For nearest-neighbour integrable spin chains, analytical quantum-quench dynamics has become exact through integrable boundary states and determinant formulas for their overlaps with Bethe eigenstates, which feed the quench-action and thermodynamic-Bethe-ansatz frameworks. For the Haldane-Shastry (HS) chain - the prototypical integrable model with inverse-square long-range interactions, realizable in trapped-ion quantum simulators - no integrable boundary states are known and no overlap formulas between its eigenstates and any nontrivial initial state exist. The problem is to (i) formulate a boundary-integrability criterion appropriate to the HS chain as a long-range integrable model and construct explicit nontrivial integrable boundary states for the HS chain, (ii) derive exact closed-form overlap formulas between such states and the Yangian highest-weight and descendant eigenstates (typically expressed through their Bethe roots in the ABA construction), with a controlled thermodynamic limit, and (iii) use these overlaps in an analytical treatment of at least one nontrivial quench in the HS chain.

Background

The Haldane-Shastry (HS) chain is the paradigmatic exactly solvable quantum spin chain with long-range interactions. It is a spin-1/2 chain on a periodic ring whose Hamiltonian is, up to an additive constant and overall scale, H=∑i<jPij/sin⁡2[π(i−j)/N]H=\sum_{i<j} P_{ij}/\sin^2[\pi(i-j)/N], where PijP_{ij} is the operator permuting the spins at sites ii and jj; the interaction decays as the inverse square of the chordal distance. It was introduced independently by Haldane and Shastry in 1988, and is remarkable because it hosts elementary excitations (spinons) obeying semionic fractional statistics (an exchange phase intermediate between that of bosons and fermions), its thermodynamics is exactly that of an ideal gas of such spinons, and it possesses a full Yangian symmetry - an infinite-dimensional quantum-group symmetry extending the chain's spin-rotation symmetry - making it a discrete analogue of a conformal field theory. Energy eigenstates are organized by Yangian highest-weight states, labeled by combinatorial data called motifs, and their degeneracies reflect this symmetry. A quantum quench is the sudden change of a system's parameters, after which an isolated many-body system evolves towards a stationary state that need not be thermal. For integrable systems - which possess infinitely many conserved quantities - the steady state is a generalized Gibbs ensemble (GGE), and for a special class of initial states the full time evolution can be computed analytically by the quench action method: a variational saddle-point over thermodynamic Bethe ansatz (TBA) macrostates whose key input is the exact overlap of the initial state with each energy eigenstate. The method became quantitative for nearest-neighbour chains through a chain of results: overlap formulas between the Néel state (the alternating up-down product state) and Bethe states (eigenstates produced by the Bethe ansatz, the exact diagonalization technique of integrable models) in the XXZ chain, the notion of an integrable quench - an initial state, such as the Néel state or integrable matrix product states (states built by contracting site-independent matrices, a compact ansatz class that goes beyond simple product states), that is annihilated by a suitable twisted or boundary transfer matrix (a product of Lax matrices, the local building blocks of the Bethe ansatz, taken around the chain) built from the chain's Lax matrices - a 'boundary integrability' criterion - and determinant (twisted-Yangian and separation-of-variables) formulas for the resulting overlaps. By now this technology is standard for nearest-neighbour chains (XXZ, Hubbard, nested gl(N) chains). The HS chain stubbornly resists this transfer. Its eigenstates are not conventional nearest-neighbour Bethe states: until recently they were accessible mainly through fractional-statistics and conformal-field-theory techniques, and no orthogonal Bethe-type eigenbasis with computable norms was available. This changed only in 2025-2026: Ferrando, Lamers, Levkovich-Maslyuk and Serban developed an algebraic Bethe ansatz (ABA) - the constructive form of the Bethe ansatz, generating eigenstates by repeated action of the creation operators of a transfer matrix - inside the Calogero-Sutherland model, which Jiang, Lamers and Miao combined with the 'freezing' construction (the HS chain is the freezing limit of the spin Calogero-Sutherland system of particles with inverse-square interactions and internal spin degrees of freedom) to construct the Yangian descendant states of the HS chain as Bethe states of a twisted transfer matrix, with Bethe roots determined by Bethe-ansatz and TQ-type relations (functional equations whose solutions are the Bethe roots). They derived Gaudin-type determinant formulas for the norms of these states and Slavnov-type determinants for on-shell/off-shell overlaps within the eigenbasis (that is, overlaps of Bethe states whose roots do or do not satisfy the Bethe equations), and in the extreme-twist limits recovered the Gelfand-Tsetlin basis (a canonical orthogonal basis of Yangian representations). What their work - and the entire literature - still lacks is the object needed for quench dynamics: a class of integrable boundary states for the HS chain and exact overlap formulas between such states and the energy eigenstates. The nearest-neighbour boundary-integrability criterion does not transfer, because it relies on the Hamiltonian being a sum of nearest-neighbour two-site gates and on boundary transfer matrices generated from local Lax matrices, whereas the HS Hamiltonian is a genuinely long-range sum tied to Dunkl operators (operators combining coordinate shifts with spin exchange, whose algebra underlies the integrability of the Calogero-Sutherland family). A parallel motivation comes from hydrodynamics: Bulchandani and Ha used a kinetic theory of Bethe quasiparticles together with the TBA to derive the spin hydrodynamics of the HS chain, showing it is ballistic with no velocity dressing (the spinon velocities are not renormalized by interactions - a free-spinon-gas feature); the microscopic derivation of their postulated Boltzmann-equation system remains open and, like quench dynamics, is expected to require the Hilbert-space and descendant-state structure that Jiang, Lamers and Miao began to map. Experimental relevance is direct: trapped-ion simulators naturally realize power-law interacting spin chains with the HS chain as the integrable limiting point.

