% %\underline{version 5.0} \\ \vspace{0.5cm} Superconductivity and Density Wave in the Quasi-One-Dimensional Systems: Renormalization Group Study
KWRG_ver5.tex
\documentclass[twocolumn]{jpsj2}
%%%\documentclass[letterpaper]{jpsj2}
% 2002/12/16
%\usepackage[mtbold]{mathtime}

\def\runtitle{Superconductivity and the density wave in the q1d systems}
\def\runauthor{Yuki \textsc{Fuseya} and Yoshikazu \textsc{Suzumura}}

















\title{%
%\underline{\texttt{version 5.0}} \\ \vspace{0.5cm}
Superconductivity and Density Wave in the Quasi-One-Dimensional Systems: 
Renormalization Group Study
}

\author{%
Yuki \textsc{Fuseya}\thanks{E-mail address: fuseya@slab.phys.nagoya-u.ac.jp} and 
Yoshikazu \textsc{Suzumura}
}

\inst{%
Department of Physics, Nagoya University, Nagoya 464-8602
}

\recdate{\today}























\abst{% 
	The anisotropic superconductivity and the density wave have been investigated by applying
	the Kadanoff-Wilson renormalization group technique to the quasi-one-dimensional system
	with finite-range interactions.
%
	It is found that a temperature ($T$) dependence of response functions is 
	proportional to $\exp(1/T)$
	in a wide region of temperature even within the one-loop approximation.
%
	Transition temperatures are calculated
	to obtain the phase diagram of the quasi-one-dimensional system, 
	which is compared with that of the pure-one-dimensional system.
%	
	Next-nearest neighbor interactions ($V_2$) 
	induce large charge fluctuations, which suppress the $d_{x^2 -y^2}$-wave 
	singlet superconducting ($d$SS) state
	and enhance the $f$-wave triplet superconducting ($f$TS)
	state. 
%	
	From this effect, the transition temperature of $f$TS becomes comparable
	to that of $d$SS for large $V_2$,
	so that field-induced $f$-wave triplet pairing could be possible.
%	
	These features are discussed to comprehend the experiments on the (TMTSF)$_2$PF$_6$ salt.
%	
	
}

\kword{%
quasi-one dimension, renormalization group, anisotropic superconductivity,
triplet superconductivity, (TMTSF)$_2$PF$_6$
}

\begin{document}

\maketitle






































%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

	The mechanism of superconductivity in quasi-one-dimensional (q1d) systems
	has been attracting renewed interest since the appearance of
	the triplet superconductivity suggested in the
	experiments on the q1d organic conductor (TMTSF)$_2$PF$_6$.
	\cite{Hc2,Knight}
%	
	In the phase diagram of (TMTSF)$_2$PF$_6$\cite{Jerome}, 
	the superconducting phase is adjacent to the spin density wave (SDW) phase.
%	
	From the knowledge of the anisotropic superconductivity,\cite{MSV,SLH,Emery}
	which have been extensively enriched for two decades,
	the $d$-wave singlet superconductivity ($d$SS) is expected to occur 
	in such superconducting phase.
%	
	However, recent experiments on (TMTSF)$_2$PF$_6$ 
	indicate a counter example of this widespread knowledge.
%	
	The unsaturated magnitude of the upper critical field $H_{{\rm c}2}$\cite{Hc2}
	and the constant Knight shift thorough the superconducting transition temperature
	$T_{\rm c}$\cite{Knight}
	strongly suggest the formation of triplet pairing.
%	
	In addition, the SDW phase turned out to be an exotic one,
	in which the $2k_F$ charge density wave (CDW) coexists 
	with the $2k_F$-SDW as seen from the X-ray experiments.\cite{PR,Kagoshima}
%	
	
	The theoretical study of q1d systems
	is complicated due to its dimensionality originated from the anisotropy,
	where the effects of the one-dimensional (1d) fluctuation 
	are essential to understand its physics.
%	
	In the system with the 1d fluctuation,
	there are 
	equally divergent
	contributions from both Cooper (electron-electron)
	and Peierls (electron-hole) channels.
%
	However, the 1d fluctuation 
	cannot be evaluated correctly by the conventional method 
	such as
	the random phase approximation (RPA)\cite{Shimahara},
	the fluctuation exchange (FLEX) approximation\cite{KK,KAA}
	or the third-order perturbation theory,\cite{NY}
	which
	are powerful for the study of three- or 
	two-dimensional anisotropic superconductivity.
%
	On the other hand, 
	the renormalization group (RG) technique based on the g-ology
	is extremely useful for the treatment of the 1d fluctuation,
	whereas it is complicated to deal with anisotropic superconducting states,
	such as $d$SS.
%
	
	Recently, Duprat and Bourbonnais extended the technique of Kadanoff-Wilson
	RG for purely 1d systems
	to that for q1d systems.\cite{DB}
%	
	This technique enabled us to treat both 1d fluctuations and 
	various kinds of anisotropic 
	superconductivity.
%	
	Meanwhile, the essential role of charge fluctuation has been pointed out 
	phenomenologically.\cite{KAA,FOKM}
%	
	Quite recently, it is shown by the RPA\cite{TK} that 
	charge fluctuations induced by next-nearest repulsive interactions
	can give rise to the $f$-wave triplet superconductivity ($f$TS), 
	though the 1d fluctuation effects are still pushed aside.
%	
	
	The purpose of the present paper is to get a deep insight into
	the interplay of the superconductivity and the density waves in q1d systems
	in terms of the Kadanoff-Wilson RG technique developed for q1d system.
%	
	Especially, we shall focus on clarifying
	i) crossover from purely 1d case into q1d case,
	ii) effects of long-range Coulomb interactions
	and iii) microscopic mechanism of the q1d superconductivity,
	considering the effect of the 1d fluctuation.
%	
	
	In \S 2, we briefly describe the Kadanoff-Wilson RG technique.
%	
	Next, the method of the present work.
%	
	In \S 3,
	it is shown that the response functions
	exhibit a noticeable property being proportional to $\exp (1/T)$.
%	
	Phase diagrams for several magnitudes of {\it inter}chain hopping
	are demonstrated on the plane of {\it intra}chain interactions.
%	
	Transition temperatures as a function of the {\it inter}chain hopping, 
	are also presented.
%	
	Then a discussion of the mechanism of the q1d superconductivity
	is given.
%	
	Section 4 is devoted to conclusion and 
	a comparison of the present result and the experiments.


























%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Formulation}
\subsection{Model}

