Deep thermalization in interacting integrable quantum systems: constructing the analogue of the deep generalized Gibbs ensemble beyond free fermions

For non-interacting integrable systems, Lucas, Piroli, De Nardis, and De Luca [Phys. Rev. A 107, 032215 (2023)] showed that the projected ensemble (PE) — the ensemble of pure states of a subsystem obtained by projectively measuring its complement — is reproduced by a 'deep generalized Gibbs ensemble' (dGGE): a representative-state ensemble built from pure Gaussian states carrying the same conserved charges as the initial state. They explicitly left open the generalization to interacting integrable models, where the Gaussian structure is lost and the extensive conserved charges are quasi-local, together with the question of whether the post-measurement ensemble can be expressed through charges of the subsystem $A$ alone when $L_A$ is much larger than the initial-state correlation length. (The $t^{-1/2}$ relaxation of the PE-dGGE discrepancy was reported in the source as a numerical observation, for which no analytic derivation is known.) Subsequent work has settled the chaotic (maximum-entropy/Scrooge ensembles under finitely many global constraints), many-body-localized (l-bit systems with extensive local charges), and free-fermion-circuit (matchgate) cases, but no construction or verification of a dGGE-like ensemble exists for genuinely interacting integrable models. The problem — to construct the charge-constrained representative-state ensemble for interacting integrable dynamics and decide whether it reproduces the full PE at all moments — remains open.

Background

Deep thermalization asks what an observer sees when, instead of discarding a many-body system's environment, they measure it. After a quantum quench — a sudden change of a Hamiltonian parameter followed by isolated unitary evolution — one performs projective measurements on a subsystem $B$ (the 'bath'), recording each outcome $z_B$; the subsystem $A$ is left in a pure state $|\Psi'(z_B)\rangle_A$ occurring with probability $p(z_B)$, and the collection $\mathcal{E}^{\rm PE}=\{p(z_B),|\Psi'(z_B)\rangle_A\}$ is the projected ensemble (PE). Averaging over the PE reproduces the reduced density matrix $\rho_A$, but the PE also carries higher moments $\rho^{(k)}=\sum_{z_B}p(z_B)(|\Psi'(z_B)\rangle\langle\Psi'(z_B)|)^{\otimes k}$, which encode information beyond conventional thermalization. For chaotic dynamics at infinite temperature, the PE was found theoretically and in Rydberg-atom experiments to behave as a quantum state design — an ensemble whose moments up to a given order match those of the Haar (uniform) measure on pure states — i.e. the maximally random distribution consistent with no constraints.

Integrable systems do not thermalize in this sense: local observables relax instead to a generalized Gibbs ensemble (GGE), $\rho_{\rm GGE}\propto {\rm tr}_B\exp(-\sum_k\beta_k I_k)$, built from the extensive family of conserved operators ('charges') $I_k$; in the spin-1/2 Heisenberg/XXZ chain these charges are quasi-local (their operator support is spread over a range but decays exponentially with distance), so only their net density is locally accessible. Lucas, Piroli, De Nardis, and De Luca posed the question of how conservation laws shape the PE in the simplest setting: free fermions (a tight-binding chain). Because projective measurements of local densities preserve Gaussianity, each post-measurement state of $A$ is a pure Gaussian state fully specified by a covariance matrix. They proposed the deep GGE: the representative-state ensemble generated by taking a pure Gaussian state $|\Phi\rangle_{AB}$ with the same conserved charges (occupation numbers $n(k)$) as the initial state and performing the same measurements on $B$. Two equivalent samplings were given: the single-eigenstate ensemble (a many-body eigenstate with momenta drawn from $n(k)$) and a generalized Haar ensemble of covariance matrices weighted by $\exp[-{\rm Tr}(\Omega FUDU^\dagger F^\dagger)]$, where $F$ maps momentum to real space, $D$ fixes the particle number, and the Lagrange multipliers $\omega_k$ in $\Omega$ enforce the charge constraints, the normalization being a Harish-Chandra-Itzykson-Zuber (HCIZ) matrix integral. Monte Carlo sampling showed excellent agreement of all moments $\rho^{(k)}$ and of the entanglement entropy between the PE and the dGGE, with the moment discrepancy $\Delta_t^{(k)}$ decaying as $t^{-1/2}$; at infinite temperature the dGGE reduces to the uniform measure over the manifold of Gaussian states (restricted by global symmetries), a universal ensemble independent of microscopic details.

Later work settled several adjacent cases, all outside interacting integrability. For chaotic dynamics with global constraints, the limiting PE is described by a maximum-entropy principle: the 'Scrooge' ensemble, the distribution of pure states that reproduces a given density matrix while being maximally difficult to interrogate; this was established as a principle for chaotic systems (Mark, Surace, Elben), verified with a rigorous proof in special cases for a single conserved U(1) charge (Chang, Shrotriya, Ho, Ippoliti), and developed into a general design framework (Mok, Haug, Ho, Preskill). For many-body-localized systems — strongly disordered phases whose extensive conserved charges are strictly local operators, the 'l-bits' (local integrals of motion) — the PE was shown numerically and semi-analytically to converge to a Scrooge ensemble (Manna, Roy, Sreejith). For matchgate — i.e. free-fermion — quantum circuits, rigorous convergence theorems in moment and Wasserstein distance now exist (Bejan, Béri, McGinley). The interacting integrable case is, however, untouched: there the representative states are not Gaussian, the charges are quasi-local rather than strictly local, and no analogue of the dGGE has been constructed or tested. The source also explicitly leaves open whether, at finite temperature (finite initial-state correlation length $\xi$), the post-measurement ensemble can be expressed solely in terms of charges of $A$ when $L_A\gg\xi$, and no derivation of the observed $t^{-1/2}$ relaxation is known.

