The exact Quench Action solution of quantum quenches in the one-dimensional Hubbard model from integrable two-site K-matrix initial states rests on the overlaps between those states and the paired nested Bethe-ansatz eigenstates. Rylands, Bertini and Calabrese (J. Stat. Mech. 2210 (2022) 103103) derived these overlaps exactly only in the low-density and infinite-repulsion limits and conjectured a closed form at arbitrary filling and interaction: a product of single-particle overlap functions, a ratio of Gaudin determinants, and an explicit combinatorial descendant factor; in the thermodynamic Quench Action they absorbed the sum over descendant sectors into a redefined parameter fixed post hoc rather than evaluating it from first principles. Both gaps remain: no rigorous derivation of the conjectured formula exists, and no exact thermodynamic treatment of the descendant combinatorial sum has replaced the post-hoc redefinition.

Background

The one-dimensional Hubbard model, H=−t∑j,σ(cj+1,σ†cj,σ†+cj,σ†cj+1,σ†)+u∑jnj,↑nj,↓,H=-\mathfrak{t}\sum_{j,\sigma}\left(c^\dagger_{j+1,\sigma}c^{\phantom{\dagger}}_{j,\sigma}+c^\dagger_{j,\sigma}c^{\phantom{\dagger}}_{j+1,\sigma}\right)+\mathfrak{u}\sum_j n_{j,\uparrow}n_{j,\downarrow}, describes spin-1/21/2 fermions hopping on a lattice with on-site interaction u\mathfrak{u}. Following the source, all overlap formulas in this statement are expressed through the reduced interaction strength U=u/(2t)U=\mathfrak{u}/(2\mathfrak{t}) (its Eq. (15)). It is integrable: Lieb and Wu diagonalized it by a nested Bethe ansatz, in which each eigenstate is labelled by NN charge momenta {kj}\{k_j\} and MM spin rapidities {λγ}\{\lambda_\gamma\} satisfying coupled quantization equations ("nested" because the spin sector is itself diagonalized by a second, XXX-spin-chain-type Bethe ansatz). In the thermodynamic limit L→∞L\to\infty the roots are described by rapidity distributions, with the spin rapidities organizing into strings (bound-state patterns of complex roots).

In a quantum quench one prepares a state ∣Ψ0⟩|\Psi_0\rangle that is not an eigenstate of HH and follows its unitary evolution. The Quench Action method of Caux and Essler gives the exact long-time steady state for integrable post-quench Hamiltonians, provided the initial state is an integrable initial state: in the practical criterion of Piroli, Pozsgay and Vernier, a state annihilated by all conserved charges odd under spatial reflection. For such states only Bethe eigenstates whose rapidities come in opposite pairs {kj,−kj}\{k_j,-k_j\}, {λγ,−λγ}\{\lambda_\gamma,-\lambda_\gamma\} have nonzero overlap, and the steady state is found as a saddle point of a functional built from the Yang-Yang entropy (the Bethe-ansatz counting entropy of the root configurations) and the extensive part of the overlaps. The method therefore hinges on knowing the exact overlaps between ∣Ψ0⟩|\Psi_0\rangle and the paired Bethe eigenstates.

