Finite-temperature extension of the exact hole dynamics in the antiferromagnetic Bethe-lattice $t$–$J_z$ model

A single hole moving in an antiferromagnetic Bethe lattice with purely Ising spin coupling $J_z$ (the $t$–$J_z$ model on a loop-free regular tree) is one of the rare interacting many-body systems whose full nonequilibrium dynamics is exactly solvable: the magnetic-polaron ground state and the string excitations form exact eigenstates with a discrete ladder and vanishing spectral width, and the quasiparticle residues obey simple strong-coupling scalings. The exact solution of Nielsen (Phys. Rev. B 106, 115144 (2022)) holds, however, only at zero temperature, at infinite lattice depth with surface effects neglected, and on the loop-free Bethe lattice without transverse spin exchange. The source paper's Conclusions and Outlook pose two future directions: extending the exact methodology to nonzero temperatures, and the pairing of two holes. The open problem addressed here is the finite-temperature extension: extend the exact methodology to $T>0$ and characterize the resulting hole dynamics and the polaron and string spectral features on the Bethe lattice, with the correct $T\to 0$ limit. Surface and finite-depth corrections enter the source only as a technical convergence remark, and the contrast with regular crystal lattices and with transverse spin exchange is drawn there as a known result that motivates the finite-temperature question; the recent identification of long-lived eigenstates beyond the Bethe-lattice ladder on the square lattice sharpens that motivation.

Background

A doped antiferromagnet is described, in its simplest form, by the $t$–$J_z$ model: spin-$1/2$ particles on a lattice with nearest-neighbor antiferromagnetic Ising coupling $J_z$ between spins, in which one spin is removed (a hole, or dopant) that hops between nearest-neighbor sites with amplitude $t$. Moving through the Néel-ordered background (neighboring spins alternating in opposite directions), the hole leaves behind a trail of misaligned spins — a string — so its motion competes with the magnetic energy cost of spin misalignment. The hole dressed by its spin distortion is called a magnetic polaron; the ladder of states in which the hole is bound to a string of increasing length gives the string excitations. The quasiparticle residue $Z_n$ of an eigenstate $n$ is the spectral weight with which a bare hole injected into the antiferromagnet contributes to that state. On the Bethe lattice — an infinite regular tree with coordination number $q$ (each site has $q$ neighbors, and there are no loops) — the $t$–$J_z$ model becomes exactly solvable: the fractal self-similarity of the tree reduces the exponentially many coefficients of the many-body wave function to one coefficient per depth $d$ away from the initial hole position, closed by recursion relations (with a self-consistency equation in the thermodynamic limit $d_{\rm tot}\to\infty$). Nielsen (Phys. Rev. B 106, 115144 (2022)) used this to compute the exact nonequilibrium quench dynamics of an initially localized hole at zero temperature, the exact magnetic-polaron ground state, and the exact string-excitation eigenstates, whose energies form a discrete ladder indexed by the zeros of the Airy function with vanishing spectral width, and whose residues scale linearly in $J_z/t$ at strong coupling. This contrasts sharply with regular finite-dimensional lattices: for a hole in the two-dimensional Ising antiferromagnet the spectrum is incoherent (Wrzosek and Wohlfeld, Phys. Rev. B 103, 035113 (2021)), long-lived anomalous eigenstates with excitation energies scaling approximately linearly in $J_z$ — absent from the Bethe-lattice ladder — have been identified on the square lattice within the self-avoiding-walk approximation and exact diagonalization (Wrzosek et al., arXiv:2512.01815 (2025)), and in the full $t$–$J$ model, where transverse spin exchange $J_\perp$ is added to $J_z$, the string states are no longer exact eigenstates. The exact Bethe-lattice solution rests on four idealizations: zero temperature (the unique Néel background), infinite depth (a finite lattice is truncated at a total depth $d_{\rm tot}$, the maximum number of hopping steps from the hole's initial site to the boundary, and corrections from that surface are neglected), purely Ising coupling, and the loop-free tree geometry. Earlier work had obtained only the exact local hole Green's function, not the full many-body wave function.