Problem Statement

Establish analytical, overlap-based control of integrable quantum quenches in the Haldane-Shastry chain - the prototypical integrable spin chain with long-range interactions - paralleling the completed treatment of nearest-neighbour chains. Concretely, for the spin-1/2 HS chain (periodic ring of N sites, Hamiltonian H=∑i<jPij/sin⁡2[π(i−j)/N]H=\sum_{i<j}P_{ij}/\sin^2[\pi(i-j)/N] up to constants), a solution should deliver, as one coherent program:

  1. A precise boundary-integrability criterion appropriate to the HS chain as a long-range integrable model - e.g., a condition that the initial state be annihilated by suitable twisted or boundary transfer matrices constructed from the HS monodromy (built from the frozen Dunkl operators), or an equivalent symmetry criterion, reducing to the standard nearest-neighbour criterion when applied to chains built from two-site gates - together with explicit nontrivial integrable boundary states for the HS chain (product-type or matrix-product-type initial states). A boundary state counts as nontrivial here if it is not an energy eigenstate of HH (in particular, not the ferromagnetic Yangian highest-weight state ∣↑⋯↑⟩|\uparrow\cdots\uparrow\rangle) and it has nonzero overlap with an extensive set of energy eigenstates in the thermodynamic limit.

  2. Exact closed-form formulas - e.g., of product/determinant type expressed through the Bethe roots of the ABA construction of the eigenstates - for the overlaps between these boundary states and the energy eigenstates (Yangian highest-weight and descendant states). The eigenstates are understood as a single complete eigenbasis of Yangian highest-weight states and their ABA descendants for the diagonally twisted transfer matrix T(x;κ)=κA(x)+κ−1D(x)T(x;\kappa)=\kappa A(x)+\kappa^{-1}D(x) at a fixed twist κ>0\kappa>0 (the regime in which the transfer matrix is hermitian and the basis orthogonal); the choice of κ\kappa - including κ=1\kappa=1 and the extreme-twist Gelfand-Tsetlin limit - is free, but one basis must be fixed and used consistently in items 2-3. The formulas must be valid at finite N, together with a controlled thermodynamic limit: root densities exist and the limiting (e.g., saddle-point) expressions for the sums over eigenstates are justified, with careful bookkeeping of the motif degeneracies and descendant states.

  3. The application of these overlap formulas in an analytical treatment (e.g., via a quench-action or TBA-type analysis) of at least one nontrivial quench in the HS chain, determining the post-quench steady state and the long-time behavior of local observables and demonstrating that the formulas are usable in practice. A nontrivial quench here means that the initial state - one of the nontrivial integrable boundary states of item 1 - is prepared and evolved under HH (a state quench, as in the nearest-neighbour Néel-into-XXZ literature), or that an equivalent coupling quench into the HS chain is performed; in either case the evolution is genuinely out of equilibrium because the initial state is not an eigenstate of HH.

Items 1 and 2 constitute the 'exact overlap formulas between eigenstates and integrable boundary states' that are still lacking for long-range spin chains; item 3 is the quench-dynamics goal that motivates them, and the three are jointly the stated objective. Two further directions - the microscopic derivation of the Boltzmann-equation system of Bulchandani and Ha, and finite-temperature correlation functions of the HS chain - are distinct open problems, not part of this completion boundary.

References

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