistent.
  4. [Section 4, Eq. (88); Introduction, Eq. (2)] The paper claims that Eq. (2) specifies the 'most general functional form' of J(t) for which the time-dependent Kondo problem admits an exact solution. The derivation establishes a sufficient condition: if g(z) satisfies g(z+L)=g(z)+κ/c, then the boundary-condition difference equations map to qKZ, leading to J(t) of the stated form. However, no argument rules out other time dependences that could be solvable by a different construction. The claim is therefore not proven. It should be weakened to 'the most general form for which the qKZ-based construction applies' or supported by a no-go theorem showing that any exact solution must satisfy the stated condition.
minor comments (3)
  1. [Section 4, Eq. (82)-(93)] The dot appearing above the square-root bracket in Eq. (2) appears to be a typographical artifact; the coupling is correctly written with a square root in Eq. (88) and in the abstract.
  2. [Abstract and Section 5] The phase function γ(z), defined through h(z)=e^{iγ(z)} and the difference equation (93), is never explicitly constructed, and the claim that it cancels from equal-time correlation functions is stated without proof. Since γ(z) is a product of functions of the individual z_i, it is not a single global phase, and the cancellation argument should be spelled out (e.g., by noting that all sector amplitudes share the same factor ∏ h(z_i)).
  3. [Various] The text contains several typos: 'low dimesional' in the abstract, 'Y ang–Baxter' with an extra space, and '..' at the end of Eq. (104). The derivation of Eq. (104) from Eqs. (47) and (87) should also be made explicit, as the phase factor contains a '±' sign that is not explained.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the J(t) family is derived from the qKZ compatibility condition g(z+L)=g(z)+κ/c, and the wavefunction is an explicit off-shell Bethe ansatz solution; the only self-citation is minor and non-load-bearing.

full rationale

None of the paper's load-bearing claims reduces to its own inputs by construction. Section 4 derives the time-dependence family rather than assuming it: matching the boundary-condition difference equations (76) to the qKZ equations (73) under the variable change (80) yields the compatibility condition g(z+L)=g(z)+κ/c, Eq. (83), whose general solution is g(z)=κz/(cL)+periodic(z), Eq. (86); inverting the defining relation (87) between g and J then gives Eq. (88), which is Eq. (2). Thus the J(t) family is an output of a stated compatibility condition, not a fitted ansatz or a renamed input. The central wavefunction (99) is obtained from the explicit off-shell Bethe ansatz solution of the qKZ system, Eq. (A.51), with the unwanted-term cancellation displayed in Eqs. (A.42)-(A.48); the method is attributed to Babujian and is generalized in the appendix, so it is an external benchmark rather than a self-imported result. The only self-citation is Ref. [91], used to say that constraints on integrable J(t) were identified previously and that Sec. 2's construction follows it, but Secs. 2-4 rederive the S-matrix relations and the integrability conditions, so the citation is not load-bearing. The claim that Eq. (2) is the most general solvable J(t) is stronger than the qKZ-mappability classification actually proven, and Eq. (A.51)'s unrestricted sum over l_j lacks a convergence or summation-order proof, while Section 5's S^z=(N+1)/2-2M conflicts with Eq. (A.23)'s S^z=(N+1)/2-M; these are correctness or rigor concerns, not circular reductions. The score of 2 reflects only the minor, non-load-bearing self-citation present in the paper.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivation relies on standard integrable model mathematics plus the specific Hamiltonian and ansatz. The main unstated assumption is the convergence of the infinite sum in the qKZ solution. No new physical entities are introduced.

free parameters (3)
  • λ
    Real parameter setting the characteristic time scale of J(t). Free in the Hamiltonian.
  • p(t)
    Arbitrary periodic function of period L appearing in J(t). Functional freedom in the coupling.
  • \tilde{u}_j
    M arbitrary complex parameters in the off-shell Bethe ansatz solution (Eq. A.51). They parameterize the initial state or solution space, not introduced to fit data.
assumptions (5)
  • domain assumption The Kondo Hamiltonian with linear dispersion and a δ-function impurity interaction (Eq. 1) is the model to solve.
    The entire paper assumes this Hamiltonian as the physical starting point.
  • domain assumption The wavefunction can be decomposed into ordering sectors with arbitrary functions of light-cone coordinates x-t, and amplitudes related by S-matrices.
    This is the standard coordinate Bethe ansatz structure, asserted in Section 2 and used throughout.
  • standard math The quantum Yang-Baxter equation holds for the R-matrices and S-matrices.
    Used to ensure consistency of the multi-particle S-matrix relations (Eqs. 37, 43, 48). Standard in integrable models.
  • standard math The qKZ equations with the XXX R-matrix admit an off-shell Bethe ansatz solution of the form Eq. (A.51), with arbitrary κ/c.
    Taken from Babujian et al. and extended here. The functional equations for η and τ are verified, but the completeness and convergence of the sum are assumed.
  • ad hoc to paper The infinite sum over integer lattice shifts in Eq. (A.51) converges or is defined by analytic continuation for the relevant parameters.
    No convergence proof or contour specification is given. This is load-bearing for the wavefunction to be well-defined.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Exact many-body wavefunction of the Kondo model with time-dependent interaction strength." pith.science (2026). https://pith.science/paper/GWYLP7ZA

@misc{pith2026250905640,
  author       = {Pith},
  title        = {Pith review of: Exact many-body wavefunction of the Kondo model with time-dependent interaction strength},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GWYLP7ZA}},
  note         = {Machine review of arXiv:2509.05640}
}
abstract

Quantum integrabilty has been applied to a large variety of low dimesional Hamiltonians in Quantum Field Theory, Condensed Matter Physics, and Statistical Mechanics to obtain exact expressions for the spectrum and thermodynamics of these systems. In most of these studies the coupling constants are constant in time. Here we present an exact solution of the nonstationary Schr\"odinger equation for the Kondo Hamiltonian with a time-dependent spin-exchange coupling $J(t)$ of the form $\lambda t + p(t) \pm \sqrt{(\lambda t + p(t))^2 + 4/3}$, where $p(t)$ is an arbitrary periodic function, under periodic boundary conditions. Unlike previously studied time-dependent integrable models, which are rooted in the classical Yang--Baxter structure and associated Knizhnik--Zamolodchikov equations, our approach is based on the quantum Knizhnik--Zamolodchikov framework and the quantum Yang--Baxter algebra. Our results broaden the domain of time-dependent integrability to a genuinely quantum class of models and provide a new tools for exploring coherent nonequilibrium dynamics in strongly correlated systems.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Time-Dependent Integrability from Gauge Theory, I

    hep-th 2026-07 accept novelty 7.5 of 10

    Spacetime-dependent 4d Chern-Simons theory generates time-dependent integrable field theories whose allowed time dependence coincides with one-loop RG flow while preserving Lax integrability.

