text:
\section{Introduction}

One-dimensional quantum spin systems exhibit fascinating physical
properties. Most of the striking features obsverved in these
systems are pure quantum effects and give rise to many
exotic ground-states or unconventional excitation spectra that
are not accessible by traditional approaches like spin-wave or
perturbation theory.
A famous example is the prediction made by Haldane\cite{haldane} that
half-integer antiferromagnetic spin chains are gapless whereas
those with integer spin have a finite energy gap in its spectrum.

The different one-dimensional spin gapped phases are usually
characterized by the nature of the broken discrete symmetries (lattice
translational symmetry for instance) in the ground-state,
the quantum numbers associated with the massive excitations, and the
structure of the dynamical magnetic susceptibility.
Many different spin gapped phases have been identified.
The spin-1 Heisenberg chain is certainly the most famous example
of a spin liquid with a gap.
It is characterized by a hidden topological order\cite{den}
related to the breakdown of a hidden Z$_2 \times$ Z$_2$ symmetry\cite{kohmoto}.
The spin excitations are optical magnons carrying spin S=1 and display
a single peak near $q=\pi$ in the dynamical magnetic susceptibility.
The low-lying excitations of a massive phase
can also have fractional quantum numbers
as in the J$_1$-J$_2$ S=1/2 Heisenberg chain for a sufficiently
large value of J$_2$: The ground-state is
spontaneously dimerized with a two-fold degeneracy and the fundamental
excitations are massive S=1/2 solitons (spinons) separating the
two ground-states\cite{haldane1,sorensen}.
Another scenario can also be realized in a two-leg spin ladder coupled
by a four spin exchange interaction where the elementary excitation
is neither an optical magnon nor a massive spinon but rather a pair of
propagating triplet or singlet solitons connecting the two spontaneously
dimerized ground-states\cite{nersesyan,kolezhuk}.
This special structure of the excitation will reveal itself in
the dynamical magnetic susceptibility which displays a two-particle
threshold instead of a sharp single magnon peak near $q=\pi$
as in the S=1 spin chain.
This incoherent background in the dynamical structure factor gives
rise to a new one-dimensional spin liquid state: the so-called
non-Haldane spin liquid\cite{nersesyan}.

Clearly, the general classification of all possible one-dimensional
spin gapped phases is still lacking.
However, the integrability of some special field theoretical models
has given a lot of insight and has been employed to extract
the degeneracies, quantum numbers of low-energy excited states of spin
liquid phases and to provide sometimes the computation of the leading
asymptotics of correlation functions.
This program has been realised for the sine-Gordon model associated
with the low-energy description of the S=1/2 Heisenberg chain with
alternating exchange\cite{essler}
and the Cu benzoate\cite{oshikawa,essler1}
and also for the O(N) (N=4,6,8) Gross-Neveu model for the description
of a special frustrated two-leg ladder\cite{allen1}, weakly two-leg
Hubbard ladder\cite{lin,konik,schulz}, (N,N) armchair
Carbon nanotubes\cite{konik}, and spin-orbital model\cite{azaria}.

The situation is totally different for critical one-dimensional quantum
spin systems like the spin-1/2 Heisenberg chain where Conformal Field
Theory (CFT) allows a complete classification of one-dimensional
quantum systems or finite temperature two-dimensional classical
systems\cite{polyakov,dms}.
The low-energy spectrum of the lattice model with a continuous
symmetry is described in terms of
representations of a certain current algebra\cite{knizhnik}.
This affine symmetry determines the operator content of the theory
and all possible scaling dimensions of the operators are in turn
fixed by the conformal invariance of the underlying
Wess-Zumino-Novikov-Witten (WZNW)
model built from the currents of the affine symmetry
with the Sugawara construction\cite{knizhnik}.
The knowledge of the scaling dimensions allows the computation of
correlation functions and the scaling behaviour of the energies of
the ground-state and low-lying states as a function of the size of
the system\cite{cardy1}.
The different critical phases are labelled, in particular, by
the central charge (c) of the underlying Virasoro algebra of the
WZNW CFT which fixes the low-temperature behaviour of the specific heat of
the model\cite{cardy}.

A simple example of this description is the Luttinger
liquids, a general class of one-dimensional interacting
electron models whose infrared (IR) behaviour is governed
by the Luttinger model\cite{haldanelutt}, which have an U(1) affine
Kac Moody (KM) symmetry and a central charge c=1.
The Luttinger liquids are usually described in a bosonized form
with the introduction of one gapless bosonic field (c=1 CFT)
and are characterized by single-particle correlation functions having
branch cuts with non-universal exponents\cite{voit}.
Such state provides an example of non-Fermi liquid behaviour\cite{voit}.
The critical theory of quantum spin systems with non-Abelian continuous
symmetries can also be analysed on the basis of their SU(2)$_k$ affine KM
algebra\cite{affleck0,affleck}.
Examples of such representation are the critical theory of
half-integer spin antiferromagnets described by the SU(2)$_1$ WZNW model
whereas the family of integrable Hamiltonian of arbitrary spin S\cite{kulish}
belongs to the SU(2)$_{2S}$ universality class\cite{affleck}.
This approach has also been employed to describe the low-energy physics
of Heisenberg antiferromagnets with a larger symmetry group like a SU(N)
symmetry\cite{sutherland} on the basis of the SU(N)$_1$ WZNW
model\cite{affleck,affleck1,itoi,assaraf}.

In a recent paper, Andrei, Douglas, and Jerez\cite{andrei} identified
a new non-Fermi-liquid class of fixed points, the so-called chiral
spin fluids, describing the IR behaviour of one-dimensional interacting
{\sl chiral} fermions.
The chiral spin liquid (CSL) behaviour appears as a stable critical state of
one-dimensional interacting fermions with {\sl unequal} numbers of right-moving
and left-moving particles.
The phenomenological Hamiltonian, in the
continuum limit, corresponding to this state reads:
\begin{eqnarray}
{\cal H}_{CSL} = -\mbox{i} v_F \left(\sum_{r=1}^{f_R}
R_{\alpha r}^{\dagger} \partial_x R_{\alpha r} -
\sum_{l=1}^{f_L}
L_{\alpha l}^{\dagger} \partial_x L_{\alpha l}\rightarrow)
+ g {\bf J}_R \cdot {\bf J}_L
\label{handrei}
\end{eqnarray}
where $R_{\alpha r}^{\dagger}$ (respectively $L_{\alpha l}^{\dagger}$)
is the creation operator of a right-moving (respectively left-moving)
fermion with spin index $\alpha=\uparrow,\downarrow$ and flavor index $r$ (respectively
$l$): $r=1,..,f_R$ (respectively $l=1,..,f_L$).
The interaction affects only the spin degrees of freedom and consists
of a marginal relevant ($g >0$) current-current term where the spin currents
write in terms of the fermions as follows:
\begin{eqnarray}
J_R^a &=& \frac{1}{2} \sum_{r=1}^{f_R}
R_{\alpha r}^{\dagger} \sigma_{\alpha \beta}^a R_{\beta r} \nonumber \\
J_L^a &=& \frac{1}{2} \sum_{l=1}^{f_L}
L_{\alpha l}^{\dagger} \sigma_{\alpha \beta}^a L_{\beta l},
\label{currand}
\end{eqnarray}
$\sigma^a$ (a=x,y,z) being the Pauli matrices
and $J_R^a$ (respectively $J_L^a$) belongs to
the SU(2)$_{f_R}$ (respectively SU(2)$_{f_L}$) KM algebra.
The Hamiltonian (\ref{handrei}), in the spin sector, describes thus
two SU(2)$_{f_R}$ and SU(2)$_{f_L}$ WZNW models
marginally coupled by a current-current interaction.
For $f_R \ne f_L$, i.e. breakdown of the $R\rightarrow L$ symmetry
implying breakdown of the time-reversal symmetry, the
Hamiltonian (\ref{handrei}) belongs to a new class of problems not
encountered in the above mentionned problems of one-dimensional
physics.
By a combination of CFT argument and Bethe Ansatz techniques,
the authors of Ref. \cite{andrei} showed that the Hamiltonian (\ref{handrei})
for $f_R \ne f_L$ flows in the IR limit to an intermediate fixed point
(chiral fixed point) characterizing by universal exponents in the leading
asymptotic of the electronic Green's functions.
From the electronic point of view, such a state provides a non-trivial
example of a non-Fermi-liquid state which differs from
the Luttinger liquids: the CSL state.
A typical example of possible realizations
of CSL, given in Ref. \cite{andrei}, is a contact between the edges of
two integer Quantum Hall Effect systems with different filing factors:
$\nu_L = f_L \ne \nu_R = f_R$.

In this paper, we shall consider another route
where the total Hamiltonian ($\cal H$) of
a possible realization of CSL is time-reversal invariant.
Indeed, in the continuum limit,
the leading part of the Hamiltonian that
imposes the strong coupling
behaviour decomposes into two commuting marginal
{\sl chirally asymmetric} parts ${\cal H}_1$ and ${\cal H}_2$:
\begin{equation}
{\cal H} = {\cal H}_1 + {\cal H}_2, \; \; [{\cal H}_1, {\cal H}_2] = 0,
\label{hgenmod}
\end{equation}
where ${\cal H}_1$ and ${\cal H}_2$ are not separately time-reversal
invariant but they transform into each other under the symmetry $t \rightarrow
-t$.
The model (\ref{hgenmod}) as a whole will display a CSL behaviour in the
IR limit.
Such scenario suggests that other realizations of this state
can be found in ladder problems.
However,
in the continuum limit of spin ladders, one has, apart from
marginal current-current interactions, backscattering terms
${\bf n}_a \cdot {\bf n}_b$ (${\bf n}_a$ being the staggered magnetizations
of a chain of index a) which have scaling dimension 1 at
the ultraviolet (UV) fixed point and thus represent strongly
relevant perturbations.
Therefore, one has clearly to kill this contribution to
have a marginal perturbation and thus to find some specific
realizations of the CSL state.
We shall follow, in this paper, two different routes to find some
CSL behaviour in the IR limit of ladder problems.
On the one hand, frustration may play its role either by suppressing
geometrically
the backscattering contribution
as in zigzag
ladders\cite{white,allen,nersesyan1,cabra} or
by allowing several independent coupling constants and
the possibility to eliminate the unwanted term by a fine
tuning mechanism\cite{azaria1,allen1}.
In that case, as we shall see later, the CSL found at this
special point will be destabilized upon switching on
weak backscattering contributions but there will exist
an intermediate low-energy region governed by the CSL behaviour.
On the other hand, doping effects can also be useful to
suppress the backscattering term by some incommensurate effects:
In the continuum limit,
the unwanted contribution will become a strongly oscillating
perturbation  that can be ignored in the long distance limit.

The remainder of the paper is organized as follows:
Different possible realizations of the CSL state
are introduced in Section II:
the (N+1)-chain cylinder model, its asymmetric doped
version, and the N-channel Kondo-Heisenberg model
away from half-filling.
The nature of the IR fixed point (chiral fixed
point) for these models is determined in
Section III.
In Section IV, we present a Toulouse point solution in the $N=2$ case
that captures all universal properties of the chiral fixed point.
In that case, the massless degrees of freedom are described
in terms of an effective S=1/2 Heisenberg spin chain and two critical
Ising models.
A brief summary of this approach has already been published\cite{azaria1}.
We shall provide here and in Section V
the technical
details of the Toulouse point solution and in particular
the determination of the physical properties of the 3-chain cylinder
model.
The nature of the phase diagram of the doped models in the N=2 case
will be addressed in a forthcoming publication\cite{azarial}.
In section VI, the stability of the chiral fixed point of the
3-chain cylinder model is analysed under the presence of
weak backscattering perturbations.
Finally, our concluding remarks are
presented in Section VII. Three Appendixes (A,B,C) give further technical
details necessary for the computation of the leading asymptotics
of the correlation functions occuring in Section V
whereas in the last one the stability of the Toulouse limit solution is
studied.

\section{Possible realizations of chiral spin liquid behaviour}

In this section, we shall present some lattice models that
are described in the continuum limit by an Hamiltonian with
only marginal current-current interactions.
In the next section, we shall show that the low-energy physics of this
Hamiltonian is governed by the chiral fixed point and thus represents
some specific realizations of a CSL state.