	We consider a system consisting of an array of $N_b$ chains of length $L$,
	which is described by the Hamiltonian,
%	
\begin{align}
	H &= \sum_{p, {\bf k}, \sigma}\xi ({\bf k})a^{\dagger}_{p{\bf k} \sigma}a_{p{\bf k} \sigma}
	 \nonumber \\
	& +\frac{\pi v_F}{LN_b}
	\sum_{{\bf k}_{i},{\bf q},\sigma}(g_2 
	\delta_{\sigma_1 \sigma_4}\delta_{\sigma_2 \sigma_3}
	-g_1 \delta_{\sigma_1 \sigma_3}\delta_{\sigma_2 \sigma_4}) \nonumber \\
	& \times a_{+, {\bf k}_1 + {\bf q} \sigma_1}^{\dagger}a_{-,{\bf k}_2 - {\bf q} \sigma_2}^{\dagger}
	a_{-, {\bf k}_2 \sigma_3}a_{+, {\bf k}_1 \sigma_4}.
	\label{Hamiltonian}
\end{align}
%
	The first term denotes the band energy with the q1d dispersion:
\begin{align}
	\xi ({\bf k})&= -2t_a \cos k_a -2t_b \cos k_b -\mu \nonumber\\
	&\simeq v_F (|k_a | -k_F^0 ) -2t_b \cos k_b + 2t_b' \cos 2k_b ,
	\label{dispersion}
\end{align}
%
	where the {\it intra}chain electron spectrum with the nearest {\it intra}chain hopping
	$t_a $ is 
	linearized at the Fermi points
	$\pm k_F^0 $ with the Fermi velocity $v_F$, and $\mu$ is the chemical potential.
%	
	In Eq. (\ref{Hamiltonian}), $a_{p{\bf k} \sigma }^{\dagger}$ ($a_{p{\bf k} \sigma}$)
	is a creation (an annihilation) operator close to 
	the right $ k_a =+k_F^0 $ $(p=+)$ and the left $k_a = -k_F^0 $ $(p=-)$ Fermi surface, respectively.
%	
	The second and third terms of Eq. (\ref{dispersion}) denote
	the nearest {\it inter}chain hopping $t_b$
	and the next-nearest {\it inter}chain hopping $t_b'$, respectively,
	where
\begin{align}
	t_b' =\frac{t_b ^2 \cos (k_F^0 )}{4t_a \sin^2 (k_F^0 )},
\end{align}
	is obtained as the nesting deviation.
%	
	For the interaction part, the second term of Eq. (\ref{Hamiltonian}), 
	we follow the standard notations of the g-ology,
	where $g_1$ corresponds to backward scattering and $g_2$ does to forward scattering.
%	
	(Note that $g_{1, 2}$ are normalized by $\pi v_F $ here.)
%	
	For the 1/4-filling case, 
	$g_1 $ and $g_2$ are given in terms of on-site $U$, nearest-neighbor $V_1$ 
	and next-nearest-neighbor $V_2$ interactions, as
\begin{align}
	\pi v_F g_1 &= U/2 - V_2 , \\
	\pi v_F g_2 &= U/2 +V_1 +V_2 .
\end{align}
%	
	In order to examine superconductivity and density wave
	in terms of RG technique,
	we utilize the path integral method\cite{DB},
	where the partition function $Z$ ($={\rm Tr}e ^{-\beta H}$) is given by
\begin{align}
	Z = \int \!\!\!\int \mathcal{D}\psi^* \mathcal{D}\psi
	\exp (S[\psi^* , \psi ]_\ell ),
\end{align}
%	
	and the effective action $S$
	is written in the form,
\begin{align}
	S[\psi^* , \psi ]_\ell &=S_0 [\psi^* , \psi ]_\ell 
	+ S_{\rm I} [\psi^* , \psi ]_\ell , 
	\label{action}\\
	S_0 [\psi^* , \psi ]_\ell 
	&=\sum_{p, \sigma , \{ \tilde{{\bf k}}\} ^*}[G_p^0 (\tilde{{\bf k}})]^{-1}
	\psi_{p, \sigma}^* (\tilde{{\bf k}})\psi_{p, \sigma} (\tilde{{\bf k}}), \\
	S_{\rm I} [\psi^* , \psi ]_\ell &= 
	-\pi v_F \sum_{\mu, \tilde{\bf Q}}J_{\mu}(k_{1b}, k_{2b}, q_b ; \ell )
	O_{\mu}^{{\rm P}*} (\tilde{\bf Q})O_{\mu}^{\rm P} (\tilde{\bf Q}), \label{Js}\\
%
	&= -\pi v_F \sum_{\mu, \tilde{\bf q}}W_{\mu}(k_{1b}, k_{2b}, q_b ; \ell )
	O_{\bar{\mu}}^{{\rm C}*} (\tilde{\bf q})O_{\bar{\mu}}^{\rm C} (\tilde{\bf q}),
	\label{Ws}
\end{align}
%	
	where the renormalization of 
	the bandwidth is defined as 
\begin{align}
	E_0 (\ell ) = E_0 \exp (-\ell ),
\end{align}
	with the bandwidth cut-off $E_0$.
%
	The quantities $\psi (\tilde{\bf k})$ and $\psi^* (\tilde{\bf k})$
	are the Grassmann fields
	and $\tilde{\bf k}=({\bf k}, \omega_n )$.
%	
	$\{ {\bf k}\}^* $ denotes the renormalized momentum space.
%	
	The operator $O_{\mu}^{{\rm P}}$ denotes Peierls fields
\begin{align}
	O_{\mu}^{{\rm P}}(\tilde{\bf Q})=
	\sqrt{\frac{T}{LN_b}}\sum_{\alpha , \beta , \{ \tilde{\bf k}\} ^*}
	\psi_{-, \alpha}^* (\tilde{\bf k}-\tilde{\bf Q})
	\sigma_{\mu}^{\alpha \beta}
	\psi_{+, \beta} (\tilde{\bf k}),
\end{align}
	describing the CDW ($\mu = 0$) and the SDW ($\mu = 1, 2, 3$) channels,
	where $\tilde{\bf Q}= (2k_F^0 + Q_0 , Q_b , \omega_m = 0, \pm 2\pi T, \cdots)$
	($Q_0\ll 2k_F^0$).
%
	The operator $O_{\bar{\mu}}^{{\rm C}}$ is Cooper fields
\begin{align}
	O_{\mu}^{{\rm C}}(\tilde{\bf q})=
	\sqrt{\frac{T}{LN_b}}\sum_{\alpha , \beta , \{ \tilde{\bf k}\} ^*}
	\alpha \psi_{-, -\alpha}^* (-\tilde{\bf k}+\tilde{\bf q})
	\sigma_{\bar{\mu}}^{\alpha \beta}
	\psi_{+, \beta} (\tilde{\bf k}),
\end{align}
	for singlet ($\bar{\mu}=0$) and triplet ($\bar{\mu}=1, 2, 3$) pairing,
	where $\tilde{\bf q}= (q_0 , q_b , \omega_m )$
	($q_0\ll 2k_F^0$).
%
	The couplings $J_{\mu}$ and $W_{\bar{\mu}}$ 
	correspond to that of the density wave and the superconductivity respectively,
	and are
	related to $g_{1, 2}$ as
	$J_{\mu =0}=g_1 -g_2 /2$, 
	$J_{\mu \neq 0}= -g_2 /2$, 
	$W_{\bar{\mu} =0}=-(g_1 + g_2 )/2$ and
	$W_{\bar{\mu} \neq 0}=(g_1 -g_2 )/2$, respectively.
%
	
%	
	
%	
	

%===============================================================================
\subsection{Renormalization of couplings $J_{\mu }(\ell )$, $W_{\bar{\mu }}(\ell )$}
%
	First we apply the RG technique to coupling constants for 
	the Peierls and the Cooper channel.
%
	By taking contraction of each channel, the one-loop flow equation 
	are obtained as\cite{DB}
%	
\begin{align}
	\frac{d}{d\ell}J_{\mu}& (Q_b -k_b , k_b ; \ell ) \nonumber \\
	&= \frac{1}{N_b}\sum _{\bar{\mu}, k_b'}
	c_{\mu , \bar{\mu}} W_{\bar{\mu}}(Q_b -k_b , k_b' ; \ell ) %\nonumber \\
	 W_{\bar{\mu}}(k_b' , k_b ; \ell ) I_{\rm C}(\ell ) \nonumber \\
	&-J_{\mu} (Q_b -k_b , k_b ; \ell )\frac{1}{N_b}\sum _{k_b'}
	J_{\mu} (Q_b -k_b' , k_b' ; \ell )I_{\rm P}(Q_b , k_b' ; \ell ),
	\label{q1dflow}
\end{align}
%
	where $c_{0,0}=-1/2, c_{0, \bar{\mu}\neq 0}=1/2, c_{\mu \neq 0, 0}=1/2$ 
	and $c_{\mu \neq 0, \bar{\mu}\neq 0}=1/6$.
%	
	The first term of the right-hand side of Eq. (\ref{q1dflow})
	corresponds to the contraction in the Cooper channel,
	where $I_{\rm C} (\ell ) d\ell =\tanh [ E_0 (\ell )/4T] d\ell$
	is the outer-shell integration.
%	
	The second term corresponds to the contraction in the Peierls channel,
	with the outer-shell integration
\begin{align}
	I_{\rm P}(Q_b , k_{b1}, \ell ) d\ell &=\frac{E_0 (\ell )d \ell}{4}\sum_{\lambda = \pm 1}
	\frac{1}{E_0 (\ell ) + \lambda A(k_{b1}, Q_b )} \nonumber \\
	&\times \biggl[ \tanh \frac{E_0 (\ell )}{4T} 
	+\tanh \frac{1}{2T}\bigl\{ \frac{E_0 (\ell )}{2}
	+\lambda A(k_{b1}, Q_b ) \bigr\} \biggr], \\
%
	A(k_b , Q_b )&=2t_b [ \cos k_b + \cos (k_b + Q_b ) ] \nonumber \\
					&+ 2t_b ' [ \cos 2k_b + \cos 2(k_b + Q_b )].
\end{align}
%
	For $t_b = 0$, 
	the flow equation (\ref{q1dflow}) yields the one-loop 
	1d flow equations.\cite{BC,Solyom}	
%
	Note that $I_{\rm C}(\ell )$ is a constant while $I_{\rm P}(Q_b , k_{b}, \ell )$
	suppressed by $t_b '$ depending on the wave vector of particles.
%	
	Thus the Cooper channel is always singular 
	(i.e., does not depend on the shape of the Fermi surface)
	while the Peierls channel is suppressed by the nesting deviation.
%
	The renormalized Cooper couplings are obtained from
	the renormalized Peierls couplings through the relation
\begin{align}
	W_{\bar{\mu}=0}(Q_b - k_{b}, k_b ; \ell )&= 
	-\frac{1}{2}J_0 (Q_b - k_{b}, k_b ; \ell ) \nonumber \\
	&\hspace{0.5cm}+ \frac{3}{2}J_{\mu \neq 0}(Q_b - k_{b}, k_b ; \ell ), \label{WsJ}\\
	W_{\bar{\mu}\neq 0}(Q_b - k_{b}, k_b ; \ell )
	&= \frac{1}{2}J_0 (Q_b - k_{b}, k_b ; \ell ) \nonumber \\
	&\hspace{0.5cm}+ \frac{1}{2}J_{\mu \neq 0}(Q_b - k_{b}, k_b ; \ell ), \label{WtJ}
\end{align}
	which is obtained from Eqs. (\ref{Js}) and (\ref{Ws}) straightforwardly.
%	
	
