The fluctuation coupled Ising model of Patil and Sandvik — two transverse-field Ising copies coupled only through the product quantum-fluctuation term σixτix\sigma^x_i\tau^x_i — fragments exactly into exponentially many dynamically disconnected blocks whose spectra follow from partitions of natural numbers, and it realizes Ashkin-Teller quantum criticality with a partially ordered phase. The source paper left five linked questions about the robustness and generality of this fragmentation-driven criticality open: (1) the finite-temperature crossover between the zero-temperature quantum transition and the classical thermal transition in spatial dimensions d≥2d\ge 2; (2) the modified universality when typical blocks contain a percolating giant fragment on lattices whose site-percolation threshold lies below 1/21/2; (3) the dynamical fate of the anomalously long non-ergodic time scales under a weak global transverse field; (4) the origin and ultimate fate of the remnant-polarization length scale behind the pseudo-first-order Binder-cumulant anomalies of the two-dimensional q=4q=4 Potts and J1J_1-J2J_2 Ising models; and (5) whether the combinatorial fragment-to-partition solution technique transfers to restricted-dynamics quantum dimer and spin-liquid systems.

Background

This problem sits at the junction of two ideas. The first is Hilbert space fragmentation: a quantum many-body Hamiltonian whose Hilbert space decomposes into an exponentially large number of dynamically disconnected sectors ("blocks"), so that a generic initial state explores only an exponentially small fraction of its energy shell. This is a mechanism of weak ergodicity breaking that requires no quenched disorder (no fixed random background couplings) and is conceptually distinct from many-body localization, in which such disorder arrests thermalization. The second is the Ashkin-Teller (AT) universality class, a one-parameter family of critical behaviors describing two coupled Ising order parameters σ\sigma and τ\tau in two dimensions, with continuously varying critical exponents and a "partially ordered" phase in which the product στ\sigma\tau develops long-range order while neither species orders by itself.

Patil and Sandvik connected the two through what they called fluctuation coupled Ising models, with Hamiltonian H=−s2∑⟨i,j⟩(σizσjz+τizτjz)−(1−s)∑iσixτixH=-\frac{s}{2}\sum_{\langle i,j\rangle}(\sigma^z_i\sigma^z_j+\tau^z_i\tau^z_j)-(1-s)\sum_i\sigma^x_i\tau^x_i on an arbitrary lattice, where the parameter s∈[0,1]s\in[0,1] trades Ising coupling against quantum fluctuation. Because the fluctuation term σixτix\sigma^x_i\tau^x_i commutes with the local product σizτiz\sigma^z_i\tau^z_i, that product is conserved at every site, and the Hilbert space splits into 2N2^N blocks labelled by the sign pattern of {σizτiz}\{\sigma^z_i\tau^z_i\}. Within a block, neighboring sites carrying opposite signs decouple (one of the two species bonds is always broken), so the block further fragments into independent Ising fragments. On a periodic chain the fragment arrangements correspond to the partitions of the natural number LL (the ways of writing LL as a sum of positive integers; under periodic boundary conditions only partitions into an even number of parts occur), so the entire block spectrum is combinatorially computable — an exact solution in a disorder-free, finite-dimensional, interacting system. Weak global transverse fields Γ∑i(σix+τix)\Gamma\sum_i(\sigma^x_i+\tau^x_i) mix the blocks weakly and produce a zero-temperature phase with ordering in the local product of the two Ising copies but no long-range order in either species, giving a Hamiltonian realization of the AT model that quantum Monte Carlo (stochastic sampling of quantum equilibrium averages) confirms quantitatively in one dimension through the known continuously varying AT exponents. In higher dimensions, a typical block corresponds to a site-percolation problem at occupation probability 1/21/2 (each site independently occupied with probability 1/21/2); on a periodic chain only the two uniform blocks percolate because the one-dimensional percolation threshold is unity.

The fragmentation picture also feeds back on a long-standing numerical puzzle. The Binder cumulant — the dimensionless ratio r=⟨M4⟩/⟨M2⟩2r=\langle M^4\rangle/\langle M^2\rangle^2 of magnetization moments, whose non-monotonic behavior with a minimum or anomalous peak is conventionally read as a signature of a first-order transition — shows pseudo-first-order anomalies at continuous transitions of the two-dimensional q=4q=4 Potts and J1J_1-J2J_2 frustrated Ising models (Ising spins with competing nearest- and next-nearest-neighbor couplings J1J_1 and J2J_2). Patil and Sandvik attributed these to remnant effects of a polarization-like (partially ordered) phase that cannot be realized explicitly in the Potts formulation: for q=4q=4 on the square lattice the Binder-ratio peak at the transition grows logarithmically with system size up to the largest lattices studied (3072×30723072\times3072) and is expected to converge only at larger scales, implying a remnant length scale of unknown origin. Restricted-dynamics systems with qualitatively similar features were previously found in quantum dimer models — lattice models whose degrees of freedom are hard-core dimers on bonds subject to a local constraint such as every site being touched by exactly one dimer, with dynamics that respects this constraint — notably the extended U(1)U(1) quantum dimer liquid of Sikora et al., a three-dimensional gapless phase with emergent photon-like excitations.

The source paper left five unresolved directions that share a single scientific objective: how robust and how transferable the exact fragmentation mechanism and the fragmentation-driven AT criticality are once the idealizations of the exactly solved model — zero temperature, one dimension or reference blocks, and exact local conservation — are relaxed or exported to other constrained systems.

