The one-dimensional interacting open lattice of spinless fermions with nearest-neighbour interaction (the t-V chain under open boundary conditions) hosts a hierarchy of localized many-body bound states, constructed by Liu and Chen (2025) through asymmetric boundary-string solutions of the open-boundary Bethe ansatz: rapidities lying on the imaginary axis but not distributed symmetrically about the real axis, generated by a one-dimensional recurrence, yielding boundary-localized N-body states that emerge only for three or more particles and can become many-body bound states in the continuum (BICs). Whether the constructed catalogue of Bethe-ansatz solutions - real-rapidity scattering states, symmetric strings, asymmetric boundary strings and their left-right composites - exhausts the N-particle spectrum for every particle number (only the two-particle sector is proven complete), and hence whether additional, qualitatively different Bethe-root patterns and further localized eigenstates exist, remains open; so does the exhaustive interaction-versus-particle-number map of which localized branches exist, are localized, and have energies embedded in the many-body continuum - currently established only for three particles, where the BIC window is −3<U<−1-3<U<-1 in the attractive regime. This problem asks for that complete characterization over the full interaction range.

Background

The model. The tt-VV chain is the simplest interacting fermionic lattice model: spinless fermions hop with amplitude t>0t>0 between nearest-neighbour sites of a one-dimensional chain of LL sites and interact with strength UU when on adjacent sites (negative UU attractive, positive UU repulsive), H^=∑x=1L−1[−t c^x†c^x+1−t c^x+1†c^x+U n^xn^x+1],\hat H=\sum_{x=1}^{L-1}\left[-t\,\hat c_x^\dagger\hat c_{x+1}-t\,\hat c_{x+1}^\dagger\hat c_x+U\,\hat n_x\hat n_{x+1}\right], where c^x†\hat c_x^\dagger creates a fermion on site xx and n^x=c^x†c^x\hat n_x=\hat c_x^\dagger\hat c_x. Under open boundary conditions (OBC) the chain simply ends at sites 11 and LL, with no periodic wrap-around. The model is integrable: through a Jordan-Wigner transformation (a standard exact mapping between fermions and spins in one dimension) it is equivalent to the spin-1/21/2 XXZ Heisenberg chain (the anisotropic nearest-neighbour spin-exchange model) with anisotropy Δ=−U/2t\Delta=-U/2t (up to boundary fields), and its spectrum follows from the coordinate Bethe ansatz - an exact solution method that writes each eigenstate as a superposition of free-particle waves whose quasimomenta are fixed by algebraic equations. Writing the eigenvalue as E=−2∑jcos⁡kjE=-2\sum_j\cos k_j (in units where t=1t=1), the quasimomenta k1,…,kNk_1,\dots,k_N of an NN-fermion eigenstate satisfy the open-chain Bethe-ansatz (BA) equations of Alcaraz et al. (1987), e−2ikj(L+1)=∏i=1(i≠j)N1+ei(kj+ki)+Ueiki1+ei(kj+ki)+Ueikj⋅1+ei(ki−kj)+Ue−ikj1+ei(ki−kj)+Ueiki.e^{-2ik_j(L+1)}=\prod_{i=1(i\neq j)}^{N}\frac{1+e^{i(k_j+k_i)}+Ue^{ik_i}}{1+e^{i(k_j+k_i)}+Ue^{ik_j}}\cdot\frac{1+e^{i(k_i-k_j)}+Ue^{-ik_j}}{1+e^{i(k_i-k_j)}+Ue^{ik_i}}.

Strings and the Hilbert-space question. In the Bethe ansatz, bound states of several particles are conventionally described by string solutions: complex rapidities (a reparametrization of the kjk_j) that share a real part and are arranged symmetrically about the real axis - an nn-string represents nn particles bound together (a dimer for n=2n=2, a trimer for n=3n=3), with string spacing β\beta (cosh⁡β=∣U∣/2\cosh\beta=|U|/2 for ∣U∣>2|U|>2). Whether string-type BA solutions actually span the whole Hilbert space of a given integrable model is a long-standing general question, and under OBC the solution structure is richer than under periodic boundary conditions.

The boundary-localization puzzle. For interacting chains under OBC, numerical work had found localized states concentrated at the chain's open ends: edge-localized states in interacting one-dimensional lattices were reported by Pinto, Haque and Flach (2009), and Sun, Zhang, Yuan and Zhang (Communications Physics 7, 299 (2024)) demonstrated boundary-localized many-body bound states - able to become bound states in the continuum (BICs), i.e. spatially localized eigenstates whose energies lie inside the band of extended scattering states - for three or more interacting bosons in the Bose-Hubbard chain (the bosonic analogue of the tt-VV chain, with on-site rather than nearest-neighbour interaction), also simulating them in classical nonlinear circuit networks. Strikingly, no boundary potential, impurity or interaction modulation is required: localization is induced purely by the interplay of the open boundary with the interaction, and in the fermionic chain it appears only for N≥3N\ge 3 - never for one or two particles, where the OBC spectrum is completely classified (scattering states and the two-particle dimer).

