The folded spin-1/2 XXZ chain arises from the Heisenberg XXZ chain in the limit of infinite anisotropy through the 'folded picture', and is a paradigmatic interacting integrable model whose conservation laws exhibit two unusual features: an infinite family of quasilocal charges acting on the particle configuration (of which the staggered magnetisation is one example), and a non-abelian set of non-commuting quasilocal conservation laws. Zadnik, Bidzhiev and Fagotti constructed a thermodynamic Bethe Ansatz (TBA) and Euler-scale generalised hydrodynamics (GHD) for this model using only the strictly local charges and a single quasilocal charge M, conjectured — without derivation — an advection equation for the staggered magnetisation, and explicitly warned that the resulting description is incomplete. The open problem is to complete it: determine the full family of quasilocal conserved operators needed to uniquely characterise general quasilocal macrostates; compute how these charges modify the Yang–Yang entropy and the TBA equations; derive the Euler-scale hydrodynamics of the enlarged set of fields — in particular prove or refute the conjectured advection law for the staggered magnetisation and the other configuration-space charges — and establish whether root densities together with finitely many additional fields form a closed hydrodynamic description; and settle how the model's non-abelian integrability structure enters, or fails to enter, the thermodynamic framework. As of 2026 the problem remains open.

Background

The spin-1/2 Heisenberg XXZ chain, H=J∑ℓ(σℓxσℓ+1x+σℓyσℓ+1y+Δ σℓzσℓ+1z)H=J\sum_\ell(\sigma^x_\ell\sigma^x_{\ell+1}+\sigma^y_\ell\sigma^y_{\ell+1}+\Delta\,\sigma^z_\ell\sigma^z_{\ell+1}) with Pauli matrices σℓα\sigma^\alpha_\ell and anisotropy Δ\Delta, is the canonical interacting integrable quantum spin chain. In the limit of large anisotropy Δ→∞\Delta\to\infty, the Ising interaction dominates and generates fast oscillations of the state. Zadnik and Fagotti introduced the 'folded picture' for this regime: the fast dynamics is factored out of the state and attached to the operators, leaving an effective folded Hamiltonian HFH_F that governs the slow dynamics at times 1≪Jt≪κ−11\ll Jt\ll\kappa^{-1}, with κ=(4Δ)−1\kappa=(4\Delta)^{-1} the small expansion parameter. Although the folded model arises as a limit of the integrable XXZ chain, its exact solution exposes structure invisible to the standard Bethe Ansatz, and it is best regarded as a distinct integrable model. A duality transformation maps HFH_F into a local operator whose density acts on three consecutive sites, and the thermodynamic theory is developed for this dual model. Up to boundary terms, the dual folded XXZ Hamiltonian on the infinite chain reads

H~F=J∑ℓ(σℓ−1xσℓ+1x+σℓ−1yσℓ+1y) 1−σℓz2,\tilde H_F=J\sum_\ell(\sigma^x_{\ell-1}\sigma^x_{\ell+1}+\sigma^y_{\ell-1}\sigma^y_{\ell+1})\,\frac{1-\sigma^z_\ell}{2},

which is a special (strong-repulsion) point of the two-component Bariev model, an integrable lattice model of two hard-core particle species.

The folded XXZ chain is diagonalised by a coordinate Bethe Ansatz in terms of two species of impenetrable particles sitting on 'macrosites' of two neighbouring lattice sites. Particles of different species cannot pass through one another, so the sequence of the two species along the chain — the configuration BNB_N — is conserved by the dynamics. Consequences include jammed, spatially inhomogeneous eigenstates in which particles are so densely packed that they cannot move, and Hilbert-space fragmentation (the splitting of the Hilbert space into dynamically disconnected sectors). A Bethe eigenstate is labelled by its momenta together with its configuration. The conserved quantities accordingly fall into two classes: charges whose eigenvalues depend only on the momenta, such as the Hamiltonian and its tower of strictly local conserved charges; and configuration-space charges — conserved operators, local or quasilocal (quasilocality meaning that the operator is a sum of exponentially localised densities, rather than having strictly bounded support), whose eigenvalues are functionals of the configuration BNB_N. Examples are the staggered magnetisation Sstz=12∑ℓ(−1)ℓσℓzS^z_{\rm st}=\frac{1}{2}\sum_\ell(-1)^\ell\sigma^z_\ell and a distinguished quasilocal charge M\mathbf{M} related to the number of domain walls in the particle configuration. Since a generic macrostate fixes arbitrary functionals of the configuration, the configuration-space charge family is infinite. In addition, the folded model is non-abelian integrable: it possesses a set of non-commuting quasilocal conservation laws, whose physical signature is that eigenstates sharing a complete set of integrals of motion can remain locally distinguishable even at late times, with visible effects on the dynamics at intermediate times.

