\documentclass{kiss} \usepackage{presentation} \usepackage{header} \begin{document} \pagestyle{empty} \hbox{}\vfil \bf \large \hfil Non-perturbative renormalization group\par \smallskip \hfil in a hierarchical Kondo model\par \vfil \hfil Ian Jauslin \rm \normalsize \vfil \small \hfil joint with {\normalsize\bf G.~Benfatto} and {\normalsize\bf G.~Gallavotti}\par \vskip10pt arXiv: {\tt1506.04381}\hfill{\tt http://ian.jauslin.org/} \eject \pagestyle{plain} \setcounter{page}{1} \title{Kondo model} \begin{itemize} \item s-d model: [P.~Anderson, 1960] [J.~Kondo, 1964]: \itemptchange{$\scriptstyle\blacktriangleright$ } \begin{itemize} \item 1D chain of non-interacting spin-1/2 fermions: {\it electrons}. \item lone spin-1/2 fermion: {\it impurity}. \item the impurity interacts with the electron at 0. \end{itemize} \itemptreset \end{itemize} \vskip0ptplus3fil \hfil\includegraphics[width=0.8\textwidth]{Figs/kondo_model.pdf}\par \vskip0ptplus3fil \eject \title{Kondo Hamiltonian} \vskip5pt \hfil\includegraphics[width=0.8\textwidth]{Figs/kondo_model.pdf}\par $$ H=H_0+V $$ \begin{itemize} \item $H_0$: kinetic term of the {\it electrons} $$ H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(-\frac{\Delta}2-1\right)\,c_\alpha(x) $$ \itemptchange{$\scriptstyle\blacktriangleright$ } \begin{itemize} \item $c_\alpha(x)$: fermionic annihilation operator \item $\alpha$: spin \item $x$: site \end{itemize} \itemptreset \end{itemize} \eject \title{Kondo Hamiltonian} \vskip5pt \hfil\includegraphics[width=0.8\textwidth]{Figs/kondo_model.pdf}\par $$ H=H_0+V $$ \begin{itemize} \item $V$: interaction with the {\it impurity} $$ V=-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}c^\dagger_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c_{\alpha_2}(0)\, d^\dagger_{\alpha_3}\sigma^j_{\alpha_3,\alpha_4}d_{\alpha_4} $$ \itemptchange{$\scriptstyle\blacktriangleright$ } \begin{itemize} \item $d_\alpha$: fermionic annihilation operator \item $\sigma^j$: Pauli matrix \item $\lambda_0>0$: {\it ferromagnetic} case \item $\lambda_0<0$: {\it anti-ferromagnetic} case \end{itemize} \itemptreset \end{itemize} \eject \title{Kondo effect: magnetic susceptibility} \begin{itemize} \item Magnetic susceptibility: response to a magnetic field $h$: $$ \chi(h,\beta):=\partial_hm(h,\beta). $$ ($m(h,\beta)$: magnetization). \item Isolated impurity: $$ \chi^{(0)}(0,\beta)=\frac\beta2\mathop{\longrightarrow}_{\beta\to\infty}\infty $$ \item Chain of electrons: Pauli paramagnetism: $$ \lim_{\beta\to\infty}\lim_{L\to\infty}\frac1L\chi_e(0,\beta)<\infty. $$ \end{itemize} \eject \title{Kondo effect: magnetic susceptibility} \begin{itemize} \item Turn on the interaction: $\lambda_0\neq0$. Impurity susceptibility $\chi^{(\lambda_0)}(h,\beta)$. \item Ferromagnetic interaction ($\lambda_0>0$): $$ \lim_{\beta\to\infty}\chi^{(\lambda_0)}(0,\beta)=\infty. $$ \item Anti-ferromagnetic interaction ($\lambda_0<0$): $$ \lim_{\beta\to\infty}\chi^{(\lambda_0)}(0,\beta)<\infty. $$ \item {\it Non-perturbative} effect: the qualitative behavior changes as soon as $\lambda_0\neq0$. \end{itemize} \eject \title{Previous results} \begin{itemize} \item [J.~Kondo, 1964]: third order Born approximation. \vskip0pt plus3fil \item [P.~Anderson, 1970], [K.