A theory of quasiballistic spin transport

A recent work [Mierzejewski et al., Phys. Rev. B 107, 045134 (2023)] observed “quasiballistic spin transport” - long-lived and transiently ballistic modes of the magnetization density - in numerical simulations of infinite-temperature XXZ chains with power-law exchange interactions. We develop an analytical theory of such quasiballistic spin transport. Previous work found that this effect was maximized along a specific locus in the space of model parameters, which interpolated smoothly between the integrable Haldane-Shastry and XX models and whose shape was estimated from numerics. We obtain an analytical estimate for the lifetime of the spin current and show that it has a unique maximum along a different locus, which interpolates more gradually between the two integrable points. We further rule out the existence of a conserved two-body operator that protects ballistic spin transport away from these integrable points by proving that a corresponding functional equation has no solutions. We discuss connections between our approach and an integrability-transport conjecture for spin.

💡 Research Summary

In this work the authors develop an analytical theory for the “quasiballistic” spin transport observed in recent numerical studies of infinite‑temperature spin‑½ XXZ chains with long‑range power‑law exchange. Starting from the most general two‑body, translation‑invariant, inversion‑symmetric Hamiltonian with global U(1) spin‑rotation symmetry, they write the spin‑current operator Ĵ and its time derivative ˙Ĵ, which is a three‑body operator. At infinite temperature the mean of ˙Ĵ vanishes but its variance is extensive; dividing this variance by the variance of Ĵ yields a dimensionful instantaneous decay rate τ_eff⁻¹ = ⟨˙Ĵ²⟩/⟨Ĵ²⟩. This quantity provides a tractable estimate of the spin‑current lifetime, assuming Ĵ has negligible overlap with conserved quantities.

Specializing to XXZ‑type anisotropy b(r)=Δ a(r), the decay rate becomes a quadratic function of Δ: τ_eff⁻¹(Δ)= (A Δ²+2B Δ+C)/⟨Ĵ²⟩, where the coefficients A, B, C are infinite sums depending only on the exchange profile a(r). Since A>0, the lifetime is maximized at the unique Δ* = –B/A. This simple result already explains why numerical studies reported a single optimal anisotropy for each interaction range.

The authors then focus on power‑law exchange a(r)=|r|⁻ᵅ with exponent α>3/2. Substituting this form into the expressions for A and B leads to an exact closed‑form for Δ*(α) as a ratio of two double‑infinite series. The resulting curve interpolates smoothly between the two integrable limits: Δ*(2)=1 (the Haldane‑Shastry model) and Δ*→0 as α→∞ (the XX model). Importantly, the asymptotic behavior differs from the previously conjectured numerical fit Δ_num(α)≈e^{2−α}; the analytical result gives Δ*(α)∼3·2⁻ᵅ for large α, i.e. a slower exponential decay.

Using the same formalism the authors compute the minimal decay rate τ_eff⁻¹(Δ*(α)). It vanishes at the integrable points and remains exponentially small for all α>2, confirming that the long‑range XXZ chain never becomes “far” from ballistic transport. In the perturbative regime α≫1 they identify the next‑nearest‑neighbour hopping term a(2)=2⁻ᵅ as the dominant integrability‑breaking mechanism. When Δ scales as k·c⁻ᵅ, the decay rate depends sensitively on the base c. For c=2 (i.e. Δ∝2⁻ᵅ) a non‑trivial cancellation between hopping and interaction contributions yields τ_eff⁻¹∼(k−3)²·2⁻ᵅ, which is minimized at k=3. Consequently the analytical optimal anisotropy leads to a decay rate roughly three times smaller than that obtained from the numerical fit, demonstrating a substantial improvement in the predicted quasiballistic lifetime.

A further important result is the proof that no non‑trivial conserved two‑body operator exists for the class of Hamiltonians considered, except at the two integrable points. By formulating a functional equation for a candidate conserved bilinear and showing it has no solutions, the authors rule out a “ballistic protection” mechanism based on two‑body conservation away from the Haldane‑Shastry and XX limits.

The discussion highlights the practical relevance of these findings for experimental platforms capable of engineering long‑range spin interactions, such as trapped‑ion chains, Rydberg atom arrays, and superconducting qubit lattices. The analytical expressions for Δ*(α) and τ_eff⁻¹ provide concrete guidelines for tuning interaction exponents and anisotropies to maximize the quasiballistic spin‑current lifetime. While the moment‑method estimate does not replace a full Kubo‑formula calculation of the diffusion constant, it captures the essential physics and offers a tractable route to predict and control transport in non‑integrable, long‑range quantum spin systems.