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diff --git a/Benfatto_Gallavotti_Jauslin_2015.tex b/Benfatto_Gallavotti_Jauslin_2015.tex index c92e26c..e5d1acf 100644 --- a/Benfatto_Gallavotti_Jauslin_2015.tex +++ b/Benfatto_Gallavotti_Jauslin_2015.tex @@ -41,24 +41,24 @@ \section{Introduction} \label{secintro} -\indent Although at high temperature the resistivity of most metals is an increasing function of the temperature, experiments carried out since the early XX\textsuperscript{th} century have shown that in metals containing trace amounts of magnetic impurities (i.e. copper polluted by iron), the resistivity has a minimum at a small but positive temperature, below which the resistivity decreases as the temperature increases. One interesting aspect of such a phenomenon, is its strong non-perturbative nature: it has been measured in samples of copper with iron impurities at a concentration as small as 0.0005\ [\cite{konZFi}], which raises the question of how such a minute perturbation can produce such an effect. Kondo introduced a toy model in 1964, see~(\ref{eqhamkondo}) below, to understand such a phenomenon, and computed electronic scattering amplitudes at third order in the Born approximation scheme [\cite{konSF}], and found that the effect may stem from an antiferromagnetic coupling between the impurities (called ``localized spins'' in [\cite{konSF}]) and the electrons in the metal. The existence of such a coupling had been proposed by Anderson [\cite{andSO}]. +\indent Although at high temperature the resistivity of most metals is an increasing function of the temperature, experiments carried out since the early XX\textsuperscript{th} century have shown that in metals containing trace amounts of magnetic impurities (i.e. copper polluted by iron), the resistivity has a minimum at a small but positive temperature, below which the resistivity decreases as the temperature increases. One interesting aspect of such a phenomenon, is its strong non-perturbative nature: it has been measured in samples of copper with iron impurities at a concentration as small as 0.0005\ [\cite{Ko05}], which raises the question of how such a minute perturbation can produce such an effect. Kondo introduced a toy model in 1964, see~(\ref{eqhamkondo}) below, to understand such a phenomenon, and computed electronic scattering amplitudes at third order in the Born approximation scheme [\cite{Ko64}], and found that the effect may stem from an antiferromagnetic coupling between the impurities (called ``localized spins'' in [\cite{Ko64}]) and the electrons in the metal. The existence of such a coupling had been proposed by Anderson [\cite{An61}]. -\indent Kondo's theory attracted great attention and its scaling properties and connection to $1D$ Coulomb gases were understood [\cite{dysSN}, \cite{andSeZ}, \cite{ayhSeZ}] (the obstacle to a complete understanding of the model (with $\lambda_0<0$) being what would later be called the growth of a relevant coupling) when in a seminal paper, published in 1975 [\cite{wilSeFi}], Wilson addressed and solved the problem by constructing a sequence of Hamiltonians that adequately represent the system on ever increasing length scales. Using ideas from his formulation of the renormalization group, Wilson showed, by a combination of numerical and perturbative methods, that only few (three) terms in each Hamiltonian, need to be studied in order to account for the Kondo effect. +\indent Kondo's theory attracted great attention and its scaling properties and connection to $1D$ Coulomb gases were understood [\cite{Dy69}, \cite{An70}, \cite{AYH70}] (the obstacle to a complete understanding of the model (with $\lambda_0<0$) being what would later be called the growth of a relevant coupling) when in a seminal paper, published in 1975 [\cite{Wi75}], Wilson addressed and solved the problem by constructing a sequence of Hamiltonians that adequately represent the system on ever increasing length scales. Using ideas from his formulation of the renormalization group, Wilson showed, by a combination of numerical and perturbative methods, that only few (three) terms in each Hamiltonian, need to be studied in order to account for the Kondo effect (or rather, a related effect on the magnetic susceptibility of the impurities, see below). -\indent The non-perturbative nature of the effect manifests itself in Wilson's formalism by the presence of a non-trivial fixed point in the renormalization group flow, at which the corresponding effective theory behaves in a way that is qualitatively different from the non-interacting one. Wilson has studied the system around the non-trivial fixed point by perturbative expansions, but the intermediate regime (in which perturbation theory breaks down) was studied by numerical methods. In fact, when using renormalization group techniques to study systems with non-trivial fixed points, oftentimes one cannot treat non-perturbative regimes analytically. The hierarchical Kondo model, which will be discussed below, is an exception to this rule. +\indent The non-perturbative nature of the effect manifests itself in Wilson's formalism by the presence of a non-trivial fixed point in the renormalization group flow, at which the corresponding effective theory behaves in a way that is qualitatively different from the non-interacting one. Wilson has studied the system around the non-trivial fixed point by perturbative expansions, but the intermediate regime (in which perturbation theory breaks down) was studied by numerical methods. In fact, when using renormalization group techniques to study systems with non-trivial fixed points, oftentimes one cannot treat non-perturbative regimes analytically. The hierarchical Kondo model, which will be discussed below, is an exception to this rule: indeed, we will show that the physical properties of the model can be obtained by iterating an {\it explicit} map, computed analytically, and called the {\it beta function}, whereas, in the current state of the art, the beta function for the full (non-hierarchical) Kondo model