Reference graph

Works this paper leans on

127 extracted references · 75 canonical work pages · cited by 1 Pith paper

  1. [91]

    P. R. Pasnoori, Integrability of the Kondo model with time-dependent interaction strength, Phys. Rev. B 112 , L060409 (2025), doi:10.1103/78xb-5lmw https://doi.org/10.1103/78xb-5lmw

    work page doi:10.1103/78xb-5lmw 2025

  2. [1]

    I. M. Georgescu, S. Ashhab and F. Nori, Quantum simulation, Rev. Mod. Phys. 86 , 153 (2014), doi:10.1103/RevModPhys.86.153 https://doi.org/10.1103/RevModPhys.86.153

    doi:10.1103/revmodphys.86.153 2014

  3. [2]

    Gross and I

    C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357 , 995 (2017), doi:10.1126/science.aal3837 https://doi.org/10.1126/science.aal3837

    doi:10.1126/science.aal3837 2017

  4. [3]

    Blatt and C

    R. Blatt and C. F. Roos, Quantum simulations with trapped ions, Nat. Phys. 8 , 277 (2012), doi:10.1038/nphys2252 https://doi.org/10.1038/nphys2252

    doi:10.1038/nphys2252 2012

  5. [4]

    A. A. Houck, H. E. T\"ureci and J. Koch, On-chip quantum simulation with superconducting circuits, Nat. Phys. 8 , 292 (2012), doi:10.1038/nphys2251 https://doi.org/10.1038/nphys2251

    doi:10.1038/nphys2251 2012

  6. [5]

    Monroe, W

    C. Monroe, W. C. Campbell, L.-M. Duan, Z.-X. Gong, A. V. Gorshkov, P. Hess, R. Islam, K. Kim, N. M. Linke, G. Pagano, P. Richerme, C. Senko and N. Y. Yao, Programmable quantum simulations of spin systems with trapped ions, Rev. Mod. Phys. 93 , 025001 (2021), doi:10.1103/RevModPhys.93.025001 https://doi.org/10.1103/RevModPhys.93.025001

    doi:10.1103/revmodphys.93.025001 2021

  7. [6]

    M. Gong et al., Quantum walks on a programmable two-dimensional 62-qubit superconducting processor, Science 372 , 948--952 (2021), doi:10.1126/science.abg7812 https://doi.org/10.1126/science.abg7812

    work page doi:10.1126/science.abg7812 2021

  8. [7]

    K. J. Satzinger et al., Realizing topologically ordered states on a quantum processor, Science 374 , 1237--1241 (2021), doi:10.1126/science.abi8378 https://doi.org/10.1126/science.abi8378

    doi:10.1126/science.abi8378 2021

Show all 127 references
  1. [8]

    I. Cong, H. Levine, A. Keesling, D. Bluvstein, S.-T. Wang and M. D. Lukin, Hardware-Efficient, Fault-Tolerant Quantum Computation with Rydberg Atoms, Phys. Rev. X 12 , 021049 (2022), doi:10.1103/PhysRevX.12.021049 https://doi.org/10.1103/PhysRevX.12.021049

    2022 doi

  2. [9]

    X. Mi et al., Stable quantum-correlated many-body states through engineered dissipation, Science 383 , 1332--1337 (2024), doi:10.1126/science.adh9932 https://doi.org/10.1126/science.adh9932

    2024 doi

  3. [10]

    Kinoshita, T

    T. Kinoshita, T. Wenger and D. S. Weiss, A quantum Newton's cradle, Nature 440 , 900--903 (2006), doi:10.1038/nature04693 https://doi.org/10.1038/nature04693

    2006 doi

  4. [11]

    Rigol, V

    M. Rigol, V. Dunjko, V. Yurovsky and M. Olshanii, Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons, Phys. Rev. Lett. 98 , 050405 (2007), doi:10.1103/PhysRevLett.98.050...

    2007 doi

  5. [12]

    M. S. Foster, E. A. Yuzbashyan and B. L. Altshuler, Quantum quench in one dimension: Coherent inhomogeneity amplification and ``supersolitons'', Phys. Rev. Lett. 105 , 135701 (2010), doi:10.1103/PhysRevLett.105.135701 https://doi.org/10.1103/PhysRevLett.105.135701

    2010 doi

  6. [13]

    Polkovnikov, K

    A. Polkovnikov, K. Sengupta, A. Silva and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83 , 863--883 (2011), doi:10.1103/RevModPhys.83.863 https://doi.org/10.1103/RevModPhys.83.863

    2011 doi

  7. [14]

    M. S. Foster, T. C. Berkelbach, D. R. Reichman and E. A. Yuzbashyan, Quantum quench spectroscopy of a Luttinger liquid: Ultrarelativistic density wave dynamics due to fractionalization in an XXZ chain, Phys. Rev. B 84 , 085146 (2011), doi:10.1103/PhysRevB.84.085146 https://doi...

    2011 doi

  8. [15]

    Matsunaga and R

    R. Matsunaga and R. Shimano, Nonequilibrium BCS state dynamics induced by intense terahertz pulses in a superconducting NbN film, Phys. Rev. Lett. 109 , 187002 (2012), doi:10.1103/PhysRevLett.109.187002 https://doi.org/10.1103/PhysRevLett.109.187002

    2012 doi

  9. [16]

    M. Beck, I. Rousseau, M. Klammer, P. Leiderer, M. Mittendorff, S. Winnerl, M. Helm, G. N. Gol'tsman and J. Demsar, Transient increase of the energy gap of superconducting NbN thin films excited by resonant narrow-band terahertz pulses, Phys. Rev. Lett. 110 , 267003 (2013), doi...

