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arXiv:cond-mat/0408598 · Physical Review B

Long-range dynamics of magnetic impurities coupled to a two-dimensional Heisenberg antiferromagnet

Andreas Lüscher, Oleg P. Sushkov

Published Feb 23, 2005• 16 citations
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Abstract

We consider a two-dimensional Heisenberg antiferromagnet on a square lattice with weakly coupled impurities, i.e., additional spins interacting with the host magnet by a small dimensionless coupling constant $g⪡1$. Using linear spin-wave theory, we find that the magnetization disturbance at distance $r$ from a single impurity behaves as $\ensuremath{\delta}{S}^{z}\ensuremath{\propto}g∕r$ for $1⪡r⪡1∕g$ and as $\ensuremath{\delta}{S}^{z}\ensuremath{\propto}1∕(g{r}^{3})$ for $r⪢1∕g$. Surprisingly the magnetization disturbance is inversely proportional to the coupling constant. The interaction between two impurities separated by a distance $r$ is $\ensuremath{\delta}ϵ\ensuremath{\propto}{g}^{2}∕r$ for $1⪡r⪡1∕g$ and $\ensuremath{\delta}ϵ\ensuremath{\propto}1∕{r}^{3}$ for $r⪢1∕g$. For large distances, the interaction is therefore universal and independent of the coupling constant. We have also found that the frequency of Rabi oscillations between two impurities is logarithmically enhanced compared to the decay width ${\ensuremath{\omega}}_{\mathrm{Rabi}}\ensuremath{\propto}{g}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}(1∕\mathit{gr})$ at $1⪡r⪡1∕g$. This leads to a logarithmic enhancement for NMR and EPR line broadening. All these astonishing results are due to the gapless spectrum of magnetic excitations in the quantum antiferromagnet.

DOI
10.1103/physrevb.71.064414

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We consider a two-dimensional Heisenberg antiferromagnet on a square lattice with weakly coupled impurities, i.e., additional spins interacting with the host magnet by a small dimensionless coupling constant g⪡1g⪡1g⪡1. Using linear spin-wave theory, we find that the magnetization disturbance at distance rrr from a single impurity behaves as \ensuremathδSz\ensuremath∝g∕r\ensuremath{\delta}{S}^{z}\ensuremath{\propto}g∕r\ensuremathδSz\ensuremath∝g∕r for 1⪡r⪡1∕g1⪡r⪡1∕g1⪡r⪡1∕g and as \ensuremathδSz\ensuremath∝1∕(gr3)\ensuremath{\delta}{S}^{z}\ensuremath{\propto}1∕(g{r}^{3})\ensuremathδSz\ensuremath∝1∕(gr3) for r⪢1∕gr⪢1∕gr⪢1∕g. Surprisingly the magnetization disturbance is inversely proportional to the coupling constant. The interaction between two impurities separated by a distance rrr is \ensuremathδϵ\ensuremath∝g2∕r\ensuremath{\delta}ϵ\ensuremath{\propto}{g}^{2}∕r\ensuremathδϵ\ensuremath∝g2∕r for 1⪡r⪡1∕g1⪡r⪡1∕g1⪡r⪡1∕g and \ensuremathδϵ\ensuremath∝1∕r3\ensuremath{\delta}ϵ\ensuremath{\propto}1∕{r}^{3}\ensuremathδϵ\ensuremath∝1∕r3 for r⪢1∕gr⪢1∕gr⪢1∕g. For large distances, the interaction is therefore universal and independent of the coupling constant. We have also found that the frequency of Rabi oscillations between two impurities is logarithmically enhanced compared to the decay width \ensuremathωRabi\ensuremath∝g2ln(1∕gr){\ensuremath{\omega}}_{\mathrm{Rabi}}\ensuremath{\propto}{g}^{2}\phantom{\rule{0.2em}{0ex}}\mathrm{ln}(1∕\mathit{gr})\ensuremathωRabi​\ensuremath∝g2ln(1∕gr) at 1⪡r⪡1∕g1⪡r⪡1∕g1⪡r⪡1∕g. This leads to a logarithmic enhancement for NMR and EPR line broadening. All these astonishing results are due to the gapless spectrum of magnetic excitations in the quantum antiferromagnet.
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We discuss the effects of nonmagnetic impurities (static holes) in the S=1/2S=1/2S=1/2 Heisenberg model on a two-dimensional (2D) square lattice, and on two- and three-leg ladders. Results of quantum Monte Carlo simulations show that a free localized moment is induced around an isolated impurity only in the spin-gapped two-leg ladder, in agreement with previous theoretical expectations. The localization length of the S=1/2S=1/2S=1/2 impurity moment in a two-leg ladder with isotropic couplings is $\ensuremath{\approx}1.5$ lattice spacings. In gapless ladders (odd number of legs) and in the 2D lattice, no free moments are induced. However, in the 2D system, which has antiferromagnetic long-range order at T=0T=0T=0 and a gap for longitudinal fluctuations, the impurity leads to a localized distortion of the magnetization in the direction of the broken symmetry, with total Sz=1/2.{S}^{z}=1/2.Sz=1/2. The shape of the distortion is in close agreement with previous spin-wave calculations. For all the systems, a staggered moment is induced by the impurity. For the gapped two-leg ladder this is exponentially localized within a length equal to the spin correlation length of the ladder ($\ensuremath{\approx}3.2$ lattice spacings). For the three-leg ladder, the integrated staggered moment diverges as the system size goes to infinity, but the staggered magnetization at a given site appears to vanish. In 2D, we discuss the behavior seen in finite systems both with and without additional symmetry-breaking mechanisms. We also discuss the effect of impurities on the NMR Knight shift and the bulk magnetic susceptibility.
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