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Ian Jauslin
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Benfatto_Gallavotti_Jauslin_2015.tex
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@@ -2,7 +2,7 @@
% load packages
\usepackage{header}
% bibliography commands
\usepackage{bibliography}
\usepackage{BBlog}
% miscellaneous commands
\usepackage{toolbox}
% main style file
@@ -56,13 +56,13 @@
\section{Kondo model and main results} \label{secmodel}
\indent Consider a {\it 1-dimensional} Fermi gas of spin-1/2 ``electrons'', and a spin-1/2 fermionic ``impurity'', with {\it no} interactions. It is well known that:
\begin{enumerate}[\ \ (1)\ \ ]
\begin{enumerate}
\item the magnetic susceptibility of the impurity diverges as $\beta=\frac1{k_B T}\to\infty$ while
\item both the total susceptibility per particle of the electron gas ({\it i.e.}\ the response to a field acting on the whole sample) [\cite{Ki76}] and the susceptibility to a magnetic field acting on a single lattice site of the chain ({\it i.e.}\ the response to a field localized on a site, say at $0$) are finite at zero temperature (see remark (1) in appendix~\ref{appXY} for a discussion of the second claim).
\item both the total susceptibility per particle of the electron gas ({\it i.e.}\ the response to a field acting on the whole sample) [\cite{Ki76}] and the susceptibility to a magnetic field acting on a single lattice site of the chain ({\it i.e.}\ the response to a field localized on a site, say at $0$) are finite at zero temperature (see remark~\ref{rkfinitesusc} in appendix~\ref{appXY} for a discussion of the second claim).
\end{enumerate}
\medskip
\indent The question that will be addressed in this work is whether a small coupling of the impurity {fermion} with the electron gas can change this behavior, that is whether the susceptibility of the impurity interacting with the electrons diverges or not.
\indent The question that will be addressed in this work is whether a small coupling of the impurity fermion with the electron gas can change this behavior, that is whether the susceptibility of the impurity interacting with the electrons diverges or not.
\indent To that end we will study a model inspired by the Kondo Hamiltonian which, expressed in second quantized form, is
@@ -73,7 +73,7 @@ V_0=&-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}c^+_{\alp
V_h=& -h\sum_{j=1,2,3}\bm\omega_j \, \sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}d^+_\alpha\sigma^j_{\alpha,\alpha'} d_{\alpha'}^-
\label{eqhamkondo}\end{array}\end{equation}
where $\lambda_0,h$ are the interaction and magnetic field strengths and
\begin{enumerate}[\ \ (1)\ \ ]
\begin{enumerate}
\item$c_\alpha^\pm(x),d^\pm_\alpha, \,\alpha=\uparrow,\downarrow$ are creation and annihilation operators corresponding respectively to electrons and the impurity,
\item$\sigma^j,\,j=1,2,3$, are the Pauli matrices,
\item$x$ is on the unit lattice and $-{L}/2$, ${L}/2$ are identified (periodic boundary),
@@ -101,14 +101,14 @@ where $\tau^j$ is the $j$-th Pauli matrix and acts on the spin of the impurity.
{\bf Remark}: The hierarchical Kondo model {\it will not be an approximation} of~(\ref{eqhamkondo}). It is a model that illustrates a simple mechanism for the control of the growth of relevant operators in a theory exhibiting a Kondo effect.
\medskip
\indent The reason why the Kondo effect is not easy to understand is that it is an intrinsically non-perturbative effect, in that the impurity susceptibility in the interacting model is qualitatively different from its non-interacting counterpart. In the sense of the renormalization group it exhibits several ``relevant'', ``marginal'' and ``irrelevant'' running couplings: this makes any naive perturbative approach hopeless because all couplings become large ({\it i.e.}\ at least of O(1)) at large scale, no matter how small the interaction is, as long as $\lambda_0<0$, and thus leave the perturbative regime. It is among the simplest cases in which asymptotic freedom {\it does not occur}. {Using the fact that the beta function of the hierarchical model can be computed exactly, its non-perturbative regime can easily be investigated.}
\indent The reason why the Kondo effect is not easy to understand is that it is an intrinsically non-perturbative effect, in that the impurity susceptibility in the interacting model is qualitatively different from its non-interacting counterpart. In the sense of the renormalization group it exhibits several ``relevant'', ``marginal'' and ``irrelevant'' running couplings: this makes any naive perturbative approach hopeless because all couplings become large ({\it i.e.}\ at least of O(1)) at large scale, no matter how small the interaction is, as long as $\lambda_0<0$, and thus leave the perturbative regime. It is among the simplest cases in which asymptotic freedom {\it does not occur}. Using the fact that the beta function of the hierarchical model can be computed exactly, its non-perturbative regime can easily be investigated.
