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diff --git a/Jauslin_Como_2015.tex b/Jauslin_Como_2015.tex new file mode 100644 index 0000000..940c256 --- /dev/null +++ b/Jauslin_Como_2015.tex @@ -0,0 +1,324 @@ +\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}: [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: +$$ +Z=\left<e^{-\int_0^\beta dt\ \mathcal V(t)}\right> +$$ +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: +$$ +\psi_\alpha^\pm(t):=\sum_{m}\psi_\alpha^{[m]\pm}(\Delta^{[m]}(t)),\quad +\varphi_\alpha^\pm(t):=\sum_{m}\varphi_\alpha^{[m]\pm}(\Delta^{[m]}(t)) +$$ +\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 $\mathcal V^{[0]}(t):=\mathcal V(t)$ +$$ +e^{-\int dt\ \mathcal V^{[m-1]}(t)}:=\left<e^{-\int dt\ \mathcal V^{[m]}(t)}\right>_m +$$ + +\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, $e^{-\mathcal V_{i,\pm}^{[m]}}$ 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 $\ell_0,\cdots,\ell_3$: +$$ +e^{-\int dt\ \mathcal V^{[m]}(t)} +=\sum_{i,\eta}\sum_{n=0}^3\ell_n^{[m]}O_{n}^{[\leqslant m]}(\Delta_{i,\eta}) +$$ +\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} + + + + + |