\subsection{The (N+1)-chain cylinder model}

The first model that we consider below can be visualized
as follows. It consists of N identical S=1/2 antiferromagnetic
Heisenberg chains on the surface of a cylinder parallel to each other
and to the axis of the cylinder.
An additional S=1/2 Heisenberg chain, that will be called
the zeroth chain or central chain in the following, resides
inside the cylinder along
its axis. The surface chains do not interact with each other but
all of them are coupled to the central one by on-rung interchain
interaction ($J_{\perp}$) and an interchain interaction along
diagonals of the elementary plaquettes ($J_{\times}$).
The Hamiltonian corresponding to this model is given by:
\begin{equation}
{\cal H} = \sum_{a=0}^{N} {\cal H}_a^{0} + {\cal H}_{int},
\label{hamiltmod1}
\end{equation}
where ${\cal H}_a^{0}$
is the Hamiltonian of a S=1/2 antiferromagnetic
Heisenberg spin chain of index a:
\begin{equation}
{\cal H}_a^{0} = J_{\parallel} \sum_j {\bf S}_{a,j}
\cdot {\bf S}_{a,j+1}, \; \; J_{\parallel} > 0
\end{equation}
${\bf S}_{a,j}$ being S=1/2 spin operators on each site $j$ of the
chain of index $a=0,1,..,N$.
These Heisenberg chains are coupled with the interaction:
\begin{eqnarray}
{\cal H}_{int} =
J_{\perp}  \sum_j \sum_{a=1}^{N} {\bf S}_{0,j}\cdot {\bf S}_{a,j}
+ J_{\times} \sum_j
\sum_{a=1}^{N} \left[{\bf S}_{a,j}
\cdot {\bf S}_{0,j+1} + {\bf S}_{a,j+1} \cdot {\bf S}_{0,j}\rightarrow].
\label{hamilintmod1}
\end{eqnarray}
In the following, it will be assumed that all interchain
exchange interactions
are positive and weak: $0 < J_{\perp},J_{\times} \ll J_{\parallel}$
to describe the model by a continuum limit approach.
Due to the presence of the plaquette interaction $J_{\times}$,
the cylinder model belongs to the class of frustrated spin
ladders. For $N=1$, this model has been investigated by
several groups\cite{oitmaa,allen1} and reveals no CSL state.
As it will be discussed in Section III, the (N+1)-chain cylinder model
with $N>1$ is a better candidate for displaying non-trivial CSL
physics.

Let us first consider the continuum limit of the (N+1)-chain cylinder model.
The continuum description of the S=1/2 antiferromagnetic
Heisenberg spin chain is based on the Sugawara representation
of the SU(2)$_1$ WZNW model
with a marginal irrelevant perturbation\cite{affleck0,affleck}:
\begin{equation}
{\cal H} ^0_a = \frac{2\pi v}{3} \; \left({\bf J}_{aR} \cdot {\bf J}_{aR} +
{\bf J}_{aL} \cdot {\bf J}_{aL} \rightarrow) +
\gamma {\bf J}_{aR} \cdot {\bf J}_{aL}, \; \; a=0,1,..,N
\label{hcin0}
\end{equation}
with $\gamma < 0$ and
${\bf J}_{aR,L}$ are the right and left spin currents
belonging to the SU(2)$_1$ KM algebra\cite{knizhnik,dms,book,book1}
and verify the following operator product expansion (OPE):
\begin{eqnarray}
J_{aL}^{\alpha}\left(z\rightarrow) J_{bL}^{\beta}\left(w\rightarrow) &\sim&
\frac{\mbox{i} \epsilon^{\alpha \beta \gamma} \delta_{ab}}{2\pi \left(z-w\rightarrow)}
J_{aL}^{\gamma}\left(w\rightarrow)
+ \frac{\delta^{\alpha \beta}\delta_{ab}}{8\pi^2\left(z-w\rightarrow)^2}
\nonumber \\
J_{aR}^{\alpha}\left(\bar z\rightarrow) J_{bR}^{\beta}\left(\bar w\rightarrow) &\sim&
\frac{\mbox{i} \epsilon^{\alpha \beta \gamma}\delta_{ab}}{2\pi
\left(\bar z- \bar w\rightarrow)}
J_{aR}^{\gamma}\left(\bar w\rightarrow)
+ \frac{\delta^{\alpha \beta}\delta_{ab}}{8\pi^2
\left(\bar z- \bar w\rightarrow)^2} \nonumber \\
J_{aL}^{\alpha}\left(z\rightarrow)
J_{bR}^{\beta}\left(\bar w\rightarrow) &\sim& 0, \; \; \alpha,\beta=x,y,z, \; \;
a,b=0,1,..,N
\label{opesu2level1}
\end{eqnarray}
where $z= v \tau + ix$, $\tau$ being the imaginary time,
and $v \sim J_{\parallel} a_0$ denotes
the spin velocity on each chain ($a_0$ being the lattice spacing).

The next step to achieve the continuum limit of
the Hamiltonian (\ref{hamiltmod1}) is to use the continuum
representation of the spin densities of each chain\cite{affleck}:
\begin{equation}
\frac{{\bf S}_{a,j}}{a_0} \rightarrow
{\bf S}_a\left(x\rightarrow) = {\bf J}_a\left(x\rightarrow) +
 \left(-1\rightarrow)^{x/a_0}{\bf n}_a\left(x\rightarrow), \; a=0,1,..,N
\label{spinrep}
\end{equation}
with $x=ja_0$ and
${\bf J}_a$, ${\bf n}_a$ denote respectively the uniform and
staggered part of the spin density.
The smooth part ${\bf J}_a$ can be expressed as:
\begin{equation}
{\bf J}_a = {\bf J}_{aR}  + {\bf J}_{aL},
\end{equation}
whereas the ${\bf n}_a$ field is the vector part of the primary
field of the SU(2)$_1$ WZNW model transforming according to the
fundamental representation of SU(2).
Assuming weak interchain couplings $J_{\perp},
J_{\times} \ll J_{\parallel}$, one can perform
the continuum limit of the cylinder model using Eq. (\ref{hcin0})
and the continuum description (\ref{spinrep}) to obtain:
\begin{equation}
{\cal H} = \frac{2\pi v}{3} \sum_{a=0}^{N}
\left({\bf J}_{aR} \cdot {\bf J}_{aR} +
{\bf J}_{aL} \cdot {\bf J}_{aL} \rightarrow) +
\gamma \sum_{a=0}^{N}{\bf J}_{aR} \cdot {\bf J}_{aL}
+ {\tilde g} {\bf n}_{0} \cdot {\bf n}_+
+  g {\bf J}_{0} \cdot {\bf I},
\label{conintmod1}
\end{equation}
where we have discarded
all irrelevant operators and oscillatory terms.
In Eq. (\ref{conintmod1}),
${\bf I}$ (respectively ${\bf n}_+$) is the sum
of all uniform (respectively staggered) magnetizations
of the surface chains:
\begin{eqnarray}
{\bf I} &=& \sum_{a=1}^{N} {\bf J}_a \nonumber \\
{\bf n}_+ &=& \sum_{a=1}^{N} {\bf n}_a .
\label{sumcurrstag}
\end{eqnarray}
Moreover, the bare coupling constants in (\ref{conintmod1})
are given by:
\begin{eqnarray}
g &=& a_0\left(J_{\perp} + 2 J_{\times}\rightarrow) \nonumber \\
{\tilde g} &=& a_0\left(J_{\perp} - 2 J_{\times}\rightarrow).
\label{barecoupmod1}
\end{eqnarray}

The interacting terms in (\ref{conintmod1})
are of different nature. A first one with the coupling
constant ${\tilde g}$ is
a relevant perturbation of scaling dimension $d=1$
and corresponds to the usual interchain interaction of
non-frustrated spin ladders.
A second one with the coupling $g$ describes
an interaction between the total spin current
of the surface chains and that of the central chain.
This interaction is marginal relevant and, as long as ${\tilde g}$
is not too small, can be discarded. As a result, for generic
values of ${\tilde g}$ and $g$, the low-energy
physics of the model will be essentially governed by
the backscattering contribution.
In that case, frustration will play no
role except for renormalization of spin velocities and mass gaps
of the modes that will become massive in the IR limit.
However, it is important to stress that, in constrast with
non-frustrated spin ladders, the two coupling
constants ${\tilde g},g$ can vary independently in the model,
and there exists a vicinity of the line
$J_{\perp} = 2 J_{\times}$ (${\tilde g}=0$) where frustration
shows up in a nontrivial way.
Along this line, the low-energy properties of the model
are mainly determined by current-current
interactions:
\begin{equation}
{\cal H} = \frac{2\pi v}{3} \sum_{a=0}^{N}
\left({\bf J}_{aR} \cdot {\bf J}_{aR} +
{\bf J}_{aL} \cdot {\bf J}_{aL} \rightarrow) +
\gamma \sum_{a=0}^{N}{\bf J}_{aR} \cdot {\bf J}_{aL}
+  g {\bf J}_{0} \cdot {\bf I},
\label{conintmod1bis}
\end{equation}
which displays in the IR limit
CSL physics as it will be shown in Section III.

\subsection{The (N+1)-chain cylinder model with doped central chain}

As already stated in the Introduction, the effect of doping might be
useful to suppress the backscattering perturbation in the long
distance limit. In particular, one can, for example, consider the previous
(N+1)-chain cylinder model where some concentration of holes are introduced
in the central chain. Clearly, the very assumption that there can
exist such model with only one chain doped needs justification.
Probably, the simplest one would be to assume that the
intrasite single-electron energies on the central ($\epsilon_0$)
and surface ($\epsilon_S$) chains are much larger than the interchain
hopping amplitude: $t_{\perp} \ll |\epsilon_0 - \epsilon_S|$
so that different chains are out of resonance. Anyway, we shall
assume in the following that one can dope the central chain only.
In that case, this special model belongs to the class of one-dimensional
electron liquid in an antiferromagnetic
environment\cite{emery1,neto,granath,lehur}
made by the surface chains.

Due to the presence of doping, the continuum description of the spin
density of the central is no longer given by Eq. (\ref{spinrep}) but takes now
the following form (see for instance chapter 16 of Ref. \cite{book1}):
\begin{eqnarray}
{\bf S}_0\left(x\rightarrow) = {\bf J}_0\left(x\rightarrow) +
 \cos\left(2k_{F,0} x + \sqrt{2\pi} \Phi_c\left(x\rightarrow)\rightarrow)
{\bf n}_0\left(x\rightarrow),
\label{spinrepcentrdop}
\end{eqnarray}
where $\Phi_c$ is a bosonic field associated with the charge degrees of
freedom, $k_{F,0}$ being the Fermi momentum of the central chain.
In the continuum limit, the backscattering contribution (${\bf n}_0
\cdot {\bf n}_+$) acquires now an oscillating
factor $\sim \exp(i(2k_{F,0} - \pi)x)$
and thus becomes suppressed in the long distance limit if the concentration
of holes is sufficiently high.
As a result, the Hamiltonian of this asymmetric doped model in the
spin sector is
given by Eq. (\ref{conintmod1bis}) with only current-current interactions.

\subsection{The multichannel Kondo-Heisenberg model away from half-filling}

The last example that we shall give in this paper is
the so-called Kondo-Heisenberg model with N-channel.
This model consists of N identical Hubbard chains (${\cal H}_U$)
interacting via a Kondo coupling with a periodic array of
localized spins described by the spin-1/2
operator ${\bf S}_{0,i}$\cite{white,sikkema,emery2,zachartse}.
The Hamiltonian of this system reads as follows:
\begin{equation}
{\cal H} = {\cal H}_U + J_K \sum_i {\bf S}_{0,i} \cdot {\bf S}_{c,i}
+ J_H \sum_i {\bf S}_{0,i} \cdot {\bf S}_{0,i+1}
\label{Nchannel}
\end{equation}
with:
\begin{equation}
{\cal H}_U  = -t \sum_{n,\sigma,i} \left(c^{\dagger}_{n\sigma,i}
c_{n\sigma,i+1} + h.c.\rightarrow) + U \sum_{n,i} c^{\dagger}_{n\uparrow,i}
c_{n\uparrow,i} c^{\dagger}_{n\downarrow,i} c_{n\downarrow,i},
\end{equation}
where $c_{n\sigma,i}$  denotes the electron annihilation operator on
the site $i$, and $\sigma = \uparrow,\downarrow$, $n =1,..,N$
are respectively the electron spin and channel indices. The
conduction electron spin operator ${\bf S}_{c,i}$ at the ith site is defined
by:
\begin{equation}
{\bf S}^{a}_{c,i} = \frac{1}{2} \sum_{n,\alpha,\beta}
c^{\dagger}_{n\alpha,i} {\sigma}^{a}_{\alpha\beta}
c_{n\beta,i}, \; a=x,y,z.
\end{equation}
In Eq. (\ref{Nchannel}), the interaction between the electronic
degrees of freedom and the localized spin is made by a Kondo
coupling with the constant $J_K >0$.
The Hamiltonian (\ref{Nchannel}) can also be viewed as a generalization
of the multichannel Kondo lattice with an exchange interaction $J_H >0$
between the impurities spins (simple RKKY interaction).