\medskip
\indent In the sections below, we will define the hierarchical Kondo model and show numerical evidence for the following claims (in principle, such claims could be proved using computer-assisted methods, though, since the numerical results are very clear and stable, it may not be worth the trouble).
\smallskip
{\it If the interactions between the electron spins and the impurity are antiferromagnetic } ({\it i.e.}\ $\lambda_0<0$ in our notations), then
\begin{enumerate}[\ \ (1)\ \ ]
\begin{enumerate}
\item The {\it existence of a Kondo effect} can be proved in spite of the lack of asymptotic freedom and formal growth of the effective Hamiltonian away from the trivial fixed point, {\it because the beta function can be computed exactly} (in particular non-pertubatively).
\item In addition, there exists an inverse temperature $\beta_K=2^{n_K(\lambda_0)}$ called the {\it Kondo} inverse temperature, such that the Kondo effect manifests itself for $\beta>\beta_K$. Asymptotically as $\lambda_0\to0$, $n_K(\lambda_0)=c_1|\lambda_0|^{-1}+O(1)$.
@@ -130,7 +130,7 @@ where $\tau^j$ is the $j$-th Pauli matrix and acts on the spin of the impurity.
\medskip
{\bf Remark:}\listparpenalty
\begin{enumerate}[\ \ (1)\ \ ]
\begin{enumerate}
\item It is worth stressing that in a system consisting of two classical spins with coupling $\lambda_0$ the susceptibility at $0$ field is $4\beta(1+e^{-2\beta\lambda_0})^ {-1}$, hence it vanishes at $T=0$ in the antiferromagnetic case and diverges in the ferromagnetic and in the free case. Therefore this simple model does not exhibit a Kondo effect.
\item In the exactly solvable XY model, which can be shown to be equivalent to a spin-less analogue of~(\ref{eqhamkondo}), the susceptibility can be shown to diverge in the $\beta\to\infty$ limit, see appendix~\ref{appXY}, \ref{appXYcomp} (at least for some boundary conditions). Therefore this model does not exhibit a Kondo effect either.
@@ -139,7 +139,7 @@ where $\tau^j$ is the $j$-th Pauli matrix and acts on the spin of the impurity.
\section{Functional integration in the Kondo model} \label{secfunint}
\indent In [\cite{Wi75}], Wilson studies the Kondo problem using renormalization group techniques in a Hamiltonian context. In the present work, our aim is to reproduce, in a simpler model, analogous results using a formalism based on functional integrals.
\indent In this section, we give a rapid review of the functional integral formalism we will use, following Refs.[\cite{BG90b}, \cite{Sh94}]. We will not attempt to reproduce all technical details, since it will merely be used as an inspiration for the definition of the hierarchical model in section~\ref{sechierk}.
\indent In this section, we give a rapid review of the functional integral formalism we will use, following [\cite{BG90}, \cite{Sh94}]. We will not attempt to reproduce all technical details, since it will merely be used as an inspiration for the definition of the hierarchical model in section~\ref{sechierk}.
\indent We introduce an extra dimension, called {\it imaginary time}, and define new creation and annihilation operators:
@@ -168,10 +168,11 @@ g_{\varphi,\alpha}(t-t'):=&
\end{array}\right..