	Following the idea of the dimensional crossover theory\cite{BC,KY},
	we adopt two-step RG procedure.
%	
	First we apply the RG technique of 1d\cite{BC} (1d RG) to the regime
	above the crossover temperature $T_{\rm cross}$,
	and then that of q1d\cite{DB} (q1d RG) to the region below $T_{\rm cross}$ 
	as the second step.
%	
	For the first step, we adopt
	the two-loop flow equations of 1d\cite{BC}
\begin{align}
	\frac{d g_1 (\ell)}{d\ell}&=-g_1 (\ell )^2 -\frac{1}{2}g_1 (\ell )^3, \\
	\frac{dG(\ell )}{d\ell} &=0,
\end{align}
%
	where $G(\ell )\equiv g_1 (\ell ) -2g_2 (\ell )$.
%	
	During the 1d RG, the {\it inter}chain coupling $j_{\mu}$ due to 
	the pair hopping is produced as follows:
\begin{align}
	j_{\mu }(\ell )&= z(\ell )^{-2}f_{\mu } (\ell ), \\
	\frac{d f_{\mu=0} (\ell )}{d \ell } &= \biggl[ 
	\frac{(g_2 (\ell ) -2g_1 (\ell ) ) z(\ell )t_b }{E_0 (\ell )} \biggr] ^2 , \\
	\frac{d f_{\mu \neq 0} (\ell )}{d \ell } &= \biggl[ 
	\frac{g_2 (\ell ) z(\ell )t_b }{E_0 (\ell )} \biggr] ^2 ,
\end{align}
%	
	where flow equations for $t_b $ and $z$ 
	(corresponding to self energy) are 
\begin{align}
	\frac{d}{d\ell}\ln t_b (\ell )&=
	1-\frac{1}{4}\bigl[ g_1 (\ell )^2 + g_2 (\ell )^2 -g_1 (\ell )g_2 (\ell )
	\bigr] , \\
	\frac{d}{d\ell}\ln z (\ell )&=
	-\frac{1}{4}\bigl[ g_1 (\ell )^2 + g_2 (\ell )^2 -g_1 (\ell )g_2 (\ell ) \bigr] ,
\end{align}
%
	respectively.
%
	The dimensional crossover temperature is given
	by $T_{\rm cross}=E_0 (\ell_{\rm cross})/2$, 
	where $\ell _{\rm cross}$ is defined as 
	$E_{0}(\ell_{\rm cross})=t_b (\ell_{\rm cross})$.
%	
	Thus the initial couplings for the second step, i.e., q1d RG,
	are given by a combination of $g_{\mu}$ and $j_{\mu} \cos Q_b$ as
\begin{align}
	J_{\mu =0} (Q_b , \ell_{\rm cross}) &=-\frac{1}{2}\bigl[ g_2 (\ell_{\rm cross})
	-2g_1 (\ell_{\rm cross}) \bigr] + j_{0} \cos Q_b ,\\
	J_{\mu \neq 0} (Q_b , \ell_{\rm cross}) &=-\frac{1}{2} g_2 (\ell_{\rm cross})
	 + j_{\mu} \cos Q_b .
\end{align}
%	































%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Response functions $\chi $}
%
	In order to calculate the response function,
	we add the source field to Eq. (\ref{action}),
	which is given by\cite{DB} 
\begin{align}
	S_h [\psi^* , \psi ]
	&=\sum_{\mu} z_{\mu} [O_{\mu}^{\rm P *}({\bf Q}_{\rm P})h_{\mu}({\bf Q}_{\rm P})
	+ {\rm c. c}] \nonumber \\
	& +\sum_{\bar{\mu}, \{ \tilde{\bf k}\} ^* }z_{\bar{\mu}}^{(n)} 
	[O_{\bar{\mu}}^{\rm C *}(\tilde{\bf k})h_{\bar{\mu}}^{(n)}(k_b )
	+ {\rm c. c}] ,
\end{align}
%
	where ${\bf Q}_{\rm P}=(2k_F^0 , \pi)$,
	and $h_{\bar{\mu}}^{(n)}(k_b )=h_{\bar{\mu}}^{(0)}\cos (nk_b )$
	or $h_{\bar{\mu}}^{(n)}(k_b )=h_{\bar{\mu}}^{(0)}\sin (nk_b )$.
%	
	The corresponding pairing symmetries are discussed later.
%	
	Here $z_{\mu (\bar{\mu})}^{(n)}$ corresponds to vertex corrections,
	and the value $[z_{\mu (\bar{\mu})}^{(n)}]^2$ is equal to the well-known auxiliary
	susceptibility $\bar{\chi}$.\cite{Solyom}
%	
	Within the linear response theory,
	only the linear and the quadratic term contribute to the action,
	so that the action at $\ell$ has the form
\begin{align}
	S[\psi ^* , \psi , h^* , h]_{\ell} &= S[\psi ^* , \psi ]_{\ell}
	+\sum_{\mu} z_{\mu}(\ell ) [O_{\mu}^{\rm P *}({\bf Q}_{\rm P})h_{\mu}({\bf Q}_{\rm P})
	+ {\rm c. c}] \nonumber \\
%	
	&\hspace{-15mm} +\sum_{\bar{\mu}, \{ \tilde{\bf k}\} ^* }z_{\bar{\mu}}^{(n)}(\ell )
	[O_{\bar{\mu}}^{\rm C *}(\tilde{\bf k})h_{\bar{\mu}}^{(n)}(k_b )
	+ {\rm c. c}] \nonumber \\
%	
	&\hspace{-15mm}+\sum_{\mu} \chi _{\mu}(\ell )h_{\mu}^*({\bf Q}_{\rm P})h_{\mu}({\bf Q}_{\rm P})
%	
	+\sum_{\bar{\mu}, k_b }
	\chi_{\bar{\mu}}^{(n)}(\ell )h_{\bar{\mu}}^{(n) *} (k_b )h_{\bar{\mu}}^{(n)} (k_b ).
\end{align}
%	
	Flow equations of each $z_{\mu (\bar{\mu})}^{(n)}$ are given by
%	
\begin{align}
	\frac{d}{d\ell}\ln z_{\mu}(\ell )=-\frac{1}{N_b}\sum_{k_b}
	J_{\mu } (\pi -k_b , k_b ; \ell ) I_{\rm P}(\pi , k_b ; \ell ), 
\end{align}
%	
	for the Peierls channel and 
%	
\begin{subequations}
\begin{align}
	\frac{d}{d\ell} \ln z_{\bar{\mu}=0}^{(n)}(\ell )
	&= \frac{1}{2}c_{\bar{\mu}=0}^{(n)} (\ell ) I_{\rm C}(\ell ), \\
	\frac{d}{d\ell}\ln z_{\bar{\mu}\neq 0}^{(n)}(\ell )
	&= \frac{1}{2}c_{\bar{\mu}\neq 0}^{(n)} (\ell ) I_{\rm C}(\ell ),
\end{align}
\end{subequations}
%
	for the Cooper channel.
%
	Then the static response functions are obtained as
%
\begin{align}
	\chi _{\mu (\bar{\mu})}^{(n)}=(\pi v_F )^{-1}
	\int_0^{\ell} \bigl[ z_{\mu (\bar{\mu})}^{(n)} (\ell ')\bigr] ^2 d\ell '.
	\label{response}
\end{align}
%
	The coefficients $c_{\bar{\mu}}^{(n)}$ represent the Fourier coefficients
	of the coupling in Cooper channel $a_{\bar{\mu}}^{(n)}$
	and $b_{\bar{\mu}}^{(n)}$, which are defined as 
%
\begin{align}
	W_{\bar{\mu}}(k_b , k_b ' ; \ell )
	=&a_{\bar{\mu}}^{(0)} (\ell )
	+\sum_{n>0}\bigl[ 
	a_{\bar{\mu}}^{(n)} (\ell )\cos (nk_b ) \cos (nk_b ' ) \nonumber \\
	&+b_{\bar{\mu}}^{(n)} (\ell )\sin (nk_b ) \sin (nk_b ' ) \bigr] .
\end{align}
%
%	(The off-diagonal elements of the coefficients are essentially zero.)
%	
	The coefficients $a_{\bar{\mu}}^{(n)}$ ($b_{\bar{\mu}}^{(n)}$) are the even 
	(odd) components with respect to the {\it inter}chain direction $k_b , k_b'$.
%	
	The parity of the gap cannot be classified only by $k_b$-dependence.
%	
	It has to be classified by the symmetry of both $k_a$- and 
	$k_b$-dependences of the gap function.
%	
	For singlet channel, considering only the even-frequency pairing, 
	the parity of the gap is even,
	so that 
%
	\begin{description}
		\item[\hspace{10mm}A)] $a_{\bar{\mu}=0}^{(n)}$: even-$k_a$ and even-$k_b$ 
		($s$, $d_{x^2 -y^2}$, ...)
%		
		\item[\hspace{10mm}B)] $b_{\bar{\mu}=0}^{(n)}$: odd-$k_a$ and odd-$k_b$
		($d_{xy}$, $g$, ...)
	\end{description}
%	
	are possible.
%	
	Similarly for triplet channel, the parity of the gap is odd,
	so that 
%
	\begin{description}
		\item[\hspace{10mm}C)] $a_{\bar{\mu}\neq 0}^{(n)}$: odd-$k_a$ and even-$k_b$ 
		($p_x$, $f_x$, ...)
%		
		\item[\hspace{10mm}D)] $b_{\bar{\mu}\neq 0}^{(n)}$: even-$k_a$ and odd-$k_b$
		($p_y$, $f_y$, ...)
	\end{description}
%	
	are possible.
%	
	Briefly speaking, the symmetry of $k_b$-dependence is classified
	by $a_{\bar{\mu}}^{(n)}$ or $b_{\bar{\mu}}^{(n)}$,
	and that of
	$k_a$-dependence is automatically determined once we choose
	the spin-symmetry $\bar{\mu}$.
%	
	