In the weak coupling limit when $J_K,U \ll t, J_H$, this Hamiltonian
can be represented in a bosonized form following the same route
as in Refs. \cite{white,sikkema} for $N=1$.
The continuum description of the
localized spin operator (${\bf S}_{0,i}$) is still given
by Eq. (\ref{spinrep})
whereas that
of the conduction spin
density operator can be expressed in terms of N bosonic fields
$\Phi_{ac}$ and
N SU(2)$_1$ spin currents ${\bf J}_{aR,L}$:
\begin{eqnarray}
{\bf S}_c\left(x\rightarrow) = \sum_{a=1}^{N} \left({\bf J}_{aR}\left(x\rightarrow)
+ {\bf J}_{aL}\left(x\rightarrow)
+\cos\left(2 k_F x + \sqrt{2\pi} \Phi_{ac}\left(x\rightarrow)\rightarrow)
{\bf n}_a\left(x\rightarrow)\rightarrow).
\label{condspin}
\end{eqnarray}

With these relations at hand, one
can derive the continuum limit of the Hamiltonian (\ref{Nchannel})
away from half-filling (incommensurate filling).
In that case, the $2k_F$ oscillation of the spin
conduction operator becomes incommensurate
with the alternating localized spin operator so that far away from
half-filling, only marginal current-current interaction survives in
the Kondo interaction in the long distance limit:
\begin{eqnarray}
{\cal H} = \frac{2\pi v_s}{3} \sum_{a=1}^{N} \left(
{\bf J}_{aR} \cdot {\bf J}_{aR} + {\bf J}_{aL} \cdot {\bf J}_{aL}\rightarrow)
+ \frac{2\pi v_0}{3} \left(
{\bf J}_{0R} \cdot {\bf J}_{0R} + {\bf J}_{0L} \cdot {\bf J}_{0L}\rightarrow)
\nonumber \\
+ g {\bf J}_0\cdot {\bf I} +
\gamma \sum_{a=1}^{N}{\bf J}_{aR} \cdot {\bf J}_{aL}
+ \gamma^{'} {\bf J}_{0R} \cdot {\bf J}_{0L}
\label{spinsector}
\end{eqnarray}
where ${\bf I} = \sum_{a=1}^{N} {\bf J}_a$, $v_s \sim 2t a_0 -
Ua_0/2\pi$, $v_0 \sim J_H a_0$, $g\sim J_K a_0$,
and $\gamma, \gamma^{'} <0$ ($U,J_H >0$).
Therefore, the
multichannel version of the Kondo-Heisenberg model
away from half-filling belongs to the class of models with marginally
coupled current-current interactions.
One should note that it is more transparent to use a different
basis, a spin-charge-flavor decomposition\cite{affleck0,affleck},
to single out the charge degrees of freedom.
In that case, the continuum description of the
Hamiltonian of the N-channel Kondo-Heisenberg model
away from half filling reads as follows in the spin sector:
\begin{eqnarray}
{\cal H}_s = \frac{2\pi v_s}{N+2}\left(
{\bf I}_{R} \cdot {\bf I}_{R} + {\bf I}_{L} \cdot {\bf I}_{L}\rightarrow)
+ \frac{2\pi v_0}{3} \left(
{\bf J}_{0R} \cdot {\bf J}_{0R} + {\bf J}_{0L} \cdot {\bf J}_{0L}\rightarrow)
\nonumber \\
+ g {\bf J}_0\cdot {\bf I} +
\gamma {\bf I}_{R} \cdot {\bf I}_{L}
+ \gamma^{'} {\bf J}_{0R} \cdot {\bf J}_{0L}
\label{spinsectorbis}
\end{eqnarray}

\section{Identification of the infrared fixed point}

We investigate now the IR properties of
the Hamiltonian (\ref{conintmod1bis}) with only
current-current interactions
which represents the continuum limit of several lattice models
as seen in the previous section.
In particular, the nature of the IR fixed point that
governed the low-energy physics of these models for $N>1$ will be identified
and belongs to the universality class of chirally stabilized fluids
introduced in Ref. \cite{andrei}.

Let us first look at the problem from a perturbative point of view
near the UV fixed point where all chains are decoupled so that
the model has the [SU(2)$_1$]$_R^{N+1}$ $\otimes$ [SU(2)$_1$]$_L^{N+1}$
symmetry. It is conformally invariant with a
total central charge: $c_{UV}=N+1$ ($N+1$ gapless
bosonic modes). The one-loop Renormalization Group (RG) equations for
the coupling constants $g$ and $\gamma$ are given by:
\begin{eqnarray}
\frac{d \gamma}{d \ln L} &=& \frac{\gamma^2}{2\pi v} \nonumber \\
\frac{d g}{d \ln L} &=& \frac{g^2}{2\pi v}.
\label{RGeq}
\end{eqnarray}
The bare coupling constant $\gamma$ is negative in all the models
presented in Section II, so
that the perturbation $\gamma \sum_a {\bf J}_{aR} \cdot {\bf J}_{aL}$
is marginally irrelevant and can therefore be neglected: it
leads to logarithmic corrections.
On the other hand, the effective interchain coupling $g$ increases
upon renormalization ($g >0$).
Usually, the development of a strong coupling regime ($g(L) \rightarrow
+\infty$) is accompanied by a dynamical mass generation and the
loss of conformal invariance as for the non-linear sigma model or
the Gross-Neveu model in 1+1 dimensions.
This is indeed the case for $N=1$ which represents a continuum
version of the two-leg zigzag ladder\cite{white,allen}
in the absence of the so-called twist
perturbation\cite{nersesyan1}.
However, for $N\ge 2$, the interaction
flows to an intermediate fixed point ($g^{*} < \infty$) where
the system displays critical properties with a smaller (and
generally non-integer) central charge than $c_{UV}$
according to the Zamolodchikov c-theorem\cite{zamolo1}.

To this end, let us rewrite the Hamiltonian (\ref{conintmod1bis})
in terms of the leading part which governs
the strong coupling behaviour:
\begin{equation}
{\cal H} = \frac{2\pi v_1}{3} \sum_{a=1}^{N}
\left({\bf J}_{aR} \cdot {\bf J}_{aR} +
{\bf J}_{aL} \cdot {\bf J}_{aL} \rightarrow) +
\frac{2\pi v_0}{3}
\left({\bf J}_{0R} \cdot {\bf J}_{0R} +
{\bf J}_{0L} \cdot {\bf J}_{0L} \rightarrow)
+ g \left({\bf J}_{0R} \cdot {\bf I}_L + {\bf J}_{0L} \cdot {\bf I}_R\rightarrow)
\label{hgenstrong}
\end{equation}
where we have omitted the marginal irrelevant contribution and the
interaction between the currents of the same chirality whose effect
will be effectively taken into account by
allowing the surface chains velocity ($v_1$)
to be different from that of the central chain ($v_0$).
We shall
explicitely check for $N=2$, in Appendix D, by our Toulouse point approach
that all these discarded interactions will not destabilize the intermediate
fixed point.

The structure (\ref{hgenstrong}) for $N>1$ suggests
that some degrees of freedom do not participate to the interaction
and thus will decouple and remain massless.
Indeed, the free part of the Hamiltonian (\ref{hgenstrong}) is a sum of $(N+1)$
critical SU(2)$_1$ WZNW models whereas the interaction describes
a coupling between the SU(2)$_1$ spin current of the central
chain (${\bf J}_{0R,L}$)
and the total surface current (${\bf I}_{R,L}$)
which is a SU(2)$_N$ current being the sum of $N$ SU(2)$_1$ currents
(see Eq. (\ref{sumcurrstag})).
To identify the nature of the non-interacting degrees of freedom,
we use the following decomposition:
\begin{equation}
\prod_{i=1}^{N} SU(2)_1|_i = SU(2)_N \otimes {\cal G}_N
\label{cosetN}
\end{equation}
where ${\cal G}_N$ is some piece.
The central charge of the critical model with symmetry ${\cal G}_N$
is:
\begin{equation}
c_{{\cal G}_N} = N - c_{SU(2)_N} =
N - \frac{3N}{N + 2}=
\frac{N \left(N - 1\rightarrow)}{N + 2}
\label{centdiscr}
\end{equation}
where we have used that the central charge of the SU(2)$_k$ WZNW
model is: $c=3k/(k+2)$\cite{knizhnik}.
Though it requires a proof, for $N=2m$, ${\cal G}_N$
can be viewed as a direct
product of $m$ $Z_{2m}$-symmetric critical models\cite{zamolo2}
whereas for $N = 2 m + 1$, it is the product of $m$ $Z_{2m}$-symmetric
critical models and the minimal model ${\cal M}_{N+1}$.
We recall here that the central charge of the minimal model series
${\cal M}_{p}$ with $p\ge 2$ is: $c_p = 1- 6/p(p+1)$\cite{polyakov,dms}.
The previous identification reproduces the value of the central charge
and also all primary fields have the correct scaling dimensions
but it is not sufficient to prove the previous CFT embedding.
Anyway, in this work, we only need the actual value of the
central charge (\ref{centdiscr}) of the critical model
with symmetry ${\cal G}_N$ which one can note that it coincides with
the sum of the central charge of the $N-1$ first minimal models:
\begin{equation}
c_{{\cal G}_N} = \frac{N\left(N-1\rightarrow)}{N+2}
= \sum_{m=2}^{N+1}\left(1 - \frac{6}{m\left(m+1\rightarrow)}\rightarrow).
\end{equation}

The actual model under consideration simplifies thus in the
following way:
\begin{equation}
{\cal H} = {\cal H}_{{\cal G}_N} +  \bar{\cal H}
\label{HsimN}
\end{equation}
where ${\cal H}_{{\cal G}_N}$ is the Hamiltonian corresponding
to the critical degrees of freedom with symmetry ${\cal G}_N$
that decouple from the interaction.
All non-trivial physics is incorporated in
the current dependent part of the Hamiltonian ($\bar{\cal H}$)
which can be decomposed into a sum of two commuting
and {\it chirally asymmetric} parts:
\begin{equation}
\bar{\cal H} = {\cal H}_1 + {\cal H}_2, ~([{\cal H}_1, {\cal H}_2] = 0)
\label{hambarN0}
\end{equation}
where
\begin{eqnarray}
{\cal H}_1 &=&  \frac{2\pi v_1}{N+2} \;  {\bf I}_{R} \cdot{\bf I}_{R} +
 \frac{2\pi v_0}{3} \; {\bf J}_{0L} \cdot{\bf J}_{0L}
+ g \;  {\bf I} _{R} \cdot {\bf J}_{0L} \nonumber \\
\nonumber \\
{\cal H}_2 &=&  \frac{2\pi v_1}{N+2} \;  {\bf I}_{L} \cdot{\bf I}_{L} +
 \frac{2\pi v_0}{3} \; {\bf J}_{0R} \cdot{\bf J}_{0R}
+ g \;  {\bf I} _{L} \cdot {\bf J}_{0R}.
\label{hamN1}
\end{eqnarray}
The Hamiltonian ${\cal H}_1$ thus describes two marginally
coupled [SU(2)$_N$]$_R$ and [SU(2)$_1$]$_L$ WZNW models
and ${\cal H}_2$ is obtained from ${\cal H}_1$ by interchanging
the indices $R$ and $L$.

The properties of chirally asymmetric models with the structure of
${\cal H}_1$ in Eq. (\ref{hamN1})
but with a
more general symmetry group [SU(2)$_{f_{R}}$]$_R$
$\otimes$ [SU(2)$_{f_{L}}$]$_L$
($f_R > f_L$) have been studied
by Andrei et al.\cite{andrei} (see Eq. (\ref{handrei})).
These authors argued that
such models flow in the IR limit to an intermediate fixed point
$g^{*}$
(chiral fixed point) whose symmetry is determined by a
WZNW coset\cite{gko}:
\begin{equation}
\left[SU(2)_{f_R - f_L}\rightarrow]_{R} \otimes
\left[\frac{SU(2)_{f_L}\times
SU(2)_{f_R - f_L}}{SU(2)_{f_R}}\rightarrow]_{L}.
\label{coset}
\end{equation}
This fixed point has been found by the authors of
Ref. \cite{andrei} using the fact that
the differences between the chiral
components of the central charge ($c_R-c_L$)
and the KM levels ($f_R-f_L$) are preserved under the RG flow.
Morevover, since the total central charge ($c_R+c_L$)
always decreases upon renormalization according to the
Zamolodchikov c-theorem\cite{zamolo1}, the identification of the IR fixed point
is equivalent to find a conformally invariant model
with the lowest total central charge consistent with
the fixed values of $c_R-c_L$ and $f_R-f_L$.
It turns out that the solution for this problem
is given by the coset model (\ref{coset}).
The structure of this fixed point has also been checked
by the computation of the low temperature behaviour of the specific heat
from the Thermodynamical Bethe ansatz approach\cite{andrei} since
this physical quantity is a direct probe of the value of the
central charge of the IR fixed point\cite{cardy}.