can only be computed numerically. -\indent In this paper, we present a hierarchical version of the Kondo model, whose renormalization group flow equations can be written out {\it exactly}, with no need for perturbative methods, and show that the flow admits a non-trivial fixed point. In this model, the transition from the fixed point can be studied by iterating an {\it explicit} map, which allows us to compute reliable numerical values for the {\it Kondo temperature}, that is the temperature at which the Kondo effect emerges, which is related to the number of iterations required to reach the non-trivial fixed point from the trivial one. This temperature has been found to obey the expected scaling relations, as predicted in [\cite{wilSeFi}]. +\indent In this paper, we present a hierarchical version of the Kondo model, whose renormalization group flow equations can be written out {\it exactly}, with no need for perturbative methods, and show that the flow admits a non-trivial fixed point. In this model, the transition from the fixed point can be studied by iterating an {\it explicit} map, which allows us to compute reliable numerical values for the {\it Kondo temperature}, that is the temperature at which the Kondo effect emerges, which is related to the number of iterations required to reach the non-trivial fixed point from the trivial one. This temperature has been found to obey the expected scaling relations, as predicted in [\cite{Wi75}]. \medskip -\indent It is worth noting that the Kondo model (or rather a linearized continuum version of it) was shown to be exactly solvable by Andrei [\cite{andEZ}] at $h=0$, as well as at $h\ne0$, [\cite{aflETh}], using Bethe Ansatz, who proved the existence of a Kondo effect in that model. The aim of the present work is to show how the Kondo effect can be understood as coming from a non-trivial fixed point in a renormalization group analysis (in the context of a hierarchical model) rather than a proof of the existence of the Kondo effect, which has already been carried out in [\cite{andEZ}, \cite{aflETh}]. +\indent It is worth noting that the Kondo model (or rather a linearized continuum version of it) was shown to be exactly solvable by Andrei [\cite{An80}] at $h=0$, as well as at $h\ne0$, [\cite{AFL83}], using Bethe Ansatz, who proved the existence of a Kondo effect in that model. The aim of the present work is to show how the Kondo effect can be understood as coming from a non-trivial fixed point in a renormalization group analysis (in the context of a hierarchical model) rather than a proof of the existence of the Kondo effect, which has already been carried out in [\cite{An80}, \cite{AFL83}]. \medskip \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''. It is well known that: +\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)\ \ ] \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{kitSeS}] 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 (1) in appendix~\ref{appXY} for a discussion of the second claim). \end{enumerate} \medskip @@ -79,18 +79,22 @@ where $\lambda_0,h$ are the interaction and magnetic field strengths and \item$x$ is on the unit lattice and $-{L}/2$, ${L}/2$ are identified (periodic boundary), \item$\Delta f(x)= f(x+1)-2f(x)+f(x-1)$ is the discrete Laplacian, \item$\bm\omega=(\bm\omega_1,\bm\omega_2,\bm\omega_3)$ is the direction of the field, which is a norm-1 vector. +\item the $-1$ term in $H_0$ is the chemical potential, set to $-1$ (half-filling) for convenience. \end{enumerate} \medskip \indent The model in~(\ref{eqhamkondo}) differs from the original Kondo model in which the interaction was -$$-\lambda_0\sum_{j=1}^3 c^+_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c^-_{\alpha_2}(0) \,\sigma^j.$$ -The two models are closely related and equivalent for our purposes (see appendix~\ref{appcmpkondo}). The technical advantage of the model~(\ref{eqhamkondo}), is that it allows us set up the problem via a functional integral to exploit fully the remark that ``since the Kondo problem of the magnetic impurity treats only a single-point impurity, the question reduces to a sum over paths in only one (``time'') dimension'' [\cite{ayhSeZ}]. The formulation in~(\ref{eqhamkondo}) was introduced in [\cite{andEZ}]. +$$-\lambda_0\sum_{j=1}^3 c^+_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c^-_{\alpha_2}(0) \,\tau^j$$ +where $\tau^j$ is the $j$-th Pauli matrix and acts on the spin of the impurity. The two models are closely related and equivalent for our purposes (see appendix~\ref{appcmpkondo}). The technical advantage of the model~(\ref{eqhamkondo}), is that it allows us set up the problem via a functional integral to exploit fully the remark that ``since the Kondo problem of the magnetic impurity treats only a single-point impurity, the question reduces to a sum over paths in only one (``time'') dimension'' [\cite{AYH70}]. The formulation in~(\ref{eqhamkondo}) was introduced in [\cite{An80}]. \medskip -\indent The model will be said to exhibit a {\it Kondo effect} if, no matter how small the coupling $\lambda_0$ is, as long as it is {\it antiferromagnetic} ({\it i.e.}\ $\lambda_0<0$), the susceptibility {\it remains finite and positive as $\beta\to\infty$ and continuous as $h\to0$}, while it diverges in presence of a ferromagnetic ({\it i.e.}\ $\lambda_0>0$) coupling. The soluble model in [\cite{andEZ}] and Wilson's version of the model in~(\ref{eqhamkondo}) do exhibit the Kondo effect. +\indent The model will be said to exhibit a {\it Kondo effect} if, no matter how small the coupling $\lambda_0$ is, as long as it is {\it antiferromagnetic} ({\it i.e.}\ $\lambda_0<0$), the susceptibility {\it remains finite and positive as $\beta\to\infty$ and continuous as $h\to0$}, while it diverges in presence of a ferromagnetic ({\it i.e.