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arXiv:2010.08425 · Physical review. B./Physical review. B

Zero-magnon sound in quantum Heisenberg ferromagnets

Raphal Goll, Andreas Rückriegel, Peter Kopietz

Published Dec 30, 2020• 16 citations
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Abstract

Using a functional renormalization-group approach, we show that at low temperatures and in a certain range of magnetic fields, the longitudinal dynamic structure factor of quantum Heisenberg ferromagnets in dimensions $D\ensuremath{\le}2$ exhibits a well-defined quasiparticle peak with linear dispersion that we identify as zero-magnon sound. In $D>2$, the larger phase space available for the decay into transverse spin waves leads only to a broad hump centered at zero frequency whose width scales linearly in momentum.

DOI
10.1103/physrevb.102.224437

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Using a functional renormalization-group approach, we show that at low temperatures and in a certain range of magnetic fields, the longitudinal dynamic structure factor of quantum Heisenberg ferromagnets in dimensions D\ensuremath≤2D\ensuremath{\le}2D\ensuremath≤2 exhibits a well-defined quasiparticle peak with linear dispersion that we identify as zero-magnon sound. In D>2D>2D>2, the larger phase space available for the decay into transverse spin waves leads only to a broad hump centered at zero frequency whose width scales linearly in momentum.
We use the spin functional renormalization group recently developed by two of us [Krieg and Kopietz, Phys. Rev. B 99, 060403(R) (2019)] to calculate the magnetization M(H,T)M(H,T)M(H,T) and the damping of magnons due to classical longitudinal fluctuations of quantum Heisenberg ferromagnets. In order to guarantee that for vanishing magnetic field $H\ensuremath{\rightarrow}0$, the magnon spectrum is gapless when the spin rotational invariance is spontaneously broken, we use a Ward identity to express the magnon self-energy in terms of the magnetization. In two dimensions, our approach correctly predicts the absence of long-range magnetic order for H=0H=0H=0 at finite temperature TTT. The magnon spectrum then exhibits a gap from which we obtain the transverse correlation length. We also calculate the wave-function renormalization factor of the magnons. As a mathematical byproduct, we derive a recursive form of the generalized Wick theorem for spin operators in frequency space which facilitates the calculation of arbitrary time-ordered connected correlation functions of an isolated spin in a magnetic field.
The author calculates the dynamical structure factors of a two-dimensional Heisenberg antiferromagnet at T=0, to order 1/(2S) 2 , taking careful account of the umklapp processes. It is shown that, for the transverse spin part, the terms of order 1/(2S) 2 give rise to a broad sideband peak of three-magnon excitations, whose intensity is small but not negligible. For the longitudinal spin part, the terms of order 1/(2S) 2 modify noticeably the spectral shape of the continuum of two-magnon excitations as a function of frequency. These results could be observed in future neutron scattering experiments.
We calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ in the paramagnetic regime of quantum Heisenberg ferromagnets for temperatures TTT close to the critical temperature Tc{T}_{c}Tc​ using our recently developed functional renormalization group approach to quantum spin systems. In d=3d=3d=3 dimensions, we find that for small momenta $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ the dynamic structure factor assumes the scaling form $S(\mathbit{k},\ensuremath{\omega})=(\ensuremath{\tau}TG(\mathbit{k})/\ensuremath{\pi})\mathrm{\ensuremath{\Phi}}(k\ensuremath{\xi},\ensuremath{\omega}\ensuremath{\tau})$, where $G(\mathbit{k})$ is the static spin-spin correlation function, $\ensuremath{\xi}$ is the correlation length, and the characteristic timescale $\ensuremath{\tau}$ is proportional to ${\ensuremath{\xi}}^{5/2}$. We explicitly calculate the dynamic scaling function $\mathrm{\ensuremath{\Phi}}(x,y)$ and find satisfactory agreement with neutron scattering experiments probing the critical spin dynamics in EuO and