    2013 doi

  10. [17]

    Caux and F

    J.-S. Caux and F. H. L. Essler, Time evolution of local observables after quenching to an integrable model, Phys. Rev. Lett. 110 , 257203 (2013), doi:10.1103/PhysRevLett.110.257203 https://doi.org/10.1103/PhysRevLett.110.257203

    2013 doi

  11. [18]

    Matsunaga, Y

    R. Matsunaga, Y. I. Hamada, K. Makise, Y. Uzawa, H. Terai, Z. Wang and R. Shimano, Light-induced collective pseudospin precession resonating with the Higgs mode in a superconductor, Science 345 , 1145--1149 (2014), doi:10.1126/science.1254697 https://doi.org/10.1126/science.1254697

    2014 doi

  12. [19]

    M. S. Foster, V. Gurarie, M. Dzero and E. A. Yuzbashyan, Quench-induced Floquet topological p -wave superfluids, Phys. Rev. Lett. 113 , 076403 (2014), doi:10.1103/PhysRevLett.113.076403 https://doi.org/10.1103/PhysRevLett.113.076403

    2014 doi

  13. [20]

    Wouters, J

    B. Wouters, J. De Nardis, M. Brockmann, D. Fioretto, M. Rigol and J.-S. Caux, Quenching the anisotropic Heisenberg chain: Exact solution and generalized Gibbs ensemble predictions, Phys. Rev. Lett. 113 , 117202 (2014), doi:10.1103/PhysRevLett.113.117202 https://doi.org/10.1103...

    2014 doi

  14. [21]

    E. A. Yuzbashyan, M. Dzero, V. Gurarie and M. S. Foster, Quantum quench phase diagrams of an s -wave BCS-BEC condensate, Phys. Rev. A 91 , 033628 (2015), doi:10.1103/PhysRevA.91.033628 https://doi.org/10.1103/PhysRevA.91.033628

    2015 doi

  15. [22]

    Langen, S

    T. Langen, S. Erne, R. Geiger, B. Rauer, T. Schweigler, M. Kuhnert, W. Rohringer, I. E. Mazets, T. Gasenzer and J. Schmiedmayer, Experimental observation of a generalized Gibbs ensemble, Science 348 , 207--211 (2015), doi:10.1126/science.1257026 https://doi.org/10.1126/science.1257026

    2015 doi

  16. [23]

    Ilievski, J

    E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F. H. L. Essler and T. Prosen, Complete generalized Gibbs ensembles in an interacting theory, Phys. Rev. Lett. 115 , 157201 (2015), doi:10.1103/PhysRevLett.115.157201 https://doi.org/10.1103/PhysRevLett.115.157201

    2015 doi

  17. [24]

    F. H. L. Essler and M. Fagotti, Quench dynamics and relaxation in isolated integrable quantum spin chains, J. Stat. Mech. 2016 , 064002 (2016), doi:10.1088/1742-5468/2016/06/064002 https://doi.org/10.1088/1742-5468/2016/06/064002

    2016 doi

  18. [25]

    Vidmar and M

    L. Vidmar and M. Rigol, Generalized Gibbs ensemble in integrable lattice models, J. Stat. Mech. 064007 (2016), doi:10.1088/1742-5468/2016/06/064007 https://doi.org/10.1088/1742-5468/2016/06/064007

    2016 doi

  19. [26]

    O. A. Castro-Alvaredo, B. Doyon and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6 , 041065 (2016), doi:10.1103/PhysRevX.6.041065 https://doi.org/10.1103/PhysRevX.6.041065

    2016 doi

  20. [27]

    E. A. Yuzbashyan, Generalized microcanonical and Gibbs ensembles in classical and quantum integrable dynamics, Ann. Phys. 367 , 288--296 (2016), doi:10.1016/j.aop.2016.02.002 https://doi.org/10.1016/j.aop.2016.02.002

    2016 doi

  21. [28]

    Matsunaga, N

    R. Matsunaga, N. Tsuji, H. Fujita, A. Sugioka, K. Makise, Y. Uzawa, H. Terai, Z. Wang, H. Aoki and R. Shimano, Polarization-resolved terahertz third-harmonic generation in a single-crystal superconductor NbN: Dominance of the Higgs mode beyond the BCS approximation, Phys. Rev....

    2017 doi

  22. [29]

    J. A. Scaramazza, P. Smacchia and E. A. Yuzbashyan, Consequences of integrability breaking in quench dynamics of pairing Hamiltonians, Phys. Rev. B 99 , 054520 (2019), doi:10.1103/PhysRevB.99.054520 https://doi.org/10.1103/PhysRevB.99.054520

    2019 doi

  23. [30]

    Shankar, E

    A. Shankar, E. A. Yuzbashyan, V. Gurarie, P. Zoller, J. J. Bollinger and A. M. Rey, Simulating dynamical phases of chiral p+i p superconductors with a trapped-ion magnet, PRX Quantum 3 , 040324 (2022), doi:10.1103/PRXQuantum.3.040324 https://doi.org/10.1103/PRXQuantum.3.040324

    2022 doi

  24. [31]

    Morvan, T

    A. Morvan, T. I. Andersen, X. Mi et al., Formation of robust bound states of interacting microwave photons, Nature 612 , 240--245 (2022), doi:10.1038/s41586-022-05348-y https://doi.org/10.1038/s41586-022-05348-y

    2022 doi

  25. [32]

    D. J. Young, A. Chu, E. Y. Song et al., Observing dynamical phases of BCS superconductors in a cavity QED simulator, Nature 625 , 679--684 (2024), doi:10.1038/s41586-023-06911-x https://doi.org/10.1038/s41586-023-06911-x

    2024 doi

  26. [33]

    Kim and A

    H. Kim and A. Polkovnikov, Integrability as an attractor of adiabatic flows, Phys. Rev. B 109 , 195162 (2024), doi:10.1103/PhysRevB.109.195162 https://doi.org/10.1103/PhysRevB.109.195162

