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\bf\Large
\hfil Kondo effect in the hierarchical $s-d$ model
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\hfil{Giovanni Gallavotti, Ian Jauslin}
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\rm

\hfil2015\par
\hugeskip

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\small
The $s-d$ model describes a chain of spin-1/2 electrons interacting magnetically with a two-level impurity. It was introduced to study the Kondo effect, in which the magnetic susceptibility of the impurity remains finite in the 0-temperature limit as long as the interaction of the impurity with the electrons is anti-ferromagnetic. A variant of this model was introduced by Andrei, which he proved was exactly solvable via Bethe Ansatz. A hierarchical version of Andrei's model was studied by Benfatto and the authors. In the present letter, that discussion is extended to a hierarchical version of the $s-d$ model.  The resulting analysis is very similar to the hierarchical Andrei model, though the result is slightly simpler.\par
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\indent The $s-d$ model was introduced by Anderson [\cite{andSO}] and used by Kondo [\cite{konSF}] to study what would subsequently be called the {\it Kondo effect}. It describes a chain of electrons interacting with a fixed spin-1/2 magnetic impurity. One of the manifestations of the effect is that when the coupling is anti-ferrmoagnetic, the magnetic susceptibility of the impurity remains finite in the 0-temperature limit, whereas it diverges for ferromagnetic and for vanishing interactions.\par

\indent A modified version of the $s-d$ model was introduced by Andrei [\cite{andEZ}], which was shown to be exactly solvable by Bethe Ansatz. In [\cite{bgjOFi}], a hierarchical version of Andrei's model was introduced and shown to exhibit a Kondo effect. In the present letter, we show how the argument can be adapted to the $s-d$ model.\par

\indent We will show that in the hierarchical $s-d$ model, the computation of the susceptibility reduces to iterating an {\it explicit} map relating 6 {\it running coupling constants} (rccs), and that this map can be obtained by restricting the flow equation for the hierarchical Andrei model [\cite{bgjOFi}] to one of its invariant manifolds. The physics of both models are therefore very closely related, as had already been argued in [\cite{bgjOFi}]. This is particularly noteworthy since, at 0-field, the flow in the hierarchical Andrei model is relevant, whereas it is marginal in the hierarchical $s-d$ model, which shows that the relevant direction carries little to no physical significance.\par
\bigskip

\indent The $s-d$ model [\cite{konSF}] represents a chain of non-interacting spin-1/2 fermions, called {\it electrons}, which interact with an isolated spin-1/2 {\it impurity} located at site 0. The Hilbert space of the system is $\mathcal F_L\otimes\mathbb C^2$ in which $\mathcal F_L$ is the Fock space of a length-$L$ chain of spin-1/2 fermions (the electrons) and $\mathbb C^2$ is the state space for the two-level impurity. The Hamiltonian, in the presence of a magnetic field of amplitude $h$ in the direction $\bm\omega\equiv(\bm\omega_1,\bm\omega_2,\bm\omega_3)$, is
\begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
H_K=&H_0+V_0+V_h=:H_0+V\\[0.3cm]
H_0=&\sum_{\alpha\in\{\uparrow,\downarrow\}}\sum_{x=-{L}/2}^{{L}/2-1} c^+_\alpha(x)\,\left(-\frac{\Delta}2-1\right)\,c^-_\alpha(x)\\[0.5cm]
V_0=&-\lambda_0\sum_{j=1,2,3\atop\alpha_1,\alpha_2} c^+_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c^-_{\alpha_2}(0)\, \tau^j\\[0.5cm]
V_h=&-h \,\sum_{j=1,2,3}\bm\omega_j \tau^j
\end{array}\label{eqhamdef}\end{equation} 
where $\lambda_0$ is the interaction strength, $\Delta$ is the discrete Laplacian $c_\alpha^\pm(x),\,\alpha=\uparrow,\downarrow$ are creation and annihilation operators acting on {\it electrons}, and $\sigma^j=\tau^j,\,j=1,2,3$, are Pauli matrices. The operators $\tau^j$ act on the {\it impurity}. The boundary conditions are taken to be periodic.\par

\indent In the {\it Andrei model} [\cite{andEZ}], the impurity is represented by a fermion instead of a two-level system, that is the Hilbert space is replaced by $\mathcal F_L\otimes\mathcal F_1$, and the Hamiltonian is defined by replacing $\tau^j$ in~(\ref{eqhamdef}) by $d^+\tau^jd^-$ in which $d_\alpha^\pm(x),\,\alpha=\uparrow,\downarrow$ are creation and annihilation operators acting on the impurity.\par
\bigskip

