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arXiv:2104.04270 · Physical review. B./Physical review. B

Dissipative spin dynamics in hot quantum paramagnets

Dmytro Tarasevych, Peter Kopietz

Published Jul 16, 2021• 17 citations
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Abstract

We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] to calculate the spin-spin correlation function $G(\mathbit{k},\ensuremath{\omega})$ of quantum Heisenberg magnets at infinite temperature. For small wave vectors $\mathbit{k}$ and frequencies $\ensuremath{\omega}$ we find that $G(\mathbit{k},\ensuremath{\omega})$ assumes in dimensions $d>2$ the diffusive form predicted by hydrodynamics. In three dimensions our result for the spin-diffusion coefficient $\mathcal{D}$ is somewhat smaller than previous theoretical predictions based on the extrapolation of the short-time expansion, but is still about $30%$ larger than the measured high-temperature value of $\mathcal{D}$ in the Heisenberg ferromagnet ${\mathrm{Rb}}_{2}{\mathrm{CuBr}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$. In reduced dimensions $d\ensuremath{\le}2$ we find superdiffusion characterized by a frequency-dependent complex spin-diffusion coefficient $\mathcal{D}(\ensuremath{\omega})$ which diverges logarithmically in $d=2$, and as a power-law $\mathcal{D}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1/3}$ in $d=1$. Our result in one dimension implies scaling with dynamical exponent $z=3/2$, in agreement with recent calculations for integrable spin chains. Our approach is not restricted to the hydrodynamic regime and allows us to calculate the dynamic structure factor $S(\mathbit{k},\ensuremath{\omega})$ for all wave vectors. We show how the short-wavelength behavior of $S(\mathbit{k},\ensuremath{\omega})$ at high temperatures reflects the relative sign and strength of competing exchange interactions.

