Impurity spin dynamics in 2D antiferromagnets and superconductors
Matthias Vojta
We discuss the universal theory of localized impurities in the paramagnetic state of 2D antiferromagnets where the spin gap is assumed to be significantly smaller than a typical exchange energy. We study the impurity spin susceptibility near the host quantum transition from a gapped paramagnet to a N\'{e}el state, and we compute the impurity-induced damping of the spin-1 mode of the gapped antiferromagnet. Under suitable conditions our results apply also to d-wave superconductors.
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% THE HOUSTON M2S CONFERENCE PAPER
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\title{Impurity spin dynamics in 2D antiferromagnets and superconductors}

\author{ Matthias Vojta, Chiranjeeb Buragohain and Subir Sachdev
\address{Department of Physics, Yale University, P.O.
         Box 208120, New Haven, CT 06520-8120, USA}%
        \thanks{Research supported by US NSF Grant No
                DMR 96--23181 and by the DFG (VO 794/1-1)}
        }

\begin{document}

\begin{abstract}

We discuss the universal theory of localized impurities in the paramagnetic state
of 2D antiferromagnets
where the spin gap is assumed to be significantly smaller than a
typical exchange energy.
We study the impurity spin susceptibility
near the host quantum transition from a gapped paramagnet to a
N\'{e}el state,
and we compute the impurity-induced damping of the spin-1
mode of the gapped antiferromagnet.
Under suitable conditions our results apply also to d-wave superconductors.
\end{abstract}

\maketitle

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Doped antiferromagnets (AF) have been the subject of intense studies in the context of the
cuprate high-temperature superconductors and other layered
transition metal compounds.
We present a quantum theory of a particular class of doped AF
where it is possible to neglect the coupling between
the spin and charge degrees of freedom and consider a theory of
the spin excitations alone.
Such a theory will apply to
(i) quasi-2D `spin gap' insulators
like ${\rm Sr Cu}_2 {\rm O}_3$ or ${\rm Na V}_2 {\rm O}_5$
in which a small fraction of the magnetic ions
(Cu or V) are replaced by non-magnetic ions like Zn or
Li and to
(ii) high-temperature superconductors like ${\rm Y} {\rm Ba}_2 {\rm
Cu}_3 {\rm O}_7$ in which a small fraction of Cu has been replaced
by non-magnetic Zn or Li.
In the first case the spin gap $\Delta$ is significantly smaller than the charge
gap justifying a theory of the spin excitations alone.
In the second situation the effect of the fermionic quasiparticles in
the superconducting state can be shown \cite{long} to be weak
due to the linearly vanishing density of states of the Fermi level.

The effect of a (magnetic or non-magnetic) impurity can be probed by
measuring the uniform spin susceptibility, which takes the form
$\chi = (g \mu_B)^2 (A \chi_b + \chi_{{\rm imp}})$ where $A$ is
the total area of the AF,
$\chi_b$ is the bulk response per unit area, and $\chi_{{\rm imp}}$ is the additional
impurity contribution.
In the paramagnetic ground state of the host
each impurity induces a distortion of the host spin arrangement
with a net magnetic moment $S$ associated with
the impurity.
The distortion is {\em confined} to the vicinity of the impurity
which implies that the impurity susceptibility follows
\begin{equation}
\chi_{{\rm imp}} = \frac{S(S+1)}{3 k_B T} ~~~~ \mbox{as}~~T\to 0.
\label{chi1}
\end{equation}
For a non-magnetic impurity in a spin-1/2 system we have $S=1/2$;
for a general impurity eq. (\ref{chi1}) can be used as definition
of $S$.