Problem Statement

Extend the charge-constrained representative-state description of the projected ensemble from free fermions to interacting integrable quantum systems. Concretely, for an interacting integrable model such as the spin-1/2 XXZ (Heisenberg) chain after a quench from a short-range-correlated initial state, with the projected ensemble of subsystem $A$ generated by projective measurements on its complement $B$:

1. Construct the analogue of the deep GGE: specify a family of representative pure states of $A\cup B$ carrying the same extensive (quasi-local) conserved charges as the initial state — or an equivalent maximum-entropy ensemble constrained by those charges — together with the measure on it, such that performing the same projective measurements on $B$ generates an ensemble of pure states on $A$. The construction must state explicitly how the resulting ensemble depends on data restricted to (or near) $A$; in particular, whether at finite initial-state correlation length $\xi$ it can be expressed solely in terms of charges of $A$ in the regime $L_A\gg\xi$.

2. Decide whether this ensemble reproduces the full projected ensemble — all higher moments $\rho^{(k)}$, $k\ge 2$, and not only the first moment (the GGE) — either by rigorous analysis in a tractable interacting integrable model or by controlled numerics with finite-size and finite-time extrapolation; or identify precisely where and why the construction fails.

3. Beyond parts 1–2, characterize the temporal approach to the limiting ensemble: the relaxation of the moment discrepancy $\Delta_t^{(k)}$ and its modification, if any, by interactions. (For free-fermion quenches the source reports a numerical decay $\Delta_t^{(k)}\sim t^{-1/2}$, independently of $k$, for which no analytic derivation is known.)

Parts (1) and (2) form one scientific objective — (1) is the object and (2) is its validation against the actual projected ensemble — mirroring the construction-plus-numerical-validation carried out for free fermions in the source; part (3) is an additional, non-mandatory characterization of the approach to the limit, in which the free-fermion $t^{-1/2}$ law serves as a reference point rather than as a required derivation.

References

1. Maxime Lucas, Lorenzo Piroli, Jacopo De Nardis, and Andrea De Luca, "Generalized deep thermalization for free fermions", Phys. Rev. A 107, 032215 (2023), DOI 10.1103/PhysRevA.107.032215, arXiv:2207.13628, https://arxiv.org/abs/2207.13628
2. Jordan S. Cotler, Daniel K. Mark, Hsin-Yuan Huang, Felipe Hernández, Joonhee Choi, Adam L. Shaw, Manuel Endres, and Soonwon Choi, "Emergent Quantum State Designs from Individual Many-Body Wave Functions", PRX Quantum 4, 010311 (2023), DOI 10.1103/PRXQuantum.4.010311, arXiv:2103.03536, https://arxiv.org/abs/2103.03536
3. Joonhee Choi, Adam L. Shaw, Ivaylo S. Madjarov, Xin Xie, Ran Finkelstein, Jacob P. Covey, Jordan S. Cotler, Daniel K. Mark, Hsin-Yuan Huang, Anant Kale, Hannes Pichler, Fernando G. S. L. Brandão, Soonwon Choi, and Manuel Endres, "Preparing random states and benchmarking with many-body quantum chaos", Nature 613, 458 (2023), DOI 10.1038/s41586-022-05442-1, arXiv:2103.03535, https://arxiv.org/abs/2103.03535
4. Marcos Rigol, Vanja Dunjko, and Maxim Olshanii, "Thermalization and its mechanism for generic isolated quantum systems", Nature 452, 854 (2008), DOI 10.1038/nature06838, arXiv:0708.1324, https://arxiv.org/abs/0708.1324
5. Enej Ilievski, Marko Medenjak, and Tomaž Prosen, "Quasilocal Conserved Operators in the Isotropic Heisenberg Spin-1/2 Chain", Phys. Rev. Lett. 115, 120601 (2015), DOI 10.1103/PhysRevLett.115.120601, https://doi.org/10.1103/PhysRevLett.115.120601
6. Daniel K. Mark, Federica Surace, Andreas Elben, Adam L. Shaw, Joonhee Choi, Gil Refael, Manuel Endres, and Soonwon Choi, "Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity", Phys. Rev. X 14, 041051 (2024), DOI 10.1103/PhysRevX.14.041051, arXiv:2403.11970, https://arxiv.org/abs/2403.11970
7. Rui-An Chang, Harshank Shrotriya, Wen Wei Ho, and Matteo Ippoliti, "Deep Thermalization under Charge-Conserving Quantum Dynamics", PRX Quantum 6, 020343 (2025), DOI 10.1103/PRXQuantum.6.020343, arXiv:2408.15325, https://arxiv.org/abs/2408.15325
8. Sandipan Manna, Sthitadhi Roy, and G. J. Sreejith, "Projected ensemble in a system with locally supported conserved charges", Phys. Rev. B 111, 144302 (2025), DOI 10.1103/PhysRevB.111.144302, arXiv:2501.01823, https://arxiv.org/abs/2501.01823
9. Mircea Bejan, Benjamin Béri, and Max McGinley, "Matchgate Circuits Deeply Thermalize", Phys. Rev. Lett. 135, 020401 (2025), DOI 10.1103/v8kp-39ry, arXiv:2412.01884, https://arxiv.org/abs/2412.01884
10. Wai-Keong Mok, Tobias Haug, Wen Wei Ho, and John Preskill, "Nature is stingy: Universality of Scrooge ensembles in quantum many-body systems", arXiv:2601.00266 (2026), https://arxiv.org/abs/2601.00266
11. Tabea Herrmann, Felix Fritzsch, and Arnd Bäcker, "Timescales for Deep and Full Thermalization", arXiv:2604.27749 (2026), https://arxiv.org/abs/2604.27749