	In this paper, we consider the following 8 symmetries
	for order parameters of superconductivity.
%	
	For singlet channel, we calculate the response functions for
	$a_{\bar{\mu}=0}^{(0)}$, $a_{\bar{\mu}=0}^{(1)}$, $b_{\bar{\mu}=0}^{(1)}$
	and $b_{\bar{\mu}=0}^{(2)}$, which can be recognized as 
	$s$-, $d_{x^2 -y^2}$-, $d_{xy}$- and $g$-wave, respectively.\cite{symmetry}
%	
	For triplet channel, the response functions for $a_{\bar{\mu}\neq 0}^{(0)}$, 
	$a_{\bar{\mu}\neq 0}^{(1)}$,
	$b_{\bar{\mu}\neq 0}^{(1)}$
	and $b_{\bar{\mu}\neq 0}^{(2)}$, which can be recognized as 
	$p$-, $f_x$-, $p_y$- and $f_y$-wave respectively,
	are calculated.
%	
	(See Fig. \ref{spdfg} for the graphical representation of 
	each symmetry.)
%	
	We examined whole 8 symmetries,
	and found the clear instability only for
	$p_x$-, $d_{x^2 -y^2}$- and $f_x$-wave.
%	
	Therefore, from now on, we simply call them $p$-, $d$-, $f$-wave, respectively.
%
\begin{figure}[t]
\begin{center}
\includegraphics[width=8cm]{spdfg.eps}
\end{center}
\caption{
	Types of the gap symmetry. The upper panels corresponds to the singlet
	pairing, whose symmetry belongs to $s$-, $d_{x^2 -y^2}$-, 
	$d_{xy}$-, 	$g$-wave respectively (from left to right).
%	
	The lower panels corresponds to the triplet
	pairing, whose symmetry belongs to $p_x$-, $f_x$-, $p_{y}$-, 
	$f_y$-wave 	respectively (from left to right).
%	
	The coefficients ($a_{\bar{\mu}}^{(n)}, b_{\bar{\mu}}^{(n)}$) above the panels denote
	the corresponding Fourier coefficients.
%
	The arrows indicate the inversion (${\bf k} \to -{\bf k}$), 
	which does not (does) change the sign of 
	the gap function for singlet (triplet) pairing.
	}
\label{spdfg}
\end{figure}






































%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Superconductivity vs. Density Wave}
\subsection{Temperature dependence of $\chi (T)$ and $T_{\rm c}$}

%\begin{minipage}
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=8cm]{res.eps}
\end{center}
\caption{Response function of (a) SDW and (b) $d$SS.
The dotted straight line is the line of $\tilde{\chi}_{\rm prop}$
drawn by fitting with $\tilde{\chi}$
at the inflection point.
The insets show $d (\ln [\pi v_F \chi ])/d(1/T)$ for the respective response function.}
\label{res}
\end{figure}
%	
	Based on the RG method derived in previous section,
	we examine the possible states in the present q1d system.
%	
	First we calculate the response functions obtained from Eq. (\ref{response}),
	which are 
	shown in Fig. \ref{res} for the SDW (a) and the $d$SS (b).
%	
	Each response function (also for other response functions) shows
	the similarity in behavior.
%	
	The important point to note is that the response function $\chi (T)$ 
	becomes proportional
	to $\exp (1/T)$ (i.e., $\ln \chi (T) \propto 1/T $)
	in the wide region of intermediate temperatures.
%	
	From the theorems of Mermin and Wagner\cite{MW},
	and Hohenberg\cite{Hohenberg},
	there is 
	no magnetic or superconducting long-range order 
	at any finite temperature in one and two dimensions.
%	
	Generally, the response functions in two dimension have the temperature dependence
	$\chi (T) \propto \exp (1/T)$ for both magnetic\cite{KT}
	and superconducting\cite{TMY,private,Kosuge} cases,
	when we do not consider the Kosterlitz-Thouless transition.\cite{KT2}
%	
	From these points, one may say that
	the present method gives a rigorous solution in such temperatures
	as $\chi (T) \propto \exp (1/T)$
	in spite of the one-loop approximation.
%	
	