Problem Statement

The exact solution of the single-hole $t$–$J_z$ model on the antiferromagnetic Bethe lattice (Nielsen, Phys. Rev. B 106, 115144 (2022)) is established at zero temperature, in the infinite-depth limit with surface effects neglected, for purely Ising spin coupling, and on the loop-free tree geometry. The open problem is the finite-temperature extension of this exact solution. The task is resolved by an exact finite-temperature extension of the methodology: extend the methodology to $T>0$ — by the full density-matrix evolution or, equivalently, by a finite-temperature field-theory formulation anticipated in the source paper — and characterize the resulting hole dynamics and the polaron and string spectral features on the Bethe lattice, with the correct $T\to 0$ limit. The two-hole extension and binding question is out of scope and not part of the open problem. Surface and finite-depth corrections and the quantitative transferability of the Bethe-lattice predictions to regular finite-dimensional lattices and to the full $t$–$J$ model with transverse exchange $J_\perp$ are related directions that the source discusses only as a technical convergence remark and as a known contrast motivating the finite-temperature question; they are background and motivation, not posed open questions.

References

1. K. Knakkergaard Nielsen, Exact nonequilibrium hole dynamics, magnetic polarons and string excitations in antiferromagnetic Bethe lattices, Phys. Rev. B 106, 115144 (2022). DOI: 10.1103/PhysRevB.106.115144; arXiv:2207.01020. https://arxiv.org/abs/2207.01020
2. K. Knakkergaard Nielsen, Omnipresent bound state of two holes in antiferromagnetic Bethe lattices, Phys. Rev. B 108, 085122 (2023). DOI: 10.1103/PhysRevB.108.085122; arXiv:2302.07304. https://arxiv.org/abs/2302.07304
3. K. Knakkergaard Nielsen, Exact dynamics of two holes in two-leg antiferromagnetic ladders, Phys. Rev. B 108, 085125 (2023). DOI: 10.1103/PhysRevB.108.085125; arXiv:2303.17381. https://arxiv.org/abs/2303.17381
4. K. Knakkergaard Nielsen, Thermally induced localization of dopants in a magnetic spin ladder, Phys. Rev. Research 6, 023325 (2024). DOI: 10.1103/PhysRevResearch.6.023325; arXiv:2310.11193. https://arxiv.org/abs/2310.11193
5. K. Knakkergaard Nielsen, Localized dopant motion across the 2D Ising phase transition, SciPost Phys. Core 7, 054 (2024). DOI: 10.21468/SciPostPhysCore.7.3.054; arXiv:2404.11608. https://arxiv.org/abs/2404.11608
6. Mingru Yang, Sajant Anand, Kristian Knakkergaard Nielsen, Beyond fragmented dopant dynamics in quantum spin lattices: Robust localization and non-Gaussian diffusion, Phys. Rev. B 112, 165129 (2025). DOI: 10.1103/clhr-4h26; arXiv:2507.16042. https://arxiv.org/abs/2507.16042
7. K. Knakkergaard Nielsen, Dopant pairing in a disordered magnetic spin ladder, Phys. Rev. Research 7, L012033 (2025). DOI: 10.1103/PhysRevResearch.7.L012033; arXiv:2407.01252. https://arxiv.org/abs/2407.01252
8. Piotr Wrzosek, Krzysztof Wohlfeld, Hole in the two-dimensional Ising antiferromagnet: Origin of the incoherent spectrum, Phys. Rev. B 103, 035113 (2021); arXiv:2012.00395. DOI: 10.1103/PhysRevB.103.035113. https://doi.org/10.1103/PhysRevB.103.035113
9. Magnus Callsen, Jens H. Nyhegn, Kristian Knakkergaard Nielsen, Georg M. Bruun, Hole and spin dynamics in an anti-ferromagnet close to half filling, arXiv:2604.14039 (2026). https://arxiv.org/abs/2604.14039
10. Piotr Wrzosek, Krzysztof Wohlfeld, Eugene A. Demler, Annabelle Bohrdt, Fabian Grusdt, Anomalous Eigenstates of a Doped Hole in the Ising Antiferromagnet, arXiv:2512.01815 (2025). https://arxiv.org/abs/2512.01815