Dressed magnon dynamics in a Bose-Hubbard bath: retardation, pairing, entanglement and two-magnon scattering

Abstract: We study one- and two-magnon dynamics in a spin-$1/2$ XX chain coupled to a tunable Bose-Hubbard bath. The XX Hamiltonian transports the magnons, while the bath dresses their motion and mediates an effective attraction. A single magnon forms a mobile polaron whose retarded bosonic cloud reduces its velocity below the static Lang-Firsov prediction and generates magnon-boson entanglement. In the weak-dressing, off-resonant mobile regime, the residual velocity deficit and entanglement are governed by the excitation weight of the cloud, as captured by perturbation theory and tensor network-based real-time simulations. Transport therefore provides a model-calibrated proxy for entanglement without joint-state reconstruction. When two clouds overlap, the bath mediates a finite-range attraction, yielding finite-size evidence for a compact two-magnon bound state and a transient post-scattering signature. On-site bath interactions suppress local boson number fluctuations and stiffen the bath response, reducing entanglement and weakening ground-state binding. Bath deformability thus emerges as a common control parameter for magnon transport, entanglement, and bath-mediated pairing.

• The paper combines polaron transformations, variational dynamics, perturbation theory, Born-Oppenheimer analysis, DMRG, and MPS simulations to study one- and two-magnon motion in a tunable Bose-Hubbard environment.
• For single magnons, the dressing weight governs both velocity reduction and magnon-boson entanglement, making transport an experimental proxy for entanglement in weakly dressed, off-resonant regimes.
• For two magnons, bath deformation generates a finite-range attraction and finite-size evidence of pairing, while increasing on-site repulsion U suppresses both entanglement and binding, with a hard-core response floor near 0.17.

The paper studies real-time dynamics of one and two magnons in a spin-$1/2$ XX chain coupled to a tunable Bose-Hubbard bath through a Holstein density coupling (2608.17980). The central results are threefold: (i) for a single magnon, the residual velocity deficit due to bath retardation and the magnon-boson entanglement entropy are both governed by the same dressing weight $\bar n$, so that transport provides an experimentally accessible proxy for entanglement; (ii) for two magnons, the dispersive bath mediates a finite-range attractive potential that yields finite-size evidence for a bound pair; and (iii) the on-site Bose-Hubbard repulsion $U$ acts as a common control parameter, suppressing entanglement and binding by stiffening the bath response. The analysis combines a Lang-Firsov polaron frame, Davydov coherent-state mean field, perturbation theory, Born-Oppenheimer integration of the bath, and converged matrix product state (MPS) real-time simulations.

Model and polaron frame

The Hamiltonian comprises an XX chain with hopping $J$, a gapped tight-binding bosonic bath ($t_{\mathrm{B}}<\omega_0/2$) with on-site interaction $U$, and a Holstein coupling $\lambda_z\sum_j(\sigma_j^+ + \hat a_j)n_j$. The total magnetization is conserved, so one- and two-magnon sectors are treated separately. A Lang-Firsov transformation with $\kappa=-\lambda_z/\omega_0$ removes the linear coupling exactly, dressing the exchange with displacement operators $e^{\kappa(n_j-\hat a_j)}$ and producing band narrowing $J\to \widetilde J=Je^{-\kappa^2}$. Expanding the transformed bath hopping terminates exactly at order $\bar n$0 and generates a nearest-neighbor magnon attraction $\bar n$1; in spin language this is an effective ferromagnetic XXZ anisotropy $\bar n$2. Crucially, this attraction vanishes for an Einstein bath ($\bar n$3), where hard-core magnons on distinct sites have non-overlapping clouds. The transformation also leaves a residual linear term that cannot be removed locally; it underlies the retardation physics and contributes to the full static potential.

Single-magnon dynamics: retardation and entanglement

Within the single-field Davydov ansatz, the mean-field equations are a Schrödinger equation with Lang-Firsov narrowing, a dynamical Peierls phase, and a potential $\bar n$4, coupled to a driven discrete Gross-Pitaevskii equation whose current-divergence source encodes how magnon motion excites the bath. In the fast-bath limit the equations reduce to a focusing nonlinear Schrödinger equation with cubic coupling $\bar n$5, valid only for broad packets near the dressed band bottom. The authors are explicit that the single-field ansatz carries zero polaron-frame entanglement identically; all quantum entanglement beyond the deterministic adiabatic contribution requires the MPS treatment.