}\ $\lambda_0>0$) coupling. The soluble model in [\cite{An80}] and Wilson's version of the model in~(\ref{eqhamkondo}) do exhibit the Kondo effect. \medskip -\indent Here the same questions will be studied in a hierarchical model defined below. The interest of this model is that various observables can be computed by iterating a map, which is explicitly computed and called the ``beta function'', involving few (nine) variables, called ``running couplings''. The possibility of computing the beta function exactly for general fermionic hierarchical models has been noticed and used in [\cite{dorNO}]. +{\bf Remark}: In the present work, the {\it Kondo effect} is defined as an effect on the susceptibility of the impurity, and not on the resistivity of the electrons of the chain, which, we recall, was Kondo's original motivation [\cite{Ko64}]. The reason for this is that the magnetic susceptibility of the impurity is easier to compute than the resistivity of the chain, but still exhibits a non-trivial effect, as discussed by Wilson~[\cite{Wi75}]. +\medskip + +\indent Here the same questions will be studied in a hierarchical model defined below. The interest of this model is that various observables can be computed by iterating a map, which is explicitly computed and called the ``beta function'', involving few (nine) variables, called ``running couplings''. The possibility of computing the beta function exactly for general fermionic hierarchical models has been noticed and used in [\cite{Do91}]. \medskip @@ -105,12 +109,12 @@ The two models are closely related and equivalent for our purposes (see appendix \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)\ \ ] -\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) and is quite simple as it expresses each running coupling, on a given scale $n$ (corresponding to a length scale $2^n$), as a ratio of two (rather large, but scale independent) polynomials in the couplings at the preceding scale, $n-1$. - -\item It will appear that perturbation theory can only work to describe properties measurable up to a length scale $2^{n_2(\lambda_0)}$, in which $n_2(\lambda_0)$ depends on the coupling $\lambda_0$ between the impurity and the electron chain and, asymptotically as $\lambda_0\to0$, $n_2(\lambda_0)=c_2\log|\lambda_0|^{-1}+O(1)$ for some $c_2>0$; at larger scales perturbation theory breaks down and the evolution of the running couplings is controlled by a non-trivial fixed point (which can be computed exactly). +\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 $\lambda_0\to0$, $n_K(\lambda_0)=c_1|\lambda_0|^{-1}+O(1)$. +\item It will appear that perturbation theory can only work to describe properties measurable up to a length scale $2^{n_2(\lambda_0)}$, in which $n_2(\lambda_0)$ depends on the coupling $\lambda_0$ between the impurity and the electron chain and, asymptotically as $\lambda_0\to0$, $n_2(\lambda_0)=c_2\log|\lambda_0|^{-1}+O(1)$ for some $c_2>0$; at larger scales perturbation theory breaks down and the evolution of the running couplings is controlled by a non-trivial fixed point (which can be computed exactly). + \item Denoting the magnetic field by $h$, if $h>0$ and $\beta_Kh\ll1$, the flow of the running couplings tends to a trivial fixed point ($h$-independent but different from $0$) which is reached on a scale $r(h)$ which, {asymptotically} as $h\to0$, is $r(h)=c_r \log h^{-1} +O(1)$. \end{enumerate} @@ -119,7 +123,7 @@ The two models are closely related and equivalent for our purposes (see appendix {\it The picture is completely different in the ferromagnetic case,} in which the susceptibility diverges at zero temperature and the flow of the running couplings is not controlled by the non trivial fixed point. \medskip -{\bf Remark}: Unlike in the model studied by Wilson [\cite{wilSeFi}], the $T=0$ nontrivial fixed point is {\it not} infinite in the hierarchical Kondo model: this shows that the Kondo effect can, in some models, be somewhat subtler than a rigid locking of the impurity spin with an electron spin [\cite{nozSeF}]. +{\bf Remark}: Unlike in the model studied by Wilson [\cite{Wi75}], the $T=0$ nontrivial fixed point is {\it not} infinite in the hierarchical Kondo model: this shows that the Kondo effect can, in some models, be somewhat subtler than a rigid locking of the impurity spin with an electron spin [\cite{No74}]. \medskip \indent Technically this is one of the few cases in which functional integration for fermionic fields is controlled by a non-trivial fixed point and can be performed rigorously and applied to a concrete problem. @@ -133,9 +137,9 @@ The two models are closely related and equivalent for our purposes (see appendix \end{enumerate}\unlistparpenalty \section{Functional integration in the Kondo model} \label{secfunint} -\indent In [\cite{wilSeFi}], 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 [\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{bgNZ}, \cite{shaNF}]. 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 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 We introduce an extra dimension, called {\it imaginary time}, and define new creation and annihilation operators: @@ -164,11 +168,11 @@ g_{\varphi,\alpha}(t-t'):=& \end{array}\right.. \end{array}\label{eqprop}\end{equation} -\indent By a direct computation [\cite{bgNZ},~(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{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 \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}(\xi,\tau) = \int\frac{dk_0}{2\pi}\,{e^{-ik_0(\tau+0^-)} \over-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'' $k_0= \frac\pi{\beta} +\frac{2\pi}\beta n_0$, $n_0\in\mathbb Z$, $2^{|n_0|}\le\frac12\beta$, and $k$ is the linear momentum $k=\frac{2\pi}L n$, $n\in [-L/2,L/2-1]\cap\mathbb Z$. +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 \indent In the functional representation, the operator $V$ of~(\ref{eqhamkondo}) is substituted with the following function of the Grassmann variables~(\ref{eqfermgrass}): @@ -177,7 +181,7 @@ 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) \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{aySN}, \cite{ayhSeZ}]. +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}]. \indent The average of a physical observable $F$ localized at $x=0$, which is a polynomial in the fields $\psi_{\alpha}^\pm(0,t)$ and $\varphi_\alpha^\pm(t)$, will be denoted by @@ -199,11 +203,11 @@ 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{bgNZ}, \cite{shaNF}] 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{BG90b}, \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{bgNZ}, \cite{shaNF}], 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{bgNZ}, \cite{bgeNF}], 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{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. \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. @@ -244,10 +248,10 @@ with $\psi_{\alpha}^{[m]}(0,t)$ and $\varphi_\alpha^{[m]}(t)$ being, respectivel \varphi_{\alpha}^{[\le m]\pm}(t):=\sum_{m'=-N_\beta}^{m}\varphi_{\alpha}^{[m']\pm}(t). \label{eqfieldlem}\end{equation} -\indent Notice that the functions $g_\psi^{[m]}(\xi,\tau),g_\varphi^{[m]}(\tau)$ decay faster than any power as $\tau$ tends to $\infty$ (as a consequence of the smoothness of the cut-off function $\chi$), so that at any fixed scale $m\le 0$, the fields $\psi^{[m]},\varphi^{[m]}$ can be regarded as (almost) independent on the scale $2^{-m}$. +\indent Notice that the functions $g_\psi^{[m]}(\xi,\tau),g_\varphi^{[m]}(\tau)$ decay faster than any power as $\tau$ tends to $\infty$ (as a consequence of the smoothness of the cut-off function $\chi$), so that at any fixed scale $m\le 0$, fields $\psi^{[m]},\varphi^{[m]}$ that are separated in time by more than $2^{-m}$ can be regarded as (almost) independent. \medskip -\indent The decomposition into scales allows us to compute the quantities in~(\ref{eqavggrass}) inductively. For instance the partition function $Z$ is given by +\indent The decomposition into scales allows us to express the quantities in~(\ref{eqavggrass}) inductively (see~(\ref{eqeffpotrec})). For instance the partition function $Z$ is given by \begin{equation} Z=\exp\Big(-\beta\sum_{m=-N_\beta}^{0} c^{[m]}\,\Big) \label{eqZmultiscale}\end{equation} @@ -282,13 +286,13 @@ for $m< 0$ and similarly for $m=0$. Thus $\Delta_{-}$ is the lower half of $\Del \indent The elementary fields used to define the hierarchical Kondo model will be {\it constant on each half-box} and will be denoted by $\psi_\alpha^{[m]\pm}(\Delta_{\eta})$ and $\varphi_{\alpha}^{[m]\pm}(\Delta_{\eta})$ for $m\in\{0,-1,\cdots,$ $-N_\beta\}$, $\Delta\in\mathcal Q_m$, $\eta\in\{-,+\}$, $\alpha\in\{\uparrow,\downarrow\}$. \medskip -\indent We now define the propagators associated with $\psi$ and $\varphi$. The idea is to define propagators that are {\it similar} [\cite{wilSFi}, \cite{wilSeZ}, \cite{dysSN}], in a sense made more precise below, to the non-hierarchical propagators defined in~(\ref{eqprop}). Bearing that in mind, we compute the value of the non-hierarchical propagators between fields at the centers of two half boxes: given a box $\Delta\in\mathcal Q_{0}$ and $\eta\in\{-,+\}$, let $\delta:=2^{-1}$ denote the distance between the centers of $\Delta_-$ and $\Delta_+$, we get +\indent We now define the propagators associated with $\psi$ and $\varphi$. The idea is to define propagators that are {\it similar} [\cite{Wi65}, \cite{Wi70}, \cite{Dy69}], in a sense made more precise below, to the non-hierarchical propagators defined in~(\ref{eqprop}). Bearing that in mind, we compute the value of the non-hierarchical propagators between fields at the centers of two half boxes: given a box $\Delta\in\mathcal Q_{0}$ and $\eta\in\{-,+\}$, let $\delta:=2^{-1}$ denote the distance between the centers of $\Delta_-$ and $\Delta_+$, we get \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l} g^{[0]}_{\psi}(0,\eta\delta)=&\eta\sum_{\omega=\pm}\int \frac{dkdk_0}{(2\pi)^2} \frac{k_0\sin(k_0\delta)}{k_0^2+k^2}\chi(k^2+k_0^2) :=\eta a\\[0.5cm] g^{[0]}_{\varphi}(\eta\delta)=&\eta\int\frac{dk_0}{2\pi} \frac{\sin(k_0\delta)}{k_0}\chi(k_0^2)\,:=\eta b \end{array} \label{eqpropcoarse}\end{equation} -in which $a$ and $b$ are constants, see [\cite{ayhSeZ}, p.4465]. We define the hierarchical propagators, drawing inspiration from~(\ref{eqpropcoarse}). In an effort to make computations more explicit, we set $a=b\equiv1$ and define +in which $a$ and $b$ are constants, see [\cite{AYH70}, p.4465]. We define the hierarchical propagators, drawing inspiration from~(\ref{eqpropcoarse}). In an effort to make computations more explicit, we set $a=b\equiv1$ and define \begin{equation} \left<\psi_{\alpha}^{[m]-}(\Delta_{-\eta})\psi_{\alpha}^{[m]+}(\Delta_{\eta})\right >:= \eta,\quad \left<\varphi_{\alpha}^{[m]-}(\Delta_{-\eta})\varphi_{\alpha}^{[m]+}(\Delta_{\eta})\right> := \eta @@ -301,7 +305,7 @@ for $m\in\{0,-1,\cdots,-N_\beta\}$, $\eta\in\{-,+\}$, $\Delta\in\mathcal Q_m$, $ \psi^{\pm}_{\alpha}(0,t):= \sum_{m=-N_\beta}^{0}2^{\frac{m}2} \psi^{[m]\pm}_{\alpha}(\Delta^{[m+1]}(t)),\quad \varphi^{\pm}_{\alpha}(t):=\sum_{m=-N_\beta}^{0} \varphi^{[m]\pm}_{\alpha}(\Delta^{[m+1]}(t)) \label{eqfieldhier}\end{equation} -(recall that $m\le0$ and $\Delta^{[m]}(t)\supset\Delta^{[m+1]}(t)$). The hierarchical model for the on-site Kondo effect so defined is such that the propagator between two fields vanishes unless both fields belong to the same box and, at the same time, to two different halves within that box. In addition, given $t$ and $t'$ that are such that $|t-t'|>2^{m-1}$, there exists one and only one scale $m_{(t-t')}$ that is such that $\Delta^{[m_{(t-t')}]}(t)=\Delta^{[m_{(t-t')}]}(t')$ and $\Delta^{[m_{(t-t')}+1]}(t)\neq\Delta^{[m_{(t-t')}+1]}(t')$. Therefore $\forall(t,t')\in R^2$, $\forall(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2$, +(recall that $m\le0$ and $\Delta^{[m]}(t)\supset\Delta^{[m+1]}(t)$). The hierarchical model for the on-site Kondo effect so defined is such that the propagator on scale $m$ between two fields vanishes unless both fields belong to the same box and, at the same time, to two different halves within that box. In addition, given $t$ and $t'$ that are such that $|t-t'|>2^{-1}$, there exists one and only one scale $m_{(t-t')}$ that is such that $\Delta^{[m_{(t-t')}]}(t)=\Delta^{[m_{(t-t')}]}(t')$ and $\Delta^{[m_{(t-t')}+1]}(t)\neq\Delta^{[m_{(t-t')}+1]}(t')$. Therefore $\forall(t,t')\in R^2$, $\forall(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2$, \begin{equation} \left<\,\psi_\alpha^-(0,t)\psi_{\alpha'}^+(0,t')\,\right>=\delta_{\alpha,\alpha'} 2^{m_{(t-t')}}\mathrm{sign}(t-t'). \label{eqprophiercmp}\end{equation} @@ -330,7 +334,7 @@ in which $\psi^\pm_\alpha(0,t)$ and $\varphi^\pm_\alpha(t)$ are now defined in~( \indent Note that since the model defined above only involves field 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}). \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 below. +\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}. \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}. @@ -358,7 +362,7 @@ and for $\Delta\in\mathcal Q_{-m},\,m<-N_\beta$, in which $c^{[m-1]}\in R$ is a constant and $V^{[m-1]}$ contains no constant term. By a straightforward induction, we then find that $Z$ is given again by~(\ref{eqZmultiscale}) with the present definition of $c^{[m]}$ (see~(\ref{eqeffpothier})). \medskip -\indent We will now prove by induction that the hierarchical Kondo model defined above is {\it integrable}, in the sense that~(\ref{eqeffpothier}) can be written out {\it explicitly} as a {\it finite} system of equations. To that end it will be shown that $V^{[m]}$ can be parameterized by only four real numbers, $\bm\alpha^{[m]}=(\alpha^{[m]}_0,\cdots,\alpha^{[m]}_3)\in R^4$ and, in the process, the equation relating $\bm\alpha^{[m]}$ and $\bm\alpha^{[m-1]}$ (called the {\it beta function}) will be computed: +\indent We will now prove by induction that the hierarchical Kondo model defined above is {\it exactly solvable}, in the sense that~(\ref{eqeffpothier}) can be written out {\it explicitly} as a {\it finite} system of equations. To that end it will be shown that $V^{[m]}$ can be parameterized by only four real numbers, $\bm\alpha^{[m]}=(\alpha^{[m]}_0,\cdots,\alpha^{[m]}_3)\in R^4$ and, in the process, the equation relating $\bm\alpha^{[m]}$ and $\bm\alpha^{[m-1]}$ (called the {\it beta function}) will be computed: \begin{equation} -V^{[m]}(\psi^{[\le m]},\varphi^{[\le m]}) =\sum_{\Delta\in\mathcal Q_m}\sum_{n=0}^3\alpha_n^{[m]} \sum_{\eta=\pm}O_{n,\eta}^{[\le m]}(\Delta) @@ -376,7 +380,7 @@ A^{[\le m]j}_\eta(\Delta):=&\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2} B^{[\le m]j}_\eta(\Delta):=&\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2} \varphi_{\alpha}^{[\le m]+}(\Delta_\eta)\sigma^j_{\alpha,\alpha'}\varphi_{\alpha'}^{[\le m]-}(\Delta_\eta). \end{array}\label{eqABdef}\end{equation} -\indent For $m=0$, by injecting~(\ref{eqhierfieldindinit}) into~(\ref{eqpothier}), we find that $V^{[0]}$ can be written as in~(\ref{eqeffpotform}) with $\bm\alpha^{[0]}=(\lambda_0,0,0,0)$. +\indent For $m=0$, by injecting~(\ref{eqhierfieldindinit}) into~(\ref{eqpothier}), we find that $V^{[0]}$ can be written as in~(\ref{eqeffpotform}) with $\bm\alpha^{[0]}=(\lambda_0,0,0,0)$. As follows from~(\ref{eqhierbeta}) below, for all initial conditions, the running couplings $\alpha^{[m]}$ remain bounded, and are attracted by a sphere whose radius is independent of the initial data. \indent We then compute $V^{[m-1]}$ using~(\ref{eqeffpothier}) and show it can be written as in~(\ref{eqeffpotform}). We first notice that the propagator in~(\ref{eqprophier}) is diagonal in $\Delta$, and does not depend on the value of $\Delta$, therefore, we can split the averaging over $\psi^{[m]}(\Delta_\pm)$ for different $\Delta$, as well as that over $\varphi^{[m]}(\Delta)$. We thereby find that \begin{equation} @@ -408,10 +412,10 @@ where with (in order to reduce the size of the following equation, we dropped all $^{[m]}$ from the right side) \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l} C^{[m]}=&1+ 3\ell_0^2+9\ell_1^2+9\ell_2^2+324\ell_3^2\\[0.3cm] -\ell_0^{[m-1]}=&\frac1C\Big(\ell_0 +18\ell_0\ell_3+3 \ell_0\ell_2+3 \ell_0\ell_1 -2\ell_0^2\Big)\\[0.3cm] -\ell_1^{[m-1]}=&\frac1C\Big( \frac12\ell_1+9\ell_2\ell_3 +\frac14\ell_0^2\Big)\\[0.3cm] -\ell_2^{[m-1]}=& \frac1C\Big(2\ell_2+36\ell_1\ell_3+ \ell_0^2\Big)\\[0.3cm] -\ell_3^{[m-1]}=& \frac1C\Big(\frac12\ell_3+\frac14\ell_1\ell_2+\frac1{24} \ell_0^2\Big). +\ell_0^{[m-1]}=&\frac1{C^{[m]}}\Big(\ell_0 +18\ell_0\ell_3+3 \ell_0\ell_2+3 \ell_0\ell_1 -2\ell_0^2\Big)\\[0.3cm] +\ell_1^{[m-1]}=&\frac1{C^{[m]}}\Big( \frac12\ell_1+9\ell_2\ell_3 +\frac14\ell_0^2\Big)\\[0.3cm] +\ell_2^{[m-1]}=& \frac1{C^{[m]}}\Big(2\ell_2+36\ell_1\ell_3+ \ell_0^2\Big)\\[0.3cm] +\ell_3^{[m-1]}=& \frac1{C^{[m]}}\Big(\frac12\ell_3+\frac14\ell_1\ell_2+\frac1{24} \ell_0^2\Big). \end{array}\label{eqhierbeta}\end{equation} \indent The $\bm\alpha^{[m-1]}$ could then be reconstructed from~(\ref{eqhierbeta}) by inverting the map $\bm\alpha\mapsto\bm\ell$ (see~(\ref{eqhieralphatoellexplicit})). It is nevertheless convenient to work with the $\bm\ell$'s as running couplings rather than with the $\bm\alpha$'s. @@ -438,11 +442,11 @@ where $x_0\approx0.15878626704216...$ is the real root of $4-19x-22x^2-107x^3=0$ {\bf Remark}: Proving that the flow converges to $\bm\ell^*$ analytically is complicated by the somewhat contrived expression of $\bm\ell^*$. It is however not difficult to prove that if the flow converges, then it must go to $\bm\ell^*$ (see appendix~\ref{appfixed}-{\bf\ref{ptfixed}}). Since the numerical iterations of the flow converge quite clearly, we will not attempt a full proof of the convergence to the fixed point. \medskip -{\bf Remark}: A simpler case that can be treated analytically is that in which the {\it irrelevant} terms ($\ell_1$ and $\ell_3$) are neglected. Indeed the map reduces to +{\bf Remark}: A simpler case that can be treated analytically is that in which the {\it irrelevant} terms ($\ell_1$ and $\ell_3$) are neglected (the flow in this case is (at least numerically) {\it close} to that of the full beta function in~(\ref{eqhierbeta}) projected onto $\ell_1=\ell_3=cst$). Indeed the map reduces to \begin{equation}\begin{array}{r@{\ }l} C^{[m]}=&1+ 3\ell_0^2+9\ell_2^2\\[0.3cm] -\ell_0^{[m-1]}=&\frac1C\Big(\ell_0 +3 \ell_0\ell_2 -2\ell_0^2\Big)\\[0.3cm] -\ell_2^{[m-1]}=& \frac1C\Big(2\ell_2+ \ell_0^2\Big)\\ +\ell_0^{[m-1]}=&\frac1{C^{[m]}}\Big(\ell_0 +3 \ell_0\ell_2 -2\ell_0^2\Big)\\[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)\ \ ] @@ -521,7 +525,7 @@ and similarly for $\partial_h^2\bm\ell_h^{[m]}$. Therefore, using~(\ref{eqbetasu \indent By a numerical study which produces results that are stable and clear we find the following results. \medskip -\point If $\lambda_0\equiv \alpha_0<0$ and $\alpha_j=0,\, j>0$, the flow tends to a nontrivial, $\lambda$--independent, fixed point $\bm\ell^*$ (see figure~\ref{figbetaplot}). +\point If $\lambda_0\equiv \alpha_0<0$, $\alpha_j=0,\, j>0$ and $h=0$, then the flow tends to a nontrivial, $\lambda$--independent, fixed point $\bm\ell^*$ (see figure~\ref{figbetaplot}). \begin{figure} \hfil\includegraphics[width=250pt]{Figs/beta_plot.pdf}\par\penalty10000 @@ -549,7 +553,7 @@ n_2(\lambda_0)=c_2|\log_2|\lambda_0||+O(1),\quad c_2\approx2. \indent When the running coupling constants are at $\bm\ell^*$, the susceptibility remains finite as $\beta\to\infty$ and positive, whereas when they are at $\bm\ell_+^*$, it grows linearly with $\beta$ (which is why $\bm\ell_+^*$ was called ``trivial'' in the introduction). -\indent In addition, when $\lambda_0<0$ the flow escapes along the unstable direction towards the neighborhood of $\bm\ell^*_+$, which is reached after $n_2(\lambda_0)$ steps, but since it is marginally unstable for $\lambda_0<0$, it flows away towards $\bm\ell^*$ after $n_K(\lambda_0)$ steps. The susceptibilty is therefore finite for $\lambda_0<0$ (see figure~\ref{figsuscbeta} (which may be compared to the exact solution [\cite{aflETh}, figure~3])). +\indent In addition, when $\lambda_0<0$ the flow escapes along the unstable direction towards the neighborhood of $\bm\ell^*_+$, which is reached after $n_2(\lambda_0)$ steps, but since it is marginally unstable for $\lambda_0<0$, it flows away towards $\bm\ell^*$ after $n_K(\lambda_0)$ steps. The susceptibility is therefore finite for $\lambda_0<0$ (see figure~\ref{figsuscbeta} (which may be compared to the exact solution [\cite{AFL83}, figure~3])). \indent If $\lambda_0>0$, then the flow approaches $\bm\ell^*_+$ from the $\lambda_0>0$ side, which is marginally stable, so the flow never leaves the vicinity of $\ell_+^*$ and the susceptibility diverges as $\beta\to\infty$. @@ -584,15 +588,15 @@ r_j(h)=c_r\log_2 h^{-1}+O(1),\quad c_r\approx2.6. \label{figsusch} \end{figure} -\point In [\cite{wilSeFi}, figure~17, p.836], there is a plot of $\frac{\beta_K}{\chi(\beta,0)}$ as a function of $\frac{\beta_K}\beta$. For the sake of comparison, we have reproduced it for the hierarchical Kondo model (see figure~\ref{figsuscwilson}). +\point In [\cite{Wi75}, figure~17, p.836], there is a plot of $\frac{\beta_K}{\chi(\beta,0)}$ as a function of $\frac{\beta_K}\beta$. For the sake of comparison, we have reproduced it for the hierarchical Kondo model (see figure~\ref{figsuscwilson}). \begin{figure} \hfil\includegraphics[width=250pt]{Figs/susc_wilson.pdf}\par\penalty10000 -\caption{plot of $\frac{\beta_K}{\chi(\beta,0)}$ as a function of $\frac{\beta_K}{\beta}$ for various values of $\lambda_0$: $\lambda_0=-0.024$ ({\color{iblue}blue}), $\lambda_0=-0.02412$ ({\color{igreen}green}), $\lambda_0=-0.05$ ({\color{ired}red}). In [\cite{wilSeFi}], $\lambda_0=-0.024$ and $-0.02412$. Note that the abscissa of the data points are $2^{-n}$ for $n\ge0$, so that there are only $3$ points in the range $[0.25,1]$. The lines are drawn for visual aid.} +\caption{plot of $\frac{\beta_K}{\chi(\beta,0)}$ as a function of $\frac{\beta_K}{\beta}$ for various values of $\lambda_0$: $\lambda_0=-0.024$ ({\color{iblue}blue}), $\lambda_0=-0.02412$ ({\color{igreen}green}), $\lambda_0=-0.05$ ({\color{ired}red}). In [\cite{Wi75}], $\lambda_0=-0.024$ and $-0.02412$. Note that the abscissa of the data points are $2^{-n}$ for $n\ge0$, so that there are only $3$ points in the range $[0.25,1]$. The lines are drawn for visual aid.} \label{figsuscwilson} \end{figure} -\indent Similarly to \cite{wilSeFi}, we find that $\frac{\beta_K}\chi$ seems to be affine as it approaches the Kondo temperature, although it is hard to tell for sure because of the scarcity of data points (by its very construction, the hierarchical Kondo model only admits inverse temperatures that are powers of 2 so the portion of figure~\ref{figsuscwilson} that appears to be affine actually only contains three data points). However, we have found that such a diagram depends on $\lambda_0$: indeed, by sampling values of $|\lambda_0|$ down to $10^{-4}$, $\frac{\beta_K}{\chi(\beta,0)}$ has been found to tend to 0 faster than $(\log\beta_K)^{-1.2}$ but slower than $(\log\beta_K)^{-1.3}$. In order to get a more precise estimate on this exponent, one would need to consider $|\lambda_0|$ that are smaller than $10^{-4}$, which would give rise to numerical values larger than $10^{5000}$, and since the numbers used to perform the numerical computations are {\it {\tt x86}-extended precision floating point numbers}, such values are too large. +\indent Similarly to \cite{Wi75}, we find that $\frac{\beta_K}\chi$ seems to be affine as it approaches the Kondo temperature, although it is hard to tell for sure because of the scarcity of data points (by its very construction, the hierarchical Kondo model only admits inverse temperatures that are powers of 2 so the portion of figure~\ref{figsuscwilson} that appears to be affine actually only contains three data points). However, we have found that such a diagram depends on $\lambda_0$: indeed, by sampling values of $|\lambda_0|$ down to $10^{-4}$, $\frac{\beta_K}{\chi(\beta,0)}$ has been found to tend to 0 faster than $(\log\beta_K)^{-1.2}$ but slower than $(\log\beta_K)^{-1.3}$. In order to get a more precise estimate on this exponent, one would need to consider $|\lambda_0|$ that are smaller than $10^{-4}$, which would give rise to numerical values larger than $10^{5000}$, and since the numbers used to perform the numerical computations are {\it {\tt x86}-extended