    2024 doi

  27. [34]

    L. D. Landau, A theory of energy transfer. II, in Collected papers of L. D. Landau, Elsevier, Amsterdam, Netherlands, ISBN 9780080105864 (1965), doi:10.1016/B978-0-08-010586-4.50014-6 https://doi.org/10.1016/B978-0-08-010586-4.50014-6

    1965 doi

  28. [35]

    Zener and R

    C. Zener and R. H. Fowler, Non-adiabatic crossing of energy levels, Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 137, 696 (1932), doi:10.1098/rspa.1932.0165 https://doi.org/10.1098/rspa.1932.0165

    1932

  29. [36]

    Majorana, Atomi orientati in campo magnetico variabile, Nuovo Cim

    E. Majorana, Atomi orientati in campo magnetico variabile, Nuovo Cim. 9, 43 (1932), doi:10.1007/BF02960953 https://doi.org/10.1007/BF02960953

    1932 doi

  30. [37]

    E. C. G. Stueckelberg, Theory of inelastic collisions between atoms, Helv. Phys. Acta 5, 369 (1932)

    1932

  31. [38]

    R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, McGraw--Hill, New York (1965)

    1965

  32. [39]

    Mukhanov and S

    V. Mukhanov and S. Winitzki, Introduction to Quantum Effects in Gravity, Cambridge University Press, Cambridge (2007)

    2007

  33. [40]

    A. M. Perelomov and V. S. Popov, Group-theoretical aspects of the variable frequency oscillator problem, Theor. Math. Phys. 1 , 360 (1969), doi:10.1007/BF01035742 https://doi.org/10.1007/BF01035742

    1969 doi

  34. [41]

    Y. N. Demkov and V. I. Osherov, Stationary and nonstationary problems in quantum mechanics that can be solved by means of contour integration, Sov. Phys. JETP 26 , 916 (1968)

    1968

  35. [42]

    V. N. Ostrovsky and H. Nakamura, Exact analytical solution of the N-level Landau--Zener-type bow-tie model, J. Phys. A: Math. Gen. 30 , 6939 (1997), doi:10.1088/0305-4470/30/19/028 https://doi.org/10.1088/0305-4470/30/19/028

    1997 doi

  36. [43]

    Y. N. Demkov and V. N. Ostrovsky, Multipath interference in a multistate Landau--Zener-type model, Phys. Rev. A 61 , 032705 (2000), doi:10.1103/PhysRevA.61.032705 https://doi.org/10.1103/PhysRevA.61.032705

    2000 doi

  37. [44]

    Y. N. Demkov and V. N. Ostrovsky, The exact solution of the multistate Landau--Zener-type model: The generalized bow-tie model, J. Phys. B: At. Mol. Opt. Phys. 34 , 2419 (2001), doi:10.1088/0953-4075/34/12/309 https://doi.org/10.1088/0953-4075/34/12/309

    2001 doi

  38. [45]

    V. Y. Chernyak, N. A. Sinitsyn and C. Sun, A large class of solvable multistate Landau--Zener models and quantum integrability, J. Phys. A: Math. Theor. 51 , 245201 (2018), doi:10.1088/1751-8121/aac3b2 https://doi.org/10.1088/1751-8121/aac3b2

    2018 doi

  39. [46]

    V. Y. Chernyak, N. A. Sinitsyn and C. Sun, Multitime Landau--Zener model: classification of solvable Hamiltonians, J. Phys. A: Math. Theor. 53 , 185203 (2020), doi:10.1088/1751-8121/ab7fdd https://doi.org/10.1088/1751-8121/ab7fdd

    2020 doi

  40. [47]

    Barik, L

    S. Barik, L. Bakker, V. Gritsev and E. A. Yuzbashyan, Knizhnik--Zamolodchikov equations and integrable hyperbolic Landau--Zener models, SciPost Phys. 18 , 212 (2025), doi:10.21468/SciPostPhys.18.6.212 https://doi.org/10.21468/SciPostPhys.18.6.212

    2025 doi

  41. [48]

    Polkovnikov, Universal adiabatic dynamics in the vicinity of a quantum critical point, Phys

    A. Polkovnikov, Universal adiabatic dynamics in the vicinity of a quantum critical point, Phys. Rev. B 72 , 161201(R) (2005), doi:10.1103/PhysRevB.72.161201 https://doi.org/10.1103/PhysRevB.72.161201

    2005 doi

  42. [49]

    W. H. Zurek, U. Dorner and P. Zoller, Dynamics of a quantum phase transition, Phys. Rev. Lett. 95 , 105701 (2005), doi:10.1103/PhysRevLett.95.105701 https://doi.org/10.1103/PhysRevLett.95.105701

    2005 doi

  43. [50]

    Dziarmaga, Dynamics of a quantum phase transition: Exact solution of the quantum Ising model, Phys

    J. Dziarmaga, Dynamics of a quantum phase transition: Exact solution of the quantum Ising model, Phys. Rev. Lett. 95 , 245701 (2005), doi:10.1103/PhysRevLett.95.245701 https://doi.org/10.1103/PhysRevLett.95.245701

    2005 doi

  44. [51]

    Galitski, Quantum-to-classical correspondence and Hubbard--Stratonovich dynamical systems: A Lie-algebraic approach, Phys

    V. Galitski, Quantum-to-classical correspondence and Hubbard--Stratonovich dynamical systems: A Lie-algebraic approach, Phys. Rev. A 84 , 012118 (2011), doi:10.1103/PhysRevA.84.012118 https://doi.org/10.1103/PhysRevA.84.012118

    2011 doi

  45. [52]

    Ringel and V

    M. Ringel and V. Gritsev, Dynamical symmetry approach to path integrals of quantum spin systems, Phys. Rev. A 88 , 062105 (2013), doi:10.1103/PhysRevA.88.062105 https://doi.org/10.1103/PhysRevA.88.062105

    2013 doi

  46. [53]

    Davidson and A

    S. Davidson and A. Polkovnikov, SU(3) semiclassical representation of quantum dynamics of interacting spins, Phys. Rev. Lett. 114 , 045701 (2015), doi:10.1103/PhysRevLett.114.045701 https://doi.org/10.1103/PhysRevLett.114.045701

