Initial commit
This commit is contained in:
Ian Jauslin
commit
1a0ba99f6b
|
|
@ -0,0 +1,38 @@
|
|||
%%
|
||||
%% BBlog bibliography related commands
|
||||
%%
|
||||
|
||||
%% length used to display the bibliography
|
||||
\newlength{\rw}
|
||||
\setlength{\rw}{1.5cm}
|
||||
|
||||
%% read header
|
||||
\IfFileExists{header.BBlog.tex}{\input{header.BBlog}}{}
|
||||
|
||||
%% cite a reference
|
||||
\def\cite#1{%
|
||||
\ref{cite#1}%
|
||||
%% add entry to citelist after checking it has not already been added
|
||||
\ifcsname if#1cited\endcsname%
|
||||
\expandafter\if\csname if#1cited\endcsname%
|
||||
\else%
|
||||
\csname if#1citedtrue\endcsname%
|
||||
\immediate\write\@auxout{\noexpand\BBlogcite{#1}}%
|
||||
\fi%
|
||||
\else%
|
||||
\expandafter\newif\csname if#1cited\endcsname%
|
||||
\csname if#1citedtrue\endcsname%
|
||||
\immediate\write\@auxout{\noexpand\BBlogcite{#1}}%
|
||||
\fi%
|
||||
}
|
||||
%% an empty definition for the aux file
|
||||
\def\BBlogcite#1{}
|
||||
|
||||
%% display the bibliography
|
||||
\long\def\BBlography{
|
||||
\newlength{\colw}
|
||||
\setlength{\colw}{\textwidth}
|
||||
\addtolength{\colw}{-\rw}
|
||||
\IfFileExists{bibliography.BBlog.tex}{
|
||||
\input{bibliography.BBlog}}{{\tt error: missing BBlog bibliography file}}
|
||||
}
|
||||
|
|
@ -0,0 +1,173 @@
|
|||
\documentclass{kiss}
|
||||
% load packages
|
||||
\usepackage{header}
|
||||
% bibliography commands
|
||||
\usepackage{BBlog}
|
||||
% miscellaneous commands
|
||||
\usepackage{toolbox}
|
||||
% main style file
|
||||
\usepackage{iansecs}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\bf\Large
|
||||
\hfil Kondo effect in the hierarchical $s-d$ model
|
||||
\normalsize
|
||||
\vskip20pt
|
||||
|
||||
\hfil{Giovanni Gallavotti, Ian Jauslin}
|
||||
\vskip20pt
|
||||
\rm
|
||||
|
||||
\hfil2015\par
|
||||
\hugeskip
|
||||
|
||||
\leftskip20pt
|
||||
\rightskip20pt
|
||||
\small
|
||||
The $s-d$ model describes a chain of spin-1/2 electrons interacting magnetically with a two-level impurity. It was introduced to study the Kondo effect, in which the magnetic susceptibility of the impurity remains finite in the 0-temperature limit as long as the interaction of the impurity with the electrons is anti-ferromagnetic. A variant of this model was introduced by Andrei, which he proved was exactly solvable via Bethe Ansatz. A hierarchical version of Andrei's model was studied by Benfatto and the authors. In the present letter, that discussion is extended to a hierarchical version of the $s-d$ model. The resulting analysis is very similar to the hierarchical Andrei model, though the result is slightly simpler.\par
|
||||
\leftskip0pt
|
||||
\rightskip0pt
|
||||
\normalsize
|
||||
|
||||
\hugeskip
|
||||
|
||||
\indent The $s-d$ model was introduced by Anderson [\cite{andSO}] and used by Kondo [\cite{konSF}] to study what would subsequently be called the {\it Kondo effect}. It describes a chain of electrons interacting with a fixed spin-1/2 magnetic impurity. One of the manifestations of the effect is that when the coupling is anti-ferrmoagnetic, the magnetic susceptibility of the impurity remains finite in the 0-temperature limit, whereas it diverges for ferromagnetic and for vanishing interactions.\par
|
||||
|
||||
\indent A modified version of the $s-d$ model was introduced by Andrei [\cite{andEZ}], which was shown to be exactly solvable by Bethe Ansatz. In [\cite{bgjOFi}], a hierarchical version of Andrei's model was introduced and shown to exhibit a Kondo effect. In the present letter, we show how the argument can be adapted to the $s-d$ model.\par
|
||||
|
||||
\indent We will show that in the hierarchical $s-d$ model, the computation of the susceptibility reduces to iterating an {\it explicit} map relating 6 {\it running coupling constants} (rccs), and that this map can be obtained by restricting the flow equation for the hierarchical Andrei model [\cite{bgjOFi}] to one of its invariant manifolds. The physics of both models are therefore very closely related, as had already been argued in [\cite{bgjOFi}]. This is particularly noteworthy since, at 0-field, the flow in the hierarchical Andrei model is relevant, whereas it is marginal in the hierarchical $s-d$ model, which shows that the relevant direction carries little to no physical significance.\par
|
||||
\bigskip
|
||||
|
||||
\indent The $s-d$ model [\cite{konSF}] represents a chain of non-interacting spin-1/2 fermions, called {\it electrons}, which interact with an isolated spin-1/2 {\it impurity} located at site 0. The Hilbert space of the system is $\mathcal F_L\otimes\mathbb C^2$ in which $\mathcal F_L$ is the Fock space of a length-$L$ chain of spin-1/2 fermions (the electrons) and $\mathbb C^2$ is the state space for the two-level impurity. The Hamiltonian, in the presence of a magnetic field of amplitude $h$ in the direction $\bm\omega\equiv(\bm\omega_1,\bm\omega_2,\bm\omega_3)$, is
