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arXiv:0909.1972 · Physical Review B

Hydrodynamic limit for the spin dynamics of the Heisenberg chain from quantum Monte Carlo calculations

Simon Grossjohann, Wolfram Brenig

Published Jan 20, 2010• 43 citations
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Abstract

We show that quantum Monte Carlo calculations of the dynamic structure factor of the isotropic-spin-1/2 antiferromagnetic chain at intermediate temperatures corroborate a picture of diffusive spin dynamics at finite frequencies in the low-energy long-wavelength limit and are in good agreement with recent predictions for this by Sirker et al. [Phys. Rev. Lett. 103, 216602 (2009)].

DOI
10.1103/physrevb.81.012404

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We show that quantum Monte Carlo calculations of the dynamic structure factor of the isotropic-spin-1/2 antiferromagnetic chain at intermediate temperatures corroborate a picture of diffusive spin dynamics at finite frequencies in the low-energy long-wavelength limit and are in good agreement with recent predictions for this by Sirker et al. [Phys. Rev. Lett. 103, 216602 (2009)].
Combining high-temperature expansions with the recursion method and quantum Monte Carlo simulations with the maximum entropy method, we study the dynamics of the spin-1/2 Heisenberg chain at temperatures above and below the coupling J. By comparing the two sets of calculations, their relative strengths are assessed. At high temperatures, we find that there is a low-frequency peak in the momentum-integrated dynamic structure factor, due to diffusive long-wavelength modes. This peak is rapidly suppressed as the temperature is lowered below J. Calculation of the complete dynamic structure factor S(k,\ensuremath{\omega}) shows how the spectral features associated with the two-spinon continuum develop at low temperatures. We extract the nuclear spin-lattice relaxation rate 1/T1{\mathrm{T}}_{1}T1​ from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for Sr2{\mathrm{Sr}}_{2}Sr2​CuO3{\mathrm{CuO}}_{3}CuO3​ and CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​. We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T)]3/2{]}^{3\mathrm{/}2}]3/2 and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T)]1/2{]}^{1\mathrm{/}2}]1/2, expected from previous theoretical studies.
We study the finite temperature, low energy, long wave-length spectrum of the dynamic structure factor of the spin-1/21/21/2 antiferromagnetic Heisenberg chain in the presence of exchange anisotropy and external magnetic fields. Using imaginary-time quantum Monte-Carlo we extract parameters, relevant to characterize a {\it renormalized} Luttinger liquid. For small momentum our results are consistent with a change from propagating spinon density waves to spin diffusion, described by a finite-frequency spin-current relaxation rate. Results for this relaxation rate as well as other Luttinger liquid parameters are presented versus temperature, momentum, magnetic field, and anisotropy, including finite-size analysis, and checks for anomalous diffusion. Our results are consistent with exact diagonalization and Bethe Ansatz, where available, and with corroborate findings of other previous studies using bosonization, transfer matrix renormalization group, and quantum Monte-Carlo.
We used a combination of the quantum Monte Carlo technique and the maximum entropy method to calculate the dynamic structure factor of the quantum Heisenberg antiferromagnet. The dynamics is governed by renormalized damped spin waves and we calculate their frequencies and linewidths. The computed spin-wave dispersion is in quantitative agreement with inelastic-neutron-scattering experiments on La2{\mathrm{La}}_{2}La2​CuO4{\mathrm{CuO}}_{4}CuO4​. The low-energy sector exhibits scaling behavior with the characteristic frequency scale, which is inversely proportional to the correlation length.
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>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 ${\mathrm{Rb}}_{2}{\mathrm{CuBr}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}2{\mathrm{H}}_{2}\mathrm{O}$. 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=2d=2d=2, and as a power-law $\mathcal{D}(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{\ensuremath{-}1/3}$ 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 $S(\mathbit{k},\ensuremath{\omega})$ 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.
The spin-lattice relaxation rate 1/T1{\mathit{T}}_{1}T1​ and the spin-echo decay rate 1/T2G{\mathit{T}}_{2\mathit{G}}T2G​ for the spin-1/2 antiferromagnetic Heisenberg chain are calculated using quantum Monte Carlo and maximum-entropy analytic continuation. The results are compared with recent analytical calculations by Sachdev. If the nuclear hyperfine form factor Aq{\mathit{A}}_{\mathit{q}}Aq​ is strongly peaked around q=\ensuremath{\pi}, the predicted low-temperature behavior [1/T1{\mathit{T}}_{1}T1​\ensuremath{\sim}ln1/2{\mathrm{ln}}^{1/2}ln1/2(1/T), 1/T2G{\mathit{T}}_{2\mathit{G}}T2G​\ensuremath{\sim}ln1/2{\mathrm{ln}}^{1/2}ln1/2(1/T)/\ensuremath{\surd}T] extends up to temperatures as high as T/J\ensuremath{\approxeq}0.5. If Aq{\mathit{A}}_{\mathit{q}}Aq​ has significant weight for q\ensuremath{\approxeq}0 there are large contributions from diffusive long-wavelength processes not taken into account in the theory, and very low temperatures are needed in order to observe the asymptotic T\ensuremath{\rightarrow}0 forms.
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