Using the result (\ref{coset}), one can
immediately deduce the structure of the IR
fixed point of the Hamiltonian (\ref{hambarN0}):
\begin{eqnarray}
\left[\left[SU(2)_{N - 1}\rightarrow]_{R} \otimes
\left[\frac{SU(2)_{1}\times
SU(2)_{N - 1}}{SU(2)_{N}}\rightarrow]_{L}\rightarrow]
\otimes
\left[\left[SU(2)_{N - 1}\rightarrow]_{L} \otimes
\left[\frac{SU(2)_{1}\times
SU(2)_{N - 1}}{SU(2)_{N}}\rightarrow]_{R}\rightarrow] \nonumber\\
\label{groupsymmhbar}
\end{eqnarray}
where the first (respectively second) part is the symmetry of the
IR fixed point associated with ${\cal H}_1$ (respectively
${\cal H}_2$).
The central charge in the IR limit (${\tilde c}_{IR}$)
corresponding to the model
described by the
Hamiltonian ($\bar{\cal H}$) of Eq. (\ref{hambarN0}) is thus given
by:
\begin{equation}
{\tilde c}_{IR}\left(N\rightarrow)
= \frac{2\left(N-1\rightarrow)\left(2N+5\rightarrow)}{\left(N+1\rightarrow)
\left(N+2\rightarrow)}.
\label{centralcharghbar}
\end{equation}
From this result, we deduce the value of the
total central at the IR fixed point of
the original Hamiltonian (\ref{hgenstrong}) taking into account the contribution
of the massless degrees of freedom (\ref{centdiscr}) that do not participate
to the interaction:
\begin{equation}
c_{IR}\left(N\rightarrow) = {\tilde c}_{IR}\left(N\rightarrow) + c_{{\cal G}_N} =
\frac{\left(N-1\rightarrow)\left(N^2+5N + 10\rightarrow)}{\left(N+1\rightarrow)
\left(N+2\rightarrow)}.
\label{totalcentralchargeNfin}
\end{equation}

In summary, the low-energy physics of the (N+1)-chain cylinder model,
its asymmetric doped version, and
the multichannel Kondo-Heisenberg model away from half-filling,
are governed by an intermediate
IR fixed point displaying CSL behaviour with a central charge
which is not integer but rational for a generic value of $N$.
The first two models, in the spin sector, have a
central charge given by
Eq. (\ref{totalcentralchargeNfin}) whereas the gapless modes
corresponding to the {\sl spin} sector of the N-channel
Kondo-Heisenberg model are fixed by Eq. (\ref{centralcharghbar}).
Moreover, from Eqs. (\ref{centralcharghbar},\ref{totalcentralchargeNfin}),
one can notice that the
central charges vanishe for $N=1$ as it should be (spin gap phase)
whereas for $N=2$ one has the simple values:
${\tilde c}_{IR}(2)= 3/2, c_{IR}(2) = 2$ which means
that a simple and independent approach should be possible
in that case.

In the following, we shall focus on this $N=2$ special case by
presenting an exact solution of the U(1) version of the $N=2$
problem using a Toulouse point solution.
This approach has been extremely fruitful especially in
Quantum Impurity problems and gives a simple description of the
two-channel Kondo model and also of the
Kondo lattice\cite{emery,zachar,georges}.
In particular, for the two-channel Kondo problem, Emery
and Kivelson\cite{emery} identified the residual zero point entropy
stemming from the decoupling of a Majorana fermion degrees of freedom
using a mapping of the model onto free fermion theory for a very
special value of the interaction (Toulouse point).
Although the position of the solvable point is non-universal, the
Toulouse limit solution captures the physical and universal properties
of the low-temperature behaviour of the model.
In our problem, the Toulouse point approach provides a {\sl direct and
transparent} reading of the spectrum of the model for $N=2$.
It also gives a
non-perturbative basis from which all leading asymptotics of spin-spin
correlations for the 3-chain cylinder model can be determined.

\section{The Toulouse point solution in the N=2 case}

In this section, we present the solution,
by a Toulouse point approach,
of the model (\ref{hgenstrong}) with $N=2$:
\begin{eqnarray}
{\cal H} = \frac{2 \pi v_1}{3} \;\left({\bf J}_{1R}.{\bf J}_{1R}
+ {\bf J}_{1L}.{\bf J}_{1L}+
{\bf J}_{2R}.{\bf J}_{2R} + {\bf J}_{2L}.{\bf J}_{2L} \rightarrow)
+ \frac{2 \pi v_0}{3} \;\left({\bf J}_{0R}.{\bf J}_{0R}
+ {\bf J}_{0L}.{\bf J}_{0L}\rightarrow)
\nonumber \\
+ \; g \;\left({\bf I}_R. {\bf J}_{0L}
+ {\bf I}_L. {\bf J}_{0R} \rightarrow).
\label{hsu2}
\end{eqnarray}

\subsection{Decoupling of a Z$_2$ non-magnetic excitation}

As discussed in section III A,
the structure of the interaction of Eq. (\ref{hsu2}) suggests an alternative
representation of the two SU(2)$_1$ WZNW models of the surface
chains in a SU(2)$\otimes$ Z$_2$ way reflecting the global
spin rotational symmetry as well as the discrete Z$_2$
related to the interchange symmetry ($1 \rightarrow 2$).
The critical model with the symmetry
group ${\cal G}_2$ in the description (\ref{cosetN}) for $N=2$
identifies with a single critical Ising model (or a free Majorana
fermion in the continuum limit) since on has from
the Goddard-Kent-Olive
construction\cite{gko}:
\begin{equation}
SU(2)_1 \otimes SU(2)_1 \sim SU(2)_2 \otimes Z_2.
\label{gkoe}
\end{equation}
A simple way to take into account this non-magnetic Z$_2$
excitation (Ising degrees of freedom)
is to use the representation of two SU(2)$_1$ spin currents
in terms of four Majorana fermions
$\xi^{0}$ and $\vec \xi$\cite{zamolo,allen,book1}:
\begin{eqnarray}
{\bf I}_{\alpha} =
{\bf J}_{1\alpha} + {\bf J}_{2\alpha}
= -\frac{\mbox{i}}{2} \;
{\vec \xi}_{\alpha} \wedge {\vec \xi}_{\alpha}, \; \alpha=R,L, \nonumber\\
{\bf J}_{1\alpha} - {\bf J}_{2\alpha} = \mbox{i}
\; {\vec \xi}_{\alpha} \xi_{\alpha}^{0}, \; \alpha=R,L
\label{j1mj2}
\end{eqnarray}
where the four right and left moving Majorana fermions have
the following anticommutation relations ($a,b=0,1,2,3$):
\begin{eqnarray}
\{\xi_{R,L}^a(x),\xi_{R,L}^b(y)\} &=& \delta^{ab}\delta(x - y) \nonumber \\
\{\xi_R^a(x),\xi_L^b(y)\} &=& 0
\end{eqnarray}
and are normalized according to:
\begin{eqnarray}
\xi^a_L\left(z\rightarrow) \xi^b_L\left(\omega\rightarrow)
&\sim& \frac{\delta^{ab}}{2\pi\left(z-\omega\rightarrow)} \nonumber \\
\xi^a_R\left(\bar z\rightarrow) \xi^b_R\left(\bar \omega\rightarrow)
&\sim& \frac{\delta^{ab}}{2\pi\left(\bar z-\bar \omega\rightarrow)} \nonumber \\
\xi^a_R\left(\bar z\rightarrow) \xi^b_L\left(\omega\rightarrow)
&\sim& 0 .
\end{eqnarray}
Using the correspondence (\ref{j1mj2}),
the Hamiltonian (\ref{hsu2}) simplifies as
\begin{equation}
{\cal H} = - \mbox{i} \frac{v_1}{2} \left(\xi_{R}^{0} \partial_x  \xi_{R}^{0} -
\xi_{L}^{0} \partial_x  \xi_{L}^{0}  \rightarrow)
+ \bar{\cal H} [{\bf I}, {\bf J}_0]
\label{htotexact}
\end{equation}
where the first part is the free Hamiltonian of the Majorana
fermion $\xi^0$ with central charge $c=1/2$ and
\begin{equation}
\bar{\cal H} =
\frac{\pi v_1}{2} \;\left({\bf I}_{R}.{\bf I}_{R} +
{\bf I}_{L}.{\bf I}_{L}\rightarrow) +
\frac{2 \pi v_0}{3} \;\left({\bf J}_{0R}.{\bf J}_{0R}
+ {\bf J}_{0L}.{\bf J}_{0L}\rightarrow)
+ \; g \;\left({\bf I}_R. {\bf J}_{0L}
+ {\bf I}_L. {\bf J}_{0R} \rightarrow).
\label{hbarsu2}
\end{equation}
All non-trivial physics is incorporated in
the current dependent part of the Hamiltonian ($\bar{\cal H}$)
which decomposes into a sum of two commuting
and chirally asymmetric parts as in
Eqs. ((\ref{hambarN0}), (\ref{hamN1})) for $N=2$:
\begin{equation}
\bar{\cal H} = {\cal H}_1 + {\cal H}_2, ~([{\cal H}_1, {\cal H}_2] = 0)
\label{hambar0}
\end{equation}
\begin{eqnarray}
{\cal H}_1 &=&  \frac{\pi v_1}{2} \;  {\bf I}_{R} \cdot{\bf I}_{R} +
 \frac{2\pi v_0}{3} \; {\bf J}_{0L} \cdot{\bf J}_{0L}
+ g \;  {\bf I} _{R} \cdot {\bf J}_{0L} \nonumber \\
{\cal H}_2 &=&  \frac{\pi v_1}{2} \;  {\bf I}_{L} \cdot{\bf I}_{L} +
 \frac{2\pi v_0}{3} \; {\bf J}_{0R} \cdot{\bf J}_{0R}
+ g \;  {\bf I} _{L} \cdot {\bf J}_{0R}.
\label{ham1}
\end{eqnarray}

\subsection{Identification of the Toulouse point}

We consider now
the SU(2) broken version of model (\ref{hambar0}) ($g
\rightarrow g_{\parallel}, g_{\perp}$) where
we allow $g_{\parallel}$ to be different from
$g_{\perp}$:
\begin{equation}
{\cal H}_1 = \frac{\pi v_1}{2} \;{\bf I}_R.{\bf I}_R
+ \frac{2 \pi v_0}{3} \;{\bf J}_{0L}.{\bf J}_{0L}
+ \; g_{\parallel} \;I_R^z J_{0L}^z
+ \frac{g_{\perp}}{2} \;\left(I_R^{+} J_{0L}^{-}
+ H.c. \rightarrow)
\label{hu1}
\end{equation}
and ${\cal H}_2$ is obtained from ${\cal H}_1$ by
inverting chiralities of all the spin currents.

In this section, we present an exact solution of
the Hamiltonian (\ref{hu1}) using
Abelian bosonization.
The solution is based on a mapping onto Majorana fermions
and exploits the existence of a Toulouse-like, exactly
solvable point,
at a special value (though non-universal) of $g_{\parallel}$
where the fermions are free.

The scaling equations for the model (\ref{hu1})
are of the Kosterlitz-Thouless
form:
\begin{equation}
\frac{d g_{\parallel}}{d \ln L} =
\frac{g_{\perp}^2}{\pi\left(v_1+v_0\rightarrow)},
\; \frac{d g_{\perp}}{d \ln L} =
\frac{g_{\parallel}g_{\perp}}{\pi\left(v_1+v_0\rightarrow)},
\label{rgxxz}
\end{equation}
indicating the increase of the coupling constants upon
renormalization for $g_{\parallel} > - |g_{\perp}|$. Since
both the exactly solvable point and the isotropic strong-coupling
separatrix $g_{\parallel} = |g_{\perp}|$ occur within this range,
the exact solution of the anisotropic model (\ref{hu1}) is expected
to exhibit generic properties of the original, SU(2) symmetric,
model (\ref{ham1}).

