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arXiv:2610.10736

Finite- and long-range spin-wave free-energy asymptotics for the two-dimensional quantum Heisenberg ferromagnet

Andreas Klippel

Published Oct 7, 2026• 0 citations
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Abstract

For the two-dimensional quantum Heisenberg ferromagnet with asymptotically radial power-law exchange, we prove the leading low-temperature free energy at every fixed spin, uniformly on the weak-field scale. We obtain fractional and logarithmically corrected laws associated with the dispersions predicted in [16, 17]. Quantum Monte Carlo simulations support the associated anomalous magnon dispersions for spin one-half [20]. For kernels with finite second moment, we also extend the nearest-neighbour theorem [14], generalizing Takahashi's quadratic law [21] and its weak-field dependence [11].

DOI
10.48550/arxiv.2610.10736

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For the two-dimensional quantum Heisenberg ferromagnet with asymptotically radial power-law exchange, we prove the leading low-temperature free energy at every fixed spin, uniformly on the weak-field scale. We obtain fractional and logarithmically corrected laws associated with the dispersions predicted in [16, 17]. Quantum Monte Carlo simulations support the associated anomalous magnon dispersions for spin one-half [20]. For kernels with finite second moment, we also extend the nearest-neighbour theorem [14], generalizing Takahashi's quadratic law [21] and its weak-field dependence [11].
We prove the leading low-temperature free-energy asymptotics of the nearest-neighbour quantum Heisenberg ferromagnet on Z2\mathbb Z^2Z2. For every fixed S∈{12,1,32,…}S\in\{\frac12,1,\frac32,\ldots\}S∈{21​,1,23​,…}, the free energy per site satisfies lim⁡β→∞β2Sf2(β,S)=−π24. \lim_{β\to\infty}β^2S f_2(β,S)=-\fracπ{24}. limβ→∞​β2Sf2​(β,S)=−24π​. This formula was predicted by Takahashi's modified spin-wave theory [15]. Napiórkowski and Seiringer proved the corresponding upper bound but left the matching lower bound in two dimensions open [13], a gap later highlighted again by Seiringer [14]. We prove this missing bound through a new comparison between the magnon exclusion dynamics and free bosons, based on extending wave functions to collision configurations with controlled kinetic cost.
Abstract We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin S≥1/2S\ge 1/2S≥1/2 S ≥ 1 / 2 . We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.
Employing large-scale quantum Monte Carlo simulations, we systematically compute the energy spectra of the two-dimensional (2D) spin-1/2 Heisenberg model with long-range interactions. With the $1/{r}^{\ensuremath{\alpha}}$ ferromagnetic and staggered antiferromagnetic interactions, we find the explicit range in $\ensuremath{\alpha}$ for the short-range Goldstone-type (gapless), anomalous Goldstone-type (gapless), and Higgs-type (gapped) spectra. Accompanied by the spin-wave analysis, our numerical results vividly reveal how the long-range interactions alter the usual linear and quadratic magnon dispersions in 2D quantum magnets and give rise to anomalous dynamical exponents. Moreover, we find the explicit case where the gapped excitation emerges at a noninteger decay exponent $\ensuremath{\alpha}$ for the antiferromagnetic Hamiltonian. This work provides the first set of unbiased dynamical data of long-range quantum many-body systems and suggests that many universally accepted low-energy customs for short-range systems need to be substantially modified for long-range ones, which are of immediate relevance to the ongoing experimental efforts from quantum simulators to 2D quantum moir\'e materials.
We show that a small fraction of antiferromagnetic bonds qualitatively alters the long-wavelength dynamics of two-dimensional Heisenberg ferromagnets. Although the classical ground state remains magnetized, weak bond frustration generates logarithmically correlated spatial fluctuations of the local spin stiffness, despite the microscopic disorder being short ranged. A replica field theory calculation shows that the effective disorder strength grows under coarse graining, while the spin stiffness decreases, yielding anomalously soft magnons with a scale-dependent dynamical exponent z>2z > 2z>2. Numerical diagonalization of the semiclassical spin-wave Hamiltonian confirms the anomalous low-energy scaling. The flow is toward an infinite-disorder, zero stiffness regime.
We prove Bloch's T3/2 law for the spontaneous magnetization of three-dimensional quantum Heisenberg ferromagnets at every fixed positive quantum spin. The result holds for every nonnegative symmetric finite-range interaction whose support generates ℤ3, including spatially anisotropic couplings. The leading magnetization deficit has the exact coefficient determined by the determinant of the quadratic one-magnon dispersion. The thermodynamic limit is taken before the right field derivative at zero, and the low-temperature limit is taken last.