EuS. Precisely at the critical point where $\ensuremath{\xi}=\ensuremath{\infty}$, our result for the dynamic structure factor can be written as $S(\mathbit{k},\ensuremath{\omega})={(\ensuremath{\pi}{\ensuremath{\omega}}_{k})}^{\ensuremath{-}1}{T}_{c}G(\mathbit{k}){\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\omega}/{\ensuremath{\omega}}_{k})$, where ${\ensuremath{\omega}}_{k}\ensuremath{\propto}{k}^{5/2}$. We find that ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ vanishes as ${\ensuremath{\nu}}^{\ensuremath{-}13/5}$ for large $\ensuremath{\nu}$, and as ${\ensuremath{\nu}}^{3/5}$ for small $\ensuremath{\nu}$. While the large-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$ is consistent with calculations based on mode-coupling theory and with perturbative renormalization group calculations to second order in $\ensuremath{\epsilon}=6\ensuremath{-}d$, our result for small frequencies disagrees with previous calculations. We argue that until now, neither experiments nor numerical simulations have been sufficiently accurate to determine the low-frequency behavior of ${\mathrm{\ensuremath{\Psi}}}_{c}(\ensuremath{\nu})$. We also calculate the low-temperature behavior of $S(\mathbit{k},\ensuremath{\omega})$ in one- and two-dimensional ferromagnets and find that it satisfies dynamic scaling with exponent z=2z=2z=2 and exhibits a pseudogap for small frequencies.
The longitudinal spin structure factor for the XXZ-chain at small wave vector q is obtained using Bethe ansatz, field theory methods, and the density matrix renormalization group. It consists of a peak with a peculiar, non-Lorentzian shape and a high-frequency tail. We show that the width of the peak is proportional to q2 for finite magnetic field compared to q3 for a zero field. For the tail we derive an analytic formula without any adjustable parameters and demonstrate that the integrability of the model directly affects the line shape.
The large-scale dynamics of quantum integrable systems is often dominated by ballistic modes due to the existence of stable quasiparticles. We here consider as an archetypical example for such a system the spin-12\frac{1}{2}21​ XXX Heisenberg chain that features magnons and their bound states. An interesting question, which we here investigate numerically, arises with respect to the fate of ballistic modes at finite temperatures in the limit of zero magnetization m=0m=0m=0. At a finite magnetization density mmm, the spin-autocorrelation function $\mathrm{\ensuremath{\Pi}}(x,t)$ (at high temperatures) typically exhibits a trimodal behavior with left- and right-moving quasiparticle modes and a broad center peak with slower dynamics. The broadening of the fastest propagating modes exhibits a subdiffusive t1/3{t}^{1/3}t1/3 scaling at large magnetization densities $m\ensuremath{\rightarrow}\frac{1}{2}$, familiar from noninteracting models; it crosses over into a diffusive scaling t1/2{t}^{1/2}t1/2 upon decreasing the magnetization to smaller values. The behavior of the center peak appears to exhibit a crossover from transient superdiffusion to ballistic relaxation at long times. In the limit $m\ensuremath{\rightarrow}0$, the weight carried by the propagating peaks tends to zero; the residual dynamics is carried only by the central peak; it is sub-ballistic and characterized by a dynamical exponent zzz close to the value 32\frac{3}{2}23​ familiar from Kardar-Parisi-Zhang (KPZ) scaling. We confirm that, employing elaborate finite-time extrapolations, that the spatial scaling of the correlator $\mathrm{\ensuremath{\Pi}}$ is in excellent agreement with KPZ-type behavior and analyze the corresponding corrections.
The influence of the nonlinearity of the spin equations of motion on the dynamic structure factor of the antiferromagnetic Heisenberg chain in an external field is calculated in the low-temperature limit. The memory functions for the longitudinal and transverse spin component are calculated rigorously to lowest nonvanishing order in the temperature. In this limit, only two-magnon processes contribute to the memory functions. Divergencies in the two-magnon density of states give rise to resonances in addition to the usual spin-wave peak in the dynamic structure factor.