~Wilson, 1975]: renormalization group approach \itemptchange{$\scriptstyle\blacktriangleright$ } \begin{itemize} \item Sequence of effective Hamiltonians at varying energy scales. \item For anti-ferromagnetic interactions, the effective Hamiltonians go to a {\it non-trivial fixed point}. \item Anderson: instability of the trivial fixed point ($H_0$). \item Wilson: numerical diagonalization at each step, and perturbative expansions around the trivial and non-trivial fixed points. \end{itemize} \end{itemize} \eject \title{Current results} \begin{itemize} \item Hierarchical Kondo model: idealization of the Kondo model that has the same scaling properties. \item It is {\it exactly solvable}: the map relating the effective theories at different scales is {\it explicit} (no perturbative expansions). \item With $\lambda_0<0$, the flow tends to a non-trivial fixed point, and $\chi^{(\lambda_0)}<\infty$ in the $\beta\to\infty$ limit (Kondo effect). \item{\tt Remark}: [N.~Andrei, 1980]: the Kondo model (suitably linearized) is exactly solvable via Bethe Ansatz. \end{itemize} \eject \title{Field theory for the Kondo model} \begin{itemize} \item Partition function $Z:=\mathrm{Tr}(e^{-\beta H})$. \vskip5pt \item By introducing an extra dimension ({\it imaginary time}), $Z$ can be expressed as the {\it Gaussian} average over a {\it Grassmann} algebra: $$ Z=\left $$ where $$ \mathcal V(t)=-\lambda_0\kern-10pt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}}\kern-10pt \psi^+_{\alpha_1}(0,t)\sigma^j_{\alpha_1,\alpha_2}\psi^-_{\alpha_2}(0,t)\, \varphi^+_{\alpha_3}(t)\sigma^j_{\alpha_3,\alpha_4}\varphi^-_{\alpha_4}(t) $$ with $\{\psi^\pm_\alpha(0,t),\psi^\pm_{\alpha'}(0,t')\}=0$, $\{\varphi^\pm_\alpha(t),\varphi^\pm_{\alpha'}(t')\}=0$. \end{itemize} \eject \title{Hierarchical fields} \hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxes.pdf}\par \begin{itemize} \item For each $m<0$, we introduce fields {\it on scale $m$} indexed by an interval: $$ \psi_\alpha^\pm(\Delta_{i,\pm}^{[m]}),\quad \varphi_\alpha^\pm(\Delta_{i,\pm}^{[m]}) $$ where $$\begin{array}l \Delta_{i,-}^{[m]}:=[2^{-m}i,2^{-m}(i+\frac12))\\[0.3cm] \Delta_{i,+}^{[m]}:=[2^{-m}(i+\frac12),2^{-m}(i+1)) \end{array}$$ \item There are 8 fields in each $\Delta_{i,\pm}^{[m]}$. \end{itemize} \eject \title{Hierarchical fields} \begin{itemize} \item Split fields over scales: $$ \psi_\alpha^\pm(t):=\sum_{m\leqslant0}\psi_\alpha^{[m]\pm}(\Delta^{[m]}(t)),\quad \varphi_\alpha^\pm(t):=\sum_{m\leqslant0}\varphi_\alpha^{[m]\pm}(\Delta^{[m]}(t)) $$ \end{itemize} \hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxtree.pdf}\par \vfil \eject \title{Hierarchical propagators} \begin{itemize} \item Moments: $$\begin{array}{r@{\ }l} \left<\psi_\alpha^{[m]-}(\Delta_{i,-\eta}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\eta}^{[m]})\right>=&\eta 2^m\\[0.3cm] \left<\varphi_\alpha^{[m]-}(\Delta_{i,-\eta}^{[m]})\varphi_\alpha^{[m]+}(\Delta_{i,\eta}^{[m]})\right>=&\eta \end{array}$$ \item Full propagator: $$ \left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>=\mathrm{sign}(t-t') 2^{m_{t,t'}},\quad \left<\varphi_\alpha^{-}(t)\varphi_\alpha^{+}(t')\right>=\mathrm{sign}(t-t') $$ \item For the (non-hierarchical) Kondo model: $$\begin{array}{r@{\ }l} \left<\psi_\alpha^{-}(0,t)\psi_\alpha^{+}(0,t')\right>\approx&\sum_m 2^{m}g_\psi^{[0]}(2^m(t-t')),\\[0.3cm] \left<\varphi_\alpha^{-}(t)\varphi_\alpha^{+}(t')\right>\approx&\sum_m g^{[0]}_\varphi(2^m(t-t')) \end{array}$$ \end{itemize} \eject \title{Hierarchical beta function} \begin{itemize} \item Compute $Z$ by $\mathcal V^{[0]}(t):=\mathcal V(t)$ $$ e^{-\int dt\ \mathcal V^{[m-1]}(t)}:=\left_m $$ \item Effective potential: $$ \int dt\ \mathcal V^{[m]}(t)=\sum_{i=1}^{2^{-m}}\mathcal V^{[m]}_{i,-}+\mathcal V^{[m]}_{i,+} $$ \item Iteration $$ \left_m=\prod_{i=1}^{2^{-m}}\left_m $$ \item By anti-commutation of the fields, $e^{-\mathcal V_{i,\pm}^{[m]}}$ is a polynomial in the fields of order $\leqslant 8$. \end{itemize} \eject \title{Hierarchical beta function} \begin{itemize} \item The computation of the beta function reduces to computing the average of a degree-$16$ polynomial. \item 4 running