The starting point of our approach is the Abelian bosonization
of the component of the SU(2)$_1$ current ${\bf J}_0$
associated with the spin excitation of the central chain.
We introduce a massless bosonic field ${\varphi}$
and write (see for instance Appendix A of Ref. \cite{shelton}
or chapter 13 of Ref. \cite{book1}):
\begin{eqnarray}
J_{0R}^z &=& \frac{1}{\sqrt{2\pi}}\; \partial_x \varphi_R, \;\; \;
\; \;\;\; \;
J_{0L}^z = \frac{1}{\sqrt{2\pi}}\; \partial_x \varphi_L \nonumber \\
J_{0R}^{\pm} &=& \frac{1}{2\pi a_0}\; \mbox{e}^{\mp \mbox{i} \sqrt{8\pi} \varphi_R},
\;\; J_{0L}^{\pm} = \frac{1}{2\pi a_0}\; \mbox{e}^{\pm \mbox{i} \sqrt{8\pi} \varphi_L}.
\label{boso1}
\end{eqnarray}
It is a simple matter to show that the representation (\ref{boso1}) indeed
reproduces
the OPEs (\ref{opesu2level1}) of the SU(2)$_1$ WZNW model

On the other hand, the SU(2)$_2$ current ${\bf I}$ can be
expressed in terms of a triplet of massless Majorana fermions $\xi^a$,
a=1,2,3 (see Eq. (\ref{j1mj2})).
Since we consider the U(1)$\otimes Z_2$-symmetric version of the
model, it is natural to discriminate $(\xi^1, \xi^2)$ from
the $\xi^3$ field. The two Majorana fields $(\xi^1, \xi^2)$ can be
combined to form a single Dirac field $\chi$ which can in turn
be bosonized with the introduction of a massless bosonic
field $\Phi$:
\begin{eqnarray}
\chi_R &=& \frac{\xi_R^2 + \mbox{i} \xi_R^1}{\sqrt 2} =
\frac{\kappa}{\sqrt{2\pi a_0}} \;
\mbox{e}^{\mbox{i} \sqrt{4\pi} \Phi_R} \nonumber \\
\chi_L &=& \frac{\xi_L^2 + \mbox{i} \xi_L^1}{\sqrt 2} =
\frac{\kappa}{\sqrt{2\pi a_0}} \;
\mbox{e}^{-\mbox{i} \sqrt{4\pi} \Phi_L},
\label{boso2}
\end{eqnarray}
where the anticommutation relation
between $\chi_R$ and $\chi_L$ is insured by the
choice $[\Phi_R, \Phi_L]=\mbox{i}/4$. On the other hand, to take into account
the correct
anticommutation with the third Majorana fermion $\xi^3$, one
needs to introduce
an additional real fermionic degree
of freedom $\kappa$ ($\kappa^2=1$). Using Eqs. (\ref{j1mj2}, \ref{boso2}),
the components of the SU(2)$_2$ spin current ${\bf I}$ transform to:
\begin{eqnarray}
I_R^z &=& \frac{1}{\sqrt{\pi}}\; \partial_x \Phi_R, \;\;\;\;\;\;
\;\;\;\;\;\;\;\;\;\;
I_L^z = \frac{1}{\sqrt{\pi}}\; \partial_x \Phi_L \nonumber \\
I_R^{\pm} &=& \frac{\mbox{i}}{\sqrt{\pi a_0}}\;\xi_R^3 \kappa
\mbox{e}^{\mp \mbox{i} \sqrt{4\pi} \Phi_R}, \;
I_L^{\pm} = \frac{\mbox{i}}{\sqrt{\pi a_0}}\;\xi_L^3 \kappa
\mbox{e}^{\pm \mbox{i} \sqrt{4\pi} \Phi_L}.
\label{boso3}
\end{eqnarray}
The chiral aymmetric parts
of the total
Hamiltonian $\bar{\cal H}$ (\ref{hambar0}) can then be written
in the following
bosonized form:
\begin{eqnarray}
{\cal H}_1 = v_0 \; \left(\partial_x \varphi_L\rightarrow)^2 +
v_1 \; \left(\partial_x \Phi_R\rightarrow)^2 -\mbox{i} \frac{v_1}{2} \;
\xi_R^3 \partial_x \xi_R^3 \nonumber \\
+ \frac{g_{\parallel}}{\sqrt 2 \pi} \; \partial_x \varphi_L
\partial_x \Phi_R + \frac{\mbox{i} g_{\perp}}{2 \left(\pi a_0\rightarrow)^{3/2}}
\; \xi_R^3 \kappa \cos \left(\sqrt{4\pi} \Phi_R + \sqrt{8\pi} \varphi_L
\rightarrow)
\label{h1boso}
\end{eqnarray}
and
\begin{eqnarray}
{\cal H}_2 = v_0 \; \left(\partial_x \varphi_R\rightarrow)^2 +
v_1 \; \left(\partial_x \Phi_L\rightarrow)^2 +\mbox{i} \frac{v_1}{2} \;
\xi_L^3 \partial_x \xi_L^3 \nonumber \\
+ \frac{g_{\parallel}}{\sqrt 2 \pi} \; \partial_x \varphi_R
\partial_x \Phi_L + \frac{\mbox{i} g_{\perp}}{2 \left(\pi a_0\rightarrow)^{3/2}}
\; \xi_L^3 \kappa \cos \left(\sqrt{4\pi} \Phi_L + \sqrt{8\pi} \varphi_R
\rightarrow).
\label{h2boso}
\end{eqnarray}

One can eliminate the cross terms $\partial_x \varphi \partial_x \Phi$
by performing a canonical transformation:
\begin{eqnarray}
\left(\begin{array}{c}
\varphi_L\\
\Phi_R
\end{array}\rightarrow) =
\left(
\begin{array}{lccr}
{\rm ch} \alpha & {\rm sh} \alpha \\
{\rm sh} \alpha & {\rm ch} \alpha
\end{array} \rightarrow) \left(\begin{array}{c}
\bar \Phi_{2L}\\
\bar \Phi_{1R}
\end{array}\rightarrow)
\label{can1}
\end{eqnarray}
and
\begin{eqnarray}
\left(\begin{array}{c}
\Phi_L\\
\varphi_R
\end{array}\rightarrow) =
\left(
\begin{array}{lccr}
{\rm ch} \alpha & {\rm sh} \alpha \\
{\rm sh} \alpha & {\rm ch} \alpha
\end{array} \rightarrow) \left(\begin{array}{c}
\bar \Phi_{1L}\\
\bar \Phi_{2R}
\end{array}\rightarrow)
\label{can2}
\end{eqnarray}
with
\begin{equation}
{\rm th} 2 \alpha = - \frac{g_{\parallel}}{\pi \sqrt 2 \left(v_1+v_2\rightarrow)}.
\label{th1}
\end{equation}
Under the transformation (\ref{can1}) and (\ref{can2}), the
arguments of the cosines in Eqs. (\ref{h1boso},\ref{h2boso})
become:
\begin{eqnarray}
\sqrt{4\pi} \Phi_R + \sqrt{8\pi} \varphi_L
\rightarrow \sqrt{4\pi} \left[\left(
\sqrt 2 {\rm ch} \alpha + {\rm sh} \alpha\rightarrow) \bar \Phi_{2L}
+ \left({\rm ch} \alpha + \sqrt 2 {\rm sh} \alpha \rightarrow) \bar \Phi_{1R}
\rightarrow] \\
\sqrt{4\pi} \Phi_L + \sqrt{8\pi} \varphi_R
\rightarrow \sqrt{4\pi} \left[\left(
\sqrt 2 {\rm ch} \alpha + {\rm sh} \alpha\rightarrow) \bar \Phi_{2R}
+ \left({\rm ch} \alpha + \sqrt 2 {\rm sh} \alpha \rightarrow) \bar \Phi_{1L}
\rightarrow].
\label{costran}
\end{eqnarray}
We immediately observe that for a special
value of $\alpha$ given by
\begin{equation}
{\rm th} \alpha = -\frac{1}{\sqrt 2},
\label{th2}
\end{equation}
we are at the free fermion point where
the two cosine terms acquire respectively conformal weights $(0,1/2)$
and $(1/2,0)$ and
can be expressed in terms of the left and right components of
a single Dirac fermion $\psi$. When $\alpha$
satisfies both Eqs. (\ref{th1}, \ref{th2}), $g_{\parallel}$ takes
a special $positive$ value $g_{\parallel}^{*}$ (the Toulouse point):
\begin{equation}
g_{\parallel}^{*} = \frac{4\pi}{3} \left(v_1+v_0\rightarrow),
\label{toulouse}
\end{equation}
and the total Hamiltonian ($\bar{\cal H}$) reads as follows:
\begin{eqnarray}
\bar {\cal H} &=& - \frac{\mbox{i} v_1}{2}\left(\xi_R^{3}\partial_x \xi_R^{3}
-\xi_L^{3}\partial_x \xi_L^{3}\rightarrow) +
\frac{u_1}{2} \left[\left(\partial_x {\bar \Phi}_1\rightarrow)^2 +
\left(\partial_x  {\bar \Theta}_1\rightarrow)^2 \rightarrow] +
\frac{u_2}{2} \left[\left(\partial_x {\bar \Phi}_2\rightarrow)^2 +
\left(\partial_x  {\bar \Theta}_2\rightarrow)^2 \rightarrow] \nonumber \\
&+& \frac{\mbox{i} g_{\perp}}{2 \left(\pi a_0\rightarrow)^{3/2}}
\; \xi_R^3 \kappa \cos \left(\sqrt{4\pi} \bar \Phi_{2L}\rightarrow)
+ \frac{\mbox{i} g_{\perp}}{2 \left(\pi a_0\rightarrow)^{3/2}}
\; \xi_L^3 \kappa \cos \left(\sqrt{4\pi} \bar \Phi_{2R}\rightarrow)
\label{htrans}
\end{eqnarray}
where we have reestablished the complete bosonic
fields $\bar \Phi_{a}= \bar \Phi_{aL} + \bar \Phi_{aR}$ ($a=1,2$)
and their dual counterpart $\bar \Theta_{a} =
\bar \Phi_{aL} - \bar \Phi_{aR}$.
The associated velocities $u_1$ and $u_2$
are given at the Toulouse point by:
\begin{equation}
u_1 = \frac{2 v_1 - v_0}{3},\ \  u_2 = \frac{2 v_0 - v_1}{3}.
\label{velo}
\end{equation}
The Toulouse point solution is thus stable provided
that all the velocities of the modes are positive, i.e.
$1/2 \le v_0/v_1 \le 2$.
Since we are at the free fermion point, we can
refermionize the two cosine terms in Eq. (\ref{htrans}).
Introducing
a pair of Majorana fields $\eta$ and $\zeta$ and using
the correspondence:
\begin{eqnarray}
\psi_{R,L} &=& \frac{\eta_{R,L}+ i \zeta_{R,L}}{\sqrt 2} \nonumber\\
           &=& \frac{\kappa}{\sqrt{2\pi a_0}} \;
\mbox{e}^{\pm \mbox{i} \sqrt{4\pi} \bar \Phi_{2R,L}},
\label{psi}
\end{eqnarray}
the two cosine terms become simply:
\begin{equation}
\frac{\kappa}{\sqrt{\pi a_0}} \cos \left(\sqrt{4\pi}
\bar \Phi_{2R,L}\rightarrow) =
\eta_{R,L}.
\end{equation}
Therefore, the Hamiltonian ($\bar{\cal H}$) takes the following final form:
\begin{eqnarray}
{\bar {\cal H}} &=& \frac{u_1}{2} \left[  (\partial_x {\bar \Phi}_1)^2 +
 (\partial_x  {\bar \Theta}_1)^2 \rightarrow] - \frac{\mbox{i} u_2}{2}
 \left[ \zeta_R\partial_x \zeta_R - \zeta_L\partial_x \zeta_L\rightarrow]
 - \frac{\mbox{i} v_1}{2}\left[ \xi_R^{3}\partial_x \xi_R^{3}
-\xi_L^{3}\partial_x \xi_L^{3}\rightarrow]\nonumber\\
&-& \frac{\mbox{i} u_2}{2}
 \left[ \eta_R\partial_x \eta_R - \eta_L\partial_x \eta_L\rightarrow] +
 \mbox{i} m \left[ \xi_R^{3}\eta_L - \eta_R\xi_L^{3}\rightarrow]
\label{hfin}
\end{eqnarray}
with $m = g_{\perp}/2 a_0 \pi$.
Taking into account the contribution of the $Z_2$
Majorana fermion $\xi^0$ which
has decoupled from the beginning,
the original Hamiltonian (\ref{htotexact}) reads at the Toulouse point:
\begin{eqnarray}
{\cal H} &=& \frac{u_1}{2} \left[  (\partial_x {\bar \Phi}_1)^2 +
 (\partial_x  {\bar \Theta}_1)^2 \rightarrow] - \frac{\mbox{i} u_2}{2}
 \left[ \zeta_R\partial_x \zeta_R - \zeta_L\partial_x \zeta_L\rightarrow]
 - \frac{\mbox{i} v_1}{2}\left[ \xi_R^{0}\partial_x \xi_R^{0}
-\xi_L^{0}\partial_x \xi_L^{0}\rightarrow]\nonumber\\
 &-& \frac{\mbox{i} v_1}{2}\left[ \xi_R^{3}\partial_x \xi_R^{3}
-\xi_L^{3}\partial_x \xi_L^{3}\rightarrow]
- \frac{\mbox{i} u_2}{2}
 \left[ \eta_R\partial_x \eta_R - \eta_L\partial_x \eta_L\rightarrow] +
 \mbox{i} m \left[ \xi_R^{3}\eta_L - \eta_R\xi_L^{3}\rightarrow].
\label{hfintot}
\end{eqnarray}

At the Toulouse point, the massless degrees of freedom are thus
described in terms of an effective S=1/2 Heisenberg spin chain
associated with the ${\bar \Phi}_1$ field and two decoupled critical
Ising models ($\xi^0, \zeta$) so that the total central
charge in the IR limit is: $c_{IR}=2$
in full agreement with Eq. (\ref{totalcentralchargeNfin}) for $N=2$.
One should also notice that the nature of the critical fields found in
the Toulouse limit approach reproduces, for $N=2$, the
structure of the symmetry
group (\ref{groupsymmhbar}) of the fixed point
corresponding to the Hamiltonian
(${\bar {\cal H}}$).
Indeed, substituting the value $N = 2$ in Eq. (\ref{groupsymmhbar}),
we see
using Eq. (\ref{gkoe}) that
this symmetry group simplifies as:
\begin{equation}
\left[\left[SU(2)_1\rightarrow]_R \otimes \left[Z_2 \rightarrow]_L \rightarrow]
\otimes
\left[\left[SU(2)_1\rightarrow]_L \otimes \left[Z_2 \rightarrow]_R \rightarrow]
\end{equation}
which is in
full agreement with the
structure of the Hamiltonian (\ref{hfin}) at the Toulouse point.