Rylands, Bertini and Calabrese constructed integrable initial states for the Hubbard model by taking the Hubbard limit of two-site boundary K-matrices that Gombor and Bajnok had derived for SU(2∣2)SU(2|2)-invariant spin chains (the K-matrix solves the reflection, or boundary Yang-Baxter, equation). The states are products ⊗l=1L/2∣K⟩l\otimes_{l=1}^{L/2}|K\rangle_l of identical two-site states whose coefficients are the entries of KK; in a convenient parametric limit the family reduces to an exponential pair condensate ∣Ψ0⟩=Nexp⁡[∑l=1L/2(αXl−α22Xl2)]∣0⟩,|\Psi_0\rangle=\mathcal{N}\exp\left[\sum_{l=1}^{L/2}\left(\alpha X_l-\frac{\alpha^2}{2}X_l^2\right)\right]|0\rangle, where XlX_l creates a spin singlet on the dimer of sites (2l−1,2l)(2l-1,2l), Xl3=0X_l^3=0, and the real parameter α\alpha fixes the conserved filling through ⟨N⟩/L=4α2/(1+4α2)\langle N\rangle/L=4\alpha^2/(1+4\alpha^2); ∣Ψ0⟩|\Psi_0\rangle is a spin singlet but not a particle-number eigenstate. The Quench Action treatment uses its translation-invariant version ∣Ψ~0⟩=1−T2∣Ψ0⟩|\tilde\Psi_0\rangle=\frac{1-T}{\sqrt{2}}|\Psi_0\rangle, with TT the one-site shift operator (its Eq. (58)); the source notes this replacement is harmless for the stationary state.

For these states the authors computed the overlaps exactly in two limits. At U→∞U\to\infty the charge and spin sectors decouple and the overlap follows from dimer-XXX overlaps. At low density N≪LN\ll L, the spin part of the wavefunction contracts against the dimer into the Gaudin-determinant formula of Gombor and Pozsgay for inhomogeneous XXX chains, and the charge part yields ∣cos⁡(k+Φk)∣2|\cos(k+\Phi_k)|^2 with the bare two-body phase Φk\Phi_k. The source displays Φk=π2−arctan⁡(2sin⁡kU)\Phi_k=\frac{\pi}{2}-\arctan\left(\frac{2\sin k}{U}\right); with this sign, ∣cos⁡(k+Φk)∣2=sk2[ck−U/2]2/(sk2+(U/2)2)|\cos(k+\Phi_k)|^2=s_k^2[c_k-U/2]^2/\left(s_k^2+(U/2)^2\right), whereas the choice Φk=π2+arctan⁡(2sin⁡kU)\Phi_k=\frac{\pi}{2}+\arctan\left(\frac{2\sin k}{U}\right) reproduces exactly the charge factor hc\mathfrak{h}_c of the conjecture below. The displayed phase therefore contains a sign slip internal to the source; the conjectured formula (its Eqs. (87) and (91)) is the object the Problem Statement asks to derive or refute. In every model where overlaps between Bethe states and integrable states have been computed exactly, the normalized squared overlap factorizes as Am∏jhc(kj)∏γhs(λγ) det⁡G+det⁡G−,\mathcal{A}_m\prod_{j}\mathfrak{h}_c(k_j)\prod_{\gamma}\mathfrak{h}_s(\lambda_\gamma)\,\frac{\det G^+}{\det G^-}, where hc,s\mathfrak{h}_{c,s} are single-particle overlap functions and G±G^\pm are Gaudin matrices (matrices of derivatives