\indent The partition function $Z={\rm Tr}\, e^{-\beta H_K}$ can be expressed formally as a functional integral:
\begin{equation}
Z=\mathrm{Tr}\int P(d\psi)\, \sum_{n=0}^\infty(-1)^n\int_{0<t_1<\cdots<t_n<\beta}\kern-50pt dt_1\cdots dt_n\, \mathcal V(t_1)\cdots\mathcal V(t_n)
\label{eqpartfn}\end{equation}
in which $\mathcal V(t)$ is obtained from $V$ by replacing $c_\alpha^\pm(0)$ in~(\ref{eqhamdef}) by a {\it Grassmann} field $\psi_\alpha^\pm(0,t)$, $P(d\psi)$ is a {\it Gaussian Grassmann measure} over the fields $\{\psi_\alpha^\pm(0,t)\}_{t,\alpha}$ whose {\it propagator} ({\it i.e.} {\it covariance}) is, in the $L\to\infty$ limit,
$$
g(t,t')=\frac1{(2\pi)^2}\int dk dk_0 \frac{e^{i k_0(t-t')}}{i k_0-\cos k},
$$
and the trace is over the state-space of the spin-1/2 impurity, that is a trace over $\mathbb C^2$.\par
\bigskip

\indent We will consider a {\it hierarchical} version of the $s-d$ model. The hierarchical model defined below is {\it inspired} by the $s-d$ model in the same way as the hierarchical model defined in [\cite{bgjOFi}] was inspired by the Andrei model. We will not give any details on the justification of the definition, as such considerations are entirely analogous to the discussion in [\cite{bgjOFi}].\par

\indent The model is defined by introducing a family of {\it hierarchical fields} and specifying a {\it propagator} for each pair of fields. The average of any monomial of fields is then computed using the Wick rule.\par

\indent Assuming $\beta=2^{N_\beta}$ with $N_\beta=\log_2\beta\in\mathbb N$, the time axis $[0,\beta)$ is paved with boxes ({\it i.e.} intervals) of size $2^{-m}$ for every $m\in\{0,-1,\ldots,-N_\beta\}$: let 
\begin{equation}
\mathcal Q_m:=\left\{[i 2^{|m|}, (i+1) 2^{|m|})\right\}_{i=0,1,\cdots,2^{N_\beta-|m|}-1}^{m=0,-1,\ldots}
\label{eqtiledef}\end{equation}
Given a box $\Delta\in{\mathcal Q}_m$, let $t_\Delta$ denote the center of $\Delta$, and given a point $t\in R$, let $\Delta^{[m]}(t)$ be the (unique) box on scale $m$ that contains $t$. We further decompose each box $\Delta\in\mathcal Q_m$ into two {\it half boxes}: for $\eta\in\{-,+\}$, let 
\begin{equation}
\Delta_{\eta}:=\Delta^{[m+1]}(t_{\Delta}+\eta2^{-m-2})
\label{eqhalfboxdef}\end{equation} 
for $m\le 0$. Thus $\Delta_{-}$ can be called the ``lower half'' of $\Delta$ and $\Delta_{+}$ the ``upper half''.\par

\indent The elementary fields used to define the hierarchical $s-d$ model will be {\it constant on each half-box} and will be denoted by $\psi_\alpha^{[m]\pm}(\Delta_{\eta})$ for $m\in\{0,-1,\cdots,$ $-N_\beta\}$, $\Delta\in\mathcal Q_m$, $\eta\in\{-,+\}$, $\alpha\in\{\uparrow,\downarrow\}$.\par

\indent The propagator of the hierarchical $s-d$ model is defined as 
\begin{equation}
\left<\psi_{\alpha}^{[m]-}(\Delta_{-\eta})\psi_{\alpha}^{[m]+}(\Delta_{\eta})\right >:= \eta
\label{eqprop}\end{equation} 
for $m\in\{0,-1,\cdots,$ $-N_\beta\}$, $\Delta\in\mathcal Q_m$, $\eta\in\{-,+\}$, $\alpha\in\{\uparrow,\downarrow\}$. The propagator of any other pair of fields is set to 0.\par

\indent Finally, we define 
\begin{equation}
\psi^\pm_\alpha(t):= \sum_{m=0}^{-N_\beta} 2^{\frac{m}2}\psi_\alpha^{[m]\pm}(\Delta^{[m+1]}(t)).
\label{eqfielddcmp}\end{equation} 