The basis of our investigations is a boundary quantum field theory
which describes a bulk AF together with arbitrary
localized deformations.
We focus on the vicinity of a quantum transition from a paramagnet to a magnetically
ordered N\'{e}el state:
Then the spin gap in the paramagnetic state is small
compared to a typical nearest-neighbor exchange, $\Delta \ll J$, which
is the situation realized in many compounds.
The field theory has been discussed in Ref. \cite{long};
it consists of $d+1$-dimensional $\phi^4$ theory for the bulk ordering
transition and a coupling to a local quantum impurity spin.
The renormalization-group (RG) analysis shows that both the bulk and the boundary couplings
are marginal for $d=3$ and flow to fixed-point values for $d<3$.
This implies that the coupling between the bulk and impurity excitations
becomes {\em universal},
and the spin dynamics in the vicinity of the impurity is completely determined
by bulk parameters, the gap $\Delta$ and the velocity of spin
excitations $c$.
Based on the RG results one obtains a number of universal properties in an
expansion in $\epsilon=3-d$; we mention here the behavior of the uniform susceptibility
at the bulk critical point.
The system shows the Curie response of an {\em irrational} spin
as $T\to 0$, $\chi_{{\rm imp}} = {\cal C}_1 / (k_B T)$,
where ${\cal C}_1$ is a {\em universal} number independent of
microscopic details.
The $\epsilon$ expansion result for ${\cal C}_1$ is
\begin{equation}
{\cal C}_1 = \frac{S(S+1)}{3} \left[ 1 + \left(\frac{33 \epsilon}{40}\right)^{1/2}
\!\! -\frac{7 \epsilon}{4} + ...   \right].
\end{equation}


% Here are the "names" for particles, constants etc.
\newcommand{\bb}  {{f}}         % the auxiliary particles for the impurity spin
\newcommand{\ttd} {{t}}         % the "diagonal" bulk bosons (after Bogoliubov), H_bulk is diag. in \ttd
\newcommand{\KK}  {{K}}         % the bulk-boundary coupling constant
\newcommand{\JJ}  {{J}}         % the bulk energy scale
\newcommand{\KKK} {{\tilde K}}  % the combination of \KK and \JJ left in the Green's function
\newcommand{\OM}  {{\epsilon}}  % the excitation energy of the bulk bosons
\newcommand{\MM}  {m}           % the (renormalized) mass of the bulk bosons (~T in QC region, =Delta for T<<Delta)
\newcommand{\DD}  {\Delta}      % the T=0 mass of the bulk bosons
\newcommand{\coup} {s}          % the bulk coupling constant which drives the bulk transition

More detailed dynamic information can be obtained by a self-consistent
diagrammatic method.
% which we will discuss in the remainder of the paper.
The paramagnetic phase of the bulk is assumed to be dimerized,
its spin-1 excitations can be described using triplet bosons $\ttd_{\bf k \alpha}$.
The impurity is represented by an additional spin $S_\alpha$ at site 0,
\begin{eqnarray}
H =
\sum_{{\bf k},\alpha} \OM_{\bf k} \ttd_{\bf k \alpha}^\dagger \ttd_{\bf k \alpha}
\,+\,
\frac{\KK}{\sqrt{N_s}}
\sum_{\bf k\alpha} S_\alpha
{ \ttd_{\bf k\alpha}^\dagger + \ttd_{\bf k\alpha}
\over
\sqrt{\OM_{\bf k} / \JJ } }
\end{eqnarray}
where $J$ is the host exchange constant, $\OM_{\bf k}$ the energy of the spin-1
mode in the bulk, $K$ the coupling constant to the impurity spin, and $N_s$ the number
of lattice sites.
The impurity spin is represented by auxiliary fermions $\bb$,
the impurity dynamics is contained in the fermion self-energy which arises
from the scattering off the $\ttd$ bosons.
We employ a self-consistent non-crossing approximation (NCA) to calculate
this self-energy; this approach follows from a saddle-point principle
after generalizing the spin symmetry to SU($N$) and taking
the limit $N\to\infty$.
The NCA equations can be solved in the scaling limit;
the value of the coupling $K$ drops out of all results for physical
observables provided that $\Delta\ll J$ -- we obtain the same universal
behavior as predicted by the RG.
In fact, the results for susceptibility and impurity spin correlations
agree with the one-loop RG result \cite{long}.