DOI
10.1103/physrevb.104.024423

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We use the functional renormalization approach for quantum spin systems developed by Krieg and Kopietz [Phys. Rev. B 99, 060403(R) (2019)] to calculate the spin-spin correlation function G(\mathbitk,\ensuremathω)G(\mathbit{k},\ensuremath{\omega})G(\mathbitk,\ensuremathω) of quantum Heisenberg magnets at infinite temperature. For small wave vectors \mathbitk\mathbit{k}\mathbitk and frequencies \ensuremathω\ensuremath{\omega}\ensuremathω we find that G(\mathbitk,\ensuremathω)G(\mathbit{k},\ensuremath{\omega})G(\mathbitk,\ensuremathω) assumes in dimensions d>2d>2d>2 the diffusive form predicted by hydrodynamics. In three dimensions our result for the spin-diffusion coefficient D\mathcal{D}D is somewhat smaller than previous theoretical predictions based on the extrapolation of the short-time expansion, but is still about 3030%30 larger than the measured high-temperature value of D\mathcal{D}D in the Heisenberg ferromagnet Rb2CuBr4\ifmmode⋅\else\textperiodcentered\fi2H2O{\mathrm{Rb}}_{2}{\mathrm{CuBr}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}Rb2​CuBr4​\ifmmode⋅\else\textperiodcentered\fi2H2​O. In reduced dimensions d\ensuremath≤2d\ensuremath{\le}2d\ensuremath≤2 we find superdiffusion characterized by a frequency-dependent complex spin-diffusion coefficient D(\ensuremathω)\mathcal{D}(\ensuremath{\omega})D(\ensuremathω) which diverges logarithmically in d=2d=2d=2, and as a power-law D(\ensuremathω)\ensuremath∝\ensuremathω\ensuremath−1/3\mathcal{D}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1/3}D(\ensuremathω)\ensuremath∝\ensuremathω\ensuremath−1/3 in d=1d=1d=1. Our result in one dimension implies scaling with dynamical exponent z=3/2z=3/2z=3/2, in agreement with recent calculations for integrable spin chains. Our approach is not restricted to the hydrodynamic regime and allows us to calculate the dynamic structure factor S(\mathbitk,\ensuremathω)S(\mathbit{k},\ensuremath{\omega})S(\mathbitk,\ensuremathω) for all wave vectors. We show how the short-wavelength behavior of S(\mathbitk,\ensuremathω)S(\mathbit{k},\ensuremath{\omega})S(\mathbitk,\ensuremathω) at high temperatures reflects the relative sign and strength of competing exchange interactions.
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.
This thesis is concerned with the investigation of static and dynamic properties of quantum Heisenberg paramagnets in the absence of a magnetic field and therefore for vanishing magnetization. For this purpose a new formulation of the spin functional renormalization group (SFRG) is employed. The first manifestations of the SFRG were developed by Krieg and Kopietz, motivated by the FRG approach to ordinary field theories and the older works of Vaks, Larkin and Pikin on diagrammatic methods for spin operators. The main idea is to study quantum spin systems by considering the evolution of correlation functions under a continuous deformation of the interaction between magnetic moments, starting from a solvable limit. This leads to nonperturbative results for quantities like the spin-spin correlation function. After a basic introduction to the phenomena and concomitant problems discussed in this thesis, a detailed description of the SFRG method in its initial formulation is given in the second chapter. We start with the generating functional of connected imaginary-time spin-correlation functions GΛ [h], for which an exact flow equation is derived. A particular issue, already pointed out by Krieg and Kopietz, arises here, namely the singular non-interacting limit of its subtracted Legendre transform ΓΛ [m]. As a consequence the initial condition of that functional does not have a proper series expansion in powers of m. This prevents us from working directly within a pure one-particle irreducible (1-PI) parametrization of the correlation functions, as is often done in the context of field theories. Thus motivated, we develop a workaround explicitly tailored to paramagnets, which provides us with a functional that has a well-behaved Legendre transform. The new approach is based on a different treatment of fluctuations at zero and finite frequencies, analogous to a previous hybrid formulation for the symmetry-broken phase. Certain properties, considered to be highly relevant for isotropic paramagnets, as well as previous observations, already made in the study of simpler spin systems like the Ising model, serve as additional justifications for choosing this construction. In the third chapter our new method is assessed by calculating the dynamic susceptibility G(k, iω) and thus the dynamic structure factor S(k, ω) in the symmetric phase. For this purpose an approximate integral equation for the dynamic polarization function Π̃(k, iω) was derived. This equation results from a truncation of the hierarchy of flow equations and contains static quantities, that are assumed to be known from another source. Our first application is the high-temperature limit T → ∞ in d ≤ 3 dimensions. Salient features, believed to be part of the spin dynamics in isotropic Heisenberg magnets are also exhibited by our solution, like (anomalous) diffusion in a suitable hydrodynamic limit. Moreover we obtain the same order of magnitude for the diffusion coefficient D as in experiments and other theoretical calculations. Other aspects do not entirely agree with previous approaches. Afterwards we continue by investigating systems close to the critical point Tc. Dynamic scaling forms for Π̃(k, iω) and S(k, ω), which, like spin diffusion, are postulated on the basis of quite general physical arguments, are reproduced. Agreement of the line-shapes 2with neutron scattering experiments at T = Tc is found to be satisfying, with deviations for ω → 0, that may be attributed to the simplicity of the approximation, like at infinite temperature. Finally, we focus our attention on the thermodynamic properties of isotropic Heisenberg paramagnets by calculating the static susceptibility G(k). For this purpose we employ simple truncation schemes of the flow equations for the static self-energy ΣΛ (k) and four-spin vertex ΓΛ , together with a basic ansatz for the dynamic polarization Π̃(k, iω) in quantum systems. As a result we obtain transition temperatures Tc of three-dimensional nonfrustrated magnets within an accuracy of 5 percent compared to established benchmark values from Quantum Monte Carlo and high temperature expansion series. We conclude this chapter by giving an outlook on the application of our method to frustrated systems, which may require a combined non-trivial calculation of static and dynamic properties.
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 spin dynamics of strongly interacting one-dimensional quantum magnets at high-temperature displays an emergent coarse-grained hydrodynamic behavior, understood as the result of conserved quantities. Different kinds of emergent hydrodynamics can be characterized by the dynamical exponent zzz governing the length-time scaling $x\ensuremath{\sim}{t}^{1/z}$. Relying on extensive numerical simulations, the authors extract the exponent zzz for various microscopic quantum spin-SSS models. They identify three universal regimes for the spin transport (superdiffusive with z=3/2z=3/2z=3/2, ballistic with z=1z=1z=1, and diffusive with z=2z=2z=2), which depend only on the isotropy and integrability of the system, but are independent of its microscopic details.
In this thesis we investigate the thermodynamic and dynamic properties of the D-dimensional quantum Heisenberg ferromagnet within the spin functional renormalization group (FRG); a formalism describing the evolution of the system’s observables as the magnetic exchange inter-action is artificially deformed. Following an introduction providing a self contained summary of the conceptual and mathematical background, we present the spin FRG as developed by Krieg and Kopietz in references [1] and [2] in chapter two. Thereto, the generating functional of the imaginary time-spin correlation functions and its exact flow equation describing the deformation process of the exchange interaction are introduced. In addition, it is highlighted that - in contrast to conventional field-theoretic FRG approaches - the related Legendre trans-formed functional cannot be defined if the exchange interaction is initially switched off. Next, we show that this limitation can be circumvented within an alternativ hybrid approach, which treats transverse and longitudinal spin fluctuations differently. The relevant functionals are introduced and the relations of the corresponding functional Taylor coefficients with the spin correlation functions are discussed. Lastly, the associated flow equations are derived and the possibility of explicit or spontaneous symmetry breaking is taken into account. In chapter three, we benchmark the hybrid formalism against a calculation of the thermo-dynamic properties of the one and two-dimensional Heisenberg model at low temperatures T and finite magnetic field H. For this purpose, we devise an anisotropic deformation scheme of the exchange interaction which allows for a controlled truncation of the infinite hierarchy of FRG flow equations. Thereby, contact with mean-field and spin-wave theory is made and the violation of the Mermin-Wagner theorem is discussed. To fulfill the latter, the truncation scheme is then complemented by a Ward identity relating the transverse self-energy and the magnetization. The resulting magnetization M (H, T ) and isothermal susceptibility χ(H, T ) are in quantitative agreement with the literature and the established behavior of the transverse correlation length and the zero-field susceptibility close to the critical point is qualitatively reproduced in the limit H → 0. Finally, we investigate the longitudinal dynamics at low temperatures. To this end, the hierarchy of flow equations is solved within the same anisotropic deformation scheme complemented by an expansion in the inverse interaction range, and the resulting longitudinal dynamic structure factor is calculated within a low-momentum expansion. In D = 3, the large phase space accessible for the decay into transverse magnons yields only a broad hump centered at zero frequency whose width scales linearly in momentum. In contrast, at low temperatures and in a certain range of magnetic fields, a well-defined quasiparticle peak with linear dispersion emerges in D ≤ 2, which we identify as zero-magnon sound. Sound velocity and damping are discussed as a function of temperature and magnetic field, and the relevant momentum-frequency window is estimated and compared to the hydrodynamic second-magnon regime.