	In the very low temperatures, on the other hand,
	$\chi (T)$ strays from the behavior of $\exp (1/T)$
	and diverges at finite temperature
	because of the divergence of the coupling $J$ or $W$.
%	
	Therefore, it is appropriate to recognize 
	the response function in the intermediate temperatures
	as a proper response function, since it is proportional to $\exp (1/T)$.
%	
	In the region, where the obtained response function 
	does not show the behavior of $\exp (1/T)$, the present RG technique 
	does not work well due to the relevant (large enough) couplings.
%	
	In the sense, the characteristic temperature, $T^*$,
	for the lower bound of the present RG technique, 
	may be taken by a condition that 
	$d (\ln [\pi v_F \chi ])/dT^{-1}$ takes a minimum, i.e., the inflection point.
%	
	In order to obtain the proper response function,
	we estimate by using an extrapolation formula
\begin{align}
	\tilde{\chi}_{\rm prop} (T) =\alpha \exp ^{\beta /T},
\end{align}
	where $\tilde{\chi}=\pi v_F \chi$.
%
	The parameter $\beta$ 
	is defined as $\beta \equiv d (\ln [\tilde{\chi} ])/dT^{-1}|_{T=T^*}$,
	which is of the order of the transition temperature
	obtained by the mean field approximation.
%
	The parameter $\alpha$ is defined as
	$\alpha \equiv \tilde{\chi} (T^*)e^{-\beta /T^*}$,
	namely, $\tilde{\chi}_{\rm prop} (T)$ is the tangential line of 
	$\ln \tilde{\chi}(1/T)$ at $T=T^*$.
%	
	Assuming that the practical transition temperature $T_{\rm c}$ is lead by
	the three-dimensionality,
	we use the conventional RPA (for the inter-plane coupling),
	$g_{\rm 3d}\tilde{\chi}_{\rm prop} (T_{\rm c})=1$,
	which yields the form 
\begin{align}
	T_{\rm c} = -\frac{\beta}{\ln (\alpha g_{\rm 3d})},
	\label{Tccondition}
\end{align}
	where $g_{\rm 3d}$ is an inter-plane coupling.
%
	Of course, we need to discuss
	the problem in the Kosterlitz-Thouless (KT) context
	in two-dimensional systems.\cite{KT2}
%	
	In the case of $T_{\rm c}\ll E_F$, however,
	the difference between $T_{\rm c}$ and 
	the KT transition temperature $T_{\rm c}^{\rm KT}$ is small
	compared to $T_{\rm c}$, i.e., $|T_{\rm c}-T_{\rm c}^{\rm KT}|/T_{\rm c}
	\sim T_{\rm c}/E_F$.\cite{Miyake}
%	
	Consequently, $T_{\rm c}$ determined by the present method
	almost corresponds to the actual transition temperature $T_{\rm c}^{\rm KT}$.
	
%	
	In this paper, $T_{\rm c}$ is calculated by taking
	$\mathcal{O}(g_{\rm 3d}) \sim 10^{-3}$.
%	
	We calculated $32\times 32\times 2$ simultaneous differential equations,
	i.e., a 32-chains system with 64 points on the Fermi surface.
%	
	The initial bandwidth is set to be $E_0 =5500K$ 
	($t_a = E_0/(2\sqrt{2}k_F^0)\simeq 2500$K ),
	which is consistent with the bandwidth of (TMTSF)$_2$PF$_6$.\cite{JTB,Grant}

































%==============================================================================
\subsection{$g_2$-$g_1$ phase diagram}
\begin{figure}[t]
\begin{center}
\includegraphics[width=9cm]{g1g2.eps}
\end{center}
\caption{
$g_2$-$g_1$ phase diagrams for $t_b =100, 200, 300, 400$ and 500K.
The transition temperatures for the CDW, SDW, $d$SS, $p$TS and $f$TS
are expressed as the tones of green, red, blue, yellow and orange, respectively.
The dashed-line indicates the phase boundary between the $p$TS and the SDW
of the g-ology.
}
\label{g1g2}
\end{figure}

%
	First, we show phase diagrams on the $g_2$-$g_1$ plane for several choices of 
	$t_b$'s
	in order to understand the variation of the phase diagram
	from pure 1d case to q1d case.
%
	For repulsive interactions $g_{1, 2}>0$,
	the SDW is dominant for $g_1 <2g_2$
	and the $p$-wave triplet superconductivity ($p$TS)
	is dominant for $g_1 > 2g_2$ as found in the g-ology.\cite{Solyom}
%	
	Figure \ref{g1g2} shows the phase diagram for $t_b =$ 100, 200, 300, 400, 500K.
%
	The dashed-lines in Fig. \ref{g1g2} denote $g_1$=$2g_2$, 
	which correspond to the phase boundary between the $p$TS and the SDW
	of the g-ology.
%	
	
	The present results show that the transition temperatures of the $p$TS and the SDW
	are extremely low (numerically, below $10^{-2}$ K) near the phase boundary $g_1 =2g_2$.
%	
	Moving from that line, 
	the $p$TS and the SDW phase appears parallel to the $g_1$=$2g_2$-line.
%	
	The salient feature is
	that, in q1d, the $d$SS phase emerges between the $g_1$=$2g_2$-line and the SDW phase,
	and this $d$SS phase extends to the SDW phase
	also parallel to $g_1$=$2g_2$,
	as $t_b$ increases.
%	
	From these features,
	it turns out that the transition temperature 
	depends on the intensity of $|g_1 -2g_2|$,
	corresponding to the intensity of density fluctuations.
%	
	The nearer the parameters move to the line of $g_1$=$2g_2$, 
	the weaker the SDW instability becomes.
%	
	Thus the $d$SS appears
	instead of the SDW state due to the deviation of the 
	nesting condition.
%	
	Just on the line of $g_1$=0, the CDW ($f$TS) state
	has the same transition temperature as the SDW ($d$SS) state.
%	
	The phase boundary between the $d$SS and the normal state 
	swerves from the line parallel to $g_1 =2g_2$ for small $g_1$,
	while that between the $d$SS and the SDW is parallel to the $g_1$=$2g_1$-line.
%	
	This can be understood as the interference effect 
	of the charge fluctuations and the spin fluctuations,
	which shall be discussed later.






















%==============================================================================
\subsection{$t_b$-dependence of transition temperature}


	Next, we show $t_b$-dependence of the transition temperature 
	in Fig. \ref{Ttb}; 
	(a) the Hubbard ($V$=0) model with
	$g_1 =g_2 =0.88$ ($U \simeq 7.8 t_a , V_1 = V_2 =0.0$),
	and (b) the extended Hubbard model with large-$V$ given by
	$g_1 = 0.01, g_2 = 0.55$ ($U \simeq 1.3 t_a,
	V_1 \simeq 0.72 t_a$ and $V_2 \simeq 0.60 t_a $, if we assume 
	$V_1 = 1.2V_2$).	
%	
	For small $t_b$ ($\lesssim $300K), the transition temperature
	of SDW, $T_{\rm SDW}$, increases linearly.
%	
	More precisely,
	it is approximately given by $T_{\rm SDW}\sim t_b \exp (-2/g_2^*)$.
%	
	Here, $g_2^* /2 $ corresponds to the strength of the spin fluctuation,
	where the asterisk denotes its value at $T_{\rm cross}$.
%	
	Such increase cannot be obtained by the mean field theory
	or the RPA-type theory,
	and consistent with the previous 1d RG theory;\cite{SF,BC}
%	
	this feature originates from the effect of 1d fluctuation,
	i.e., the equally divergent contributions from 
	both the Cooper and the Peierls channel.
%	
	The transition temperature has a maximum followed by the decrease
	due to the nesting deviation.
%	
%	Since the {\it inter}chain hopping gives a maximum of $T_{\rm SDW}$
%	depends on $|g_1 -2g_2 |$.
%	
	When $T_{\rm SDW}$ is sufficiently suppressed due to the nesting deviation,
	the {\it hidden} superconductivity, $d$SS in the present case, emerges.
%	
	These features are seen for almost every couplings in the parameter region
	$0<g_1 < 2g_2$, corresponding to the SDW phase of the g-ology.
%	
\begin{figure}[tb]
\begin{center}
\includegraphics[width=8cm]{Ttb.eps}
\end{center}
\caption{Transition temperature as a function of the {\it inter}chain hopping $t_b$
for (a) the Hubbard ($V$=0) model $g_1 =g_2 =0.88$ ($U \simeq 7.8 t_a , V_1 = V_2 =0.0$),
and for (b) the large-$V$ model $g_1 = 0.01, g_2 = 0.55$ ($U \simeq 1.3 t_a,
V_1 \simeq 0.72 t_a , V_2 \simeq 0.60 t_a $).
The closed circles and triangles are the transition temperature of SDW and $d$SS,
respectively. The open circles and diamonds indicates the temperature
where the response functions of CDW and $f$TS satisfy the condition
(\ref{Tccondition}), respectively.}
\label{Ttb}
\end{figure}
	