\end{array}\label{eqprop}\end{equation}
\indent By a direct computation [\cite{BG90b},~(2.7)], we find that in the limit $L,\beta\to\infty$, if $e(k):=(1-\cos k) -1\equiv -\cos k$ (assuming the Fermi level is set to $1$, {\it i.e.}\ the Fermi momentum to $\pm\frac\pi2$) then
\indent By a direct computation [\cite{BG90},~(2.7)], we find that in the limit $L,\beta\to\infty$, if $e(k):=(1-\cos k) -1\equiv -\cos k$ (assuming the Fermi level is set to $1$, {\it i.e.}\ the Fermi momentum to $\pm\frac\pi2$) then
\begin{equation}
g_{\psi,\alpha}(\xi,\tau) =\int\frac{dk_0 dk}{(2\pi)^2}\,{e^{-ik_0(\tau+0^-)-ik\xi} \over-ik_0+e(k) },\quad
g_{\varphi,\alpha}(\tau) = \int\frac{dk_0}{2\pi}\,{e^{-ik_0(\tau+0^-)} \over-ik_0}.\label{eqpropk}\end{equation}
g_{\psi,\alpha}(\xi,\tau) =\int\frac{dk_0 dk}{(2\pi)^2}\,\frac{e^{-ik_0(\tau+0^-)-ik\xi}}{-ik_0+e(k)},\quad
g_{\varphi,\alpha}(\tau) = \int\frac{dk_0}{2\pi}\,\frac{e^{-ik_0(\tau+0^-)}}{-ik_0}.
\label{eqpropk}\end{equation}
If $\beta,L$ are finite, $\int\,\frac{dk_0 dk}{(2\pi)^2}$ in~(\ref{eqpropk}) has to be understood as $\frac1\beta \sum_{k_0} \frac1L \sum_k$, where $k_0$ is the ``Matsubara momentum'' $k_0= \frac\pi{\beta} +\frac{2\pi}\beta n_0$, $n_0\in\mathbb Z$, $|n_0|\le\frac12\beta$, and $k$ is the linear momentum $k=\frac{2\pi}L n$, $n\in [-L/2,L/2-1]\cap\mathbb Z$.
\medskip
@@ -179,7 +180,7 @@ If $\beta,L$ are finite, $\int\,\frac{dk_0 dk}{(2\pi)^2}$ in~(\ref{eqpropk}) has
\begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
V(\psi,\varphi)=&
-\lambda_0 \sum_{{j\in\{1,2,3\}}\atop{\alpha_1,\alpha'_1,\alpha_2,\alpha_2'}}\int dt \,(\psi^+_{\alpha_1}(0,t)\sigma^j_{\alpha_1,\alpha'_1} \psi^-_{\alpha'_1}(0,t)) (\varphi^+_{\alpha_2}(t)\sigma^j_{\alpha_2,\alpha_2'} \varphi^-_{\alpha_2'}(t))\\
&-h \sum_{j\in\{1,2,3\}}\bm\omega_j \int dt\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}\varphi^+_{\alpha}(t)\sigma^j_{\alpha,\alpha'} \varphi^-_{\alpha'}(t)
&-h \sum_{j\in\{1,2,3\}}\bm\omega_j \int dt\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}\varphi^+_{\alpha}(t)\sigma^j_{\alpha,\alpha'} \varphi^-_{\alpha'}(t).
\end{array}\label{eqpotgrass}\end{equation}
Notice that $V$ only depends on the fields located at the site $x=0$. This is important because it will allow us to reduce the problem to a 1-dimensional one [\cite{AY69}, \cite{AYH70}].
@@ -203,11 +204,11 @@ g_\varphi^{[\mathrm{uv}]}(\tau):=&g_\varphi(\tau)-\sum_{m=-N_\beta}^{m_0}g_\varp
where $m_0$ is an integer of order one (see below).