precision floating point numbers}, such values are too large. \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: @@ -608,7 +612,7 @@ $\chi(\beta,0)$ is a thermodynamical quantity of the model. \point In the hierarchical model defined in section~\ref{sechierk}, quantities other than the magnetic susceptibility of the impurity can be computed, although all observables must only involve fields localized at $x=0$. For instance, the response to a magnetic field acting on all sites of the fermionic chain as well as the impurity cannot be investigated in this model, since the sites of the chain with $x\neq0$ are not accounted for. \medskip -\subpoint We have attempted to extend the definition of the hierarchical model to take the chain into account, by paving the space-time plane with square boxes (instead of paving the time axis with intervals, see section~\ref{sechierk}), defining hierarchical fields for each quarter box and postulating a propagator between them by analogy with the non-hierarchical model. The magnetic susceptibility of the impurity is defined as the response to a magnetic field acting on every site of the chain and on the impurity, to which the susceptibility of the non-interacting chain is subtracted. We have found, iterating the flow numerical, that for such a model {\it there is no Kondo effect}, that is the impurity susceptibility diverges as $\beta$ when $\beta\to\infty$. +\subpoint We have attempted to extend the definition of the hierarchical model to allow observables on the sites of the chain at $x\neq0$, by paving the space-time plane with square boxes (instead of paving the time axis with intervals, see section~\ref{sechierk}), defining hierarchical fields for each quarter box and postulating a propagator between them by analogy with the non-hierarchical model. The magnetic susceptibility of the impurity is defined as the response to a magnetic field acting on every site of the chain and on the impurity, to which the susceptibility of the non-interacting chain is subtracted. We have found, iterating the flow numerically, that for such a model {\it there is no Kondo effect}, that is the impurity susceptibility diverges as $\beta$ when $\beta\to\infty$. \medskip \subpoint A second approach has yielded better results, although it is not completely satisfactory. The idea is to incorporate the effect of the magnetic field $h$ acting on the fermionic chain into the propagator of the non-hierarchical model, after which the potential $V$ only depends on the site at $x=0$, so that the hierarchical model can be defined in the same way as in section~\ref{sechierk} but with {\it an $h$-dependent propagator}. In this model, we have found that {\it there is a Kondo effect}. @@ -634,7 +638,7 @@ m^0_K(\beta,\lambda_0,h)=&\kappa\, m_K(\beta,\lambda_0,h)\\[0.3cm] \chi_K^0(\beta,\lambda_0,h)=&\kappa\, \chi_K(\beta,\lambda_0,h)-(\kappa-1)\beta m_K(\beta,\lambda_0,h)^2. \end{array}\label{eqmagsusckondoandrei}\end{equation} -\indent In addition $1\le\kappa\le2$: indeed the first inequality is trivial and the second follows from the variational principle (see [\cite{rueSN}, theorem 7.4.1, p.188]): +\indent In addition $1\le\kappa\le2$: indeed the first inequality is trivial and the second follows from the variational principle (see [\cite{Ru99b}, theorem 7.4.1, p.188]): \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l} \log Z^0_K(\beta,\lambda_0,h)=&\max_\mu(s(\mu)-\mu(H_0+V))\\ \ge& s(\mu_0)-\mu_0(H_0)+\mu_0(V)=s(\mu_0)-\mu_0(H_0) @@ -659,23 +663,23 @@ where the lower case $\mathbf a$ denotes $\left<\,\mathbf A_1\,\right>\equiv\lef \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l} C^{[m]}=&1+ 2\ell_0^2+(\ell_0+\ell_6)^2 +9\ell_1^2 +9\ell_2^2 +324\ell_3^2 +\frac12\ell_4^2 +\frac12\ell_5^2 +18\ell_7^2 +18\ell_8^2\\[0.3cm] -\ell_0^{[m-1]}=&\frac1{C}\Big(\ell_0 -2\ell_0\ell_6 +18\ell_0\ell_3 +3 \ell_0\ell_2 +3 \ell_0\ell_1-2\ell_0^2\Big)\\[0.3cm] -\ell_1^{[m-1]}=&\frac1{C}\Big(\frac12\ell_1 +9\ell_2\ell_3 +\frac32\ell_8^2 +\frac1{12}\ell_6^2 +\frac12\ell_5\ell_7+\frac1{24}\ell_4^2 +\frac16\ell_0\ell_6 +\frac14\ell_0^2\Big)\\[0.3cm] -\ell_2^{[m-1]}=&\frac1{C}\Big( 2\ell_2+36\ell_1\ell_3 + \ell_0^2 +6\ell_7^2 +\frac13\ell_6^2 +\frac1{6}\ell_5^2 +2\ell_4\ell_8 +\frac23\ell_0\ell_6\Big)\\[0.3cm] -\ell_3^{[m-1]}=&\frac1{C}\Big( \frac12\ell_3 +\frac14\ell_1\ell_2 +\frac1{24} \ell_0^2 +\frac1{36}\ell_0\ell_6 +\frac1{72}\ell_6^2 +\frac1{12}\ell_5\ell_7 + \frac1{12}\ell_4\ell_8\Big)\\[0.3cm] -\ell_4^{[m-1]}=&\frac1{C}\Big( \ell_4 +6\ell_6\ell_7 +\ell_5\ell_6 +108\ell_3\ell_8 +18\ell_2\ell_8 +3\ell_1\ell_4 +6\ell_0\ell_7 +\ell_0\ell_5\Big)\\[0.3cm] -\ell_5^{[m-1]}=&\frac1{C}\Big( 2\ell_5 +12 \ell_6\ell_8 +2\ell_4\ell_6 +216 \ell_3\ell_7 +\ell_0^{[m-1]}=&\frac1{C^{[m]}}\Big(\ell_0 -2\ell_0\ell_6 +18\ell_0\ell_3 +3 \ell_0\ell_2 +3 \ell_0\ell_1-2\ell_0^2\Big)\\[0.3cm] +\ell_1^{[m-1]}=&\frac1{C^{[m]}}\Big(\frac12\ell_1 +9\ell_2\ell_3 +\frac32\ell_8^2 +\frac1{12}\ell_6^2 +\frac12\ell_5\ell_7+\frac1{24}\ell_4^2 +\frac16\ell_0\ell_6 +\frac14\ell_0^2\Big)\\[0.3cm] +\ell_2^{[m-1]}=&\frac1{C^{[m]}}\Big( 2\ell_2+36\ell_1\ell_3 + \ell_0^2 +6\ell_7^2 +\frac13\ell_6^2 +\frac1{6}\ell_5^2 +2\ell_4\ell_8 +\frac23\ell_0\ell_6\Big)\\[0.3cm] +\ell_3^{[m-1]}=&\frac1{C^{[m]}}\Big( \frac12\ell_3 +\frac14\ell_1\ell_2 +\frac1{24} \ell_0^2 +\frac1{36}\ell_0\ell_6 +\frac1{72}\ell_6^2 +\frac1{12}\ell_5\ell_7 + \frac1{12}\ell_4\ell_8\Big)\\[0.3cm] +\ell_4^{[m-1]}=&\frac1{C^{[m]}}\Big( \ell_4 +6\ell_6\ell_7 +\ell_5\ell_6 +108\ell_3\ell_8 +18\ell_2\ell_8 +3\ell_1\ell_4 +6\ell_0\ell_7 +\ell_0\ell_5\Big)\\[0.3cm] +\ell_5^{[m-1]}=&\frac1{C^{[m]}}\Big( 2\ell_5 +12 \ell_6\ell_8 +2\ell_4\ell_6 +216 \ell_3\ell_7 +6\ell_2\ell_5 +36\ell_1\ell_7 +12\ell_0\ell_8 +2\ell_0\ell_4\Big)\\[0.3cm] -\ell_6^{[m-1]}=&\frac1{C}( \ell_6 +18\ell_7\ell_8 +3\ell_5\ell_8 +3\ell_4\ell_7 +\frac12\ell_4\ell_5 +18\ell_3\ell_6 +3\ell_2\ell_6 +\ell_6^{[m-1]}=&\frac1{C^{[m]}}( \ell_6 +18\ell_7\ell_8 +3\ell_5\ell_8 +3\ell_4\ell_7 +\frac12\ell_4\ell_5 +18\ell_3\ell_6 +3\ell_2\ell_6 +3\ell_1\ell_6 +2\ell_0\ell_6\Big)\\[0.3cm] -\ell_7^{[m-1]}=&\frac1{C}\Big( \frac12\ell_7 +\frac12\ell_6\ell_8 +\frac1{12}\ell_4\ell_6 +\frac32\ell_3\ell_5 +\frac32\ell_2\ell_7 +\frac14\ell_1\ell_5 +\frac12\ell_0\ell_8 +\ell_7^{[m-1]}=&\frac1{C^{[m]}}\Big( \frac12\ell_7 +\frac12\ell_6\ell_8 +\frac1{12}\ell_4\ell_6 +\frac32\ell_3\ell_5 +\frac32\ell_2\ell_7 +\frac14\ell_1\ell_5 +\frac12\ell_0\ell_8 +\frac1{12}\ell_0\ell_4\Big)\\[0.3cm] -\ell_8^{[m-1]}=&\frac1{C}\Big( \ell_8 +\ell_6\ell_7 +\frac1{6}\ell_5\ell_6 +3\ell_3\ell_4 +\frac12\ell_2\ell_4 +3\ell_1\ell_8 +\ell_0\ell_7 +\ell_8^{[m-1]}=&\frac1{C^{[m]}}\Big( \ell_8 +\ell_6\ell_7 +\frac1{6}\ell_5\ell_6 +3\ell_3\ell_4 +\frac12\ell_2\ell_4 +3\ell_1\ell_8 +\ell_0\ell_7 +\frac1{6}\ell_0\ell_5\Big) \end{array} \label{eqbetasusc}\end{equation} \preblock -in which we dropped the $^{[m]}$ exponent on the right side. By considering the linearized flow equation, we find that $\ell_0,\ell_4,\ell_6,\ell_8$ are {\it marginal}, $\ell_2,\ell_5$ {\it relevant} and $\ell_1,\ell_3,\ell_7$ {\it irrelevant}. The consequent linear flow is {\it very different} from the full flow discussed in section~\ref{secbetakondo}. +in which we dropped the $^{[m]}$ exponent on the right side. By considering the linearized flow equation (around $\ell_j=0$), we find that $\ell_0,\ell_4,\ell_6,\ell_8$ are {\it marginal}, $\ell_2,\ell_5$ {\it relevant} and $\ell_1,\ell_3,\ell_7$ {\it irrelevant}. The consequent linear flow is {\it very different} from the full flow discussed in section~\ref{secbetakondo}. \indent The vector $\bm\ell$ is related to $\bm\alpha$ and via the following map: \preblock @@ -807,11 +811,11 @@ for all $m\le0$, which implies that the set $\{\bm\ell\ |\ \ell_0<0,\ \ell_2\ge0 \end{figure} \section{Kondo effect, XY-model, free fermions} \label{appXY} -\indent In [\cite{abeSeO}], given $\nu\in [1,\ldots,L]$, the Hamiltonian $\mathcal H_h=\mathcal H_0 {-h} \,\sigma_\nu^z$, with +\indent In [\cite{ABe71}], given $\nu\in [1,\ldots,L]$, the Hamiltonian $\mathcal H_h=\mathcal H_0 {-h} \,\sigma_\nu^z$, with \begin{equation} \mathcal H_0={- \frac 14} \sum_{n=1}^L (\sigma^x_n\sigma^x_{n+1}+\sigma^y_n\sigma^y_{n+1}). \label{eqhamXY}\end{equation} -has been considered with suitable boundary conditions, under which $H_0$ and ${\sigma^z_0} +1$ are unitarily equivalent to $\sum_{q}{(-\cos q)} \, a^+_qa^-_q$ and, respectively, to {$\frac2L \sum_{q,q'} a^+_q a^-_{q'} e^{i\nu(q-q')}$} in which $a^\pm_q$ are fermionic creation and annihilation operators and the sums run over $q$'s that are such that $e^{iq L}=-1$. It has been shown (see [\cite{abeSeO},~(3.18)] which, after integration by parts is equivalent to what follows; since the scope of [\cite{abeSeO}] was somewhat different we give here a complete self-contained account of the derivation of~(\ref{eqdiagFXY}) and the following ones, see appendix~\ref{appXYcomp}), that, by defining +has been considered with suitable boundary conditions (see appendix~\ref{appXYcomp}), under which $H_0$ and ${\sigma^z_0} +1$ are unitarily equivalent to $\sum_{q}{(-\cos q)} \, a^+_qa^-_q$ and, respectively, to {$\frac2L \sum_{q,q'} a^+_q a^-_{q'} e^{i\nu(q-q')}$} in which $a^\pm_q$ are fermionic creation and annihilation operators and the sums run over $q$'s that are such that $e^{iq L}=-1$. It has been shown (see [\cite{ABe71},~(3.18)] which, after integration by parts is equivalent to what follows; since the scope of [\cite{ABe71}] was somewhat different we give here a complete self-contained account of the derivation of~(\ref{eqdiagFXY}) and the following ones, see appendix~\ref{appXYcomp}), that, by defining \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l} F_L(\zeta)=&1+\frac{2 h}{L} \sum_q \frac1{\zeta+\cos q}\\ \indent F(z)=& \lim_{L\to\infty} F_L(z)=1+\frac{2\,h}{\pi}\,\int_0^\pi \frac{dq}{(z+\cos q)} @@ -841,7 +845,7 @@ is obtained. Does it exhibit a Kondo effect? \indent Since $\bm\tau_0$ commutes with the $\bm\sigma_n$ and, hence, with $H_0$, the average magnetization and susceptibility, $m^{int}(\beta,h,\lambda)$ and $\chi^{int}(\beta,h,\lambda)$, responding to a field $h$ acting only on the site $0$, can be expressed in terms of the functions $\zeta(\beta,h)$ and its derivatives $\zeta'(\beta,h)$ and $\zeta''(\beta,h)$. By using the fact that $\zeta(\beta,h)$ and $\zeta''(\beta,h)$ are even in $h$, while $\zeta'(\beta,h)$ is odd, we get: \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l} -\chi^{int}(\beta,h)&=\beta^{-1}\partial^2_h\log{\rm Tr}\,\sum_{\tau=\pm1} \Big(e^{-\beta H_0+\beta\lambda\sigma^z\tau+\beta h(\sigma^z+\tau)}\Big)\Big|_{h=0}\\ +\chi^{int}(\beta,0)&=\beta^{-1}\partial^2_h\log{\rm Tr}\,\sum_{\tau=\pm1} \Big(e^{-\beta H_0+\beta\lambda\sigma^z\tau+\beta h(\sigma^z+\tau)}\Big)\Big|_{h=0}\\ &:=\beta^{-1}\partial^2_h\log Z^{int}(\beta,h,\lambda)\Big|_{h=0}\\ &=\beta^{-1}\left.\Big( \sum_\tau\frac{\zeta''+\zeta' \beta \tau +(\zeta'+\beta\tau\zeta)\beta\tau}{Z^{int}} -\Big(\sum_\tau\frac{(\zeta'+\beta\tau\zeta)}{Z^{int}}\Big)^2\Big)\right|_{h=0}\\ &=\chi(\beta,|\lambda|)+\beta(m(\beta,|\lambda|)+1)^2\mathop{\longrightarrow}_{\beta\to\infty}+\infty @@ -896,7 +900,7 @@ hence \begin{equation} (\lambda_j+\cos q)U_{jq}e^{-iq\nu}= -\frac{2h}L\sum_{q''} e^{-iq''\nu} U_{jq''} \label{eqeigvalsolXY}\end{equation} -$\forall q\in I$, where we used the fact that $A^-_p A^+_q \left|0\right> =\delta_{p,q}\left|0\right>$. Therefore either $\lambda_j\ne -\cos q$ for all $q\in I$ or $\lambda=-\cos q_0$ for some $q_0\in I$. +$\forall q\in I$, where we used the fact that $A^-_p A^+_q \left|0\right> =\delta_{p,q}\left|0\right>$. We consider the two cases $\lambda_j\ne -\cos q$ for all $q\in I$ or $\lambda=-\cos q_0$ for some $q_0\in I$. \medskip \indent In the first case: @@ -907,7 +911,7 @@ provided \begin{equation} F_L(\lambda_j):= 1+\frac{2h}L\sum_q\frac1{\lambda_j+\cos q}=0 \label{eqFdiagdefXY}\end{equation} -or, in the second case, +where $N(\lambda_j)$ is set in such a way that $U$ is unitary, or, in the second case, \begin{equation} \lambda_j=-\cos q_0,\quad U_{jq}=\frac{e^{iq\nu}}{\sqrt2}(\delta_{q,q_0}-\delta_{q,-q_0}) \label{eqeigstrivXY}\end{equation} |