|
||||
\begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
|
||||
H_K=&H_0+V_0+V_h=:H_0+V\\[0.3cm]
|
||||
H_0=&\sum_{\alpha\in\{\uparrow,\downarrow\}}\sum_{x=-{L}/2}^{{L}/2-1} c^+_\alpha(x)\,\left(-\frac{\Delta}2-1\right)\,c^-_\alpha(x)\\[0.5cm]
|
||||
V_0=&-\lambda_0\sum_{j=1,2,3\atop\alpha_1,\alpha_2} c^+_{\alpha_1}(0)\sigma^j_{\alpha_1,\alpha_2}c^-_{\alpha_2}(0)\, \tau^j\\[0.5cm]
|
||||
V_h=&-h \,\sum_{j=1,2,3}\bm\omega_j \tau^j
|
||||
\end{array}\label{eqhamdef}\end{equation}
|
||||
where $\lambda_0$ is the interaction strength, $\Delta$ is the discrete Laplacian $c_\alpha^\pm(x),\,\alpha=\uparrow,\downarrow$ are creation and annihilation operators acting on {\it electrons}, and $\sigma^j=\tau^j,\,j=1,2,3$, are Pauli matrices. The operators $\tau^j$ act on the {\it impurity}. The boundary conditions are taken to be periodic.\par
|
||||
|
||||
\indent In the {\it Andrei model} [\cite{andEZ}], the impurity is represented by a fermion instead of a two-level system, that is the Hilbert space is replaced by $\mathcal F_L\otimes\mathcal F_1$, and the Hamiltonian is defined by replacing $\tau^j$ in~(\ref{eqhamdef}) by $d^+\tau^jd^-$ in which $d_\alpha^\pm(x),\,\alpha=\uparrow,\downarrow$ are creation and annihilation operators acting on the impurity.\par
|
||||
\bigskip
|
||||
|
||||
\indent The partition function $Z={\rm Tr}\, e^{-\beta H_K}$ can be expressed formally as a functional integral:
|
||||
\begin{equation}
|
||||
Z=\mathrm{Tr}\int P(d\psi)\, \sum_{n=0}^\infty(-1)^n\int_{0<t_1<\cdots<t_n<\beta}\kern-50pt dt_1\cdots dt_n\, \mathcal V(t_1)\cdots\mathcal V(t_n)
|
||||
\label{eqpartfn}\end{equation}
|
||||
in which $\mathcal V(t)$ is obtained from $V$ by replacing $c_\alpha^\pm(0)$ in~(\ref{eqhamdef}) by a {\it Grassmann} field $\psi_\alpha^\pm(0,t)$, $P(d\psi)$ is a {\it Gaussian Grassmann measure} over the fields $\{\psi_\alpha^\pm(0,t)\}_{t,\alpha}$ whose {\it propagator} ({\it i.e.} {\it covariance}) is, in the $L\to\infty$ limit,
|
||||
$$
|
||||
g(t,t')=\frac1{(2\pi)^2}\int dk dk_0 \frac{e^{i k_0(t-t')}}{i k_0-\cos k},
|
||||
$$
|
||||
and the trace is over the state-space of the spin-1/2 impurity, that is a trace over $\mathbb C^2$.\par
|
||||
\bigskip
|
||||
|
||||
\indent We will consider a {\it hierarchical} version of the $s-d$ model. The hierarchical model defined below is {\it inspired} by the $s-d$ model in the same way as the hierarchical model defined in [\cite{bgjOFi}] was inspired by the Andrei model. We will not give any details on the justification of the definition, as such considerations are entirely analogous to the discussion in [\cite{bgjOFi}].\par
|
||||
|
||||
\indent The model is defined by introducing a family of {\it hierarchical fields} and specifying a {\it propagator} for each pair of fields. The average of any monomial of fields is then computed using the Wick rule.\par
|
||||
|
||||
\indent Assuming $\beta=2^{N_\beta}$ with $N_\beta=\log_2\beta\in\mathbb N$, the time axis $[0,\beta)$ is paved with boxes ({\it i.e.} intervals) of size $2^{-m}$ for every $m\in\{0,-1,\ldots,-N_\beta\}$: let
|
||||
\begin{equation}
|
||||
\mathcal Q_m:=\left\{[i 2^{|m|}, (i+1) 2^{|m|})\right\}_{i=0,1,\cdots,2^{N_\beta-|m|}-1}^{m=0,-1,\ldots}
|
||||
\label{eqtiledef}\end{equation}
|
||||
Given a box $\Delta\in{\mathcal Q}_m$, let $t_\Delta$ denote the center of $\Delta$, and given a point $t\in R$, let $\Delta^{[m]}(t)$ be the (unique) box on scale $m$ that contains $t$. We further decompose each box $\Delta\in\mathcal Q_m$ into two {\it half boxes}: for $\eta\in\{-,+\}$, let
|
||||
\begin{equation}
|
||||
\Delta_{\eta}:=\Delta^{[m+1]}(t_{\Delta}+\eta2^{-m-2})
|
||||
\label{eqhalfboxdef}\end{equation}
|
||||
for $m\le 0$. Thus $\Delta_{-}$ can be called the ``lower half'' of $\Delta$ and $\Delta_{+}$ the ``upper half''.\par
|
||||
|
||||
\indent The elementary fields used to define the hierarchical $s-d$ model will be {\it constant on each half-box} and will be denoted by $\psi_\alpha^{[m]\pm}(\Delta_{\eta})$ for $m\in\{0,-1,\cdots,$ $-N_\beta\}$, $\Delta\in\mathcal Q_m$, $\eta\in\{-,+\}$, $\alpha\in\{\uparrow,\downarrow\}$.\par
|
||||
|
||||
\indent The propagator of the hierarchical $s-d$ model is defined as
|
||||
\begin{equation}
|
||||
\left<\psi_{\alpha}^{[m]-}(\Delta_{-\eta})\psi_{\alpha}^{[m]+}(\Delta_{\eta})\right >:= \eta
|
||||
\label{eqprop}\end{equation}
|
||||
for $m\in\{0,-1,\cdots,$ $-N_\beta\}$, $\Delta\in\mathcal Q_m$, $\eta\in\{-,+\}$, $\alpha\in\{\uparrow,\downarrow\}$. The propagator of any other pair of fields is set to 0.\par
|
||||
|
||||
\indent Finally, we define
|
||||
\begin{equation}
|
||||
\psi^\pm_\alpha(t):= \sum_{m=0}^{-N_\beta} 2^{\frac{m}2}\psi_\alpha^{[m]\pm}(\Delta^{[m+1]}(t)).