The remaining part of the Hamiltonian (\ref{hfintot})
has a spectral gap and
describe the hybridization of the Majorana $\xi^3$ and $\eta$ fields with
different chiralities. Indeed, it can be written as a sum of two commuting
part ${\cal H}_{+}+ {\cal H}_{-}$ with
\begin{eqnarray}
{\cal H}_{+} &=& \frac{\mbox{i} v_1}{2}\xi_L^{3}\partial_x \xi_L^{3}
 -\frac{\mbox{i} u_2}{2}
\eta_R\partial_x \eta_R
-\frac{\mbox{i} m}{2} \left(\eta_R\xi_L^{3} - \xi_L^{3} \eta_R\rightarrow) \nonumber \\
{\cal H}_{-} &=& -\frac{\mbox{i} v_1}{2}\xi_R^{3}\partial_x \xi_R^{3}
+\frac{\mbox{i} u_2}{2}
\eta_L\partial_x \eta_L
+\frac{\mbox{i} m}{2} \left(\xi_R^{3}\eta_L - \eta_L\xi_R^{3}\rightarrow).
\label{hybrid}
\end{eqnarray}
The resulting spectrum is given by:
\begin{eqnarray}
E_{+}\left(k\rightarrow) &=& k u_1 \pm \sqrt{k^2\left(\frac{v_1+v_0}{3}\rightarrow)^2 +
m^2} \label{hybrispecp}\\
\nonumber\\
E_{-}\left(k\rightarrow) &=& -k u_1 \pm \sqrt{k^2\left(\frac{v_1+v_0}{3}\rightarrow)^2 +
m^2}, \label{hybrispecm}
\end{eqnarray}
which, apart from the velocity anisotropy, is reminiscent
of the structure of excitations found in the
Majorana approach for the Kondo lattice\cite{coleman}.
One should stress that each Majorana fermion hybridizes with the
other with the opposite chirality and that there is no coupling between
Majorana fermion of the same chirality. This reflects the chiral
nature of the fixed point.
One should finally note that our solution
contains the SU(2) point when $g_{\perp} =
g_{\parallel}$ and the mass at the Toulouse
point reads in that case: $m = g_{\parallel}^{*}/2\pi a_0 =
2(v_1+v_0)/3a_0$.

The exact solution based on the existence
of a Toulouse point allows us to get a simple understanding
of the structure of the critical elementary excitations and
to estimate the leading asymptotics of
the correlation functions.
In the following, we shall realize this program by directly focusing
on the physical properties of the 3-chain cylinder model.

\section{Physical properties of the 3-chain cylinder model}

In this section, the spectral
properties of the 3-chain cylinder model
are discussed and  we
compute the leading asymptotics of all spin-spin
correlation functions at the chiral fixed point.
To this end, we first express the smooth and staggered
magnetization of each chain in terms of the suitable
fields describing the long distance physics at the
chiral fixed point.

\subsection{Representation of the spin currents and
staggered spin fields at the Toulouse point}

In this subsection, the spin currents (${\bf J}_{aR,L}$)
and staggered spin fields (${\bf n}_a$) of the
chain $a=0,1,2$ are related
to the two critical Majorana fermions ($\xi^0, \zeta$)
and the massless bosonic field ($\bar \Phi_1$) at the Toulouse
point. To get a better insight in the structure of excitations,
we introduce the smooth part (${\cal J}^a$) and
staggered part (${\cal N}^a$) corresponding to
the effective S=1/2 Heisenberg spin chain described by
the bosonic $\bar \Phi_1$ field. Using Eq. (\ref{boso1}),
one has the following correspondence:
\begin{eqnarray}
{\cal J}_R^z &=& \frac{1}{\sqrt{2\pi}}\; \partial_x \bar \Phi_{1R}, \;\; \;
\; \;\;\; \;
{\cal J}_L^z = \frac{1}{\sqrt{2\pi}}\; \partial_x \bar \Phi_{1L} \nonumber \\
{\cal J}_R^{\pm} &=& \frac{1}{2\pi a_0}\; \mbox{e}^{\mp \mbox{i} \sqrt{8\pi}
\bar \Phi_{1R}},
\;\; {\cal J}_L^{\pm} =
\frac{1}{2\pi a_0}\; \mbox{e}^{\pm \mbox{i} \sqrt{8\pi} \bar \Phi_{1L}}.
\label{bosophi1}
\end{eqnarray}
The staggered part of a SU(2) spin density can also be expressed in terms of
the bosonic $\bar \Phi_1$ and its dual field $\bar \Theta_1$
by (see Ref.\cite{shelton} or chapter 13 of Ref. \cite{book1}):
\begin{eqnarray}
{\cal N}^z &=& -\frac{\lambda}{\pi a_0} \sin\left(\sqrt{2\pi}\bar\Phi_1\rightarrow)
\nonumber\\
{\cal N}^{\pm} &=& \frac{\lambda}{\pi a_0}
\mbox{e}^{\pm \mbox{i} \sqrt{2\pi} \bar \Theta_1}
\label{stagbosophi1}
\end{eqnarray}
where $\lambda$ is non-universal constant related to
the band gap for the charge excitations of the underlying
Hubbard model.
The WZNW primary operator $g$ transforming according to
the fundamental representation then writes:
\begin{equation}
g = \frac{\pi}{\lambda \sqrt{2}}\left(\epsilon +
\vec {\cal N} \cdot \vec{\sigma}\rightarrow)
\label{prim}
\end{equation}
where $\epsilon = \lambda \cos(\sqrt{2\pi}\bar\Phi_1)/\pi a_0$
is the dimerization operator.
With the bosonic representation (\ref{bosophi1}, \ref{stagbosophi1}),
one can check that the primary field $g$ (\ref{prim}) verifies the following
OPE:
\begin{eqnarray}
{\cal J}_L^a\left(z\rightarrow) g\left(\omega, \bar \omega\rightarrow) &\sim&
-\frac{1}{2\pi\left(z-\omega\rightarrow)} \left(\frac{\sigma^a}{2}\rightarrow)
g\left(\omega, \bar \omega\rightarrow) \\
{\cal J}_R^a\left(\bar z\rightarrow) g\left(\omega, \bar \omega\rightarrow) &\sim&
\frac{1}{2\pi\left(\bar z-\bar \omega\rightarrow)} g\left(\omega, \bar \omega\rightarrow)
\left(\frac{\sigma^a}{2}\rightarrow)
\end{eqnarray}
which corresponds (up to the normalization factor $1/2\pi$) to the
definition of a WZNW primary operator of Ref. \cite{dms} (chapter 15).

\subsubsection{Representation of the spin currents}

We begin our analysis by considering the SU(2)$_1$ spin
currents ${\bf J}_{0R,L}$ of the central chain. Using the
canonical transformation (\ref{can1}, \ref{can2}) with
the special value (\ref{th2}), we obtain:
\begin{eqnarray}
\varphi_R &=& -\bar \Phi_{1L} + \sqrt{2} \bar \Phi_{2R} \nonumber\\
\varphi_L &=& \sqrt{2} \bar \Phi_{2L} - \bar \Phi_{1R}
\label{transphichir}
\end{eqnarray}
from which together with Eq. (\ref{boso1}) we deduce the
transformation of the spin currents
of the central chain at the Toulouse point:
\begin{eqnarray}
J_{0R,L}^z &=& - {\cal J}_{L,R}^z + \frac{1}{\sqrt \pi} \partial_x
\bar \Phi_{2R,L}\nonumber\\
J_{0R}^{\pm} &=& {\cal J}_L^{\pm}
\mbox{e}^{\mp \mbox{i} \sqrt{16\pi} \bar\Phi_{2R}}\nonumber\\
J_{0L}^{\pm} &=& {\cal J}_R^{\pm}
\mbox{e}^{\pm \mbox{i} \sqrt{16\pi} \bar\Phi_{2L}}
\label{spincentrcur}
\end{eqnarray}
where we have used (\ref{bosophi1}) to express the spin currents
of the central chain in terms of the ``physical'' current ${\cal J}^a$.
Using the correspondence (\ref{psi}),
$\partial_x \bar \Phi_{2R,L}$ can be related to
the Majorana fermions $\eta_{R,L}$ and $\zeta_{R,L}$:
\begin{equation}
\partial_x \bar \Phi_{2R,L} = \mbox{i} \sqrt \pi \eta_{R,L} \zeta_{R,L}.
\label{parphi2rl}
\end{equation}
Consequently, at the Toulouse point, we deduce the expression of
the z-part of the spin current of the central chain:
\begin{eqnarray}
J_{0R}^z &=& - {\cal J}_{L}^z + \mbox{i} \eta_R \zeta_R \nonumber\\
J_{0L}^z &=& - {\cal J}_{R}^z + \mbox{i} \eta_L \zeta_L.
\label{centrcurz1}
\end{eqnarray}
Since the bilinear fields $\eta_{R,L} \zeta_{R,L}$ are
short-ranged,  we can simplify further (\ref{centrcurz1})
and obtain the following correspondence
valid for the estimation of the leading contribution
of correlation functions:
\begin{equation}
J_{0R,L}^z \sim - {\cal J}_{L,R}^z.
\end{equation}
Notice that the chiralities of the central chain and the physical
spin current are opposite. We shall return to this important
point when discussing the nature
of elementary excitations of the model. For the transverse part,
the bosonic operator $\exp(\pm \mbox{i}
\sqrt{16\pi} \bar \Phi_{2R,L})$ has a non-zero expectation value,
one thus obtains ignoring short-ranged pieces:
\begin{eqnarray}
J_{0R}^{+} &\sim& \gamma_R^{*} {\cal J}_L^{+}\nonumber\\
J_{0R}^{-} &\sim& \gamma_R {\cal J}_L^{-}\nonumber\\
J_{0L}^{+} &\sim& \gamma_L {\cal J}_R^{+}\nonumber\\
J_{0L}^{-} &\sim& \gamma_L^{*} {\cal J}_R^{-}
\label{centrcurpm1}
\end{eqnarray}
where $\gamma_{R,L}$ are defined as:
\begin{equation}
\langle \mbox{e}^{\mbox{i}
\sqrt{16\pi} \bar \Phi_{2R,L}} \rangle = \gamma_{R,L}.
\label{expec1}
\end{equation}
The actual
value of $\gamma_{R,L}$ is computed in Appendix A where one
can show that $\gamma_{R} = \gamma_{L}=\gamma_m$ and $\gamma_m$
is real. One thus concludes that:
\begin{equation}
J_{0R,L}^{\pm} \sim \gamma_m {\cal J}_{L,R}^{\pm}.
\label{centrcurpm2}
\end{equation}