\medskip
{\bf Remark}: The $\omega=\pm$ label refers to the ``quasi particle'' momentum $\omega p_F$, where $p_F$ is the Fermi momentum. The usual approach [\cite{BG90b}, \cite{Sh94}] is to decompose the field $\psi$ into quasi-particle fields:
{\bf Remark}: The $\omega=\pm$ label refers to the ``quasi particle'' momentum $\omega p_F$, where $p_F$ is the Fermi momentum. The usual approach [\cite{BG90}, \cite{Sh94}] is to decompose the field $\psi$ into quasi-particle fields:
\begin{equation}
\psi^\pm_{\alpha}(0,t)=\sum_{\omega=\pm} \psi^\pm_{\omega,\alpha}(0,t),
\label{eqquasipartdcmp} \end{equation}
indeed, the introduction of quasi particles [\cite{BG90b}, \cite{Sh94}], is key to separating the oscillations on the Fermi scale $p_F^{-1}$ from the propagators thus allowing a ``naive'' renormalization group analysis of fermionic models in which multiscale phenomena are important (as in the theory of the ground state of interacting fermions [\cite{BG90b}, \cite{BGe94}], or as in the Kondo model). In this case, however, since the fields are evaluated at $x=0$, such oscillations play no role, so we will not decompose the field.
indeed, the introduction of quasi particles [\cite{BG90}, \cite{Sh94}], is key to separating the oscillations on the Fermi scale $p_F^{-1}$ from the propagators thus allowing a ``naive'' renormalization group analysis of fermionic models in which multiscale phenomena are important (as in the theory of the ground state of interacting fermions [\cite{BG90}, \cite{BGe94}], or as in the Kondo model). In this case, however, since the fields are evaluated at $x=0$, such oscillations play no role, so we will not decompose the field.
\medskip
\indent We set $m_0$ to be small enough ({\it i.e.}\ negative enough) so that $2^{m_0}p_F\le1$ and introduce a first {\it approximation}: we neglect $g_{\psi}^{[\mathrm{uv}]}$ and $g_\varphi^{[\mathrm{uv}]}$, and replace $e(k)$ in~(\ref{eqpropk}) by its first order Taylor expansion around $\omega p_F$, that is by $\omega k$. As long as $m_0$ is small enough, for all $m\le m_0$ the supports {of the two functions $\chi(2^{-2m}((k-\omega\pi/2)^2+k_0^2))$, $\omega=\pm1$}, which appear in the first of~(\ref{eqpropdcmp}) do not intersect, and approximating $e(k)$ by $\omega k$ is reasonable. We shall hereafter fix $m_0=0$ thus avoiding the introduction of a further length scale and keeping only two scales when no impurity is present.
@@ -236,10 +237,10 @@ with $\psi_{\alpha}^{[m]}(0,t)$ and $\varphi_\alpha^{[m]}(t)$ being, respectivel
\int P_0(d\psi^{[m]})\psi_{\alpha}^{[m]-}(0,t)\psi_{\alpha'}^{[m]+}(0,t')
=:&\delta_{\alpha,\alpha'}g_{\psi}^{[0]}(0,2^{m}(t-t'))\\
\int P(d\varphi^{[m]})\varphi_{\alpha}^{[m]-}(t)\varphi_{\alpha'}^{[m]+}(t')
=:&\delta_{\alpha,\alpha'}g_{\varphi}^{[0]}(2^{m}(t-t'))
=:&\delta_{\alpha,\alpha'}g_{\varphi}^{[0]}(2^{m}(t-t')).
\end{array}\label{eqfieldprop}\end{equation}
{\bf Remark:} by \label.(\ref{eqpropscale}) this is equivalent to stating that the propagators associated with the $\psi^{[m]},\varphi^{[m]}$ fields are $2^{-m}g^{[m]}$ and $g^{[m]}$, respectively.
{\bf Remark:} by (\ref{eqpropscale}) this is equivalent to stating that the propagators associated with the $\psi^{[m]},\varphi^{[m]}$ fields are $2^{-m}g^{[m]}$ and $g^{[m]}$, respectively.
\medskip
\indent Finally, we define
@@ -331,10 +332,10 @@ V(\psi,\varphi)=&
in which $\psi^\pm_\alpha(0,t)$ and $\varphi^\pm_\alpha(t)$ are now defined in~(\ref{eqfieldhier}).