|
||||
\label{eqfielddcmp}\end{equation}
|
||||
|
||||
\indent The partition function for the hierarchical $s-d$ model is
|
||||
\begin{equation}
|
||||
Z=\mathrm{Tr}\left< \sum_{n=0}^\infty(-1)^n\int_{0<t_1<\cdots<t_n<\beta}\kern-50pt dt_1\cdots dt_n\, \mathcal V(t_1)\cdots\mathcal V(t_n) \right>
|
||||
\label{eqhierpartfn}\end{equation}
|
||||
in which the $\psi^\pm_\alpha(0,t)$ in $\mathcal V(t)$ have been replaced by the $\psi_\alpha^\pm(t)$ defined in~(\ref{eqfielddcmp}):
|
||||
\begin{equation}
|
||||
\mathcal V(t):=-\lambda_0\sum_{j=1,2,3\atop\alpha_1,\alpha_2} \psi^+_{\alpha_1}(t)\sigma^j_{\alpha_1,\alpha_2}\psi^-_{\alpha_2}(t)\, \tau^j -h \,\sum_{j=1,2,3}\bm\omega_j \tau^j.
|
||||
\label{eqhierpot}\end{equation}
|
||||
This concludes the definition of the hierarchical $s-d$ model.\par
|
||||
\bigskip
|
||||
|
||||
|
||||
\indent We will now show how to compute the partition function~(\ref{eqhierpartfn}) using a renormalization group iteration. We first rewrite
|
||||
\begin{equation}
|
||||
\sum_{n=0}^\infty(-1)^n\int_{0<t_1<\cdots<t_n<\beta}\kern-50pt dt_1\cdots dt_n\, \mathcal V(t_1)\cdots\mathcal V(t_n) =\prod_{\Delta\in\mathcal Q_0}\prod_{\eta=\pm}\left(\sum_{n=0}^\infty\frac{(-1)^n}{2^nn!}\mathcal V(t_{\Delta_\eta})^n\right)
|
||||
\label{eqtrotthier}\end{equation}
|
||||
and find that
|
||||
\begin{equation}
|
||||
\sum_{n=0}^\infty\frac{(-1)^n}{2^nn!}\mathcal V(t_{\Delta_\eta^{[0]}})^n =C\left(1+\sum_{p}\ell_p^{[0]}O_{p,\eta}^{[\le 0]}(\Delta^{[0]})\right)
|
||||
\label{eqexpV}\end{equation}
|
||||
with
|
||||
\begin{equation}\begin{array}{r@{\quad}l}
|
||||
O_{0,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf A^{[\le 0]}_\eta(\Delta)\cdot\bm\tau,& O_{1,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf A^{[\le 0]}_\eta(\Delta)^2,\\[0.3cm]
|
||||
O_{4,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf A^{[\le 0]}_\eta(\Delta)\cdot\bm\omega,& O_{5,\eta}^{[\le 0]}(\Delta):=\frac12\mathbf \bm\tau\cdot\bm\omega,\\[0.3cm]
|
||||
O_{6,\eta}^{[\le 0]}(\Delta):=\frac12(\mathbf A^{[\le 0]}_\eta(\Delta)\cdot\bm\omega)(\bm\tau\cdot\bm\omega),& O_{7,\eta}^{[\le 0]}(\Delta):=\frac12(\mathbf A^{[\le 0]}_\eta(\Delta)^2)(\bm\tau\cdot\bm\omega)
|
||||
\end{array}\label{eqOdef}\end{equation}
|
||||
(the numbering is meant to recall that in [\cite{bgjOFi}]) in which $\bm\tau=(\tau^1,\tau^2,\tau^3)$ and $\mathbf A_\eta^{[\le 0]}(\Delta)$ is a vector of polynomials in the fields whose $j$-th component for $j\in\{1,2,3\}$ is
|
||||
\begin{equation}
|
||||
A_\eta^{[\le 0]j}(\Delta):=\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2} \psi_\alpha^{[\le 0]+}(\Delta_\eta)\sigma^j_{\alpha,\alpha'}\psi_{\alpha'}^{[\le 0]-}(\Delta_\eta)
|
||||
\label{eqAdef}\end{equation}
|
||||
$\psi_\alpha^{[\le 0]\pm}:=\sum_{m\le0}2^{\frac m2}\psi_\alpha^{[m]\pm}$, and
|
||||
\begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
|
||||
C=&\cosh(\tilde h),\quad \ell_0^{[0]}=\frac1C\frac{\lambda_0}{\tilde h}\sinh(\tilde h),\quad
|
||||
\ell_1^{[0]}=\frac1C\frac{\lambda_0^2}{12\tilde h}(\tilde h\cosh(\tilde h)+2\sinh(\tilde h))\\[0.3cm]
|
||||
\ell_4^{[0]}=&\frac1C\lambda_0\sinh(\tilde h),\quad \ell_5^{[0]}=\frac1C\sinh(\tilde h),\quad
|
||||
\ell_6^{[0]}=\frac1C\frac{\lambda_0}{\tilde h}(\tilde h\cosh(\tilde h)-\sinh(\tilde h))\\[0.3cm]
|
||||
\ell_7^{[0]}=&\frac1C\frac{\lambda_0^2}{12\tilde h^2}(\tilde h^2\sinh(\tilde h)+2\tilde h\cosh(\tilde h)-2\sinh(\tilde h))
|
||||
\end{array}\label{eqinitcd}\end{equation}
|
||||
in which $\tilde h:=h/2$.\par
|
||||
|
||||
\indent By a straightforward induction, we find that the partition function~(\ref{eqhierpartfn}) can be computed by defining
|
||||
\begin{equation}
|
||||
C^{[m]}\mathcal W^{[m-1]}(\Delta^{[m]}):=\left<\prod_\eta\left(\mathcal W^{[m]}(\Delta^{[m]}_\eta)\right)\right>_m
