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arXiv:2503.15756 · Physical review. B./Physical review. B

Theory of quasiballistic spin transport

Jeffrey Z. Song, Hyunsoo Ha, Wen Wei Ho, Vir B. Bulchandani

Published Nov 21, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/khpy-d9s7

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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.
Purely ballistic transport is a rare feature even for integrable models. By numerically studying the Heisenberg chain with the power-law exchange, $J\ensuremath{\propto}1/{r}^{\ensuremath{\alpha}}$, where rrr is a distance, we show that for spin anisotropy $\mathrm{\ensuremath{\Delta}}\ensuremath{\simeq}exp(\ensuremath{-}\ensuremath{\alpha}+2)$ the system exhibits a quasiballistic spin transport and the presence of fermionic excitation, which do not decay up to extremely long times $\ensuremath{\sim}{10}^{3}/J$. This conclusion is reached on the basis of the dynamics of spin domains, the dynamical spin conductivity, inspection of the matrix elements of the spin-current operator, and analysis of most conserved operators. Our results smoothly connect two models in which fully ballistic transport is present: free particles with nearest-neighbor hopping and the isotropic Haldane-Shastry model.
We study the ballistic-to-diffusive transition induced by the weak breaking of integrability in a boundary-driven XXZ spin chain. Studying the evolution of the spin current density Js{\mathcal{J}}^{s}Js as a function of the system size LLL, we show that, accounting for boundary effects, the transition has a nontrivial universal behavior close to the XX limit. It is controlled by the scattering length ${L}^{*}\ensuremath{\propto}{V}^{\ensuremath{-}2}$, where VVV is the strength of the integrability-breaking term. In the XXZ model, the interplay of interactions controls the emergence of a transient ``quasiballistic'' regime at length scales much shorter than L∗{L}^{*}L∗. This parametrically large regime is characterized by a strong renormalization of the current which forbids a universal scaling, unlike the XX model. Our results are based on matrix product operator numerical simulations and agree with perturbative analytical calculations.
In integrable one-dimensional quantum systems an infinite set of local conserved quantities exists which can prevent a current from decaying completely. For cases like the spin current in the XXZ\mathit{XXZ}XXZ model at zero magnetic field or the charge current in the attractive Hubbard model at half filling, however, the current operator does not have overlap with any of the local conserved quantities. We show that in these situations transport at finite temperatures is dominated by a diffusive contribution with the Drude weight being either small or even zero. For the XXZ\mathit{XXZ}XXZ model we discuss in detail the relation between our results, the phenomenological theory of spin diffusion, and measurements of the spin-lattice relaxation rate in spin chain compounds. Furthermore, we study the Haldane-Shastry model where a conserved spin current exists.
Abstract We consider the integrable one-dimensional spin-1 chain defined by the Zamolodchikov–Fateev (ZF) Hamiltonian. The latter is parametrized, analogously to the XXZ spin-1/2 model, by a continuous anisotropy parameter and at the isotropic point coincides with the well-known spin-1 Babujian–Takhtajan Hamiltonian. Following a procedure recently developed for the XXZ model, we explicitly construct a continuous family of quasi-local conserved operators for the periodic spin-1 ZF chain. Our construction is valid for a dense set of commensurate values of the anisotropy parameter in the gapless regime where the isotropic point is excluded. Using the Mazur inequality, we show that, as for the XXZ model, these quasi-local charges are enough to prove that the high-temperature spin Drude weight is non-vanishing in the thermodynamic limit, thus establishing ballistic spin transport at high temperature.
Many experimentally relevant quantum spin chains are approximately integrable, and support long-lived quasiparticle excitations. A canonical example of integrable model of quantum magnetism is the XXZ spin chain, for which energy spreads ballistically, but, surprisingly, spin transport can be diffusive or superdiffusive. We review the transport properties of this model using an intuitive quasiparticle picture that relies on the recently introduced framework of generalized hydrodynamics. We discuss how anomalous linear response properties emerge from hierarchies of quasiparticles both in integrable and near-integrable limits, with an emphasis on the role of hydrodynamic fluctuations. We also comment on recent developments including non-linear response, full-counting statistics and far-from-equilibrium transport. We provide an overview of recent numerical and experimental results on transport in XXZ spin chains.
The nonequilibrium dynamics of isolated many-body quantum systems leads generically to the emergence of standard thermal ensembles. However, some models, denoted as integrable, can display nontrivial steady states with a large memory of the initial state. This is due to the presence of conserved quantities constraining the unitary dynamics. Here, the authors show that in interacting integrable spin chains there exist also conserved operators that are responsible for ballistic spin transport. Such conserved operators highly influence both the equilibrium and nonequilibrium physics: at equilibrium they are responsible for a nonvanishing spin Drude weight even at half filling, while in the nonequilibrium case they are responsible for out-of-equilibrium steady states characterized by persistent spin currents or dynamically expanding magnetic domains.