%	
	
	There are some differences between the $V$=0
	and the large-$V$ model.
%	
	The most noticeable difference exists in the subdominant phase 
	for the large-$V$,
	which is shown by open squares (CDW) and diamonds ($f$TS) in Fig. \ref{Ttb} (b).
%	
	Here, the transition temperature for the subdominant state
	are also obtained from the condition (\ref{Tccondition}).
%	
	The transition temperature for the subdominant CDW ($f$TS)
	is exactly the same as that of SDW ($d$SS) when
	$g_1 =0$ ($U=2V_2$), i.e., just on the phase boundary between
	SDW and CDW of the g-ology.
%	
	The actual phase transition, of course, occurs 
	only for the most divergent response function,
	so that these subdominant phases are masked practically.
%	
	Therefore the subdominant transition temperature
	is better considered as an emergence of fluctuations.
%	
	In the case of large-$V$,
	not only the spin fluctuations but also the charge fluctuations
	develop, while only the spin fluctuations 
	develop for the $V$=0.
%	
	This enhancement of the charge fluctuation promotes the $f$TS state;
%	
	especially, $f$TS is realized for $g_1 = 0$.
%	
	
	Another significant difference is that the superconducting 
	transition temperature $T_{\rm c}$ for large-$V$ is much lower 
	than that for $V$=0,
	while $T_{\rm SDW}$ of both cases are almost the same.
%	
	This comes from the difference of the strength of the charge fluctuations.
%
	It is clear from the eqs. (\ref{WsJ}) and (\ref{WtJ}) 
	that the charge fluctuation $J_0$ suppresses the singlet channel
	$W_{0}$, and enhances the triplet channel $W_{\bar{\mu}\neq 0}$.
%	
	When $J_0 = J_{\mu \neq 0}$, the amplitude of the triplet channel
	becomes just the same as that of singlet one,
	which corresponds to the $g_1 =0$ ($U=2V_2$) case.
%	
	Thus the low $T_{\rm c}$ of the large-$V$ model also suggests
	the enhancement of the charge fluctuation,
	which promotes the $f$TS.
%	
	The $t_b$-dependence of $T_{\rm c}$ would be given by
	$T_{\rm c} \sim t_b \exp (-1/\tilde{g}_{\bar{\mu}}^*)$
	in the same way as the SDW.
%	
	(Here, $\tilde{g}_{\bar{\mu}}^*$ is slightly smaller than $g_{\bar{\mu}}^*$
	because of the nodes of the gap.)
%	
	Therefore, the increasing $T_{\rm c}$ for the Hubbard model
	is due to the increasing $t_b$.
%	
	The reason why $T_{\rm c}$ decreases for the large-$V$
	is the rapid decrease of the spin fluctuation due to the cancellation 
	with the charge fluctuation.
%	
	
	Note that the possible $f$TS state is a characteristic of 
	systems with open-Fermi-surface
	such as q1d.
%	
	On the other hand, in the case of the closed Fermi surface, 
	such as the square lattice,
	$f$TS cannot have the comparable transition temperature
	even for large-$V$,
	but $d_{xy}$-wave singlet superconductivity can
	become a dominant state.\cite{OAKA,KTOS}
%	
	This comes from a property of the open Fermi surface.
%	
	The number of nodes of $f$TS is 4 in the case of q1d open Fermi surface,
	which is the same as $d$SS.
%	
	Here, the number is 6 for the closed Fermi surface.
%	
	Hence, the q1d system possesses a property which is favorable for 
	the triplet pairing.
%	
	As far as the pairing symmetry is concerned,
	the present results is consistent with the RPA results.\cite{TK}
%	
	The RPA, however, does not take into account the 1d fluctuations,
	so that its $T_{\rm c}$ is much higher than the present one.
%
	



































%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}
	We have calculated the response functions and the transition temperatures 
	for the density wave (CDW and SDW) and 
	the superconductivity 
	($s$-, $p_{x}$-, $p_{y}$-, $d_{x^2 - y^2}$-, $d_{xy}$-, $f_x$-, $f_y$- and $g$-wave) 
	of q1d systems by the Kadanoff-Wilson renormalization group 
	technique, which has been developed by Duprat and Bourbonnais.\cite{DB}
%	
	The present method is able to investigate a system in pure 1d and q1d
	in the same footing.
%	
	Each response function exhibits the temperature dependence, 
	$\chi (T) \propto \exp (1/T)$,
	in a wide region of temperature, which 
	satisfies the theorems of the Hohenberg\cite{Hohenberg}, 
	and Mermin and Wagner.\cite{MW} 
%	
	Also, the temperature dependence is consistent 
	with that obtained by 2d fluctuation theories.
	\cite{KT,TMY,private,Kosuge}
%	
	This may suggest that the present method gives a rigorous solution
	to some extent.
%	
	
	In the phase diagram of the $g_2$-$g_1$ plane,
	the $p$TS and the SDW are locating in the region parallel to $g_1$=$2g_2$.
%	
	The transition temperature depends on the intensity $|g_1 -2g_2 |$,
	which corresponds to the coupling constant of density-fluctuations.
%	
	The $d$SS phase, which does not appear in purely 1d case,
	exists next to the SDW phase in q1d case.
%	
	The $d$SS phase expands into the SDW region parallel to $g_1$=$2g_2$
	as $t_b$ increases.
%	
	For small $t_b$, $T_{\rm SDW}$ increases as 
	$T_{\rm SDW}\sim t_b \exp (-2/g_2^*)$.
%	
	The transition temperature of the SDW has a maximum and decreases
	at finite $t_b$ ($\sim 300$K for $(g_1 , g_2 ) = (0.88, 0.88)$ and 
	$(g_1 , g_2 )=(0.01, 0.55)$), due to the nesting deviation.
%	
	The singlet superconductivity of $d_{x^2 - y^2}$-wave ($d$SS) 
	emerges when the SDW state is suppressed
	by the moderately large $t_b$.
%	
	In the large-$V$ model, the subdominant CDW ($f$-wave triplet) state
	is comparable to the dominant SDW ($d_{x^2 -y^2}$-wave singlet) state.
%	
	Especially, the transition temperature of $f$TS (CDW)
	becomes equivalent to that of $d$SS (SDW)
	with $V_2 =U/2$.
%	
	This is because the large $V$ (practically $V_2$) leads to 
	the enhancement of the charge fluctuations.
%	
	Such large charge-fluctuation suppresses the $d$SS
	and enhances the $f$TS,
	so that $T_{\rm c}$ of $d$SS in the large-$V$ model 
	is much lower than that in the $V$=0 model.
%	
	Consequently, the long-range Coulomb interaction 
	can give rise to the large charge fluctuation, which suppresses $d$SS
	and enhances $f$TS {\it particularly} in q1d systems.
%	
	
	
	The experiments on (TMTSF)$_2$PF$_6$ compounds
	suggest the coexistence of $2k_F$-CDW and $2k_F$-SDW,\cite{PR,Kagoshima}
	and exhibit about ten times lower $T_{\rm c}$ than $T_{\rm SDW}$.\cite{Jerome}
%	
	Considering these experimental properties,
	the large-$V$ model ($V_2 /U\sim 0.5$) is appropriate for the (TMTSF)$_2$PF$_6$ compounds
	on the point of
	the phase diagram, the $2k_F$-charge-fluctuation in SDW phase,
	and the realization of the $f$TS.
%	
	Actually, there is an estimation that $V_1 /U \sim $ 0.5-0.75 
	($V_2 /U \sim$ 0.4-0.6, if we assume $V_1 = 1.2 V_2$)
	on a basis of a Valence-Bond/Hartree-Fock method for molecules
	derived of TTF.\cite{CFD}
%	
	
	Even if $V_2$ is not so large,
	$f$TS is possibly realized under moderately 
	large Zeeman magnetic field,
	which suppresses the singlet pairing of $d$SS
	due to the paramagnetic effect\cite{FOKM}.
%	
	In such a case, since the ground state
	is $d$SS state,
	the Knight shift decreases at $H=0$ but 
	does not vary under large magnetic field.
%	
	Noting that the upper critical field can be estimated as 
	$H_{{\rm c}2}=(\Delta_0 /\sqrt{2} \mu _{\rm B})(1-T^2/T_{\rm c}^2 )$,\cite{Clogston}
	we can evaluate that
	1.7 T is sufficient to realize the field-induced $f$-wave triplet pairing
	for $T_{c}(H=0)=1.2$K, even for $T_{\rm c}^{f}\simeq T_{\rm c}^d /2$.\cite{Zeeman}
%	
	Here, $\Delta_0$ is the magnitude of the superconducting gap,
	and the BCS relation $2\Delta_0 =3.5 k_{\rm B}T_{\rm c}$ is assumed.
%	
	