|
||||
\label{eqindW}\end{equation}
|
||||
in which $\left<\cdot\right>_m$ denotes the average over $\psi^{[m]}$, $C^{[m]}>0$ and
|
||||
\begin{equation}
|
||||
\mathcal W^{[m-1]}(\Delta^{[m]})=1+\sum_p\ell_p^{[m]}O_p^{[\le m]}(\Delta^{[m]})
|
||||
\label{eqexprW}\end{equation}
|
||||
in terms of which
|
||||
\begin{equation}
|
||||
Z=C^{2|\mathcal Q_0|}\prod_{m=-N(\beta)+1}^0(C^{[m]})^{|\mathcal Q_{m-1}|}
|
||||
\label{eqZind}\end{equation}
|
||||
in which $|\mathcal Q_m|=2^{N(\beta)-|m|}$ is the cardinality of $\mathcal Q_m$. In addition, similarly to [\cite{bgjOFi}], the map relating $\ell_p^{[m]}$ to $\ell_p^{[m-1]}$ and $C^{[m]}$ can be computed explicitly from~(\ref{eqindW}):
|
||||
\begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
|
||||
C^{[m]} =& 1 +\frac{3}{2}\ell_{0}^2 +\ell_{0}\ell_{6} +9\ell_{1}^2 +\frac{\ell_{4}^2}{2} +\frac{\ell_{5}^2}{4} +\frac{\ell_{6}^2}{2} +9\ell_{7}^2 \\[0.3cm]
|
||||
\ell^{[m-1]}_{0} =& \frac1C\left(\ell_{0} -\ell_{0}^2 +3\ell_{0}\ell_{1} -\ell_{0}\ell_{6} \right)\\[0.3cm]
|
||||
\ell^{[m-1]}_{1} =& \frac1C\left(\frac{\ell_{1}}{2} +\frac{\ell_{0}^2}{8} +\frac{\ell_{0}\ell_{6}}{12} +\frac{\ell_{4}^2}{24} +\frac{\ell_{5}\ell_{7}}{4} +\frac{\ell_{6}^2}{24} \right)\\[0.3cm]
|
||||
\ell^{[m-1]}_{4} =& \frac1C\left(\ell_{4} +\frac{\ell_{0}\ell_{5}}{2} +3\ell_{0}\ell_{7} +3\ell_{1}\ell_{4} +\frac{\ell_{5}\ell_{6}}{2} +3\ell_{6}\ell_{7} \right)\\[0.3cm]
|
||||
\ell^{[m-1]}_{5} =& \frac1C\left(2\ell_{5} +2\ell_{0}\ell_{4} +36\ell_{1}\ell_{7} +2\ell_{4}\ell_{6} \right)\\[0.3cm]
|
||||
\ell^{[m-1]}_{6} =& \frac1C\left(\ell_{6} +\ell_{0}\ell_{6} +3\ell_{1}\ell_{6} +\frac{\ell_{4}\ell_{5}}{2} +3\ell_{4}\ell_{7} \right)\\[0.3cm]
|
||||
\ell^{[m-1]}_{7} =& \frac1C\left(\frac{\ell_{7}}{2} +\frac{\ell_{0}\ell_{4}}{12} +\frac{\ell_{1}\ell_{5}}{4} +\frac{\ell_{4}\ell_{6}}{12} \right)
|
||||
\end{array}\label{eqbetafun}\end{equation}
|
||||
in which the $^{[m]}$ have been dropped from the right hand side.\par
|
||||
\bigskip
|
||||
|
||||
|
||||
\indent The flow equation~(\ref{eqbetafun}) can be recovered from that of the hierarchical Andrei model studied in [\cite{bgjOFi}] (see in particular [\cite{bgjOFi}, (C1)] by restricting the flow to the invariant submanifold defined by \begin{equation} \ell_2^{[m]}=\frac13,\quad \ell_3^{[m]}=\frac16\ell_1^{[m]},\quad \ell_8^{[m]}=\frac16\ell_4^{[m]}. \label{e18}\end{equation} This is of particular interest since $\ell_2^{[m]}$ is a relevant coupling and the fact that it plays no role in the $s-d$ model indicates that it has little to no physical relevance.\par
|
||||
|
||||
\indent The qualitative behavior of the flow is therefore the same as that described in [\cite{bgjOFi}] for the hierarchical Andrei model. In particular the susceptibility, which can be computed by deriving $-\beta^{-1}\log Z$ with respect to $h$, remains finite in the 0-temperature limit as long as $\lambda_0<0$, that is as long as the interaction is anti-ferromagnetic.\par
|
||||
\hugeskip
|
||||
|
||||
{\bf Acknowledgements}: We are grateful to G.~Benfatto for many enlightening discussions on the $s-d$ and Andrei's models.
|
||||
|
||||
\hugeskip
|
||||
|
||||
\small
|
||||
\BBlography
|
||||
|
||||
\vfill
|
||||
\eject
|
||||
|
||||
\end{document}
|
||||
|
|
@ -0,0 +1,49 @@
|
|||
* Typeset
|
||||
|
||||
In order to typeset the LaTeX document, run
|
||||
pdflatex Gallavotti_Jauslin_2015.tex
|
||||
pdflatex Gallavotti_Jauslin_2015.tex
|
||||
|
||||
|
||||
* Files
|
||||
|
||||
Gallavotti_Jauslin_2015.tex :
|
||||
body of the paper.
|
||||
|
||||
bibliography.BBlog.tex :
|
||||
list of references.
|
||||
|
||||
BBlog.sty :
|
||||
bibliography related commands.
|
||||
|
||||
header.sty :
|
||||
list of packages.