The analytic construction. Liu and Chen (2025) supplied the mechanism in the tt-VV chain. Solving the open-chain BA equations in the L→∞L\to\infty limit, they identified asymmetric boundary-string solutions: rapidities lying on the imaginary axis at half-odd-integer multiples of β\beta but not distributed symmetrically about the real axis (for N=3N=3: λ(3)=i 3β/2, iβ/2, −iβ/2\lambda^{(3)}=i\,3\beta/2,\ i\beta/2,\ -i\beta/2), which had previously been discarded as meaningless. The corresponding wavefunctions are supported on the fixed planes xi=xi+1±1x_i=x_{i+1}\pm 1 with exponential decay away from one edge - boundary-localized many-body bound states. With γj=eikj\gamma_j=e^{ik_j}, the solutions obey the recurrence η1=γ1=−U\eta_1=\gamma_1=-U, ηj+1=−1/ηj−U\eta_{j+1}=-1/\eta_j-U, which yields 2×⌊(N−1)/2⌋2\times\lfloor(N-1)/2\rfloor localized NN-body bound states for N≥3N\ge 3 (edge-localized for N=3,4N=3,4, with additional branches localized away from the edges appearing at N=5N=5), plus composite states with independent left and right clusters for N≥6N\ge 6. Their energies are Elb,N,i=−∑l(ηl+1/ηl)E_{lb,N,i}=-\sum_l(\eta_l+1/\eta_l) (summed over both sides of the localization plane), and localization holds whenever all partial products obey ∣η1⋯ηm∣>1|\eta_1\cdots\eta_m|>1; for U<−2U<-2 localized states exist for all N≥3N\ge 3. For N=3N=3 the energy Elb,3,1=1/U+2UE_{lb,3,1}=1/U+2U falls inside the three-particle continuum exactly when −3<U<−1-3<U<-1: a many-body BIC.

Adjacent results. Imaginary-rapidity edge states are known in a sibling model: Alba, Saha and Haque (2013) described edge-locking of one and two overturned spins for anisotropy Δ>1\Delta>1 in the free-end XXZ chain. The tt-VV chain corresponds under Jordan-Wigner to the XXZ chain plus boundary fields, so the two problems are genuinely different (edge states already at N=1,2N=1,2 there; only N≥3N\ge 3 here). Boundary-string patterns also arise in the thermodynamics of open XXZ/XYZ chains with boundary fields and in integrable boundary field theory (Skorik and Saleur, sine-Gordon model with Dirichlet boundaries), and root-pattern classifications exist for the XXX chain with non-diagonal boundaries - all with boundary parameters, unlike the field-free OBC chain here. Two-particle and impurity-induced BICs are known in several other lattice settings.

What remains. The construction is a catalogue, not a classification: except for the two-particle sector, no exhaustiveness or counting argument is given, and the BIC windows are tabulated only for N=3N=3. The open problem is exactly that completion.

Problem Statement

For the open-boundary tt-VV chain (the Hamiltonian and Bethe-ansatz equations given in the background), determine whether the catalogue of Bethe-ansatz solution types constructed by Liu and Chen - (i) real-quasimomentum scattering solutions; (ii) symmetric nn-string solutions describing bulk bound states (dimers, trimers, and higher); (iii) the asymmetric boundary strings generated by η1=−U\eta_1=-U, ηj+1=−1/ηj−U\eta_{j+1}=-1/\eta_j-U, giving 2×⌊(N−1)/2⌋2\times\lfloor(N-1)/2\rfloor localized NN-body bound states for N≥3N\ge 3; and (iv) their left-right composites for N≥6N\ge 6 - is exhaustive for every particle number NN. Specifically: either prove that these solution types account for the full NN-fermion spectrum (at finite LL, or in the L→∞L\to\infty limit in which the boundary strings are exact, stating clearly in which convention the claim is made), or exhibit additional, qualitatively different Bethe-root patterns that solve the Bethe-ansatz equations and identify the eigenstates they represent, in particular any localized states beyond the constructed hierarchy. As a direct consequence of that classification, derive the exhaustive map in the (U,N)(U,N) plane - over the full interaction range, attractive and repulsive - of the localized branches: for each branch, the parameter windows in which it exists and is localized (generalizing the partial-product conditions ∣η1⋯ηm∣>1|\eta_1\cdots\eta_m|>1), and the sub-windows in which its energy lies inside the many-body continuum, i.e. the branch becomes a bound state in the continuum, generalizing the known three-particle window −3<U<−1-3<U<-1. The two deliverables form a single objective - the complete characterization of the localized many-body bound states of the interacting open lattice: the classification fixes the set of branches, and the phase map follows from their energies measured against the continuum band edges. The stability of these states under perturbations, and extensions to related integrable models, are neighbouring open directions but are not part of this problem's completion boundary.

References

  1. Yanxia Liu and Shu Chen, "Hierarchy of many-body localized bound states in an interacting open lattice", Physical Review B 112, 245404 (2025), DOI: 10.1103/j8px-m6tn, https://doi.org/10.1103/j8px-m6tn; preprint arXiv:2505.23255 (arXiv v1 title: "Hierarchy of localized many-body bound states in an interacting open lattice"), https://arxiv.org/abs/2505.23255
  2. Na Sun, Weixuan Zhang, Hao Yuan, Xiangdong Zhang, "Boundary-localized many-body bound states in the continuum", Communications Physics 7, 299 (2024), DOI: 10.1038/s42005-024-01777-5, arXiv:2409.09925, https://arxiv.org/abs/2409.09925
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