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diff --git a/Jauslin_Rutgers_2015.tex b/Jauslin_Rutgers_2015.tex new file mode 100644 index 0000000..a4b0f63 --- /dev/null +++ b/Jauslin_Rutgers_2015.tex @@ -0,0 +1,280 @@ +\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: \parbox[b]{1cm}{\tt1506.04381\par1507.05678}\hfill{\tt http://ian.jauslin.org/} +\eject + +\pagestyle{plain} +\setcounter{page}{1} + + +\title{Kondo model} +\begin{itemize} +\item [P.~Anderson, 1960], [J.~Kondo, 1964]: +$$ +H=H_0+V\quad\mathrm{on\ }\mathcal H=\mathcal F_L\otimes\mathbb C^2 +$$ +\itemptchange{$\scriptstyle\blacktriangleright$} +\begin{itemize} +\item $H_0$: kinetic term of the {\it electrons} +$$ +H_0:=\sum_{x}\sum_{\alpha=\uparrow,\downarrow}c^\dagger_\alpha(x)\,\left(\left(-\frac{\Delta}2-1\right)\,c_\alpha\right)(x)\otimes\mathds1 +$$ +\item $V$: interaction with the {\it impurity} +$$ +V=-\lambda_0\sum_{j=1,2,3}\sum_{\alpha_1,\alpha_2}c^\dagger_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c_{\alpha_2}(0)\otimes \tau^j +$$ +\end{itemize} +\itemptreset +\end{itemize} +\hfil\includegraphics[width=0.8\textwidth]{Figs/kondo_model.pdf}\par +\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: +$$ +\chi^{(0)}(0,\beta)=\beta\mathop{\longrightarrow}_{\beta\to\infty}\infty +$$ +\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}. +\end{itemize} +\itemptreset +\item{\tt Remark}: [N.~Andrei, 1980]: the Kondo model (suitably linearized) is exactly solvable via Bethe Ansatz (which breaks down under any perturbation of the model). +\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 For $\lambda_0<0$, the flow tends to a non-trivial fixed point, and $\chi^{(\lambda_0)}<\infty$ in the $\beta\to\infty$ limit (Kondo effect). +\end{itemize} +\eject + +\title{Field theory for the Kondo model} +\begin{itemize} +\item By introducing an extra dimension ({\it imaginary time}), the partition function $Z:=\mathrm{Tr}(e^{-\beta H})$ can be expressed as the {\it Gaussian} average over a {\it Grassmann} algebra: +$$ +Z=\left<e^{-\int_0^\beta dt\ \mathcal V(t)}\right> +$$ +\vskip-10pt +\item Potential: +$$ +\mathcal V(t)=-\lambda_0\kern-10pt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2,\alpha_3,\alpha_4}}\kern-10pt +\psi^+_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^-_{\alpha_2}(t)\tau^j +$$ +with $\{\psi^\pm_\alpha(t),\psi^\pm_{\alpha'}(t')\}=0$. +\item $\left<\cdot\right>$ is defined by its second moment $\left<\psi^-_{\alpha_1}(t_1)\psi^+_{\alpha_2}(t_2)\right>$. +\end{itemize} +\eject + +\title{Hierarchical model} +\begin{itemize} +\item Replace $\psi_\alpha^\pm(t)$ in $\mathcal V(t)$ by +$$ +\psi_\alpha^\pm(t):=\sum_{m\leqslant0}\psi_\alpha^{[m]\pm}(t) +$$ +where $\psi_\alpha^{[m]\pm}(t)$ is {\it constant} over the ``time'' intervals $\Delta_{i,\pm}^{[m]}$:\par +\vfil +\hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxes.pdf}\par +\item There are 4 fields in each $\Delta_{i,\pm}^{[m]}$. +\item Moments: +$$ +\left<\psi_\alpha^{[m]-}(\Delta_{i,\mp}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\pm}^{[m]})\right>=\pm 2^m +$$ +\end{itemize} +\eject + +\title{Full propagator} +\begin{itemize} +\item Moments: +$$ +\left<\psi_\alpha^{[m]-}(\Delta_{i,\mp}^{[m]})\psi_\alpha^{[m]+}(\Delta_{i,\pm}^{[m]})\right>=\pm 2^m +$$ +\item Full propagator: +$$ +\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>= 2^{m_{t,t'}}\mathrm{sign}(t-t') +$$ +\hfil\includegraphics[width=0.8\textwidth]{Figs/hierarchical_boxtree.pdf}\par +\vfil +\eject + +\title{Comparison with the Kondo model} +\item Hierarchical model: +$$ +\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>= 2^{m_{t,t'}}\mathrm{sign}(t-t') +$$ +\item For the (non-hierarchical) Kondo model: +$$ +\left<\psi_\alpha^{-}(t)\psi_\alpha^{+}(t')\right>\approx\sum_m 2^{m}g_\psi^{[0]}(2^m(t-t')) +$$ +where $g^{[0]}_\psi$ is odd and decays faster than any power. +\end{itemize} +\eject + +\title{Hierarchical beta function} +\begin{itemize} +\item Compute $Z$ by $\mathcal V^{[0]}(t):=\mathcal V(t)$ +$$ +e^{-\int dt\ \mathcal V^{[m-1]}(t)}:=\left<e^{-\int dt\ \mathcal V^{[m]}(t)}\right>_m +$$ + +\item Effective potential: +$$ +\int dt\ \mathcal V^{[m]}(t)=\sum_{i=1}^{2^{-m}}\mathcal V^{[m]}_{i,-}+\mathcal V^{[m]}_{i,+} +$$ +\item Iteration +$$ +\left<e^{-\int dt\ \mathcal V^{[m]}(t)}\right>_m=\prod_{i=1}^{2^{-m}}\left<e^{-(\mathcal V^{[m]}_{i,-}+\mathcal V_{i,+}^{[m]})}\right>_m +$$ +\item By anti-commutation of the fields, $e^{-(\mathcal V_{i,-}^{[m]}+\mathcal V_{i,+}^{[m]})}$ is a polynomial in the fields of order $\leqslant 8$. +\end{itemize} +\eject + +\title{Hierarchical beta function} +\begin{itemize} +\item $\mathcal V^{[m]}$ is parametrized by 2 real numbers ({\it running coupling constants}) $\ell_0^{[m]},\ell_1^{[m]}$: +$$\begin{array}{r@{\ }>{\displaystyle}l} +\frac{e^{-\scriptstyle\int dt\ \mathcal V^{[m]}(t)}}{C^{[m]}} +=&1+\frac{\ell_0^{[m]}}2\int dt\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2}} +\psi^{[\le m]+}_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^{[\le m]-}_{\alpha_2}(t)\tau^j\\[0.5cm] +&+\frac{\ell_1^{[m]}}2\int dt\left(\sum_{\displaystyle\mathop{\scriptstyle j=1,2,3}_{\alpha_1,\alpha_2}} +\psi^{[\le m]+}_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^{[\le m]-}_{\alpha_2}(t)\right)^2 +\end{array}$$ +\end{itemize} +\eject + +\title{Hierarchical beta function} +\begin{itemize} +\item Beta function ({\it exact}) +$$\begin{array}{r@{\ }l} +C^{[m]}=&\displaystyle1+ \frac32(\ell_0^{[m]})^2+9(\ell_1^{[m]})^2\\[0.3cm] \ell_0^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\ell_0^{[m]} +3 \ell_0^{[m]}\ell_1^{[m]} -(\ell_0^{[m]})^2\Big)\\[0.5cm] +\ell_1^{[m-1]}=&\displaystyle\frac1{C^{[m]}}\Big(\frac12\ell_1^{[m]}+\frac18(\ell_0^{[m]})^2\Big) +\end{array}$$ +\end{itemize} +\eject + +\title{Flow} +\vfil +\hfil\includegraphics[width=0.8\textwidth]{Figs/sd_phase.pdf}\par +Fixed points: 0 (stable), $\bm\ell^*$ (marginal in $\ell_0$ and stable in $\ell_1$) +\eject + +\title{Susceptibility} +\begin{itemize} +\item Add magnetic field $h$ on the impurity. +\item New term in the potential: +$$ +-h \sum_{j\in\{1,2,3\}}\bm\omega_j \tau^j +$$ +\item 6 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 $0$, the susceptibility diverges as $\beta$. +\item At $\bm\ell^*$, the susceptibility remains finite in the $\beta\to\infty$ limit. +\end{itemize} +\eject + +\title{Susceptibility} +\begin{itemize} +\item $\lambda_0=-0.28$ +\end{itemize} +\hfil\includegraphics[width=200pt]{Figs/sd_susc_0_28.pdf}\par +\eject + +\addtocounter{page}{-1} +\title{Susceptibility} +\begin{itemize} +\item $\lambda_0=-0.02$ +\end{itemize} +\hfil\includegraphics[width=200pt]{Figs/sd_susc_0_02.pdf}\par +\eject + +\addtocounter{page}{-1} +\title{Susceptibility} +\begin{itemize} +\item $\lambda_0=-0.005$ +\end{itemize} +\hfil\includegraphics[width=200pt]{Figs/sd_susc_0_005.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 many Feynman diagrams ($\approx100$). +\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} + + + + + |