\medskip
\indent Note that since the model defined above only involves fields localized at the impurity site, that is at $x=0$, we only have to deal with $1$-dimensional fermionic fields. {\it This does not mean} that the lattice supporting the electrons plays no role: on the contrary it will show up, and in an essential way, because the ``dimension'' of the electron field will be different from that of the impurity, as made already manifest by the factor $2^m\mathop{\longrightarrow}_{m\to-\infty}0$ in~(\ref{eqprophiercmp}).
\indent Note that since the model defined above only involves fields localized at the impurity site, that is at $x=0$, we only have to deal with $1$-dimensional fermionic fields. {\it This does not mean} that the lattice supporting the electrons plays no role: on the contrary it will show up, and in an essential way, because the ``dimension'' of the electron field will be different from that of the impurity, as made already manifest by the factor $2^m\displaystyle\mathop{\longrightarrow}_{m\to-\infty}0$ in~(\ref{eqprophiercmp}).
\medskip
\indent Clearly several properties of the non-hierarchical propagators,~(\ref{eqpropapprox}), are not reflected in~(\ref{eqprophiercmp}). However it will be seen that even so simplified the model exhibits a ``Kondo effect'' in the sense outlined in section~\ref{secintroduction}.
\indent Clearly several properties of the non-hierarchical propagators,~(\ref{eqpropapprox}), are not reflected in~(\ref{eqprophiercmp}). However it will be seen that even so simplified the model exhibits a ``Kondo effect'' in the sense outlined in section~\ref{secintro}.
\section{Beta function for the partition function.} \label{secbetapart}
\indent In this section, we show how to compute the partition function $Z$ of the hierarchical Kondo model (see~(\ref{eqhieravg})), and introduce the concept of a {\it renormalization group flow} in this context. We will first restrict the discussion to the $h=0$ case, in which $V=V_0$; the case $h\ne0$ is discussed in section~\ref{secbetakondo}.
@@ -351,8 +352,9 @@ and for $\Delta\in\mathcal Q_{-m},\,m<-N_\beta$,
\indent Notice that the fields $\psi_{\alpha}^{[\le m-1]\pm}(\Delta)$ and $\varphi_{\alpha}^{[\le m-1]\pm}(\Delta)$ play (temporarily) the role of {\it external fields} as they do not depend on the index $\eta$, and are therefore independent of the half box in which the {\it internal fields} $\psi_{\alpha}^{[\le m]\pm}(\Delta_\eta)$ and $\varphi_{\alpha}^{[\le m]\pm}(\Delta_\eta)$ are defined. In addition, by iterating~(\ref{eqhierfieldind}), we can rewrite~(\ref{eqfieldhier}) as
\begin{equation}
\psi_\alpha^\pm(t)\equiv\psi_{\alpha}^{[\le0]\pm}(\Delta^{[1]}(t)),\quad \varphi_\alpha^\pm(t)\equiv\varphi_\alpha^{[\le0]\pm}(\Delta^{[1]}(t))
\psi_\alpha^\pm(t)\equiv\psi_{\alpha}^{[\le0]\pm}(\Delta^{[1]}(t)),\quad \varphi_\alpha^\pm(t)\equiv\varphi_\alpha^{[\le0]\pm}(\Delta^{[1]}(t)).
\label{eqhierfieldindinit}\end{equation}
\indent We then define, for $m\in\{0,-1,\cdots,-N_\beta\}$,
\begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
\beta c^{[m]}+V^{[m-1]}(\psi^{[\le m-1]},\varphi^{[\le m-1]})
@@ -449,7 +451,7 @@ C^{[m]}=&1+ 3\ell_0^2+9\ell_2^2\\[0.3cm]
\ell_2^{[m-1]}=& \frac1{C^{[m]}}\Big(2\ell_2+ \ell_0^2\Big)\\
\end{array}\label{eqbetareduced}\end{equation}
which can be shown to have $4$ fixed points:
\begin{enumerate}[\ \ (1)\ \ ]
\begin{enumerate}
\item$f_0=(0,0)$, unstable in the $\ell_2$ direction and marginal in the $\ell_0$ direction (repelling if $\ell_0<0, \ell_2=0$), this is the {\it trivial fixed point};
\item $f_+=(0,\frac13)$, stable in the $\ell_2$ direction and marginal in the $\ell_0$ direction (repelling if $\ell_0<0, \ell_2=\frac13$), which we call the {\it ferromagnetic fixed-point} (because the flow converges to $f_+$ in the ferromagnetic case, see below);
\item $f_-=(0,-\frac13)$ stable in both directions;
@@ -466,7 +468,7 @@ $f_+$ (see appendix~\ref{appfixed}-{\bf\ref{ptfixedreduced}}).