    2015 doi

  47. [54]

    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), doi:10.1088/1751-8113/48/24/245303 https://doi.org/10.1088/1751-8113/48/24/245303

    2015 doi

  48. [55]

    Gritsev and A

    V. Gritsev and A. Polkovnikov, Integrable Floquet dynamics, SciPost Phys. 2 , 021 (2017), doi:10.21468/SciPostPhys.2.3.021 https://doi.org/10.21468/SciPostPhys.2.3.021

    2017 doi

  49. [56]

    Malikis and V

    S. Malikis and V. Cheianov, Exact S-matrices for higher dimensional representations of generalized Landau--Zener Hamiltonians, arXiv:2505.06048 (2025), doi:10.48550/arXiv.2505.06048 https://doi.org/10.48550/arXiv.2505.06048

    2025 doi arXiv

  50. [57]

    Hirota, The B\"acklund and inverse scattering transform of the KdV equation with nonuniformities, J

    R. Hirota, The B\"acklund and inverse scattering transform of the KdV equation with nonuniformities, J. Phys. Soc. Jpn. 46 , 1681 (1979), doi:10.1143/JPSJ.46.1681 https://doi.org/10.1143/JPSJ.46.1681

    1979 doi

  51. [58]

    Nirmala, M

    N. Nirmala, M. J. Vedan and B. V. Baby, Auto-B\"acklund transformation, Lax pairs, and Painlev\'e property of a variable coefficient Korteweg--de Vries equation, J. Math. Phys. 27 , 2640 (1986), doi:10.1063/1.527282 https://doi.org/10.1063/1.527282

    1986 doi

  52. [59]

    W. L. Chan and K.-S. Li, Nonpropagating solitons of the variable coefficient and nonisospectral Korteweg--de Vries equation, J. Math. Phys. 30 , 2521 (1989), doi:10.1063/1.528533 https://doi.org/10.1063/1.528533

    1989 doi

  53. [60]

    Ganguly and A

    A. Ganguly and A. Das, Schr\"odinger equation with time-dependent mass function and associated generalized KdV equation, Phys. Scr. 90 , 055204 (2015), doi:10.1088/0031-8949/90/5/055204 https://doi.org/10.1088/0031-8949/90/5/055204

    2015 doi

  54. [61]

    Hlavat\'y, The Painlev\'e analysis of damped KdV equation, J

    L. Hlavat\'y, The Painlev\'e analysis of damped KdV equation, J. Phys. Soc. Jpn. 55 , 1405 (1986), doi:10.1143/JPSJ.55.1405 https://doi.org/10.1143/JPSJ.55.1405

    1986 doi

  55. [62]

    Joshi, Painlev\'e property of general variable-coefficient versions of the Korteweg--de Vries and non-linear Schr\"odinger equations, Phys

    N. Joshi, Painlev\'e property of general variable-coefficient versions of the Korteweg--de Vries and non-linear Schr\"odinger equations, Phys. Lett. A 125 , 456 (1987), doi:10.1016/0375-9601(87)90184-8 https://doi.org/10.1016/0375-9601(87)90184-8

    1987 doi

  56. [63]

    Brugarino, Painlev\'e property, auto-B\"acklund transformation, Lax pairs, and reduction to the standard form for the Korteweg--de Vries equation with nonuniformities, J

    T. Brugarino, Painlev\'e property, auto-B\"acklund transformation, Lax pairs, and reduction to the standard form for the Korteweg--de Vries equation with nonuniformities, J. Math. Phys. 30 , 1013 (1989), doi:10.1063/1.528368 https://doi.org/10.1063/1.528368

    1989 doi

  57. [64]

    V. M. P\'erez-Garc\'ia, P. J. Torres and V. V. Konotop, Similarity transformations for nonlinear Schr\"odinger equations with time-dependent coefficients, Physica D 221 , 31 (2006), doi:10.1016/j.physd.2006.07.002 https://doi.org/10.1016/j.physd.2006.07.002

    2006 doi

  58. [65]

    Malikis and V

    S. Malikis and V. Cheianov, An ideal rapid-cycle Thouless pump, SciPost Phys. 12 , 203 (2022), doi:10.21468/SciPostPhys.12.6.203 https://doi.org/10.21468/SciPostPhys.12.6.203

    2022 doi

  59. [66]

    Hoare, N

    B. Hoare, N. Levine and A. A. Tseytlin, Sigma models with local couplings: a new integrability--RG flow connection, J. High Energ. Phys. 2020 , 020 (2020), doi:10.1007/JHEP11(2020)020 https://doi.org/10.1007/JHEP11(2020)020

    2020 doi

  60. [67]

    N. A. Sinitsyn, E. A. Yuzbashyan, V. Y. Chernyak, A. Patra and C. Sun, Integrable Time- Dependent Quantum Hamiltonians, Phys. Rev. Lett. 120, 190402 (2018), doi:10.1103/PhysRevLett.120.190402 https://doi.org/10.1103/PhysRevLett.120.190402

    2018 doi

  61. [68]

    E. A. Yuzbashyan, Integrable time-dependent Hamiltonians, solvable Landau--Zener models and Gaudin magnets, Ann. Phys. 392 , 323 (2018), doi:10.1016/j.aop.2018.01.017 https://doi.org/10.1016/j.aop.2018.01.017

    2018 doi

  62. [69]

    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), doi:10.1103/PhysRevB.106.104513 https://doi.org/10.1103/PhysRevB.106.104513

    2022 doi

  63. [70]

    Suzuki, R

    F. Suzuki, R. K. Malla, and N. A. Sinitsyn, Competing bosonic reactions: Insight from exactly solvable time-dependent models, Phys. Rev. A 112 , 013307 (2025), doi:10.1103/68zs-gv5y https://doi.org/10.1103/68zs-gv5y

    2025 doi

  64. [71]

    Gaudin, The Bethe wavefunction, Cambridge University Press, Cambridge, UK (2014), ISBN 9781107045859, doi:10.1017/CBO9781107053885 https://doi.org/10.1017/CBO9781107053885