|
||||
|
||||
iansecs.sty :
|
||||
main style file.
|
||||
|
||||
kiss.cls :
|
||||
barebones class file
|
||||
|
||||
toolbox.sty :
|
||||
collection of useful commands.
|
||||
|
||||
|
||||
* Coding style
|
||||
|
||||
In the body of the paper, an effort has been made to keep the LaTeX code
|
||||
'standard', avoiding self-defined commands whenever possible, and sticking to
|
||||
TeX and basic LaTeX commands. In some instances however, such an approach would
|
||||
have been too restrictive, and commands defined in the style files listed above
|
||||
were used.
|
||||
|
||||
Many of the commands defined in 'iansecs.sty' are drop-in replacements for
|
||||
standard LaTeX commands, though some functionality may be lost.
|
||||
|
||||
|
||||
* Bibliography
|
||||
|
||||
The bibliography was generated by BBlog, which produced the
|
||||
bibliography.BBlog.tex file. All of the required code to adequately typeset
|
||||
the reference list and define the commands used to cite them are contained in
|
||||
that file.
|
||||
|
|
@ -0,0 +1,20 @@
|
|||
\hrefanchor
|
||||
\outdef{citeandSO}{And61}
|
||||
\hbox{\parbox[t]{\rw}{[\cite{andSO}]}\parbox[t]{\colw}{P.~Anderson - {\it Localized magnetic states in metals}, Physical Review, Vol.~124, n.~1, p.~41-53, 1961.}}\par
|
||||
\bigskip
|
||||
|
||||
\hrefanchor
|
||||
\outdef{citeandEZ}{And80}
|
||||
\hbox{\parbox[t]{\rw}{[\cite{andEZ}]}\parbox[t]{\colw}{N.~Andrei - {\it Diagonalization of the Kondo Hamiltonian}, Physical Review Letters, Vol.~45, n.~5, 1980.}}\par
|
||||
\bigskip
|
||||
|
||||
\hrefanchor
|
||||
\outdef{citebgjOFi}{BGJ15}
|
||||
\hbox{\parbox[t]{\rw}{[\cite{bgjOFi}]}\parbox[t]{\colw}{G.~Benfatto, G.~Gallavotti, I.~Jauslin - {\it Kondo effect in a Fermionic hierarchical model}, arXiv 1506.04381, 2015.}}\par
|
||||
\bigskip
|
||||
|
||||
\hrefanchor
|
||||
\outdef{citekonSF}{Kon64}
|
||||
\hbox{\parbox[t]{\rw}{[\cite{konSF}]}\parbox[t]{\colw}{J.~Kondo - {\it Resistance minimum in dilute magnetic alloys}, Progress of Theoretical Physics, Vol.~32, n.~1, 1964.}}\par
|
||||
\bigskip
|
||||
|
||||
|
|
@ -0,0 +1,11 @@
|
|||
%%
|
||||
%% Load packages
|
||||
%%
|
||||
|
||||
\usepackage{color}
|
||||
\usepackage[hidelinks]{hyperref}
|
||||
\usepackage{amsfonts}
|
||||
\usepackage{bm}
|
||||
\usepackage{array}
|
||||
\usepackage{etoolbox}
|
||||
|
||||
|
|
@ -0,0 +1,402 @@
|
|||
%%
|
||||
%% This file contains the main style commands
|
||||
%%
|
||||
%% Some options can be set by changing the \loaddefaults command
|
||||
%%
|
||||
|
||||
\usepackage{color}
|
||||
\usepackage{marginnote}
|
||||
|
||||
\def\loaddefaults{
|
||||
\sectionsfalse
|
||||
\subseqcountfalse
|
||||
\def\seqskip{\vskip1.5cm}
|
||||
\def\subseqskip{\vskip1cm}
|
||||
\resetpointattheofalse
|
||||
\parindent=0pt
|
||||
\def\indent{\hskip20pt}
|
||||
}
|
||||
|
||||
% false if there are no sections
|
||||
\newif\ifsections
|
||||
% true if equation numbers should include the subsection number
|
||||
\newif\ifsubseqcount
|
||||
% true if there is a table of contents
|
||||
\newif\iftoc
|
||||
% true if point counting should reset at each theorem
|
||||
\newif\ifresetpointattheo
|
||||
|
||||
% a prefix to put before the section number, e.g. A for appendices
|
||||
\def\sectionprefix{}
|
||||
|
||||
\loaddefaults
|
||||
|
||||
%% style for the equation number
|
||||
\def\eqnumstyle{}
|
||||
|
||||
%% correct vertical alignment at the end of a document
|
||||
\AtEndDocument{
|
||||
\vfill
|
||||
\eject
|
||||
}
|
||||
|
||||
%% hyperlinks
|
||||
% hyperlinkcounter
|
||||
\newcounter{lncount}
|
||||
% hyperref anchor
|
||||
\def\hrefanchor{%
|
||||
\stepcounter{lncount}%
|
||||
\hypertarget{ln.\thelncount}{}%
|
||||
}
|
||||
|
||||
%% define a command and write it to aux file
|
||||
\def\outdef#1#2{%
|
||||
% define command
|
||||
\expandafter\xdef\csname #1\endcsname{#2}%
|
||||
% hyperlink number
|
||||
\expandafter\xdef\csname #1@hl\endcsname{\thelncount}%
|
||||
% write command to aux
|
||||
\immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1\endcsname{\csname #1\endcsname}}%
|
||||
\immediate\write\@auxout{\noexpand\expandafter\noexpand\gdef\noexpand\csname #1@hl\endcsname{\thelncount}}%
|
||||
}
|
||||
|
||||
|
||||
%% define a label for the latest tag
|
||||
%% label defines a command containing the string stored in \tag
|
||||
\AtBeginDocument{
|
||||