\indent The partition function for the hierarchical $s-d$ model is 
\begin{equation}
Z=\mathrm{Tr}\left< \sum_{n=0}^\infty(-1)^n\int_{0<t_1<\cdots<t_n<\beta}\kern-50pt dt_1\cdots dt_n\, \mathcal V(t_1)\cdots\mathcal V(t_n) \right>
\label{eqhierpartfn}\end{equation} 
in which the $\psi^\pm_\alpha(0,t)$ in $\mathcal V(t)$ have been replaced by the $\psi_\alpha^\pm(t)$ defined in~(\ref{eqfielddcmp}): 
\begin{equation}
\mathcal V(t):=-\lambda_0\sum_{j=1,2,3\atop\alpha_1,\alpha_2} \psi^+_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^-_{\alpha_2}(t)\, \tau^j -h \,\sum_{j=1,2,3}\bm\omega_j \tau^j.
\label{eqhierpot}\end{equation} 
This concludes the definition of the hierarchical $s-d$ model.\par
\bigskip


\indent We will now show how to compute the partition function~(\ref{eqhierpartfn}) using a renormalization group iteration. We first rewrite 
\begin{equation}
\sum_{n=0}^\infty(-1)^n\int_{0<t_1<\cdots<t_n<\beta}\kern-50pt dt_1\cdots dt_n\, \mathcal V(t_1)\cdots\mathcal V(t_n) =\prod_{\Delta\in\mathcal Q_0}\prod_{\eta=\pm}\left(\sum_{n=0}^\infty\frac{(-1)^n}{2^nn!}\mathcal V(t_{\Delta_\eta})^n\right)
\label{eqtrotthier}\end{equation} 
and find that 
\begin{equation}
\sum_{n=0}^\infty\frac{(-1)^n}{2^nn!}\mathcal V(t_{\Delta_\eta^{[0]}})^n =C\left(1+\sum_{p}\ell_p^{[0]}O_{p,\eta}^{[\le 0]}(\Delta^{[0]})\right)
\label{eqexpV}\end{equation} 
with 
\begin{equation}\begin{array}{r@{\quad}l}
O_{0,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf A^{[\le 0]}_\eta(\Delta)\cdot\bm\tau,& O_{1,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf A^{[\le 0]}_\eta(\Delta)^2,\\[0.3cm]
O_{4,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf A^{[\le 0]}_\eta(\Delta)\cdot\bm\omega,& O_{5,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf \bm\tau\cdot\bm\omega,\\[0.3cm]
O_{6,\eta}^{[\le 0]}(\Delta):=\frac12(\mathbf A^{[\le 0]}_\eta(\Delta)\cdot\bm\omega)(\bm\tau\cdot\bm\omega),& O_{7,\eta}^{[\le 0]}(\Delta):=\frac12(\mathbf A^{[\le 0]}_\eta(\Delta)^2)(\bm\tau\cdot\bm\omega)
\end{array}\label{eqOdef}\end{equation} 
(the numbering is meant to recall that in [\cite{bgjOFi}]) in which $\bm\tau=(\tau^1,\tau^2,\tau^3)$ and $\mathbf A_\eta^{[\le 0]}(\Delta)$ is a vector of polynomials in the fields whose $j$-th component for $j\in\{1,2,3\}$ is 
\begin{equation}
A_\eta^{[\le 0]j}(\Delta):=\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2} \psi_\alpha^{[\le 0]+}(\Delta_\eta)\sigma^j_{\alpha,\alpha'}\psi_{\alpha'}^{[\le 0]-}(\Delta_\eta)
\label{eqAdef}\end{equation} 
$\psi_\alpha^{[\le 0]\pm}:=\sum_{m\le0}2^{\frac m2}\psi_\alpha^{[m]\pm}$, and 
\begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
C=&\cosh(\tilde h),\quad \ell_0^{[0]}=\frac1C\frac{\lambda_0}{\tilde h}\sinh(\tilde h),\quad
\ell_1^{[0]}=\frac1C\frac{\lambda_0^2}{12\tilde h}(\tilde h\cosh(\tilde h)+2\sinh(\tilde h))\\[0.3cm]
\ell_4^{[0]}=&\frac1C\lambda_0\sinh(\tilde h),\quad \ell_5^{[0]}=\frac1C\sinh(\tilde h),\quad
\ell_6^{[0]}=\frac1C\frac{\lambda_0}{\tilde h}(\tilde h\cosh(\tilde h)-\sinh(\tilde h))\\[0.3cm]
\ell_7^{[0]}=&\frac1C\frac{\lambda_0^2}{12\tilde h^2}(\tilde h^2\sinh(\tilde h)+2\tilde h\cosh(\tilde h)-2\sinh(\tilde h))
\end{array}\label{eqinitcd}\end{equation} 
in which $\tilde h:=h/2$.\par