The representation of the SU(2)$_1$ spin currents
of the surface chains (${\bf J}_{aR,L}, a=1,2$)
proceeds in the same way. To each chain $a$, we associate
a massless bosonic field $\varphi_a$ and its dual fields
$\vartheta_a$ to write
the currents as in Eq. (\ref{boso1}):
\begin{eqnarray}
J_{aR}^z &=& \frac{1}{\sqrt{2\pi}}\; \partial_x \varphi_{aR} \;\; ,\;
\; \;\;\; \;
J_{aL}^z = \frac{1}{\sqrt{2\pi}}\; \partial_x \varphi_{aL} \nonumber \\
J_{aR}^{\pm} &=& \frac{1}{2\pi a_0}\; \mbox{e}^{\mp \mbox{i} \sqrt{8\pi} \varphi_{aR}},
\;\; J_{aL}^{\pm} =
\frac{1}{2\pi a_0}\; \mbox{e}^{\pm \mbox{i} \sqrt{8\pi} \varphi_{aL}}.
\label{bosoout}
\end{eqnarray}
To relate these fields to the bosonic field $\Phi$ of
Eq. (\ref{boso3}), let us introduce the symmetric
and antisymmetric combinations:
\begin{equation}
\varphi_{\pm} = \frac{1}{\sqrt2} \left( \varphi_1 \pm \varphi_2\rightarrow),\ \
\vartheta_{\pm} = \frac{1}{\sqrt2} \left( \vartheta_1 \pm \vartheta_2\rightarrow),
\label{phipm}
\end{equation}
so that
\begin{eqnarray}
J_{aR,L}^z &=& \frac{1}{2\sqrt{\pi}}\left(\partial_x \varphi_{+R,L}
+ \tau_a \partial_x \varphi_{-R,L}\rightarrow)\nonumber\\
J_{aR}^{\pm} &=& \frac{1}{2\pi a_0}\; \mbox{e}^{\mp \mbox{i} \sqrt{4\pi} \varphi_{+R}}
\mbox{e}^{\mp \mbox{i} \tau_a\sqrt{4\pi} \varphi_{-R}}\nonumber\\
J_{aL}^{\pm} &=& \frac{1}{2\pi a_0}\; \mbox{e}^{\pm \mbox{i} \sqrt{4\pi} \varphi_{+L}}
\mbox{e}^{\pm \mbox{i} \tau_a\sqrt{4\pi} \varphi_{-L}}
\label{bosoout1}
\end{eqnarray}
where $\tau_1=1$ and $\tau_2=-1$. Moreover,
since we have the relation:
\begin{equation}
I_{\alpha}^{\pm} = J^{\pm}_{1\alpha}
+ J^{\pm}_{2\alpha}, \;\; \alpha = R,L
\end{equation}
and the expressions (\ref{boso3}) for the bosonization
of the SU(2)$_2$ current ${\bf I}$, we obtain the
following correspondence:
\begin{eqnarray}
\varphi_{+R,L} &=& \Phi_{R,L}\nonumber\\
\cos\left(\sqrt{4\pi} \varphi_{-\alpha}\rightarrow) &=& \sqrt{\pi a_0} \; \mbox{i}
\xi_{\alpha}^3 \kappa , \;\; \alpha=R,L.
\label{cosphim}
\end{eqnarray}
The difference between the SU(2)$_1$ currents of the surface
chains can also be expressed in terms of the Majorana fermions
$\xi^a, a=0,1,2,3$ (see Eq. (\ref{j1mj2})).
Using Eqs. (\ref{boso2}, \ref{bosoout1}), one thus finds
the identification:
\begin{eqnarray}
\partial_x \varphi_{-\alpha} &=& \mbox{i} \sqrt \pi \xi_{\alpha}^3 \xi_{\alpha}^0,
\;\; \alpha = R,L \nonumber \\
\sin\left(\sqrt{4\pi} \varphi_{-R}\rightarrow) &=& \sqrt{\pi a_0} \; \mbox{i}
\xi_{R}^0 \kappa \nonumber \\
\sin\left(\sqrt{4\pi} \varphi_{-L}\rightarrow) &=& -\sqrt{\pi a_0} \; \mbox{i}
\xi_{L}^0 \kappa .
\label{sinphim}
\end{eqnarray}
With these relations at hand, we can now relate the spin
currents of the surface chains (\ref{bosoout1}) as a function of the Majorana
fermions $\xi^0, \xi^3$ and the bosonic field $\Phi$:
\begin{eqnarray}
J_{aR,L}^z &=&
\frac{1}{2\sqrt{\pi}}\partial_x \Phi_{R,L}
+ \frac{\mbox{i}}{2} \tau_a \xi_{R,L}^3 \xi_{R,L}^0 \nonumber\\
J_{aR}^{\pm} &=& \frac{1}{2\sqrt{\pi a_0}}\;
\mbox{e}^{\mp \mbox{i} \sqrt{4\pi} \Phi_{R}}
\left(\mbox{i} \xi_{R}^3\kappa \pm \tau_a \xi_{R}^0\kappa\rightarrow)
\nonumber\\
J_{aL}^{\pm} &=& \frac{1}{2\sqrt{\pi a_0}}\;
\mbox{e}^{\pm \mbox{i} \sqrt{4\pi} \Phi_{L}}
\left(\mbox{i} \xi_{L}^3\kappa \pm \tau_a \xi_{L}^0\kappa\rightarrow).
\end{eqnarray}
Under the canonical transformation (\ref{can1}, \ref{can2}),
the Majorana fermions $\xi^0, \xi^3$ remain unchanged while
the chiral bosonic fields $\Phi_{R,L}$ transform at
the Toulouse point according to:
\begin{eqnarray}
\Phi_{R} &=& \sqrt2 \bar \Phi_{1R}
- \bar \Phi_{2L} \nonumber \\
\Phi_{L} &=& \sqrt2 \bar \Phi_{1L} - \bar \Phi_{2R}.
\label{transphirl}
\end{eqnarray}
Therefore, using the relation (\ref{parphi2rl})
between the Majorana fields $\zeta, \eta$ and the
bosonic field $\bar \Phi_2$, we obtain for the z-component:
\begin{eqnarray}
J_{aR}^z &=& {\cal J}_R^z - \frac{\mbox{i}}{2}
\left(\eta_L\zeta_L - \tau_a \xi_R^3 \xi_R^0\rightarrow)
\nonumber\\
J_{aL}^z &=& {\cal J}_L^z - \frac{\mbox{i}}{2}
\left(\eta_R\zeta_R - \tau_a \xi_L^3 \xi_L^0\rightarrow),
\label{outercurz}
\end{eqnarray}
and ignoring the short-ranged contributions, we get:
\begin{equation}
J_{aR,L}^z \sim {\cal J}_{R,L}^z .
\label{zpartcurrenouter}
\end{equation}
In the same way, the transverse parts of
the spin currents of the surface chains simplify as
follows:
\begin{eqnarray}
J_{aR}^{\pm} &=& -\pi a_0 {\cal J}_R^{\pm}\left(
\mbox{i} \eta_L\xi_R^3 \pm \zeta_L \xi_R^3 \pm \tau_a \eta_L
\xi_R^0 - \mbox{i} \tau_a \zeta_L \xi_R^0\rightarrow) \nonumber\\
J_{aL}^{\pm} &=& -\pi a_0 {\cal J}_L^{\pm}\left(
\mbox{i} \eta_R\xi_L^3 \pm \zeta_R \xi_L^3 \pm \tau_a \eta_R
\xi_L^0 - \mbox{i} \tau_a \zeta_R \xi_L^0\rightarrow).
\label{outercurpm}
\end{eqnarray}
Majorana bilinears, in this equation, built from
$\xi^3_{R,L}$ and $\xi^3_{R,L}$ are short-ranged
and from the hybridization (\ref{hybrid}), the only
non-zero vacuum expectation values are:
\begin{equation}
{\bar \gamma}_{R,L} = \mbox{i} \pi a_0
\langle \xi^3_{R,L} \eta_{L,R} \rangle.
\label{expec2}
\end{equation}
The value of this expectation value is estimated
in Appendix A and one finds: ${\bar \gamma}_{R} =
{\bar \gamma}_{L} = {\bar \gamma}_{m}$ (${\bar \gamma}_{m}$
being real). With the same accuracy than in Eq. (\ref{centrcurpm2}),
the transverse spin currents of the surface chains are  thus
given by:
\begin{eqnarray}
J_{aR}^{\pm} &\sim& {\cal J}_R^{\pm}\left(
{\bar \gamma}_{m} + \mbox{i} a_0 \tau_a\pi\zeta_L \xi_R^0\rightarrow) \nonumber\\
J_{aL}^{\pm} &\sim& {\cal J}_L^{\pm}\left(
{\bar \gamma}_{m} + \mbox{i} a_0 \tau_a\pi
\zeta_R \xi_L^0\rightarrow).
\label{outercurpmfin}
\end{eqnarray}

\subsubsection{Representation of the staggered spin fields}

Let us first analyse the staggered spin field of the central
chain ${\bf n}_0$. As in Eq. (\ref{stagbosophi1}), this field
can be expressed in terms of the bosonic field $\varphi$
and its dual field $\vartheta$:
\begin{eqnarray}
n_0^z &=& -\frac{\lambda}{\pi a_0} \sin\left(\sqrt{2\pi}\varphi\rightarrow)
\nonumber\\
n_0^{\pm} &=& \frac{\lambda}{\pi a_0}
\mbox{e}^{\pm \mbox{i} \sqrt{2\pi} \vartheta}.
\label{stagbosovarphi}
\end{eqnarray}
Using the transfomation of the chiral bosonic fields (\ref{transphichir}),
one finds at the Toulouse point:
\begin{eqnarray}
\varphi &=& \sqrt 2 \bar \Phi_2 - \bar \Phi_1 \nonumber \\
\vartheta &=& \sqrt 2 \bar \Theta_2 + \bar \Theta_1.
\label{transtouvar}
\end{eqnarray}
We deduce then the following result:
\begin{eqnarray}
n_0^z &=& -\frac{\lambda}{\pi a_0} \left(
\sin\left(\sqrt{4\pi}\bar\Phi_2\rightarrow)
\cos\left(\sqrt{2\pi}\bar\Phi_1\rightarrow)
-\sin\left(\sqrt{2\pi}\bar\Phi_1\rightarrow)
\cos\left(\sqrt{4\pi}\bar\Phi_2\rightarrow)\rightarrow)
\nonumber\\
n_0^{\pm} &=& \frac{\lambda}{\pi a_0}
\mbox{e}^{\pm \mbox{i} \sqrt{2\pi} \bar \Theta_1}
\mbox{e}^{\pm \mbox{i} \sqrt{4\pi} \bar \Theta_2}.
\label{stagbosovarphi1}
\end{eqnarray}
One can simplify these expressions
using the correspondence (\ref{psi})
to obtain the relations:
\begin{eqnarray}
\cos\left(\sqrt{4\pi}\bar \Phi_2\rightarrow) &=&
\mbox{i} \pi a_0 \left(\eta_R \eta_L + \zeta_R \zeta_L\rightarrow) \nonumber \\
\sin\left(\sqrt{4\pi}\bar \Phi_2\rightarrow) &=&
\mbox{i} \pi a_0 \left(-\eta_R \zeta_L + \zeta_R \eta_L\rightarrow) \nonumber \\
\mbox{e}^{\pm \mbox{i} \sqrt{4\pi} \bar \Theta_2} &=&
\mbox{i} \pi a_0 \left(-\eta_R \eta_L \pm \mbox{i} \zeta_R \eta_L \pm \mbox{i}
\eta_R \zeta_L + \zeta_R \zeta_L\rightarrow),
\label{referformphi2}
\end{eqnarray}
from which, we deduce the leading part of the
representation of the staggered spin fields ${\bf n}_0$
associated with the central chain at the Toulouse point:
\begin{eqnarray}
n_0^z &\sim& - \mbox{i} \pi a_0 \zeta_R \zeta_L {\cal N}^z \nonumber\\
n_0^{\pm} &\sim& \mbox{i} \pi a_0 \zeta_R \zeta_L {\cal N}^{\pm}
\label{staggeredrepcent}
\end{eqnarray}
where ${\vec {\cal N}}$ corresponds to the staggered magnetization
of the effective S=1/2 Heisenberg spin
chain (see Eq. (\ref{stagbosophi1})).

The same correspondence for the staggered magnetizations (${\bf n}_a$)
of the surface chains is more difficult to obtain. The reason
for this is that it
involves nonlocal strings of the Majorana
fermions $\xi^a,\; \eta$ and $\zeta$. As we shall see,
the trick will be to relate
all these quantities to the order and disorder parameters
of the underlying Ising models
present in the problem. The Appendix B gives a review of
the basic definitions for the bosonization of two Ising
models and introduces also the different fields which
will be useful in this subsection.

The staggered part (${\bf n}_a, a=1,2$) of the spin density
of the surface chains can be expressed in terms of
the bosonic field $\varphi_a$ and its dual $\vartheta_a$ as in
Eq. (\ref{stagbosovarphi}):
\begin{eqnarray}
n_a^z &=& -\frac{\lambda}{\pi a_0} \sin\left(\sqrt{2\pi}\varphi_a\rightarrow)
\nonumber\\
n_a^{\pm} &=& \frac{\lambda}{\pi a_0}
\mbox{e}^{\pm \mbox{i} \sqrt{2\pi} \vartheta_a}.
\label{stagbosophia}
\end{eqnarray}
Introducing the symmetric and antisymmetric combinations (\ref{phipm})
of the bosonic fields, we get:
\begin{eqnarray}
n_a^z &=& -\frac{\lambda}{\pi a_0} \left(
\sin\left(\sqrt{\pi}\Phi\rightarrow)
\cos\left(\sqrt{\pi}\varphi_-\rightarrow)
+\tau_a\cos\left(\sqrt{\pi}\Phi\rightarrow)
\sin\left(\sqrt{\pi}\varphi_-\rightarrow)\rightarrow)
\nonumber\\
n_a^{\pm} &=& \frac{\lambda}{\pi a_0}
\mbox{e}^{\pm \mbox{i} \sqrt{\pi} \Theta}
\mbox{e}^{\pm \mbox{i} \tau_a \sqrt{\pi} \vartheta_-},
\label{stagbosovarphia1}
\end{eqnarray}
where we have used the fact that $\varphi_{+R,L}$ identifies to
the bosonic fields $\Phi_{R,L}$.
According to the canonical transformation (\ref{transphirl}) at the Toulouse
point, the bosonic fields $\Phi, \Theta$ transform as:
\begin{eqnarray}
\Phi &=& \sqrt 2 \bar \Phi_1 - \bar \Phi_2 \nonumber \\
\Theta &=& \sqrt 2 \bar \Theta_1 + \bar \Theta_2
\label{transtouPhi}
\end{eqnarray}
whereas the fields $\varphi_-, \vartheta_-$ are not
affected by this transformation.
Using Eqs. (\ref{Isibosigmu03}, \ref{Isibosigmu45})
of the Appendix B, one can then relate
the staggered spin fields ${\bf n}_a$ of
the surface chains to the diffferent order-disorder
operators of the underlying Ising models of the problem:
\begin{eqnarray}
n_a^z = -\frac{\lambda}{\pi a_0} \left(
\sin\left(\sqrt{2\pi}{\bar \Phi}_1\rightarrow)
\left(\mu_5 \mu_4 \mu_3 \mu_0 + \tau_a \sigma_5 \sigma_4
\sigma_3 \sigma_0\rightarrow) \rightarrow. \nonumber \\
\left. + \cos\left(\sqrt{2\pi}{\bar \Phi}_1\rightarrow)
\left(-\sigma_5 \sigma_4 \mu_3 \mu_0 + \tau_a \mu_5 \mu_4
\sigma_3 \sigma_0\rightarrow) \rightarrow)
\nonumber\\
n_a^{\pm} = \frac{\lambda}{\pi a_0}
\mbox{e}^{\pm \mbox{i} \sqrt{2\pi} {\bar \Theta}_1}
\left(\sigma_5 \mu_4 \sigma_3 \mu_0 - \tau_a
\mu_5 \sigma_4 \mu_3 \sigma_0 \rightarrow. \nonumber \\
\left. \pm \mbox{i}
\mu_5 \sigma_4 \sigma_3 \mu_0 \pm \mbox{i} \tau_a
\sigma_5 \mu_4 \mu_3 \sigma_0\rightarrow).
\label{stagbosovarphia2}
\end{eqnarray}
The Ising models labelled ``3,5'' are non-critical due to
the hybridization (\ref{hybrid}) of the Majorana fermions $\eta, \xi^3$
at the Toulouse point.
As shown in Appendix C, the order-disorder operators
($\sigma_{3,5}, \mu_{3,5}$)
of the corresponding
Ising models
are short-ranged and have the following vaccum expectation values:
\begin{eqnarray}
\langle \mu_3 \mu_5 \rangle = \langle \sigma_3 \sigma_5 \rangle = \mu \ne 0
\nonumber \\
\langle \sigma_3 \mu_5 \rangle = \langle \mu_3 \sigma_5 \rangle = 0.
\label{sigmamuvacc}
\end{eqnarray}
Consequently, ignoring short-ranged
contribution in Eq. (\ref{stagbosovarphia2}), we finally obtain
the correspondence between
the staggered magnetization of the surface chains and
the critical fields at the chiral fixed point:
\begin{eqnarray}
n_a^z &\sim& \mu {\cal N}^z\left(
\mu_4 \mu_0 + \tau_a \sigma_4 \sigma_0\rightarrow)
\nonumber\\
n_a^{\pm} &\sim& \mu {\cal N}^{\pm}
\left(\mu_4 \mu_0 - \tau_a
\sigma_4 \sigma_0 \rightarrow).
\label{nsurfacefin}
\end{eqnarray}