coupling constants $\ell_0,\cdots,\ell_3$: $$ e^{-\int dt\ \mathcal V^{[m]}(t)} =1+\sum_{i,\eta}\sum_{n=0}^3\ell_n^{[m]}O_{n}^{[\leqslant m]}(\Delta_{i,\eta}) $$ \end{itemize} \eject \title{Hierarchical beta function} \begin{itemize} \item Beta function ({\it exact}) $$\begin{array}{r@{\ }l} C^{[m]}=&\displaystyle1+ 3\ell_0^2+9\ell_1^2+9\ell_2^2+324\ell_3^2\\[0.3cm] \ell_0^{[m-1]}=&\displaystyle\frac1C\Big(\ell_0 +18\ell_0\ell_3+3 \ell_0\ell_2+3 \ell_0\ell_1 -2\ell_0^2\Big)\\[0.5cm] \ell_1^{[m-1]}=&\displaystyle\frac1C\Big( \frac12\ell_1+9\ell_2\ell_3 +\frac14\ell_0^2\Big)\\[0.5cm] \ell_2^{[m-1]}=&\displaystyle\frac1C\Big(2\ell_2+36\ell_1\ell_3+ \ell_0^2\Big)\\[0.5cm] \ell_3^{[m-1]}=&\displaystyle\frac1C\Big(\frac12\ell_3+\frac14\ell_1\ell_2+\frac1{24} \ell_0^2\Big). \end{array}$$ \end{itemize} \eject \title{Hierarchical beta function} \begin{itemize} \item Beta function ({\it exact}) $$\begin{array}{r@{\ }l} C^{[m]}=&\displaystyle1+ 3\ell_0^2+9\ell_1^2+9\ell_2^2+324\ell_3^2\\[0.3cm] \color{blue}\ell_0^{[m-1]}=&\color{blue}\displaystyle\frac1C\Big(\ell_0 +18\ell_0\ell_3+3 \ell_0\ell_2+3 \ell_0\ell_1 -2\ell_0^2\Big)\\[0.5cm] \color{darkgreen}\ell_1^{[m-1]}=&\color{darkgreen}\displaystyle\frac1C\Big( \frac12\ell_1+9\ell_2\ell_3 +\frac14\ell_0^2\Big)\\[0.5cm] \color{red}\ell_2^{[m-1]}=&\color{red}\displaystyle\frac1C\Big(2\ell_2+36\ell_1\ell_3+ \ell_0^2\Big)\\[0.5cm] \color{darkgreen}\ell_3^{[m-1]}=&\color{darkgreen}\displaystyle\frac1C\Big(\frac12\ell_3+\frac14\ell_1\ell_2+\frac1{24} \ell_0^2\Big). \end{array}$$ {\color{red}relevant}, {\color{blue}marginal}, {\color{darkgreen}irrelevant} \end{itemize} \eject \title{Flow} \vfil \hfil\includegraphics[width=0.8\textwidth]{Figs/beta_phase.pdf}\par Fixed points: $\bm\ell^{(0)}$, $\bm\ell^{(+)}$, $\bm\ell^*$, $\bm\ell^{(-)}$. \eject \title{Fixed points} \begin{itemize} \item $\bm\ell^{(0)}$: unstable. \item $\bm\ell^{(+)}$: ferromagnetic ($\lambda_0>0$). \item $\bm\ell^*$: anti-ferromagnetic ($\lambda_0<0$). \end{itemize} \eject \title{Susceptibility} \begin{itemize} \item Add magnetic field $h$ on the impurity. \item New term in the potential: $$ -h \sum_{j\in\{1,2,3\}}\bm\omega_j \int dt\sum_{\alpha,\alpha'}\varphi^+_{\alpha}(t)\sigma^j_{\alpha,\alpha'} \varphi^-_{\alpha'}(t). $$ \item 9 running coupling constants. \item The susceptibility can be computed by deriving $C^{[m]}$ with respect to $h$. \end{itemize} \eject \title{Kondo effect} \begin{itemize} \item Fix $h=0$. \item At $\bm\ell^{(+)}$, the susceptibility diverges as $\beta$. \item At $\bm\ell^*$, the susceptibility remains finite in the $\beta\to\infty$ limit. \end{itemize} \hfil\includegraphics[width=150pt]{Figs/susc_plot_temp.pdf}\par \eject \title{Open questions} \begin{itemize} \item Magnetic field on the chain as well. This requires defining the hierarchical model to reflect the $x$-dependence of $\psi(x,t)$. \item Rigorous renormalization group analysis for the Kondo model (non-hierarchical). \item The exact solvability of the hierarchical Kondo model is merely a consequence of the fermionic nature of the system. Other fermionic hierarchical models can be studied to investigate other non-perturbative phenomena, e.g. high-$T_c$ superconductivity. \end{itemize} \eject \title{Epilogue: {\tt meankondo}} \begin{itemize} \item The computation in the $h$-dependent case requires computing 100 Feynman diagrams. \item By adding the field on the entire chain (open problem), this number increases to 1089. \item Software to perform the computation: {\tt meankondo}. \item {\tt meankondo} can be configured to study any fermionic hierarchical model. \end{itemize} \hfil{\tt http://ian.jauslin.org/software/meankondo/} \end{document}