\indent By a straightforward induction, we find that the partition function~(\ref{eqhierpartfn}) can be computed by defining 
\begin{equation}
C^{[m]}\mathcal W^{[m-1]}(\Delta^{[m]}):=\left<\prod_\eta\left(\mathcal W^{[m]}(\Delta^{[m]}_\eta)\right)\right>_m
\label{eqindW}\end{equation} 
in which $\left<\cdot\right>_m$ denotes the average over $\psi^{[m]}$, $C^{[m]}>0$ and 
\begin{equation}
\mathcal W^{[m-1]}(\Delta^{[m]})=1+\sum_p\ell_p^{[m]}O_p^{[\le m]}(\Delta^{[m]})
\label{eqexprW}\end{equation} 
in terms of which 
\begin{equation}
Z=C^{2|\mathcal Q_0|}\prod_{m=-N(\beta)+1}^0(C^{[m]})^{|\mathcal Q_{m-1}|}
\label{eqZind}\end{equation} 
in which $|\mathcal Q_m|=2^{N(\beta)-|m|}$ is the cardinality of $\mathcal Q_m$. In addition, similarly to [\cite{bgjOFi}], the map relating $\ell_p^{[m]}$ to $\ell_p^{[m-1]}$ and $C^{[m]}$ can be computed explicitly from~(\ref{eqindW}):
\begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
C^{[m]} =& 1 +\frac{3}{2}\ell_{0}^2 +\ell_{0}\ell_{6} +9\ell_{1}^2 +\frac{\ell_{4}^2}{2} +\frac{\ell_{5}^2}{4} +\frac{\ell_{6}^2}{2} +9\ell_{7}^2 \\[0.3cm]
\ell^{[m-1]}_{0} =& \frac1C\left(\ell_{0} -\ell_{0}^2 +3\ell_{0}\ell_{1} -\ell_{0}\ell_{6} \right)\\[0.3cm]
\ell^{[m-1]}_{1} =& \frac1C\left(\frac{\ell_{1}}{2} +\frac{\ell_{0}^2}{8} +\frac{\ell_{0}\ell_{6}}{12} +\frac{\ell_{4}^2}{24} +\frac{\ell_{5}\ell_{7}}{4} +\frac{\ell_{6}^2}{24} \right)\\[0.3cm]
\ell^{[m-1]}_{4} =& \frac1C\left(\ell_{4} +\frac{\ell_{0}\ell_{5}}{2} +3\ell_{0}\ell_{7} +3\ell_{1}\ell_{4} +\frac{\ell_{5}\ell_{6}}{2} +3\ell_{6}\ell_{7} \right)\\[0.3cm]
\ell^{[m-1]}_{5} =& \frac1C\left(2\ell_{5} +2\ell_{0}\ell_{4} +36\ell_{1}\ell_{7} +2\ell_{4}\ell_{6} \right)\\[0.3cm]
\ell^{[m-1]}_{6} =& \frac1C\left(\ell_{6} +\ell_{0}\ell_{6} +3\ell_{1}\ell_{6} +\frac{\ell_{4}\ell_{5}}{2} +3\ell_{4}\ell_{7} \right)\\[0.3cm]
\ell^{[m-1]}_{7} =& \frac1C\left(\frac{\ell_{7}}{2} +\frac{\ell_{0}\ell_{4}}{12} +\frac{\ell_{1}\ell_{5}}{4} +\frac{\ell_{4}\ell_{6}}{12} \right)
\end{array}\label{eqbetafun}\end{equation}
in which the $^{[m]}$ have been dropped from the right hand side.\par
\bigskip


\indent The flow equation~(\ref{eqbetafun}) can be recovered from that of the hierarchical Andrei model studied in [\cite{bgjOFi}] (see in particular [\cite{bgjOFi}, (C1)] by restricting the flow to the invariant submanifold defined by \begin{equation} \ell_2^{[m]}=\frac13,\quad \ell_3^{[m]}=\frac16\ell_1^{[m]},\quad \ell_8^{[m]}=\frac16\ell_4^{[m]}. \label{e18}\end{equation} This is of particular interest since $\ell_2^{[m]}$ is a relevant coupling and the fact that it plays no role in the $s-d$ model indicates that it has little to no physical relevance.\par

\indent The qualitative behavior of the flow is therefore the same as that described in [\cite{bgjOFi}] for the hierarchical Andrei model. In particular the susceptibility, which can be computed by deriving $-\beta^{-1}\log Z$ with respect to $h$, remains finite in the 0-temperature limit as long as $\lambda_0<0$, that is as long as the interaction is anti-ferromagnetic.\par
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{\bf Acknowledgements}: We are grateful to G.~Benfatto for many enlightening discussions on the $s-d$ and Andrei's models.

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