	Finally, we comment on the effects of the dimerization
	and the Umklapp scattering, which are not considered in the present work.
%	
	Following effect is expected with the Umklapp scattering $g_3$
	as $t_b$ increases.\cite{KY,BD}
%	
	First the Umklapp scattering causes the antiferromagnetic transition
	($g_3$ is relevant while $g_1$ is irrelevant).
%	
	Next, the SDW transition is realized
	due to the nesting property ($g_1$ becomes relevant).
%	
	Finally, the $f$TS state results from the charge fluctuation 
	enhanced by the off-site interaction,
	which is not affected by the Umklapp scattering;
%	
	the essential points of the present paper would be still valid.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	\section*{Acknowledgements}
	The authors thank M. Tsuchiizu for valuable discussions
	at the early stage of the present work.
%	
	They are also grateful to
	C. Bourbonnais for a lot of useful comments.
%	
	Y. F. acknowledges helpful discussions with K. Miyake
	on the two-dimensional fluctuations.
%	
	The present work has been financially supported by 
	a Grant-in-Aid for Scientific Research on Priority Areas of
	Molecular Conductors (No. 15073103 and 15073213) from
	the Ministry of Education, Culture, Sports, Science and Technology, Japan.
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
	
































%\begin{figure}[t]
%\begin{center}
%\includegraphics[width=12cm]{dummy.eps}
%\end{center}
%\caption{You can put EPS files into the document.}
%\label{f1}
%\end{figure}


\begin{thebibliography}{9}

%\bibitem{blue} L. Lamport: \textit{\LaTeX: A Document Preparation System} (Addison-Wesley, Reading, 1994) 2nd ed., Translation by H. Abe (Pearson Education, Tokyo, 1999) [in Japanese].

\bibitem{Hc2} 
	I. J. Lee, P. M. Chaikin and M. J. Naughton:
 	Phys. Rev. Lett. {\bf 78} (1997) 3555;
 	Phys. Rev. B {\bf 62} (2000) R14669;
 	{\it ibid}. {\bf 65} (2002) 180502.

\bibitem{Knight} 
	I. J. Lee, S. E. Brown, W. G. Clark, M. J. Strouse, 
	M. J. Naughton, W. Kang and P. M. Chaikin: 
	Phys. Rev. Lett. {\bf 88} (2002) 017004;
 	I. J. Lee, D. S. Chow, W. G. Clark, M. J. Strouse, 
 	M. J. Naughton, P. M. Chaikin and S. E. Brouwn: 
 	Phys. Rev. B {\bf 68} (2003) 092510.

\bibitem{Jerome} 
	D. Jerome: 
	Mol. Cryst. Liq. Cryst. {\bf 79} (1982) 155.

\bibitem{MSV}
	K. Miyake, S. Schmitt-Rink and C. M. Varma: Phys. Rev. B \textbf{34}
	(1986) 6554.
	
\bibitem{SLH}
	D. J. Scalapino, E. Loh, Jr. and J. E. Hirsch:
	Phys. Rev. B \textbf{34} (1986) 8190.
	
\bibitem{Emery}
	V. J. Emery: 
	Synthetic Metals \textbf{13} (1986) 21.

\bibitem{PR}
	J. P. Pouget and S. Ravy:
	J. Phys. I {\bf 6} (1996) 1501.
	
\bibitem{Kagoshima}
	S. Kagoshima, Y. Saso, M. Maesato, R. Kondo and T. Hasegawa:
	Solid State Comm. {\bf 110} (1999) 479.
	
\bibitem{Shimahara}
	H. Shimahara:
	J. Phys. Soc. Jpn. {\bf 58} (1989) 1735.
	
\bibitem{KK}
	H. Kino and H. Kontani:
	J. Phys. Soc. Jpn. {\bf 68} (1999) 1481.
	
\bibitem{KAA}
	K. Kuroki, R. Arita and H. Aoki:
	Phys. Rev. B {\bf 63} (2001) 094509.
	
\bibitem{NY}
	T. Nomura and K. Yamada:
	J. Phys. Soc. Jpn. {\bf 70} (2001) 2694.

\bibitem{DB} 
	R. Duprat and C. Bourbonnais: 
	Eur. Phys. J. B {\bf 21} (2001) 219.

\bibitem{FOKM} 
	Y. Fuseya, Y. Onishi, H. Kohno and K. Miyake: J. Phys.:
	Condens. Matter {\bf 14} (2002) L655.
	
\bibitem{TK}
	Y. Tanaka and K. Kuroki:
	Phys. Rev. B {\bf 70} (2004) 060502(R).
	
%\bibitem{FKM}
%	Y. Fuseya, H. Kohno and K. Miyake: 
%	to be published.
	
\bibitem{BC} 
	C. Bourbonnais and L. G. Caron: 
	Int. J. Mod. Phys. B {\bf 5} (1991) 1033;
	C. Bourbonnais, B. Guay and R. Wortis: 
	in {\it Theoretical Methods for Strongly 		Correlated Electrons} 
	ed. D. S$\acute{e}$n$\acute{e}$chal, A. M. Tremblay and C. Bourbonnais 
	(Springer-Verlag, New York, 2004) p. 77.

\bibitem{Solyom} 
	See, for example, J. Solyom: 
	Adv. Phys. {\bf 28} (1979) 201.

\bibitem{KY} 
	J. Kishine and K. Yonemitsu: 
	J. Phys. Soc. Jpn. {\bf 67} (1998) 2590;
	{\it ibid} {\bf 68} (1999) 2790.
	
\bibitem{symmetry}
	Of course, in q1d systems, we cannot classify
	the gap symmetry by the spherical-wave basis
	or the irreducible representation of the tetragonal.
	Here, we use an idiomatic representation
	such as $s$-, $p_{x, y}$-, $d_{x^2-y^2 , xy}$,
	$f_{x,y}$ and $g$ displayed in Fig. \ref{spdfg}.

\bibitem{MW} 
	N. D. Mermin and H. Wagner: 
	Phys. Rev. Lett. {\bf 17} (1966) 1133.

\bibitem{Hohenberg} 
	P. C. Hohenberg: 
	Phys. Rev. {\bf 158} (1967) 383.

\bibitem{KT} 
	See, for example, J. M. Kosterlitz and D. J. Thouless:
	in {\it Progress in Low Temperature Physics} {\bf VII B},
	ed. D. F. Brewer (North-Holland, Amsterdam, 1978) p. 371.

\bibitem{TMY} 
	A. Tokumitu, K. Miyake and K. Yamada: 
	J. Phys. Soc. Jpn. {\bf 60} (1991) 380.

\bibitem{private} 
	private communication with K. Miyake.

\bibitem{Kosuge} 
	Y. Kosuge, A. Kobayashi, T. Matsuura and Y. Kuroda: 
	J. Phys. Soc. Jpn. {\bf 72} (2003) 1166;
	Y. Kosuge: doctoral thesis, Nagoya University, (2003).
	
\bibitem{KT2}
	J. M. Kosterlitz and D. Thouless: J. Phys. {\bf C6} (1973) 1181;
	J. M. Kosterlitz: {\it ibid}. {\bf C7} (1974) 1046.
	
\bibitem{Miyake}
	K. Miyake:
	Prog. Theor. Phys. {\bf 69} (1983) 1794.
	
\bibitem{JTB}
	C. S. Jacobsen, D. B. Tanner and K. Bechgaard:
	Phys. Rev. B {\bf 28} (1983) 7019.
	
\bibitem{Grant}
	P. M. Grant:
	J. Physique {\bf 44} (1983) C3-847.

\bibitem{SF} 
	Y. Suzumura and H. Fukuyama: 
	J. Low Temp. Phys. {\bf 31} (1978) 273.
	
\bibitem{OAKA} 
	S. Onari, R. Arita, K. Kuroki and H. Aoki: 
	cond-mat/0312314.
	