\def\label#1{\expandafter\outdef{#1}{\safe\tag}}
|
||||
|
||||
\def\ref#1{%
|
||||
% check whether the label is defined (hyperlink runs into errors if this check is ommitted)
|
||||
\ifcsname #1@hl\endcsname%
|
||||
\hyperlink{ln.\csname #1@hl\endcsname}{\safe\csname #1\endcsname}%
|
||||
\else%
|
||||
\ifcsname #1\endcsname%
|
||||
\csname #1\endcsname%
|
||||
\else%
|
||||
{\bf ??}%
|
||||
\fi%
|
||||
\fi%
|
||||
}
|
||||
}
|
||||
|
||||
%% counters
|
||||
\newcounter{sectioncount}
|
||||
\newcounter{subsectioncount}
|
||||
\newcounter{pointcount}
|
||||
\newcounter{subpointcount}
|
||||
\newcounter{subsubpointcount}
|
||||
\newcounter{seqcount}
|
||||
\newcounter{figcount}
|
||||
\newcounter{Theocount}
|
||||
\newcounter{tocsectioncount}
|
||||
\newcounter{tocsubsectioncount}
|
||||
|
||||
%% section command
|
||||
\newlength\secnumwidth
|
||||
\newlength\sectitlewidth
|
||||
\def\section#1{%
|
||||
% reset counters
|
||||
\stepcounter{sectioncount}%
|
||||
\setcounter{subsectioncount}{0}%
|
||||
\setcounter{pointcount}{0}%
|
||||
\setcounter{subpointcount}{0}%
|
||||
\setcounter{subsubpointcount}{0}%
|
||||
\setcounter{figcount}{0}%
|
||||
\setcounter{Theocount}{0}%
|
||||
\setcounter{seqcount}{0}%
|
||||
% space before section (if not first)
|
||||
\ifnum\thesectioncount>1%
|
||||
\seqskip%
|
||||
\penalty-1000%
|
||||
\fi%
|
||||
% hyperref anchor
|
||||
\hrefanchor%
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\sectionprefix\thesectioncount}%
|
||||
% get widths
|
||||
\def\@secnum{{\bf\Large\sectionprefix\thesectioncount.\hskip10pt}}%
|
||||
\settowidth\secnumwidth{\@secnum}%
|
||||
\setlength\sectitlewidth\textwidth%
|
||||
\addtolength\sectitlewidth{-\secnumwidth}%
|
||||
% print name
|
||||
\parbox{\textwidth}{%
|
||||
\@secnum%
|
||||
\parbox[t]{\sectitlewidth}{\Large\bf #1}}%
|
||||
% write to table of contents
|
||||
\iftoc%
|
||||
% save lncount in aux variable which is written to toc
|
||||
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@sec.\thesectioncount\endcsname{\thelncount}}%
|
||||
\write\tocoutput{\noexpand\tocsection{#1}{\thepage}}%
|
||||
\fi%
|
||||
\par\penalty10000%
|
||||
\bigskip\penalty10000%
|
||||
}
|
||||
|
||||
%% subsection
|
||||
\def\subsection#1{
|
||||
% counters
|
||||
\stepcounter{subsectioncount}%
|
||||
\setcounter{pointcount}{0}%
|
||||
\setcounter{subpointcount}{0}%
|
||||
\setcounter{subsubpointcount}{0}%
|
||||
\ifsubseqcount%
|
||||
\setcounter{seqcount}0%
|
||||
\fi%
|
||||
% space before subsection (if not first)
|
||||
\ifnum\thesubsectioncount>1%
|
||||
\subseqskip%
|
||||
\penalty-500%
|
||||
\fi%
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\sectionprefix\thesectioncount.\thesubsectioncount}%
|
||||
% get widths
|
||||
\def\@secnum{{\bf\large\hskip.5cm\sectionprefix\thesectioncount.\thesubsectioncount.\hskip5pt}}%
|
||||
\settowidth\secnumwidth{\@secnum}%
|
||||
\setlength\sectitlewidth\textwidth%
|
||||
\addtolength\sectitlewidth{-\secnumwidth}%
|
||||
% print name
|
||||
\parbox{\textwidth}{%
|
||||
\@secnum%
|
||||
\parbox[t]{\sectitlewidth}{\large\bf #1}}%
|
||||
% write to table of contents
|
||||
\iftoc%
|
||||
% save lncount in aux variable which is written to toc
|
||||
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@subsec.\thesectioncount.\thesubsectioncount\endcsname{\thelncount}}%
|
||||
\write\tocoutput{\noexpand\tocsubsection{#1}{\thepage}}%
|
||||
\fi%
|
||||
\par\penalty10000%
|
||||
\medskip\penalty10000%
|
||||
}
|
||||
|
||||
%% itemize
|
||||
\newlength\itemizeskip
|
||||
% left margin for items
|
||||
\setlength\itemizeskip{20pt}
|
||||
% item symbol
|
||||
\def\itemizept{\textbullet}
|
||||
\newlength\itemizeseparator
|
||||
% space between the item symbol and the text
|
||||
\setlength\itemizeseparator{5pt}
|
||||
|
||||
\newlength\current@itemizeskip
|
||||
\setlength\current@itemizeskip{0pt}
|
||||
\def\itemize{
|
||||
\par\medskip
|
||||
\addtolength\current@itemizeskip{\itemizeskip}
|
||||
\leftskip\current@itemizeskip
|
||||
}
|
||||
\def\enditemize{
|
||||
\addtolength\current@itemizeskip{-\itemizeskip}
|
||||
\par\leftskip\current@itemizeskip
|
||||
\medskip
|
||||
}
|
||||
\newlength\itempt@total
|
||||
\def\item{
|
||||
\settowidth\itempt@total{\itemizept}
|
||||
\addtolength\itempt@total{\itemizeseparator}
|
||||
\par
|
||||
\medskip
|
||||
\hskip-\itempt@total\itemizept\hskip\itemizeseparator
|
||||
}
|