\indent We introduce a magnetic field of amplitude $h\in R$ and direction $\bm\omega\in\mathcal S_2$ (in which $\mathcal S_2$ denotes the $2$-sphere) acting on the impurity. As a consequence, the potential $V$ becomes
\begin{equation}
V(\psi,\varphi)=V_0(\psi,\varphi)
V_h(\psi,\varphi)=V_0(\psi,\varphi)
-h\sum_{\displaystyle\mathop{\scriptstyle(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}_{j\in\{1,2,3\}}} \int dt(\varphi_{\alpha}^+(t)\sigma_{\alpha,\alpha'}^j\varphi_{\alpha'}^-(t))\, \omega_j
\label{eqpoth}\end{equation}
@@ -490,8 +492,8 @@ where $O_{n,\eta}^{[\le m]}(\Delta)$ for $n\in\{0,1,2,3\}$ was defined in~(\ref{
\begin{equation}\begin{array}{>{\displaystyle}c}
O^{[\le m]}_{4,\eta}(\Delta):=\frac12\mathbf A^{[\le m]}_\eta(\Delta)\cdot\bm\omega,\quad
O^{[\le m]}_{5,\eta}(\Delta):=\frac12\mathbf B^{[\le m]}_\eta(\Delta)\cdot\bm\omega,\\[0.5cm]
O^{[\le m]}_{6,\eta}(\Delta):=\frac12\Big(\mathbf A^{[\le m]}_\eta(\Delta) \cdot\bm\omega\Big)\Big(\mathbf B^{[\le m]}_\eta(\Delta)\cdot\bm\omega\Big)\\[0.5cm]
O^{[\le m]}_{7,\eta}(\Delta):= \frac12\Big(\mathbf A^{[\le m]}_\eta(\Delta) \cdot\mathbf A^{[\le m]}_\eta(\Delta)\Big)\Big(\mathbf B^{[\le m]}_\eta(\Delta) \cdot\bm\omega\Big)\\[0.5cm]
O^{[\le m]}_{6,\eta}(\Delta):=\frac12\Big(\mathbf A^{[\le m]}_\eta(\Delta) \cdot\bm\omega\Big)\Big(\mathbf B^{[\le m]}_\eta(\Delta)\cdot\bm\omega\Big),\\[0.5cm]
O^{[\le m]}_{7,\eta}(\Delta):= \frac12\Big(\mathbf A^{[\le m]}_\eta(\Delta) \cdot\mathbf A^{[\le m]}_\eta(\Delta)\Big)\Big(\mathbf B^{[\le m]}_\eta(\Delta) \cdot\bm\omega\Big),\\[0.5cm]
O^{[\le m]}_{8,\eta}(\Delta):=\frac12\Big(\mathbf B^{[\le m]}_\eta(\Delta) \cdot\mathbf B^{[\le m]}_\eta(\Delta)\Big)\Big(\mathbf A^{[\le m]}_\eta(\Delta) \cdot\bm\omega\Big).
\end{array}\label{eqOrcch}\end{equation}
@@ -567,7 +569,7 @@ n_2(\lambda_0)=c_2|\log_2|\lambda_0||+O(1),\quad c_2\approx2.