\bibitem{KTOS} 
	A. Kobayashi, Y. Tanaka, M. Ogata and Y. Suzumura:
	J. Phys. Soc. Jpn. {\bf 73} (2004) 1115.
	
\bibitem{CFD}
	F. Castet, A. Fritsch and L. Ducasse:
	J. Phys. I France, {\bf 6} (1996) 583.
	
\bibitem{Clogston} 
	A. M. Clogston: 
	Phys. Rev. Lett. {\bf 9} (1962) 266.
	
\bibitem{Zeeman}
	The evaluated Zeeman magnetic field here
	is the magnetic field where
	the singlet pairings are completely broken.
	The reversal from $d$SS to $f$TS 
	is expected below the $H_{\rm c2}$ of the $d$SS.

\bibitem{BD}
	C. Bourbonnais and R. Duprat:
	J. Phys. IV France, {\bf 114} (2004) 3.
	



\end{thebibliography}
\end{document}

cond-mat0411013.extracted_bib
\begin{thebibliography}{9}

%\bibitem{blue} L. Lamport: \textit{\LaTeX: A Document Preparation System} (Addison-Wesley, Reading, 1994) 2nd ed., Translation by H. Abe (Pearson Education, Tokyo, 1999) [in Japanese].

\bibitem{Hc2} 
	I. J. Lee, P. M. Chaikin and M. J. Naughton:
 	Phys. Rev. Lett. {\bf 78} (1997) 3555;
 	Phys. Rev. B {\bf 62} (2000) R14669;
 	{\it ibid}. {\bf 65} (2002) 180502.

\bibitem{Knight} 
	I. J. Lee, S. E. Brown, W. G. Clark, M. J. Strouse, 
	M. J. Naughton, W. Kang and P. M. Chaikin: 
	Phys. Rev. Lett. {\bf 88} (2002) 017004;
 	I. J. Lee, D. S. Chow, W. G. Clark, M. J. Strouse, 
 	M. J. Naughton, P. M. Chaikin and S. E. Brouwn: 
 	Phys. Rev. B {\bf 68} (2003) 092510.

\bibitem{Jerome} 
	D. Jerome: 
	Mol. Cryst. Liq. Cryst. {\bf 79} (1982) 155.

\bibitem{MSV}
	K. Miyake, S. Schmitt-Rink and C. M. Varma: Phys. Rev. B \textbf{34}
	(1986) 6554.
	
\bibitem{SLH}
	D. J. Scalapino, E. Loh, Jr. and J. E. Hirsch:
	Phys. Rev. B \textbf{34} (1986) 8190.
	
\bibitem{Emery}
	V. J. Emery: 
	Synthetic Metals \textbf{13} (1986) 21.

\bibitem{PR}
	J. P. Pouget and S. Ravy:
	J. Phys. I {\bf 6} (1996) 1501.
	
\bibitem{Kagoshima}
	S. Kagoshima, Y. Saso, M. Maesato, R. Kondo and T. Hasegawa:
	Solid State Comm. {\bf 110} (1999) 479.
	
\bibitem{Shimahara}
	H. Shimahara:
	J. Phys. Soc. Jpn. {\bf 58} (1989) 1735.
	
\bibitem{KK}
	H. Kino and H. Kontani:
	J. Phys. Soc. Jpn. {\bf 68} (1999) 1481.
	
\bibitem{KAA}
	K. Kuroki, R. Arita and H. Aoki:
	Phys. Rev. B {\bf 63} (2001) 094509.
	
\bibitem{NY}
	T. Nomura and K. Yamada:
	J. Phys. Soc. Jpn. {\bf 70} (2001) 2694.

\bibitem{DB} 
	R. Duprat and C. Bourbonnais: 
	Eur. Phys. J. B {\bf 21} (2001) 219.

\bibitem{FOKM} 
	Y. Fuseya, Y. Onishi, H. Kohno and K. Miyake: J. Phys.:
	Condens. Matter {\bf 14} (2002) L655.
	
\bibitem{TK}
	Y. Tanaka and K. Kuroki:
	Phys. Rev. B {\bf 70} (2004) 060502(R).
	
%\bibitem{FKM}
%	Y. Fuseya, H. Kohno and K. Miyake: 
%	to be published.
	
\bibitem{BC} 
	C. Bourbonnais and L. G. Caron: 
	Int. J. Mod. Phys. B {\bf 5} (1991) 1033;
	C. Bourbonnais, B. Guay and R. Wortis: 
	in {\it Theoretical Methods for Strongly 		Correlated Electrons} 
	ed. D. S$\acute{e}$n$\acute{e}$chal, A. M. Tremblay and C. Bourbonnais 
	(Springer-Verlag, New York, 2004) p. 77.

\bibitem{Solyom} 
	See, for example, J. Solyom: 
	Adv. Phys. {\bf 28} (1979) 201.

\bibitem{KY} 
	J. Kishine and K. Yonemitsu: 
	J. Phys. Soc. Jpn. {\bf 67} (1998) 2590;
	{\it ibid} {\bf 68} (1999) 2790.
	
\bibitem{symmetry}
	Of course, in q1d systems, we cannot classify
	the gap symmetry by the spherical-wave basis
	or the irreducible representation of the tetragonal.
	Here, we use an idiomatic representation
	such as $s$-, $p_{x, y}$-, $d_{x^2-y^2 , xy}$,
	$f_{x,y}$ and $g$ displayed in Fig. \ref{spdfg}.

\bibitem{MW} 
	N. D. Mermin and H. Wagner: 
	Phys. Rev. Lett. {\bf 17} (1966) 1133.

\bibitem{Hohenberg} 
	P. C. Hohenberg: 
	Phys. Rev. {\bf 158} (1967) 383.

\bibitem{KT} 
	See, for example, J. M. Kosterlitz and D. J. Thouless:
	in {\it Progress in Low Temperature Physics} {\bf VII B},
	ed. D. F. Brewer (North-Holland, Amsterdam, 1978) p. 371.

\bibitem{TMY} 
	A. Tokumitu, K. Miyake and K. Yamada: 
	J. Phys. Soc. Jpn. {\bf 60} (1991) 380.

\bibitem{private} 
	private communication with K. Miyake.

\bibitem{Kosuge} 
	Y. Kosuge, A. Kobayashi, T. Matsuura and Y. Kuroda: 
	J. Phys. Soc. Jpn. {\bf 72} (2003) 1166;
	Y. Kosuge: doctoral thesis, Nagoya University, (2003).
	
\bibitem{KT2}
	J. M. Kosterlitz and D. Thouless: J. Phys. {\bf C6} (1973) 1181;
	J. M. Kosterlitz: {\it ibid}. {\bf C7} (1974) 1046.
	
\bibitem{Miyake}
	K. Miyake:
	Prog. Theor. Phys. {\bf 69} (1983) 1794.
	
\bibitem{JTB}
	C. S. Jacobsen, D. B. Tanner and K. Bechgaard:
	Phys. Rev. B {\bf 28} (1983) 7019.
	
\bibitem{Grant}
	P. M. Grant:
	J. Physique {\bf 44} (1983) C3-847.

\bibitem{SF} 
	Y. Suzumura and H. Fukuyama: 
	J. Low Temp. Phys. {\bf 31} (1978) 273.
	
\bibitem{OAKA} 
	S. Onari, R. Arita, K. Kuroki and H. Aoki: 
	cond-mat/0312314.
	
\bibitem{KTOS} 
	A. Kobayashi, Y. Tanaka, M. Ogata and Y. Suzumura:
	J. Phys. Soc. Jpn. {\bf 73} (2004) 1115.
	
\bibitem{CFD}
	F. Castet, A. Fritsch and L. Ducasse:
	J. Phys. I France, {\bf 6} (1996) 583.
	
\bibitem{Clogston} 
	A. M. Clogston: 
	Phys. Rev. Lett. {\bf 9} (1962) 266.
	
\bibitem{Zeeman}
	The evaluated Zeeman magnetic field here
	is the magnetic field where
	the singlet pairings are completely broken.
	The reversal from $d$SS to $f$TS 
	is expected below the $H_{\rm c2}$ of the $d$SS.

\bibitem{BD}
	C. Bourbonnais and R. Duprat:
	J. Phys. IV France, {\bf 114} (2004) 3.
	



\end{thebibliography}