Problem Statement

Determine how robust and how transferable the exact fragmentation mechanism and the fragmentation-driven Ashkin-Teller criticality of the fluctuation coupled Ising model are. The source paper posed five open directions toward this objective: (1) Finite-temperature crossover: for the fluctuation coupled Ising model in spatial dimension d≥2d\ge 2, characterize the crossover between the zero-temperature continuous quantum phase transition and the classical thermal phase transition — in particular whether and how thermal fluctuations (or coupling to a bath) couple the blocks — whether or not the block structure is washed out — and what crossover length and time scales govern this process. (2) Giant-fragment universality: on lattices whose site-percolation threshold is below 1/21/2, determine how the percolating giant fragment present in typical blocks modifies the critical behavior — whether the universality class relates to that of diluted Ising models above the percolation threshold or is new — and whether fragmentation produces finite-temperature or quantum critical phenomena beyond the Ashkin-Teller mapping. (3) Weak-field dynamics: determine whether the extremely long non-ergodic time scales that appear under a weak global transverse field persist generically in d≥2d\ge 2. (4) Binder anomaly: identify the remnant polarization-like length scale responsible for the pseudo-first-order Binder-cumulant anomalies of the two-dimensional q=4q=4 Potts (and related) models, decide whether it persists to arbitrarily large system sizes or is eventually suppressed, and establish what controls its magnitude. (5) Transfer of the solution method: determine whether restricted-dynamics quantum dimer and spin-liquid systems with the Sikora et al. features — a local dimer constraint and an extended three-dimensional U(1)U(1) dimer-liquid phase with emergent photon-like excitations — admit analogous fragment-to-partition mappings with combinatorially computable spectra. The common objective is the robustness and generality of fragmentation-driven criticality. Resolving any one of these five directions is a valid resolution of the task; resolving the full set is not required. A direction counts as resolved only when all of its stated sub-parts are answered.

References

  1. Pranay Patil and Anders W. Sandvik, Hilbert Space Fragmentation and Ashkin-Teller Criticality in Fluctuation Coupled Ising Models, Phys. Rev. B 101, 014453 (2020), DOI 10.1103/PhysRevB.101.014453, arXiv:1910.03714, https://arxiv.org/abs/1910.03714
  2. Yang Zhao, Rajesh Narayanan, and Jaeyoon Cho, Signatures of many-body localization and metastability by weak perturbation, Phys. Rev. B 102, 094201 (2020), DOI 10.1103/PhysRevB.102.094201, arXiv:1906.12020, https://arxiv.org/abs/1906.12020
  3. Atsuki Yoshinaga, Hideaki Hakoshima, Takashi Imoto, Yuichiro Matsuzaki, and Ryusuke Hamazaki, Emergence of Hilbert Space Fragmentation in Ising Models with a Weak Transverse Field, Phys. Rev. Lett. 129, 090602 (2022), DOI 10.1103/PhysRevLett.129.090602, arXiv:2111.05586, https://arxiv.org/abs/2111.05586
  4. Pranay Patil, Ayushi Singhania, and Jad C. Halimeh, Protecting Hilbert space fragmentation through quantum Zeno dynamics, Phys. Rev. B 108, 195109 (2023), DOI 10.1103/PhysRevB.108.195109, arXiv:2305.06371, https://arxiv.org/abs/2305.06371
  5. Jhao-Hong Peng and Fu-Jiun Jiang, Phase transitions of the four-state Potts model on the square and honeycomb lattices, Prog. Theor. Exp. Phys. 2024, 013A04 (2024), DOI 10.1093/ptep/ptad158, arXiv:2212.12272, https://arxiv.org/abs/2212.12272
  6. Yuan-Heng Tseng, Shang-Wei Li, and Fu-Jiun Jiang, A Bond-Weighted Tensor Renormalization Group Study of the q-State Ferromagnetic Potts Models on the Square Lattice, Prog. Theor. Exp. Phys. 2025, 073A01 (2025), DOI 10.1093/ptep/ptaf086, arXiv:2503.12361, https://arxiv.org/abs/2503.12361
  7. Yihua Sun and Yuchen Fan, Relevant-marginal field mixing resolves the four-state Potts endpoint in the square-lattice J1J_1-J2J_2 Ising model, Phys. Rev. B 114, L080406 (2026), DOI 10.1103/8yb7-rs6t, https://doi.org/10.1103/8yb7-rs6t, arXiv:2605.25946, https://arxiv.org/abs/2605.25946
  8. Zheng Yan, Zi Yang Meng, David A. Huse, and Amos Chan, Height-conserving quantum dimer models, Phys. Rev. B 106, L041115 (2022), DOI 10.1103/PhysRevB.106.L041115, arXiv:2204.01740, https://arxiv.org/abs/2204.01740
  9. Nils Niggemann, Meghadeepa Adhikary, Yannik Schaden-Thillmann, and Johannes Reuther, Classical fracton spin liquid and Hilbert space fragmentation in a 2D spin-1/2 model, Phys. Rev. B 113, 165118 (2026), DOI 10.1103/b2q3-mgvf, https://doi.org/10.1103/b2q3-mgvf
  10. Yiqiu Han, Oliver Hart, Alexey Khudorozhkov, and Rahul Nandkishore, Quantum Fragmentation, arXiv:2604.06461, https://arxiv.org/abs/2604.06461
  11. Sumiran Pujari, A theorem on extensive ground state entropy, spin liquidity and some related models, arXiv:2407.06236, https://arxiv.org/abs/2407.06236
  12. Olga Sikora, Nic Shannon, Frank Pollmann, Karlo Penc, and Peter Fulde, Extended quantum U(1)U(1)-liquid phase in a three-dimensional quantum dimer model, Phys. Rev. B 84, 115129 (2011), DOI 10.1103/PhysRevB.84.115129, https://doi.org/10.1103/PhysRevB.84.115129