Integrable systems out of equilibrium are described by generalised Gibbs ensembles (GGEs) — density matrices that are exponentials of the conserved charges considered relevant — rather than by thermal Gibbs ensembles. For the folded XXZ model, Zadnik, Bidzhiev and Fagotti constructed a thermodynamic Bethe Ansatz (TBA), the standard thermodynamic framework for Bethe-solvable models in which a macrostate is parametrised by a root density ρ(p)\rho(p) — the density of occupied quasiparticle momenta — or, equivalently, by the filling function n(p)=ρ(p)/ρtn(p)=\rho(p)/\rho_{\rm t}, where ρt\rho_{\rm t} is the momentum-independent total density of particles and holes, and the entropy is the Yang–Yang entropy, the entropy functional generated by the distribution of occupied momenta, for GGEs built from the strictly local charges and the single quasilocal charge M\mathbf{M}; the macrostate is then parametrised by the filling function together with the ratio μ=M/N\mu=M/N of the eigenvalue of M\mathbf{M} to the particle number. For macrostates built from local charges alone, μ\mu is not independent but a functional of the staggered magnetisation and the root density. Turning to inhomogeneous situations, they developed generalised hydrodynamics (GHD), the hydrodynamic theory of integrable systems at the Euler scale (the ballistic scale, on which space and time are scaled together, x/tx/t fixed), in which an evolving state is described by space-time-dependent root densities (here supplemented by a few additional fields) obeying continuity equations with quasiparticle velocities computed through the TBA. For the junction of two local macrostates they found that the ballistic profiles of essentially any local charge or current are discontinuous, and they verified the predictions against time-dependent density-matrix renormalisation group (tDMRG) simulations, a standard numerical method for the real-time dynamics of one-dimensional quantum systems. The conceptual basis is the locally quasistationary state (LQSS) — the inhomogeneous macrostate capturing late-time local expectation values; for noninteracting chains, Fagotti showed how to formalise it through the smallest family of operators, including charge densities and their currents, that is closed under time evolution, the state being the exponential of an element of that family. For interacting models, including this one, no such construction exists.

The theory so obtained is explicitly incomplete, in two ways. First, the GGEs retain only one configuration-space charge (M\mathbf{M}): the remaining infinitely many quasilocal charges that characterise general configurations are absent, so general quasilocal macrostates are not uniquely characterised, and the way the full set modifies the Yang–Yang entropy and the TBA equations is unknown; even the Euler-scale equation of motion of the staggered magnetisation is only conjectured. Second, no thermodynamic or hydrodynamic framework accounts for the non-abelian (non-commuting) quasilocal charges of the folded point. The situation contrasts with the standard XXZ chain, where the inclusion of the quasilocal conservation laws yields a complete GGE; the folded XXZ model is the outstanding interacting case, made more significant by its role as the exactly solvable representative of constrained and fragmented quantum dynamics.

Problem Statement

Complete the thermodynamic and hydrodynamic description of the dual folded spin-1/2 XXZ model (Hamiltonian H~F\tilde H_F above) in the thermodynamic limit. The source theory describes macrostates only through the strictly local charges, the filling function n(p)n(p), and the single quasilocal charge M\mathbf{M}; the completion requires all of the following, jointly constituting one objective:

  1. Complete charge family. Determine the full set of quasilocal conserved operators of the folded XXZ chain — in particular the configuration-space charges whose eigenvalues are functionals of the particle configuration BNB_N (of which the staggered magnetisation SstzS^z_{\rm st} and the quasilocal charge M\mathbf{M} are examples) — and establish which subfamily is necessary and sufficient to uniquely characterise general quasilocal macrostates.

  2. Complete TBA. Construct the thermodynamic Bethe Ansatz for generalised Gibbs ensembles built from the complete charge family: compute how the additional configuration-space charges modify the Yang–Yang entropy and the TBA equations, and determine the resulting equations of state for the additional thermodynamic fields (the ratio μ\mu associated with M\mathbf{M}, the staggered magnetisation mstzm^z_{\rm st}, and any further independent fields).

  3. Closed Euler-scale hydrodynamics. Derive, rather than conjecture, the Euler-scale generalised hydrodynamics of the enlarged set of fields. In particular, settle the conjecture of Eq. (109) of the source, that the staggered magnetisation — and every other configuration-space quasilocal charge — is advected with velocity V=⟨E′⟩/⟨1⟩V=\langle E'\rangle/\langle 1\rangle, where ⟨q⟩=∫−ππdp ρ(p) q(p)\langle q\rangle=\int_{-\pi}^{\pi}\mathrm{d}p\,\rho(p)\,q(p) is the TBA expectation value per macro-site, with ρ(p)\rho(p) the root density, and E(p)E(p) the single-particle energy:

∂t ⁣(mst;x,tz⟨1⟩x,t)+⟨E′⟩x,t⟨1⟩x,t∂x ⁣(mst;x,tz⟨1⟩x,t)=O(∂x2).\partial_t\!\left(\frac{m^{z}_{{\rm st};x,t}}{\langle 1\rangle_{x,t}}\right)+\frac{\langle E'\rangle_{x,t}}{\langle 1\rangle_{x,t}}\partial_x\!\left(\frac{m^{z}_{{\rm st};x,t}}{\langle 1\rangle_{x,t}}\right)=\mathcal{O}(\partial_x^2).