    M. Gaudin, The Bethe wavefunction, Cambridge University Press, Cambridge, UK (2014), ISBN 9781107045859, doi:10.1017/CBO9781107053885 https://doi.org/10.1017/CBO9781107053885

    2014 doi

  65. [72]

    E. K. Sklyanin, Separation of variables in the Gaudin model, J. Math. Sci. 47 , 2473 (1989), doi:10.1007/BF01840429 https://doi.org/10.1007/BF01840429

    1989 doi

  66. [73]

    Ortiz, R

    G. Ortiz, R. Somma, J. Dukelsky and S. Rombouts, Exactly-solvable models derived from a generalized Gaudin algebra, Nucl. Phys. B 707 , 421 (2005), doi:10.1016/j.nuclphysb.2004.11.008 https://doi.org/10.1016/j.nuclphysb.2004.11.008

    2005 doi

  67. [74]

    Skrypnyk, Generalized Gaudin spin chains, non-skew-symmetric r-matrices, and reflection equation algebras, J

    T. Skrypnyk, Generalized Gaudin spin chains, non-skew-symmetric r-matrices, and reflection equation algebras, J. Math. Phys. 48 , 113521 (2007), doi:10.1063/1.2816256 https://doi.org/10.1063/1.2816256

    2007 doi

  68. [75]

    Skrypnyk, Generalized Gaudin systems in a magnetic field and non-skew-symmetric r-matrices, J

    T. Skrypnyk, Generalized Gaudin systems in a magnetic field and non-skew-symmetric r-matrices, J. Phys. A: Math. Theor. 40 , 13337 (2007), doi:10.1088/1751-8113/40/44/014 https://doi.org/10.1088/1751-8113/40/44/014

    2007 doi

  69. [76]

    Fioretto, J.-S

    D. Fioretto, J.-S. Caux and V. Gritsev, Exact out-of-equilibrium central spin dynamics from integrability, New J. Phys. 16 , 043024 (2014), doi:10.1088/1367-2630/16/4/043024 https://doi.org/10.1088/1367-2630/16/4/043024

    2014 doi

  70. [77]

    Barik, L

    S. Barik, L. Bakker, V. Gritsev, J. Min\'a r and E. A. Yuzbashyan, Higher spin Richardson--Gaudin model with time-dependent coupling: Exact dynamics, arXiv:2507.10856 (2025), doi:10.48550/arXiv.2507.10856 https://doi.org/10.48550/arXiv.2507.10856

    2025 doi arXiv

  71. [78]

    F. Li, V. Y. Chernyak and N. A. Sinitsyn, Quantum annealing and thermalization: Insights from integrability, Phys. Rev. Lett. 121 , 190601 (2018), doi:10.1103/PhysRevLett.121.190601 https://doi.org/10.1103/PhysRevLett.121.190601

    2018 doi

  72. [79]

    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), doi:10.1103/PhysRevLett.129.033201 https://doi.org/10.1103/PhysRevLett.129.033201

    2022 doi

  73. [80]

    N. A. Sinitsyn and F. Li, Solvable multistate model of Landau--Zener transitions in cavity QED, Phys. Rev. A 93 , 063859 (2016), doi:10.1103/PhysRevA.93.063859 https://doi.org/10.1103/PhysRevA.93.063859

    2016 doi

  74. [81]

    Sun and N

    C. Sun and N. A. Sinitsyn, Landau--Zener extension of the Tavis--Cummings model: Structure of the solution, Phys. Rev. A 94 , 033808 (2016), doi:10.1103/PhysRevA.94.033808 https://doi.org/10.1103/PhysRevA.94.033808

    2016 doi

  75. [82]

    C. Sun, V. Y. Chernyak, A. Piryatinski and N. A. Sinitsyn, Cooperative light emission in the presence of strong inhomogeneous broadening, Phys. Rev. Lett. 123 , 123605 (2019), doi:10.1103/PhysRevLett.123.123605 https://doi.org/10.1103/PhysRevLett.123.123605

    2019 doi

  76. [83]

    Torrielli, Classical integrability, J

    A. Torrielli, Classical integrability, J. Phys. A: Math. Theor. 49 , 323001 (2016), doi:10.1088/1751-8113/49/32/323001 https://doi.org/10.1088/1751-8113/49/32/323001

    2016 doi

  77. [84]

    Loebbert, Lectures on Yangian symmetry, J

    F. Loebbert, Lectures on Yangian symmetry, J. Phys. A: Math. Theor. 49 , 323002 (2016), doi:10.1088/1751-8113/49/32/323002 https://doi.org/10.1088/1751-8113/49/32/323002

    2016 doi

  78. [85]

    A. L. Retore, Introduction to classical and quantum integrability, J. Phys. A: Math. Theor. 55 , 173001 (2022), doi:10.1088/1751-8121/ac5a8e https://doi.org/10.1088/1751-8121/ac5a8e

    2022 doi

  79. [86]

    J. B. McGuire, Study of exactly soluble one-dimensional N -body problems, J. Math. Phys. 5 , 622--636 (1964), doi:10.1063/1.1704156 https://doi.org/10.1063/1.1704156

    1964 doi

  80. [87]

    C. N. Yang, Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 , 1312--1315 (1967), doi:10.1103/PhysRevLett.19.1312 https://doi.org/10.1103/PhysRevLett.19.1312

    1967 doi

  81. [88]

    A. B. Zamolodchikov and A. B. Zamolodchikov, Factorized S -matrices in two dimensions as the exact solutions of certain relativistic quantum field theory models, Ann. Phys. 120 , 253--291 (1979), doi:10.1016/0003-4916(79)90391-9 https://doi.org/10.1016/0003-4916(79)90391-9

    1979 doi

  82. [89]

    R. J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London (1982), ISBN: 978-0-12-083180-7

    1982

  83. [90]

    Jimbo, Introduction to the Yang--Baxter equation, Int

    M. Jimbo, Introduction to the Yang--Baxter equation, Int. J. Mod. Phys. A 4 , 3759--3777 (1989), doi:10.1142/S0217751X89001503 https://doi.org/10.1142/S0217751X89001503

    1989 doi

  84. [92]

    P. W. Anderson, G. Yuval and D. R. Hamann, Exact results in the Kondo problem. II. Scaling theory, qualitatively correct solution, and some new results on one-dimensional classical statistical models , Phys. Rev. B 1 , 4464 (1970), doi:10.1103/PhysRevB.1.4464 https://doi.org/1...