||||
|
||||
%% points
|
||||
\def\point{
|
||||
\stepcounter{pointcount}
|
||||
\setcounter{subpointcount}{0}
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
\indent{\bf \thepointcount\ - }
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\thepointcount}
|
||||
}
|
||||
\def\subpoint{
|
||||
\stepcounter{subpointcount}
|
||||
\setcounter{subsubpointcount}0
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
\indent\hskip.5cm{\bf \thepointcount-\thesubpointcount\ - }
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\thepointcount-\thesubpointcount}
|
||||
}
|
||||
\def\subsubpoint{
|
||||
\stepcounter{subsubpointcount}
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
\indent\hskip1cm{\bf \thepointcount-\thesubpointcount-\thesubsubpointcount\ - }
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\thepointcount-\thesubpointcount-\thesubsubpointcount}
|
||||
}
|
||||
% reset points
|
||||
\def\resetpointcounter{
|
||||
\setcounter{pointcount}{0}
|
||||
\setcounter{subpointcount}{0}
|
||||
\setcounter{subsubpointcount}{0}
|
||||
}
|
||||
|
||||
%% equation numbering
|
||||
\def\seqcount{
|
||||
\stepcounter{seqcount}
|
||||
% the output
|
||||
\edef\seqformat{\theseqcount}
|
||||
% add subsection number
|
||||
\ifsubseqcount
|
||||
\let\tmp\seqformat
|
||||
\edef\seqformat{\thesubsectioncount.\tmp}
|
||||
\fi
|
||||
% add section number
|
||||
\ifsections
|
||||
\let\tmp\seqformat
|
||||
\edef\seqformat{\sectionprefix\thesectioncount.\tmp}
|
||||
\fi
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\seqformat}
|
||||
% write number
|
||||
\marginnote{\eqnumstyle\hfill(\seqformat)}
|
||||
}
|
||||
%% equation environment compatibility
|
||||
\def\equation{\hrefanchor$$\seqcount}
|
||||
\def\endequation{$$\@ignoretrue}
|
||||
|
||||
%% figures
|
||||
\newlength\figwidth
|
||||
\setlength\figwidth\textwidth
|
||||
\addtolength\figwidth{-2.5cm}
|
||||
|
||||
\def\figcount#1{%
|
||||
\stepcounter{figcount}%
|
||||
% hyperref anchor
|
||||
\hrefanchor%
|
||||
% the number of the figure
|
||||
\edef\figformat{\thefigcount}%
|
||||
% add section number
|
||||
\ifsections%
|
||||
\let\tmp\figformat%
|
||||
\edef\figformat{\sectionprefix\thesectioncount.\tmp}%
|
||||
\fi%
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\figformat}%
|
||||
% write
|
||||
\hfil fig \figformat: \parbox[t]{\figwidth}{\small#1}%
|
||||
\par\bigskip%
|
||||
}
|
||||
|
||||
%% environment
|
||||
\def\figure{
|
||||
\par\penalty-500
|
||||
}
|
||||
\def\endfigure{
|
||||
\par\penalty-1000
|
||||
}
|
||||
\let\caption\figcount
|
||||
|
||||
%% delimiters
|
||||
\def\delimtitle#1{\par \leavevmode\raise.3em\hbox to\hsize{\lower0.3em\hbox{\vrule height0.3em}\hrulefill\ \lower.3em\hbox{#1}\ \hrulefill\lower0.3em\hbox{\vrule height0.3em}}\par\penalty10000}
|
||||
\def\delim{\par\leavevmode\raise.3em\hbox to\hsize{\vrule height0.3em\hrulefill\vrule height0.3em}\par\penalty10000}
|
||||
\def\enddelim{\par\penalty10000\leavevmode\raise.3em\hbox to\hsize{\vrule height0.3em\hrulefill\vrule height0.3em}\par}
|
||||
|
||||
%% theorem headers
|
||||
\def\theo#1{
|
||||
\stepcounter{Theocount}
|
||||
% reset points
|
||||
\ifresetpointattheo\resetpointcounter\fi
|
||||
% hyperref anchor
|
||||
\hrefanchor
|
||||
% the number
|
||||
\def\formattheo{\theTheocount}
|
||||
% add section number
|
||||
\ifsections
|
||||
\let\tmp\formattheo
|
||||
\edef\formattheo{\sectionprefix\thesectioncount.\tmp}
|
||||
\fi
|
||||
% define tag (for \label)
|
||||
\xdef\tag{\formattheo}
|
||||
% write
|
||||
\delimtitle{\bf #1 \formattheo}
|
||||
}
|
||||
\let\endtheo\enddelim
|
||||
|
||||
%% start appendices
|
||||
\def\appendix{%
|
||||
\vfill
|
||||
\pagebreak
|
||||
% counter
|
||||
\setcounter{sectioncount}0%
|
||||
% prefix
|
||||
\def\sectionprefix{A}%
|
||||
% write
|
||||
{\bf \LARGE Appendices}\par\penalty10000\bigskip\penalty10000%
|
||||
% add a mention in the table of contents
|
||||
\iftoc%
|
||||
\immediate\write\tocoutput{\noexpand\tocappendices}\penalty10000%
|
||||
\fi%
|
||||
%% uncomment for new page for each appendix
|
||||
%\def\seqskip{\vfill\pagebreak}
|
||||
}
|
||||
|
||||
%% start references
|
||||
\def\references{%
|
||||
\hrefanchor%
|
||||
% write
|
||||
{\bf \LARGE References}\par\penalty10000\bigskip\penalty10000%
|
||||
% add a mention in the table of contents
|
||||
\iftoc%
|
||||
% save lncount in aux variable which is written to toc