\begin{figure}
\hfil\includegraphics[width=280pt]{Figs/susc_beta_plot.pdf}\par\penalty10000
\caption{plot of $\frac{\bm\ell}{\bm\ell^*}$ as a function of the iteration step $N_\beta$ for $\lambda_0=-0.01$ and $h=2^{-40}$. Here $\ell_0^*$ through $\ell_3^*$ are the components of the non-trivial fixed point $\bm\ell^*$ and $\ell_4^*$ through $\ell_8^*$ are the values reached by $\ell_4$ through $\ell_8$ of largest absolute value. The flow behaves similarly to that at $h=0$ until $\ell_4$ through $\ell_8$ become large, at which point the couplings decay to 0, except for $\ell_5$ and $\ell_2$.}
\caption{plot of $\frac{\bm\ell}{\bm\ell^*}$ as a function of the iteration step $N_\beta$ for $\lambda_0=-0.125$ and $h=2^{-40}$. Here $\ell_0^*$ through $\ell_3^*$ are the components of the non-trivial fixed point $\bm\ell^*$ and $\ell_4^*$ through $\ell_8^*$ are the values reached by $\ell_4$ through $\ell_8$ of largest absolute value. The flow behaves similarly to that at $h=0$ until $\ell_4$ through $\ell_8$ become large, at which point the couplings decay to 0, except for $\ell_5$ and $\ell_2$.}
\label{figbetasuscplot}
\end{figure}
@@ -601,7 +603,7 @@ r_j(h)=c_r\log_2 h^{-1}+O(1),\quad c_r\approx2.6.
\section{Concluding remarks} \label{secconc}
\point The hierarchical Kondo model defined in section~\ref{sechierk} is a well defined statistical mechanics model, for which the partition function and correlation functions are unambiguously defined and finite as long as $\beta$ is finite. In addition, since the magnetic susceptibility of the impurity can be rewritten as a correlation function:
\begin{equation}
\chi(\beta,0)=\int_0^\beta dt\, \left<\,(\varphi^+(0)\bm\sigma\varphi^-(0))(\varphi^+(t)\bm\sigma\varphi^-(t))\,\right>_{h=0},
\chi(\beta,0)=\int_0^\beta dt\, \left<\,((\varphi^+(0)\bm\sigma\varphi^-(0))\cdot\bm\omega)((\varphi^+(t)\bm\sigma\varphi^-(t))\cdot\bm\omega)\,\right>_{h=0},
\label{eqsuscavg}\end{equation}
$\chi(\beta,0)$ is a thermodynamical quantity of the model.
\bigskip
@@ -856,12 +858,12 @@ is obtained. Does it exhibit a Kondo effect?
{\bf Remarks:}\listparpenalty
\begin{enumerate}[\ \ (1)\ \ ]
\item Finally an analysis essentially identical to the above can be performed to study the model in~(\ref{eqhamkondo}) {\it without impurity} (and with or without spin) to check that the magnetic susceptibility to a field $h$ acting only at a single site is finite: the result is the same as that of the XY model above: the single site susceptibility is finite and, up to a factor $2$, given by the same formula $\chi(\beta,0)=\frac{4\sinh \beta}{1+\cosh\beta}$.
\begin{enumerate}
\item\label{rkfinitesusc} Finally an analysis essentially identical to the above can be performed to study the model in~(\ref{eqhamkondo}) {\it without impurity} (and with or without spin) to check that the magnetic susceptibility to a field $h$ acting only at a single site is finite: the result is the same as that of the XY model above: the single site susceptibility is finite and, up to a factor $2$, given by the same formula $\chi(\beta,0)=\frac{4\sinh \beta}{1+\cosh\beta}$.
\item The latter result makes clear both the essential roles for the Kondo effect of the spin and of the noncommutativity of the impurity spin components.
\end{enumerate}\unlistparpenalty
\section{Some details on appendix \expandonce\appXY} \label{appXYcomp}
\section{Some details on appendix \expandonce{\ref{appXY}}} \label{appXYcomp}
\indent The definition of $H_h$ has to be supplemented by a boundary condition to give a meaning to $\bm\sigma_{L+1}$. If $\sigma^\pm_n=(\sigma^x\pm i\sigma^y_n)/2$ define $\mathcal N_{<n}$ as $\sum_{i<n}\sigma^+_i\sigma^-_i=\sum_{i<n}\mathcal N_i$ and $\mathcal N=\mathcal N_{\le L}$. Then set as boundary condition
\begin{equation}
\sigma_{L+1}^\pm:= -(-1)^\mathcal N\sigma^\pm_{1}