Correct typos

Benfatto_Gallavotti_Jauslin_2015.tex

@@ -55,7 +55,7 @@
 
 
 \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''  with {\it no} interactions. 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{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).
@@ -171,7 +171,7 @@ g_{\varphi,\alpha}(t-t'):=&
 \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}
+g_{\varphi,\alpha}(\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'' $k_0= \frac\pi{\beta} +\frac{2\pi}\beta n_0$, $n_0\in\mathbb Z$, $|n_0|\le\frac12\beta$, and $k$ is the linear momentum $k=\frac{2\pi}L n$, $n\in [-L/2,L/2-1]\cap\mathbb Z$.
 \medskip
 
@@ -248,7 +248,7 @@ 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 $m\le 0$, fields $\psi^{[m]},\varphi^{[m]}$ that are separated in time by more than $2^{-m}$ can be regarded as (almost) independent.
+\indent Notice that the functions $g_\psi^{[0]}(0,\tau),g_\varphi^{[0]}(\tau)$ 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 express the quantities in~(\ref{eqavggrass}) inductively (see~(\ref{eqeffpotrec})). For instance the partition function $Z$ is given by
@@ -292,7 +292,7 @@ for $m< 0$ and similarly for $m=0$. Thus $\Delta_{-}$ is the lower half of $\Del
 :=\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{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
+in which $a$ and $b$ are constants. 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
@@ -331,7 +331,7 @@ V(\psi,\varphi)=&
 in which $\psi^\pm_\alpha(0,t)$ and $\varphi^\pm_\alpha(t)$ are now defined in~(\ref{eqfieldhier}).
 \medskip
 
-\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}).
+\indent Note that since the model defined above only involves fields 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 in section~\ref{secintroduction}.
@@ -382,9 +382,9 @@ B^{[\le m]j}_\eta(\Delta):=&\sum_{(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}
 
 \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
+\indent We then compute $V^{[m-1]}$ using~(\ref{eqeffpothier}) and show that 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}
-\left<\,e^{\sum_{\Delta} \sum_{n,\eta}\alpha_n^{[m]}O_{n,\Delta}^{[\le m]}}\,\right>_m =\prod_{\Delta}\left<\,e^{\sum_{n,\eta} \alpha_n^{[m]} O_{n,\eta}^{[\le m]}(\Delta)}\,\right>_m
+\left<\,e^{\sum_{\Delta} \sum_{n,\eta}\alpha_n^{[m]}O_{n,\Delta}^{[\le m]}}\,\right>_m =\prod_{\Delta}\left<\,e^{\sum_{n,\eta} \alpha_n^{[m]} O_{n,\eta}^{[\le m]}(\Delta)}\,\right>_m.
 \label{eqhierfactDelta}\end{equation}
 
 \indent In addition, we rewrite
@@ -427,7 +427,7 @@ c^{[m]}=-2^{N_\beta+m}\log(C^{[m]})
 which is well defined: it follows from~(\ref{eqhierbeta}) that $C^{[m]}\ge1$.
 \medskip
 
-\indent The dynamical system defined by the map $\mathcal R$ in~(\ref{eqhierbeta}) admits a few non trivial fixed points. A numerical analysis shows that if the initial data $\lambda_0\equiv\alpha_0$ is small and $<0$ the flow converges to a fixed point $\bm\ell^*$
+\indent The dynamical system defined by the map $\mathcal R$ in~(\ref{eqhierbeta}) admits a few non trivial fixed points. A numerical analysis shows that, if the initial data $\lambda_0\equiv\alpha_0$ is small and $<0$, then the flow converges to a fixed point $\bm\ell^*$
 \begin{equation}
 \ell^*_0=-x_0\frac{1+5x_0}{1-4x_0},\quad \ell^*_1=\frac{x_0}3,\quad
 \ell^*_2=\frac{1}3, \quad \ell^*_3=\frac{x_0}{18}
@@ -467,7 +467,7 @@ $f_+$ (see appendix~\ref{appfixed}-{\bf\ref{ptfixedreduced}}).
 \indent We introduce a magnetic field of amplitude $h\in R$ and direction $\bm\omega\in\mathcal S_2$ (in which $\mathcal S_2$ denotes the $2$-sphere) acting on the impurity. As a consequence, the potential $V$ becomes
 \begin{equation}
 V(\psi,\varphi)=V_0(\psi,\varphi)
--h\sum_{j\in\{1,2,3\}} \int dt(\varphi_{\alpha}^+(t)\sigma_{\alpha,\alpha'}^j\varphi_{\alpha'}^-(t))\, \omega_j
+-h\sum_{\displaystyle\mathop{\scriptstyle(\alpha,\alpha')\in\{\uparrow,\downarrow\}^2}_{j\in\{1,2,3\}}} \int dt(\varphi_{\alpha}^+(t)\sigma_{\alpha,\alpha'}^j\varphi_{\alpha'}^-(t))\, \omega_j
 \label{eqpoth}\end{equation}
 
 \indent The corresponding partition function is denoted by $Z_h:=\left<\,e^{-V_h}\,\right>$ and the free energy of the system by $f_h:=-\beta^{-1}\log Z_h$. The {\it impurity susceptibility} is then defined as
@@ -510,7 +510,7 @@ and, using the lemma~\ref{lemmaO}, we rewrite
 where $\ell_{n,h}^{[m]}$ is related to $\alpha_{n,h}^{[m]}$ by~(\ref{eqalphaellexprh}). Inserting~(\ref{eqalphatoellh}) into~(\ref{eqexpdexph}) the average is evaluated, although the computation is even longer than that in section~\ref{secbetapart}, but can be performed easily using a computer (see appendix~\ref{appmeankondo}). The result of the computation is a map $\widetilde{\mathcal R}$ which maps $\ell_{n,h}^{[m]}$ to $\ell_{n,h}^{[m-1]}$, as well as the expression for the constant $C_h^{[m]}$. Their explicit expression is somewhat long, and is deferred to~(\ref{eqbetasusc}).
 \medskip
 