|
||||
\immediate\write\tocoutput{\noexpand\expandafter\noexpand\edef\noexpand\csname toc@references\endcsname{\thelncount}}%
|
||||
\write\tocoutput{\noexpand\tocreferences{\thepage}}\penalty10000%
|
||||
\fi%
|
||||
}
|
||||
|
||||
|
||||
%% table of contents
|
||||
\newif\iftocopen
|
||||
\def\tableofcontents{
|
||||
{\bf \large Table of contents:}\par\penalty10000\bigskip\penalty10000%
|
||||
% copy content from file
|
||||
\IfFileExists{\jobname.toc}{\input{\jobname.toc}}{{\tt error: table of contents missing}}
|
||||
% open new toc
|
||||
\newwrite\tocoutput
|
||||
\immediate\openout\tocoutput=\jobname.toc
|
||||
\toctrue
|
||||
}
|
||||
%% close file
|
||||
\AtEndDocument{
|
||||
% close toc
|
||||
\iftoc
|
||||
\immediate\closeout\tocoutput
|
||||
\fi
|
||||
}
|
||||
|
||||
|
||||
%% fill line with dots
|
||||
\def\leaderfill{\leaders\hbox to 1em {\hss. \hss}\hfill}
|
||||
|
||||
%% same as sectionprefix
|
||||
\def\tocsectionprefix{}
|
||||
|
||||
%% toc formats
|
||||
\def\tocsection #1#2{
|
||||
\stepcounter{tocsectioncount}
|
||||
\setcounter{tocsubsectioncount}{0}
|
||||
% write
|
||||
\smallskip\hyperlink{ln.\csname toc@sec.\thetocsectioncount\endcsname}{{\bf \tocsectionprefix\thetocsectioncount}.\hskip5pt #1\leaderfill#2}\par
|
||||
}
|
||||
\def\tocsubsection #1#2{
|
||||
\stepcounter{tocsubsectioncount}
|
||||
% write
|
||||
{\hskip10pt\hyperlink{ln.\csname toc@subsec.\thetocsectioncount.\thetocsubsectioncount\endcsname}{{\bf \thetocsubsectioncount}.\hskip5pt {\small #1}\leaderfill#2}}\par
|
||||
}
|
||||
\def\tocappendices{
|
||||
\medskip
|
||||
\setcounter{tocsectioncount}0
|
||||
{\bf Appendices}\par
|
||||
\smallskip
|
||||
\def\tocsectionprefix{A}
|
||||
}
|
||||
\def\tocreferences#1{
|
||||
\medskip
|
||||
{\hyperlink{ln.\csname toc@references\endcsname}{{\bf References}\leaderfill#1}}\par
|
||||
\smallskip
|
||||
}
|
||||
|
|
@ -0,0 +1,42 @@
|
|||
%%
|
||||
%% Barebones class declaration
|
||||
%%
|
||||
|
||||
\NeedsTeXFormat{LaTeX2e}[1995/12/01]
|
||||
\ProvidesClass{kiss}
|
||||
|
||||
\setlength\paperheight {297mm}
|
||||
\setlength\paperwidth {210mm}
|
||||
|
||||
%% fonts
|
||||
\input{size11.clo}
|
||||
\DeclareOldFontCommand{\rm}{\normalfont\rmfamily}{\mathrm}
|
||||
\DeclareOldFontCommand{\sf}{\normalfont\sffamily}{\mathsf}
|
||||
\DeclareOldFontCommand{\tt}{\normalfont\ttfamily}{\mathtt}
|
||||
\DeclareOldFontCommand{\bf}{\normalfont\bfseries}{\mathbf}
|
||||
\DeclareOldFontCommand{\it}{\normalfont\itshape}{\mathit}
|
||||
\DeclareOldFontCommand{\sl}{\normalfont\slshape}{\@nomath\sl}
|
||||
\DeclareOldFontCommand{\sc}{\normalfont\scshape}{\@nomath\sc}
|
||||
|
||||
%% something is wrong with \thepage, redefine it
|
||||
\gdef\thepage{\the\c@page}
|
||||
|
||||
%% default offsets: 1in, correct with \hoffset and \voffset
|
||||
\hoffset=-50pt
|
||||
\voffset=-72pt
|
||||
%% horizontal margins
|
||||
%\oddsidemargin=31pt
|
||||
%\evensidemargin=31pt
|
||||
%% vertical margin
|
||||
%\topmargin=20pt
|
||||
%% body size
|
||||
\textwidth=460pt
|
||||
\textheight=704pt
|
||||
%% header size and margin
|
||||
%\headheight=12pt
|
||||
%\headsep=25pt
|
||||
%% footer size
|
||||
%\footskip=30pt
|
||||
%% margin size and margin
|
||||
\marginparwidth=35pt
|
||||
%\marginparsep=10pt
|
||||
|
|
@ -0,0 +1,41 @@
|
|||
%%
|
||||
%% A collection of useful commands
|
||||
%%
|
||||
|
||||
%% can call commands even when they are not defined
|
||||
\def\safe#1{%
|
||||
\ifdefined#1%
|
||||
#1%
|
||||
\else%
|
||||
{\color{red}\bf?}%
|
||||
\fi%
|
||||
}
|
||||
|
||||
|
||||
%% larger skip
|
||||
\newskip\hugeskipamount
|
||||
\hugeskipamount=24pt plus8pt minus8pt
|
||||
\def\hugeskip{\vskip\hugeskipamount}
|
||||
|
||||
|
||||
%% penalty before large blocks
|
||||
\def\preblock{
|
||||
\penalty-500
|
||||
}
|
||||
|
||||
%% listparpenalty prevents page breaks before lists
|
||||
\newcount\prevparpenalty
|
||||
\def\listparpenalty{
|
||||
\prevparpenalty=\@beginparpenalty
|
||||
\@beginparpenalty=10000
|
||||
}
|
||||
%% back to previous value
|
||||
\def\unlistparpenalty{
|
||||
\@beginparpenalty=\prevparpenalty
|
||||
}
|
||||
|
||||
|
||||
%% array spanning the entire line
|
||||
\def\largearray{\begin{array}{@{}>{\displaystyle}l@{}}\hphantom{\hspace{\textwidth}}\\[-.5cm]}
|
||||
\def\endlargearray{\end{array}}
|
||||
|
||||
Loading…
Reference in New Issue