For an Einstein bath at $\bar n$6 and $\bar n$7, second-order perturbation theory gives closed-form results with no fitted parameters:

$\bar n$8

and a dephased entropy $\bar n$9. Both observables are therefore controlled primarily by the same dressing weight, up to slowly varying coefficients fixed by packet shape and bath frequency. The numerical data collapse onto the perturbative curves over the off-resonant mobile regime, deteriorating near $U$0 (where on-shell emission becomes possible) and near self-trapping. The practical implication is that a measurable velocity deficit serves as a calibrated proxy for magnon-boson entanglement without reconstructing the joint state — but only within this regime, a restriction the paper states plainly.

Bath-mediated attraction and binding

In the heavy-magnon (Born-Oppenheimer) limit with $U$1, integrating out the quadratic bath gives an exact potential

$U$2

with decay length $U$3 that diverges as $U$4 and vanishes in the Einstein limit. For $U$5, DMRG evaluation of the mixed second difference of relaxed bath energies shows that $U$6 approximately rescales the potential depth by a factor $U$7 organized by $U$8. A notable result is that $U$9 does not vanish in the hard-core limit: at $J$0, $J$1, explained analytically via the Jordan-Wigner free-fermion mapping, where the leading-order response retains the same lattice Green's function while saturation of the local transverse response reduces the depth.

Ground-state DMRG at $J$2 using $J$3 shows positive finite-size binding energy across the explored range, with onset occurring at smaller effective anisotropy than the nearest-neighbor antiadiabatic threshold $J$4 — consistent with the finite range of the actual potential and the fact that the simulated parameters lie outside the asymptotically antiadiabatic regime. The paper is careful to label this as finite-size evidence rather than a thermodynamic bound state, since no finite-size scaling to the thermodynamic limit is performed. Increasing $J$5 reduces both $J$6 (becoming negative by $J$7 at $J$8) and the magnon-boson entropy, which rises from zero to roughly $J$9 as $t_{\mathrm{B}}<\omega_0/2$0 goes from $t_{\mathrm{B}}<\omega_0/2$1 to $t_{\mathrm{B}}<\omega_0/2$2 at $t_{\mathrm{B}}<\omega_0/2$3. The parallel suppression reflects their common dependence on bath deformability, although the entropy alone is not a binding diagnostic.

Two-magnon scattering

MPS collisions of counterpropagating wavepackets at $t_{\mathrm{B}}<\omega_0/2$4 show that, unlike the uncoupled chain where packets pass through each other, the bath produces a compact post-collision component ("captured weight") propagating with a shared deformation. The authors deliberately avoid claiming a stationary bound state when the antiadiabatic threshold is not met (e.g., $t_{\mathrm{B}}<\omega_0/2$5 at $t_{\mathrm{B}}<\omega_0/2$6), restricting interpretation to the simulated time window. They also note that the transient peak of the adjacent-pair weight $t_{\mathrm{B}}<\omega_0/2$7 contains a purely kinematic contribution from the free relative wavefunction at $t_{\mathrm{B}}<\omega_0/2$8; the retained post-collision value is the more informative indicator. Spatial spin negativity across cuts is strongly suppressed with increasing $t_{\mathrm{B}}<\omega_0/2$9, attributed jointly to pair compactness and bath-induced decoherence of spin configurations, though the data do not separate these mechanisms.

Limitations and open questions

Several limitations are acknowledged or evident. The velocity–entanglement proxy holds only in the weak-dressing, off-resonant, mobile regime and fails near resonance and self-trapping. Binding evidence is finite-size ($U$0) without thermodynamic scaling, and the captured post-collision component is not established as a stationary eigenstate. The factorization $U$1 is empirical and not universal in $U$2 alone. The Holstein coupling does not conserve boson number, restricting physical realizations to platforms whose bath quanta are excitations (circuit-QED resonators, trapped ions, Rydberg arrays); assembling all required ingredients into a full experimental realization remains open. Whether the compact scattering component survives at long times, and whether positive $U$3 persists under finite-size scaling, are specific questions left unanswered.

Conclusion

This work establishes bath deformability — tunable through $U$4, $U$5, and $U$6 — as a unified control parameter linking polaronic transport slowdown, magnon-boson entanglement, and bath-mediated pairing in a minimal spin-boson model. The parameter-free perturbative collapse of velocity deficit and entropy, the exact exponential induced potential, and the nonzero hard-core-bath floor $U$7 are the principal quantitative results, each delivered with clearly stated domains of validity.