    1970 doi

  85. [93]

    K. G. Wilson, The renormalization group: Critical phenomena and the Kondo problem , Rev. Mod. Phys. 47 , 773, (1975), doi:10.1103/RevModPhys.47.773 https://doi.org/10.1103/RevModPhys.47.773

    1975 doi

  86. [94]

    Nozi\`eres and A

    Ph. Nozi\`eres and A. Blandin, Kondo effect in real metals, J. Phys. France 41 , 193 (1980), doi:10.1051/jphys:01980004103019300 https://doi.org/10.1051/jphys:01980004103019300

    1980 doi

  87. [95]

    Andrei, Diagonalization of the Kondo Hamiltonian, Phys

    N. Andrei, Diagonalization of the Kondo Hamiltonian, Phys. Rev. Lett. 45 , 379 (1980), doi:10.1103/PhysRevLett.45.379 https://doi.org/10.1103/PhysRevLett.45.379

    1980 doi

  88. [96]

    P. B. Wiegmann, Exact solution of the s-d exchange model (Kondo problem), J. Phys. C: Solid State Phys. 14 , 1463 (1981), doi:10.1088/0022-3719/14/10/014 https://doi.org/10.1088/0022-3719/14/10/014

    1981 doi

  89. [97]

    I. V. Cherednik, Generalized braid groups and local r-matrix systems, Dokl. Akad. Nauk SSSR 307 , 49 (1989); English transl.: Dokl. Math. 40 , 43 (1990)

    1989

  90. [98]

    P. I. Etingof, Representations of affine Lie algebras, elliptic r-matrix systems, and special functions, Commun. Math. Phys. 159 , 471 (1994), doi:10.1007/BF02099981 https://doi.org/10.1007/BF02099981

    1994 doi

  91. [99]

    Relaxing these assumptions does not appear to substantially alter the overall framework Torrielli2016,Skrypnyk2007b

    Cherednik made the standard assumptions that r_ ij (u_1,u_2) depends only on the difference of its arguments, r_ ij (u_1,u_2)=r_ ij (u_1-u_2) , and that it is skew-symmetric, r_ ij (u_1-u_2)=-r_ ij (u_2-u_1) . Relaxing these assumptions does not appear to substantially alter t...

  92. [100]

    A. A. Belavin and V. G. Drinfeld, Solutions of the classical Yang--Baxter equations for simple Lie algebras, Funct. Anal. Appl. 16 , 159 (1982), doi:10.1007/BF01081585 https://doi.org/10.1007/BF01081585

    1982 doi

  93. [101]

    A. A. Belavin and V. G. Drinfeld, Triangle equations for simple Lie algebras, in Mathematical Physics Reviews, ed. S. P. Novikov et al., Harwood Academic Publishers, New York (1984), pp. 93--165

    1984

  94. [102]

    Stolin, On rational solutions of the Yang--Baxter equation for sl (n) , Math

    A. Stolin, On rational solutions of the Yang--Baxter equation for sl (n) , Math. Scand. 69 , 57 (1991), doi:10.7146/math.scand.a-12369 https://doi.org/10.7146/math.scand.a-12369

    1991 doi

  95. [103]

    Hikami, Gaudin magnet with boundary and generalized Knizhnik-Zamolodchikov equation, J

    K. Hikami, Gaudin magnet with boundary and generalized Knizhnik-Zamolodchikov equation, J. Phys. A: Math. Gen. 28 , 4997 (1995), doi:10.1088/0305-4470/28/17/028 https://doi.org/10.1088/0305-4470/28/17/028

    1995 doi

  96. [104]

    H. M. Babujian and A. V. Kitaev, Generalized Knizhnik--Zamolodchikov equations and isomonodromy quantization of the equations integrable via the Inverse Scattering Transform: Maxwell--Bloch system with pumping, J. Math. Phys. 39 , 2499 (1998), doi:10.1063/1.532404 https://doi....

    1998 doi

  97. [105]

    Kurak and A

    V. Kurak and A. Lima-Santos, The A(2)_2 Gaudin model and its associated Knizhnik--Zamolodchikov equation, J. Phys. A: Math. Gen. 38 , 333 (2004), doi:10.1088/0305-4470/38/2/004 https://doi.org/10.1088/0305-4470/38/2/004

    2004 doi

  98. [106]

    Lima-Santos and W

    A. Lima-Santos and W. Utiel, Gaudin magnet with impurity and its generalized Knizhnik--Zamolodchikov equation, Int. J. Mod. Phys. B 20 , 2175 (2006), doi:10.1142/S0217979206034595 https://doi.org/10.1142/S0217979206034595

    2006 doi

  99. [107]

    T. A. Sedrakyan and V. Galitski, Boundary Wess-Zumino-Novikov-Witten model from the pairing Hamiltonian, Phys. Rev. B 82 , 214502 (2010), doi:10.1103/PhysRevB.82.214502 https://doi.org/10.1103/PhysRevB.82.214502

    2010 doi

  100. [108]

    Skrypnyk, Generalized Knizhnik--Zamolodchikov equations, off-shell Bethe ansatz and non-skew-symmetric classical r -matrices, Nucl

    T. Skrypnyk, Generalized Knizhnik--Zamolodchikov equations, off-shell Bethe ansatz and non-skew-symmetric classical r -matrices, Nucl. Phys. B 824 , 436 (2010), doi:10.1016/j.nuclphysb.2009.08.005 https://doi.org/10.1016/j.nuclphysb.2009.08.005

    2010 doi

  101. [109]

    V. G. Knizhnik and A. B. Zamolodchikov, Current algebra and Wess-Zumino model in two dimensions, Nucl. Phys. B 247, 83 (1984), doi:10.1016/0550-3213(84)90374-2 https://doi.org/10.1016/0550-3213(84)90374-2

    1984 doi

  102. [110]