of the Bethe-equation scattering phases, restricted to the paired roots; for the Hubbard model they are the 34N×34N\frac{3}{4}N\times\frac{3}{4}N matrices mixing the N/2N/2 pair momenta and the N/4N/4 pair spin rapidities, with explicit entries given by the source's Eqs. (88)-(90) in terms of ϕn±(λ,μ)=ϕn(λ−μ)±ϕn(λ+μ)\phi_n^\pm(\lambda,\mu)=\phi_n(\lambda-\mu)\pm\phi_n(\lambda+\mu) and ϕn(x)=nU(nU/2)2+x2\phi_n(x)=\frac{nU}{(nU/2)^2+x^2}). Assuming this structure and reading off hc,hs\mathfrak{h}_c,\mathfrak{h}_s from the low-density computation, the source conjectured, for arbitrary filling and finite interaction, the formula (its Eqs. (87) and (91)) ∣⟨Ψ0∣{kj}{λγ};m⟩∣2∥ ∣{kj}{λγ};m⟩ ∥2=Am αN(1+4α2)L/2∏l=1N/2skl2 [ckl+U/2]2skl2+(U/2)2∏γ=1N/4U4λγ2(λγ2+(U/2)2) det⁡G+det⁡G−,\frac{|\langle\Psi_0|\{k_j\}\{\lambda_\gamma\};m\rangle|^2}{\|\,|\{k_j\}\{\lambda_\gamma\};m\rangle\,\|^2}=\frac{\mathcal{A}_m\,\alpha^{N}}{(1+4\alpha^2)^{L/2}}\prod_{l=1}^{N/2}\frac{s_{k_l}^2\,[c_{k_l}+U/2]^2}{s_{k_l}^2+(U/2)^2}\prod_{\gamma=1}^{N/4}\frac{U^4}{\lambda_\gamma^2\left(\lambda_\gamma^2+(U/2)^2\right)}\,\frac{\det G^+}{\det G^-}, with sk=sin⁡ks_k=\sin k, ck=cos⁡kc_k=\cos k, and the descendant combinatorial factor (its Eq. (92)) Am=(2α)2m (L−N−m)!m! (L−N)![(L−N2)!(L−N2−m)!]2.\mathcal{A}_m=(2\alpha)^{2m}\,\frac{(L-N-m)!}{m!\,(L-N)!}\left[\frac{\big(\frac{L-N}{2}\big)!}{\big(\frac{L-N}{2}-m\big)!}\right]^2 . The label mm counts descendant states: the Bethe states of the model organize into multiplets of the spin and η\eta-pairing SU(2)SU(2) symmetries, and the states obtained from a paired highest-weight state by the action of the lowering operators — the spin lowering S−S^-, or η−\eta^- (Yang's η\eta-pairing operator, which adds a doubly occupied site) — are their descendants. Since ∣Ψ0⟩|\Psi_0\rangle is a spin singlet but not a highest-weight state of the η\eta-pairing symmetry, these descendants also have nonzero overlap with it; the ones labelled by mm are the η−\eta^- descendants, defined by the source (its Eq. (25)) as ∣{kj}{λγ};m⟩=(L−N−m)!m! (L−N)! (η−)m∣{kj}{λγ}⟩,|\{k_j\}\{\lambda_\gamma\};m\rangle=\sqrt{\frac{(L-N-m)!}{m!\,(L-N)!}}\,(\eta^-)^m|\{k_j\}\{\lambda_\gamma\}\rangle, so each application of η−\eta^- adds two particles. The definition (its Eq. (25)) covers 0≤m≤L−N0\le m\le L-N, while the conjectured overlap formula is stated for generic m≤(L−N)/2m\le(L-N)/2, the range also required by the factorial structure of Am\mathcal{A}_m. The derivation of the conjectured formula is a bootstrap in the sense of Gombor and Bajnok: it fixes the factorized form by symmetry and limits, not by direct evaluation of the Bethe-state wavefunction at arbitrary density.