-\indent By~(\ref{eqhiercst}), we rewrite~(\ref{eqsuscdef}) as
+\indent By~(\ref{eqhiercst}) and~(\ref{eqfreeench}), we rewrite~(\ref{eqsuscdef}) as
 \begin{equation}
 \chi(h,\beta)=\sum_{m=-N_\beta}^02^m\Big(\frac{\partial_h^2 C_h^{[m]}}{C_h^{[m]}}-\frac{(\partial_h C_h^{[m]})^2}{(C_h^{[m]})^2}\Big).
 \label{eqhiersusc}\end{equation}
@@ -555,7 +555,7 @@ n_2(\lambda_0)=c_2|\log_2|\lambda_0||+O(1),\quad c_2\approx2.
 
 \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 $\beta\to\infty$.
+\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 $\bm\ell_+^*$ and the susceptibility diverges as $\beta\to\infty$.
 
 \begin{figure}
 \hfil\includegraphics[width=250pt]{Figs/susc_plot_temp.pdf}\par\penalty10000
@@ -681,7 +681,7 @@ C^{[m]}=&1+ 2\ell_0^2+(\ell_0+\ell_6)^2 +9\ell_1^2 +9\ell_2^2 +324\ell_3^2 +\fra
 \preblock
 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:
+\indent The vector $\bm\ell$ is related to $\bm\alpha$ via the following map:
 \preblock
 \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
 \ell_0=&\alpha_0,\quad \ell_1=\alpha_1+\frac1{12}\alpha_4^2,\quad \ell_2=\alpha_2+\frac1{12}\alpha_5^2\\[0.3cm]
@@ -762,7 +762,7 @@ which we inject into~(\ref{eqfixedltred}) to find that $\ell_0<0$ and
 \label{eqfixedlo}\end{equation}
 Finally, we notice that $\frac1{12}$ is a solution of~(\ref{eqfixedlo}), which implies that
 \begin{equation}
-4-19(3\ell_1)-22(3\ell_1)^2-107(3\ell_1)^3
+4-19(3\ell_1)-22(3\ell_1)^2-107(3\ell_1)^3=0
 \label{eqfixedlot}\end{equation}
 which has a unique real solution. Finally, we find that if $\ell_1$ satisfies~(\ref{eqfixedlot}), then
 \begin{equation}
@@ -811,9 +811,9 @@ 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{ABe71}], 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 $H_h=H_0 -h\sigma_\nu^z$, 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}).
+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 (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}
@@ -825,9 +825,9 @@ the partition function is equal to $Z_L^0\zeta_L$ in which $Z_L^0$ is the partit
 \log {\zeta_L (\beta,h)}=-\beta h
 +\frac1{2\pi i}\oint_C \log (1+{e^{-\beta z}} ) \Big[\frac{\partial_zF_L(z)}{F_L(z)} \Big]\,dz
 \label{eqZXY}\end{equation}
-where the contour $C$ is a closed curve which contains the zeros of $F_L(\zeta)$ ({\it e.g.}\ {for $L\to\infty$,} a curve around the real interval $[-1,\sqrt{1+4h^2}]$ if {$h<0$} and $[-\sqrt{1+4h^2},1]$ if {$h>0$}) but not around those of {$1+e^{-\beta\zeta}$} (which are on the imaginary axis and away from $0$ by at least $\frac\pi\beta$). In addition, it follows from a straightforward computation that $(F(z)-1)/h$ is equal to the analytical continuation of {$2 (z^2-1)^{-\frac12}$} from $(1,\infty)$ to $C\setminus[-1,1]$.
+where the contour $C$ is a closed curve in the complex plane which contains the zeros of $F_L(\zeta)$ ({\it e.g.}\ {for $L\to\infty$,} a curve around the real interval $[-1,\sqrt{1+4h^2}]$ if {$h<0$} and $[-\sqrt{1+4h^2},1]$ if {$h>0$}) but not around those of {$1+e^{-\beta\zeta}$} (which are on the imaginary axis and away from $0$ by at least $\frac\pi\beta$). In addition, it follows from a straightforward computation that $(F(z)-1)/h$ is equal to the analytical continuation of {$2 (z^2-1)^{-\frac12}$} from $(1,\infty)$ to $C\setminus[-1,1]$.
 
-\indent At fixed $\beta<\infty$ the partition function $\zeta_L(\beta,h)$ has a non extensive limit $\zeta(\beta,h)$ as $L\to\infty$; the $\zeta(\beta,h)$ and the susceptibility and magnetization values $m(\beta,h)$ and $\chi(\beta,h)$, are given {\it in the thermodynamic limit} by
+\indent At fixed $\beta<\infty$ the partition function $\zeta_L(\beta,h)$ has a non extensive limit $\zeta(\beta,h)$ as $L\to\infty$; $\zeta(\beta,h)$ and the susceptibility and magnetization values $m(\beta,h)$ and $\chi(\beta,h)$, are given {\it in the thermodynamic limit} by
 \begin{equation}\begin{array}{r@{\ }>{\displaystyle}l}
  \log \zeta(\beta,h)=&-\beta h {+\frac\beta{2\pi i}} \oint_C\frac{dz}{1+{e^{\beta z}}} \log(1 {+} \frac{2h}{(z^2-1)^{\frac12}})\\
 m(\beta,h)=&-1+\frac1{\pi i}\oint_C \frac1{1+{e^{\beta z}}} \frac{dz}{(z^2-1)^{\frac12}{+2h} }\\