    I. B. Frenkel and N. Yu. Reshetikhin, Quantum affine algebras and holonomic difference equations, Commun. Math. Phys. 146 , 1 (1992), doi:10.1007/BF02099206 https://doi.org/10.1007/BF02099206

    1992 doi

  103. [111]

    Reshetikhin, Jackson-type integrals, Bethe vectors, and solutions to a difference analog of the Knizhnik--Zamolodchikov system, Lett

    N. Reshetikhin, Jackson-type integrals, Bethe vectors, and solutions to a difference analog of the Knizhnik--Zamolodchikov system, Lett. Math. Phys. 26 , 153 (1992), doi:10.1007/BF00420749 https://doi.org/10.1007/BF00420749

    1992 doi

  104. [112]

    F. A. Smirnov, Dynamical symmetries of massive integrable models 1: Form factor bootstrap equations as a special case of deformed Knizhnik--Zamolodchikov equations, Int. J. Mod. Phys. A 07 , 813 (1992), doi:10.1142/S0217751X92004063 https://doi.org/10.1142/S0217751X92004063

    1992 doi

  105. [113]

    F. A. Smirnov, Completely integrable models of quantum field theory, in Form Factors in Completely Integrable Models of Quantum Field Theory, pp. 1--5, World Scientific (1992), doi:10.1142/9789812798312\_0001 https://doi.org/10.1142/9789812798312_0001

    1992 doi

  106. [114]

    Jimbo and T

    M. Jimbo and T. Miwa, Algebraic analysis of solvable lattice models, CBMS Regional Conference Series in Mathematics 85 , AMS, Providence, RI (1994)

    1994

  107. [115]

    Babujian, M

    H. Babujian, M. Karowski and A. Zapletal, Matrix difference equations and a nested Bethe ansatz, J. Phys. A: Math. Gen. 30 , 6425 (1997), doi:10.1088/0305-4470/30/18/019 https://doi.org/10.1088/0305-4470/30/18/019

    1997 doi

  108. [116]

    Etingof and O

    P. Etingof and O. Schiffmann, Lectures on the dynamical Yang--Baxter equations, in: A. Pressley (ed.), Quantum Groups and Lie Theory, London Mathematical Society Lecture Note Series 290 , pp. 89--129, Cambridge University Press (2002), doi:10.1017/CBO9780511542848.007 https://...

    2002 doi

  109. [117]

    Later in the text, we also write the same R -matrix in the form R^ ij (y) = i y\, I^ ij + c\, P^ ij i y + c . This representation follows from R_intro by setting z_i - z_j = i y and using the fact that the permutation operator for two spin- 1 2 degrees of freedom s _i and s _j...

  110. [118]

    Nakayashiki, S

    A. Nakayashiki, S. Pakuliak and V. Tarasov, On solutions of the KZ and qKZ equations at level zero, https://www.numdam.org/item/AIHPA_1999__71_4_459_0/ Ann. Inst. H. Poincar\'e Phys. Th\'eor. 71 , 459 (1999)

    1999

  111. [119]

    M. A. Semenov-Tyan-Shanskii, Classical r-matrices and quantization, J. Math. Sci. 31 , 3411 (1985), doi:10.1007/BF02107242 https://doi.org/10.1007/BF02107242

    1985 doi

  112. [120]

    Andrei, Integrable models in condensed matter physics, in: Series in Modern Condensed Matter Physics, Vol

    N. Andrei, Integrable models in condensed matter physics, in: Series in Modern Condensed Matter Physics, Vol. 6, pp. 457--551, World Scientific, Singapore (1995), doi:10.1142/9789814447027\_0008 https://doi.org/10.1142/9789814447027_0008, ISBN: 978-981-02-2140-9

    1995 doi

  113. [121]

    Grobis, I

    M. Grobis, I. G. Rau, R. M. Potok, H. Shtrikman and D. Goldhaber-Gordon, Universal Scaling in Non-equilibrium Transport Through a Single-Channel Kondo Dot, Phys. Rev. Lett. 100 , 246601 (2008), doi:10.1103/PhysRevLett.100.246601 https://doi.org/10.1103/PhysRevLett.100.246601

    2008 doi

  114. [122]

    Bauer, C

    J. Bauer, C. Salomon and E. Demler, Realizing a Kondo-Correlated State with Ultracold Atoms, Phys. Rev. Lett. 111 , 215304 (2013), doi:10.1103/PhysRevLett.111.215304 https://doi.org/10.1103/PhysRevLett.111.215304

    2013 doi

  115. [123]

    Amaricci, A

    A. Amaricci, A. Richaud, M. Capone, N. D. Oppong and F. Scazza, Engineering the Kondo impurity problem with alkaline-earth atom arrays, arXiv:2505.14630 [cond-mat.quant-gas] (2025), doi:10.48550/arXiv.2505.14630 https://doi.org/10.48550/arXiv.2505.14630

    2025 doi

  116. [124]

    H. B. Thacker, Exact integrability in quantum field theory and statistical systems, Rev. Mod. Phys. 53 , 253?285 (1981), doi:10.1103/RevModPhys.53.253 https://doi.org/10.1103/RevModPhys.53.253

    1981 doi

  117. [125]

    F. H. L. Essler and R. M. Konik, Application of massive integrable quantum field theories to problems in condensed matter physics, in: From Fields to Strings: Circumnavigating Theoretical Physics, Vol. 1, pp. 684--830, World Scientific, Singapore (2005), doi: https://doi.org/1...

    2005 doi

  118. [126]

    E. K. Sklyanin, L. A. Takhtadzhyan and L. D. Faddeev, Quantum inverse problem method. I., Theor. Math. Phys. 40 , 688 (1979), doi:10.1007/BF01018718 https://doi.org/10.1007/BF01018718

    1979 doi

  119. [127]

    Wang, W.-L

    Y. Wang, W.-L. Yang, J. Cao and K. Shi, Off-diagonal Bethe ansatz for exactly solvable models, Springer, Berlin, Heidelberg (2016), doi:10.1007/978-3-662-46756-5 https://doi.org/10.1007/978-3-662-46756-5

    2016 doi

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.