In the Quench Action construction the descendant sectors must be summed over, with each weighted by −log⁡Am-\log\mathcal{A}_m, subject to the constraint (N+2m)/L=4α2/(1+4α2)(N+2m)/L=4\alpha^2/(1+4\alpha^2) coming from the conserved particle number. The Stirling asymptotics of Am\mathcal{A}_m make this constrained double sum unwieldy, so the authors instead absorbed its effect into a redefinition α→α~\alpha\to\tilde{\alpha} of the parameter entering the overlap, fixing α~\tilde{\alpha} post hoc so that the saddle point reproduces the conserved ⟨N⟩\langle N\rangle. This works numerically but is not a first-principles treatment: the representative eigenstate is then parametrized by α~\tilde{\alpha} rather than derived from the physical initial-state parameter α\alpha.

Since the source appeared, a rigorous derivation program for overlap formulas has advanced rapidly, but only for rational spin chains. Gombor proved the two-site boundary-state overlap formulas for gl(N)\mathfrak{gl}(N)-symmetric chains, then obtained the overlap functions for all integrable matrix product states (states built by contracting a chain of finite-dimensional auxiliary matrices) of the gl(N)\mathfrak{gl}(N), o(N)\mathfrak{o}(N) and sp(N)\mathfrak{sp}(N) symmetric chains, and finally gave a representation-independent proof of the universal MPS overlap formula for gl(N)\mathfrak{gl}(N) chains using algebraic Bethe ansatz techniques (a framework in which eigenstates are constructed through the R-matrix, the operator encoding two-particle scattering) and KT-relations (identities relating the boundary K-matrix to the transfer matrix, the generating function of the conserved charges). These proofs rest on the action of off-shell Bethe vectors (Bethe-ansatz vectors built from rapidities that do not satisfy the Bethe equations), on recursion and on coproduct formulas that have not been developed for the Hubbard model, whose R-matrix (constructed by Shastry) is not of rational gl(N)\mathfrak{gl}(N) type; the extension of the results "to graded spin chains ... or the Hubbard model" is explicitly listed as an open future direction in that program. The overlap conjecture for the Hubbard model, and the first-principles thermodynamic treatment of its descendant sum, therefore remain the missing pieces for a fully rigorous Quench Action solution of quenches in the paradigmatic interacting lattice-fermion model.

Problem Statement

Establish, from first principles, the exact overlaps between the integrable two-site K-matrix initial states of the one-dimensional Hubbard model and the nested Bethe-ansatz eigenstates, at arbitrary particle density and arbitrary finite interaction strength UU.

First, provide a rigorous derivation of the normalized squared overlap between the initial state ∣Ψ0⟩|\Psi_0\rangle and the paired Bethe eigenstates ∣{kj,−kj}{λγ,−λγ}⟩|\{k_j,-k_j\}\{\lambda_\gamma,-\lambda_\gamma\}\rangle, together with their descendant states ∣{kj,−kj}{λγ,−λγ};m⟩|\{k_j,-k_j\}\{\lambda_\gamma,-\lambda_\gamma\};m\rangle (the η−\eta^- descendants defined in the Background), valid for all system sizes LL, particle numbers NN, descendant indices m≤(L−N)/2m\le(L-N)/2 and finite UU. The derivation must prove the conjectured factorized closed form of Rylands-Bertini-Calabrese (their Eqs. (87) and (91), with Am\mathcal{A}_m from Eq. (92), reproduced in the Background) — or refute it by exhibiting where it fails and derive the correct closed form. The load-bearing content is the full finite-size statement: the single-particle overlap functions hc(k)=sk2[ck+U/2]2sk2+(U/2)2\mathfrak{h}_c(k)=\frac{s_k^2[c_k+U/2]^2}{s_k^2+(U/2)^2} and hs(λ)=U4λ2(λ2+(U/2)2)\mathfrak{h}_s(\lambda)=\frac{U^4}{\lambda^2(\lambda^2+(U/2)^2)}, the mixed charge-spin Gaudin-determinant ratio det⁡G+/det⁡G−\det G^+/\det G^- for the paired roots, and the combinatorial descendant factor Am\mathcal{A}_m must all follow from the Bethe-ansatz wavefunction rather than from the factorization ansatz.

Second, treat the descendant sectors from first principles in the thermodynamic-limit Quench Action: evaluate the constrained sum over mm and NN (with weight −log⁡Am-\log\mathcal{A}_m and the constraint (N+2m)/L=4α2/(1+4α2)(N+2m)/L=4\alpha^2/(1+4\alpha^2) from the conserved particle number) in the saddle point, so that the representative eigenstate's rapidity distributions are determined by the physical parameter α\alpha of the initial state. This replaces the current redefinition α→α~\alpha\to\tilde{\alpha} with post hoc fitting of the conserved particle number.

The two parts concern the same overlap object and jointly close the gap between the conjecture-based exact quench solution of the Hubbard model and a rigorous one: part one makes the finite-size formula a theorem, part two makes its thermodynamic Quench Action consequence a derivation rather than a fitted construction. No restriction to special fillings, interaction values, or subsets of the descendant sectors is admissible; the U→∞U\to\infty and low-density limits should emerge as corollaries.

References

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