Establish or refute closure: whether the space-time-dependent root density together with finitely many additional hydrodynamic fields forms a closed description of the locally quasistationary states; the source frames a route to such a description in the identification of the smallest family of operators — including the (quasilocal) conserved charges together with their densities and currents — that is closed under time evolution and on which those states are supported.

  1. Non-abelian structure. Determine how the non-commuting quasilocal conservation laws of the folded point (the non-abelian integrability of the model) enter this framework: do they impose additional constraints on the thermodynamics and hydrodynamics beyond the abelian charge family — in particular in the interacting sector with a mixed population of both particle species — or are they immaterial at thermodynamic and Euler scale?

A complete answer must reproduce the existing theory in the appropriate limit: retaining only the strictly local charges and M\mathbf{M} must recover the TBA and GHD of Zadnik, Bidzhiev and Fagotti, including its thermal asymptotics and the discontinuous ballistic profiles at the junction of two local macrostates.

References

  1. Lenart Zadnik, Kemal Bidzhiev, Maurizio Fagotti, 'The folded spin-1/2 XXZ model: II. Thermodynamics and hydrodynamics with a minimal set of charges', SciPost Phys. 10, 099 (2021). DOI: 10.21468/SciPostPhys.10.5.099; arXiv:2011.01159; https://arxiv.org/abs/2011.01159
  2. Lenart Zadnik, Maurizio Fagotti, 'The Folded Spin-1/2 XXZ Model: I. Diagonalisation, Jamming, and Ground State Properties', SciPost Phys. Core 4, 010 (2021). DOI: 10.21468/SciPostPhysCore.4.2.010; arXiv:2009.04995; https://arxiv.org/abs/2009.04995
  3. Lenart Zadnik, Juan P. Garrahan, 'Slow heterogeneous relaxation due to constraints in dual XXZ models', Phys. Rev. B 108, L100304 (2023). DOI: 10.1103/PhysRevB.108.L100304; arXiv:2304.10394; https://arxiv.org/abs/2304.10394
  4. Johannes Feldmeier, William Witczak-Krempa, Michael Knap, 'Emergent tracer dynamics in constrained quantum systems', Phys. Rev. B 106, 094303 (2022). DOI: 10.1103/PhysRevB.106.094303; arXiv:2205.07901; https://arxiv.org/abs/2205.07901
  5. Márton Borsi, Balázs Pozsgay, 'Volume changing symmetries by matrix product operators', J. Phys. A: Math. Theor. 58, 065203 (2025). DOI: 10.1088/1751-8121/adaf83; arXiv:2408.15659; https://arxiv.org/abs/2408.15659
  6. J. Pawłowski, P. Łydżba, M. Mierzejewski, 'Local integrals of motion encoded in a few eigenstates', arXiv:2603.01859 (2026). https://arxiv.org/abs/2603.01859
  7. Maurizio Fagotti, 'Locally quasi-stationary states in noninteracting spin chains', SciPost Phys. 8, 048 (2020). DOI: 10.21468/SciPostPhys.8.3.048; arXiv:1910.01046; https://arxiv.org/abs/1910.01046
  8. E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F. H. L. Essler, 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
  9. Žiga Krajnik, 'Single-file dynamics with general charge measures', Phys. Rev. E 110, 024118 (2024). DOI: 10.1103/PhysRevE.110.024118; arXiv:2401.14378; https://arxiv.org/abs/2401.14378
  10. Jacopo De Nardis, Sarang Gopalakrishnan, Romain Vasseur, Brayden Ware, 'Subdiffusive hydrodynamics of nearly-integrable anisotropic spin chains', Proc. Natl. Acad. Sci. USA 119, e2202823119 (2022). DOI: 10.1073/pnas.2202823119; arXiv:2109.13251; https://arxiv.org/abs/2109.13251
  11. Roberto Ruiz, Alejandro Sopena, Balázs Pozsgay, Esperanza López, 'Efficient eigenstate preparation in an integrable model with Hilbert space fragmentation', PRX Quantum 6, 030316 (2025). DOI: 10.1103/g9f9-p8ks; arXiv:2411.15132; https://arxiv.org/abs/2411.15132