\begin{figure}
\centerline{\includegraphics[width=2.9in]{fig2}}
\vspace{-0.3in}
\caption{Left: Universal lineshape ${\rm Im} \Phi$
% as a function of $\hbar\omega/\Delta$
for $\Gamma/\Delta = 0.125$.
Right: The same, but convoluted with a Gaussian corresponding to the experimental
resolution of \protect\cite{keimer}, together with the data points of
Ref. \protect\cite{keimer}.
We have used $\Delta = 43$ meV -- this small shift may be attributed to
perturbations being irrelevant in the RG sense.
}
\vspace{-0.1in}
\label{fig2}
\end{figure}

The diagrammatic approach can be easily applied to a system with a finite
density of impurities $n_{\rm imp}$.
The important observation is that the impact of the impurities is
determined by a single energy scale
$\Gamma \equiv n_{\rm imp} (\hbar c)^d / \Delta^{d-1}$.
The AF in the absence of impurities shows a pole in the
dynamic susceptibility $\chi_{{\bf Q}} ( \omega )$ at the
AF wavevector $\bf Q$.
Our main concern is the fate of this collective peak upon the introduction
of impurities.
Scaling arguments predict that the susceptibility takes the form
\begin{equation}
\chi_{{\bf Q}} ( \omega ) = \frac{\mathcal{A}}{\Delta^2}
\Phi \left( \frac{\hbar \omega}{\Delta}, \frac{\Gamma}{\Delta}
\right)
\label{polescale} ~~~(T=0)\,,
\end{equation}
where $\Phi$ is a universal function, and $\cal A$ denotes the
quasiparticle weight.
In the absence of impurities we have
$\Phi ( \overline{\omega}, 0) = 1/(1 - (\overline{\omega} + i 0^{+})^2)$.
%
The self-energy of the spin-1 bosons caused by the scattering at
randomly distributed impurities is calculated using
a self-consistent Born approximation.
The equations for the Green's functions can be entirely
written in terms of scaling functions with arguments
$\hbar\omega/\Delta$ and $\Gamma/\Delta$, consistent with
the scaling prediction (\ref{polescale}).
A numerical result for $\Phi$ is shown in Fig.~\ref{fig2}.
The quasiparticle pole is broadened to an asymmetric line,
with a tail at high frequencies.
Our theory can be applied to a recent experiment \cite{keimer}
where the inpurity-induced broadening of the spin-1 `resonance peak'
at energy $\Delta = 40$ meV in ${\rm Y} {\rm Ba}_2 {\rm Cu}_3 {\rm O}_7$
has been observed.
This experiment has $n_{\rm imp}=0.005$, $\Gamma = 5$ meV, $\Gamma/\Delta = 0.125$.
The half-width of the line is approximately $\Gamma$, and this is
in excellent accord with the measured linewidth of 4.25 meV, see
Fig.~\ref{fig2}.
More tests of the predictions of our theory should be possible
in the future.



\begin{thebibliography}{9}

\bibitem{long}
%        M.~Vojta, C.~Buragohain, S.~Sachdev, cond-mat/9912020 (1999),
%        S.~Sachdev, C.~Buragohain, and M.~Vojta, Science
%        {\bf 286}, 2479 (1999).
        M.~Vojta et al., cond-mat/9912020 (1999),
        S.~Sachdev et al., Science {\bf 286}, 2479 (1999).

\bibitem{keimer} H.~F.~Fong et al., Phys. Rev. Lett. {\bf 82}, 1939 (1999).

\end{thebibliography}


\end{document}

cond-mat0002316.extracted_bib
\begin{thebibliography}{9}

\bibitem{long}
%        M.~Vojta, C.~Buragohain, S.~Sachdev, cond-mat/9912020 (1999),
%        S.~Sachdev, C.~Buragohain, and M.~Vojta, Science
%        {\bf 286}, 2479 (1999).
        M.~Vojta et al., cond-mat/9912020 (1999),
        S.~Sachdev et al., Science {\bf 286}, 2479 (1999).

\bibitem{keimer} H.~F.~Fong et al., Phys. Rev. Lett. {\bf 82}, 1939 (1999).

\end{thebibliography}