\hfil Kondo effect in the hierarchical $s-d$ model
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\hfil{Giovanni Gallavotti, Ian Jauslin}
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\hfil2015\par
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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
\indent The $s-d$ model was introduced by Anderson [\cite{An61}] and used by Kondo [\cite{Ko64}] 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{An80}], which was shown to be exactly solvable by Bethe Ansatz. In [\cite{BGJ15}], 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{BGJ15}] to one of its invariant manifolds. The physics of both models are therefore very closely related, as had already been argued in [\cite{BGJ15}]. 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
\indent The $s-d$ model [\cite{Ko64}] 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
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{An80}], 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
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,
\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{BGJ15}] 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{BGJ15}].\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
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
for $m\le0$. 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,$$-N_\beta\}$, $\Delta\in\mathcal Q_m$, $\eta\in\{-,+\}$, $\alpha\in\{\uparrow,\downarrow\}$.\par
\indent The propagator of the hierarchical $s-d$ model is defined as
for $m\in\{0,-1,\cdots,$$-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
(the numbering is meant to recall that in [\cite{BGJ15}]) in which $\bm\tau=(\tau^1,\tau^2,\tau^3)$ and $\mathbf A_\eta^{[\le0]}(\Delta)$ is a vector of polynomials in the fields whose $j$-th component for $j\in\{1,2,3\}$ is
in which $|\mathcal Q_m|=2^{N(\beta)-|m|}$ is the cardinality of $\mathcal Q_m$. In addition, similarly to [\cite{BGJ15}], the map relating $\ell_p^{[m]}$ to $\ell_p^{[m-1]}$ and $C^{[m]}$ can be computed explicitly from~(\ref{eqindW}):
\indent The flow equation~(\ref{eqbetafun}) can be recovered from that of the hierarchical Andrei model studied in [\cite{BGJ15}] (see in particular [\cite{BGJ15}, (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{BGJ15}] 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