Let us summarize our results obtained in this subsection.
Using our Toulouse point approach,
the spin currents and staggered fields of
the three chains have been expressed in terms
of the critical fields at the Toulouse point.
Up to short-ranged pieces, the spin currents are given
by:
\begin{eqnarray}
J_{0R,L}^z &\sim& - {\cal J}_{L,R}^z \nonumber \\
J_{0R,L}^{\pm} &\sim& \gamma_m {\cal J}_{L,R}^{\pm} \nonumber \\
J_{aR,L}^z &\sim& {\cal J}_{R,L}^z \nonumber \\
J_{aR}^{\pm} &\sim& {\cal J}_R^{\pm}\left(
{\bar \gamma}_{m} + \mbox{i} a_0 \tau_a\pi\zeta_L \xi_R^0\rightarrow) \nonumber\\
J_{aL}^{\pm} &\sim& {\cal J}_L^{\pm}\left(
{\bar \gamma}_{m} + \mbox{i} a_0 \tau_a\pi
\zeta_R \xi_L^0\rightarrow)
\label{spincurrrepfinal}
\end{eqnarray}
whereas the staggered magnetizations of the different chains
write as follows
\begin{eqnarray}
n_0^z &\sim& - \mbox{i} \pi a_0 \zeta_R \zeta_L {\cal N}^z \nonumber\\
n_0^{\pm} &\sim& \mbox{i} \pi a_0 \zeta_R \zeta_L {\cal N}^{\pm} \nonumber \\
n_a^z &\sim& \mu {\cal N}^z\left(
\mu_4 \mu_0 + \tau_a \sigma_4 \sigma_0\rightarrow)
\nonumber\\
n_a^{\pm} &\sim& \mu {\cal N}^{\pm}
\left(\mu_4 \mu_0 - \tau_a
\sigma_4 \sigma_0 \rightarrow).
\label{stagrepfinal}
\end{eqnarray}
These results enable us to
discuss the spectral properties of the 3-chain cylinder model
as well as the computation
of the leading
asymptotics of the spin-spin correlation functions
at the chiral fixed point as described in
the following section.

\subsection{Spectral properties}

There are two different kinds of massless elementary excitations
at the Toulouse point:
magnetic excitation described
by the field $\bar{\Phi}_1$, and non-magnetic, singlet, excitations
associated with the two Majorana fermions $\xi^0$ and $\zeta$.
These two groups of excitations are decoupled at the chiral fixed
point. Let us first discuss the nature of the magnetic excitations.

\subsubsection{Magnetic excitations}

The magnetic elementary excitations of the system correspond
to the spinons of the effective S=1/2 Heisenberg chain described by
the bosonic field $\bar{\Phi}_1$.
These spinons appear only in pairs in all physical states and
carry a spin S=1/2.
It is very important to notice that, due to the mixing
of different degrees of freedom
reflected in the canonical transformation (\ref{can1},\ref{can2}),
the ``physical'' spinons, $i.e.$ those defined as
$\sqrt{\pi/2}$-kinks of the field $\bar{\Phi}_1$
should not be misleadingly identified as
the spinons of the central chain.
To get a better understanding of the
structure of the spin excitations at the chiral fixed point,
let us recall the
expressions (\ref{centrcurz1}, \ref{outercurz}) of the currents
$J^z _{1,2}$ and $J^z _0$
in terms of the ``physical'' current
${\cal J}^z = (1/\sqrt{2 \pi}) \partial_x\bar{\Phi}_1$ at the Toulouse
point obtained in the previous subsection:
\begin{eqnarray}
J^z _{1R(L)} &=& {\cal J}^z _{R(L)} - \frac{\mbox{i}}{2}
\left(\eta_{L(R)} \zeta_{L(R)} - \xi^3 _{R(L)} \xi^0 _{R(L)} \rightarrow),
\nonumber\\
J^z _{2R(L)} &=& {\cal J}^z _{R(L)} - \frac{\mbox{i}}{2}
\left(\eta_{L(R)} \zeta_{L(R)} + \xi^3 _{R(L)} \xi^0 _{R(L)} \rightarrow),
\nonumber \\
J^z _{0R(L)} &=& - {\cal J}^z _{L(R)} + \mbox{i} \eta_{R(L)} \zeta_{R(L)}.
\label{curr}
\end{eqnarray}
At energies $|\omega| \gg m$, where all Majorana
fields can be considered
as
massless, Eqs. (\ref{curr}) transform back to the
standard definitions of the currents
of the three decoupled chains. In this UV limit, one has a
picture
of three groups of independently propagating spinons. However, in the IR
limit ($|\omega| \ll m$), all Majorana bilinears in
Eq. (\ref{curr}) are characterized by short-ranged correlations (since the
Majorana fermions $\eta$ and $\xi^3$ are massive),
implying that strongly fluctuating
parts of the currents of individual chains are no longer independent.
The spinons of the individual chains turn out to be strongly
correlated in the IR limit to participate at
the formation of a single, physical,
current ${\cal J}^z$.
In fact,
the physical spinon represents a chirally asymmetric,
strongly correlated
state of {\it three} spinons.
Indeed,
consider, for instance, a right-moving
$\sqrt{\pi/2}$-kink of the field $\bar{\Phi}_1$, representing a physical
spinon with the spin projection $S^z = 1/2$. According to the exact
relation
\begin{equation}
{\cal J}^z _{R(L)} = J^z _{1R(L)} + J^z _{2R(L)} + J^z _{0L(R)}
\label{rel}
\end{equation}
following from Eq. (\ref{curr}), such
an excitation is a combination of two right-moving spinons of the surface
chains, each carrying the spin $S^z = 1/2$, and a left-moving antispinon
of the
central chain, with $S^z = - 1/2$.
The rigidity of such a state is ensured by a finite mass gap in the
$(\eta-\xi^3)$
sector of the model.
Therefore, in physical excitations, two spinons of the surface
chains form a bound state with an antispinon of the central chain.
Otherwisely stated, the polarization of the long-wavelength
magnetic excitations of the surface chains is always the same but
opposite to the polarization of the central chain.
This peculiar structure of the elementary
spin excitations at the chiral fixed point is clearly reflected by the
expression for the spin velocity $u_1$ of the bosonic field
${\bar{\Phi}}_1$: $u_1 = (2v_1 - v_0)/3$.

These excitations correspond to the magnetic excitations of the
model since a uniform magnetic field $H$ along the z direction
only couples to the ${\bar \Phi}_1$ field.
Indeed,
in the continuum limit, this magnetic field couples to the smooth part
of the total spin density of the three chains:
\begin{equation}
{\cal H}_H = - \frac{H}{\sqrt{2\pi}} \left(\partial_x \varphi_1 +
\partial_x \varphi_1 + \partial_x \varphi_1\rightarrow),
\label{magnecham}
\end{equation}
and performing the Toulouse
transformation (\ref{transtouvar}, \ref{transtouPhi}),
one obtains:
\begin{equation}
{\cal H}_H = - \frac{H}{\sqrt{2\pi}} \; \partial_x {\bar \Phi}_1.
\label{magnechamtou}
\end{equation}
The uniform susceptibility $\chi$ of the model corresponds thus
to that of a single Heisenberg S=1/2 spin chain
and it is given by (in units of $g\mu_B$):
\begin{equation}
\chi = \frac{1}{2\pi u_1} = \frac{3}{2\pi \left(2v_1 - v_0\rightarrow)}.
\label{unifsuscept}
\end{equation}
Notice that due to the non-trivial structure
of the
spinon at the Toulouse point,
the uniform susceptibility of each chain does not add in the
case of a truly uniform field. In contrast, the susceptibilities
would add coherently if one considers a staggered
magnetic field ($H_a$) in the transverse direction:
$H_a = H(\delta_{a1} + \delta_{a2} -
\delta_{a0})$, $a$ being the chain index.
In that case, one finds the contribution:
\begin{equation}
\chi_{\perp}^{stag} = \frac{3}{2\pi u_1}.
\label{unifstagsuscept}
\end{equation}

\subsubsection{Pseudo-charge degrees of freedom}

Apart from the non-trivial nature of the spinon, the chiral
fixed point manifests
itself in the existence of massless Z$_2$ singlet excitations.
A first non-magnetic excitation stems from the Majorana fermion
$\xi^0$ that decouples from the rest of the spectrum.
This Majorana fermion describes collective
excitations of singlet pairs
formed on the surface chains.
The nature of the other massless
Majorana fermion $\zeta$ at the chiral fixed point
is less transparent: It is a highly nonlocal
object when expressed
in terms of the original spin operators.
A simple way to understand the role
of these singlet excitations is to combine the
Majorana $\xi^0$ and $\zeta$ fields
into a single Dirac $\Psi_c$
fermion and then bosonize it with the introduction of
a bosonic field ${\tilde \Phi}_c$:
\begin{eqnarray}
\Psi_{cR} &=& \frac{\xi^0_R + \mbox{i} \zeta_R}{\sqrt{2}}
= \frac{1}{\sqrt{2\pi a_0}}\mbox{e}^{\mbox{i} \sqrt{4\pi} {\tilde \Phi}_{cR}}
\nonumber \\
\Psi_{cL} &=& \frac{\xi^0_L + \mbox{i} \zeta_L}{\sqrt{2}}
= \frac{1}{\sqrt{2\pi a_0}}\mbox{e}^{-\mbox{i} \sqrt{4\pi} {\tilde \Phi}_{cL}}.
\label{pseudodef}
\end{eqnarray}
The corresponding
massless
bosonic field ${\tilde \Phi}_c$ ressembles
the scalar field describing the charge
degrees of freedom in the one-dimensional SU(2)
Hubbard model away from half-filling
in the limit  $U=\infty$.
This analogy can be seen by looking at the representation
(\ref{stagrepfinal}) of the staggered fields at the Toulouse point.
Using the bosonization result of two Ising models (\ref{Isibosigmu}),
the total staggered magnetization ${\bf n}_{+}$ of the surface chains
reads as follows in terms of the bosonic field  ${\tilde \Phi}_c$:
\begin{eqnarray}
{\bf n}_{+} = {\bf n}_{1} + {\bf n}_{2}
            &\sim& 2 \mu \mu_4 \mu_0 {\vec {\cal N}} \nonumber \\
            &\sim& \cos\left(\sqrt{\pi} {\tilde \Phi}_c \rightarrow)
{\vec {\cal N}}.
\label{hubbardrepuinf}
\end{eqnarray}
This bosonized version of ${\bf n}_{+}$ is
reminiscent of the staggered magnetization of the Hubbard
model away from half-filling at $U=\infty$ ($K_c =1/2$ in that
case, see for instance
Refs.\cite{book,book1}).
Apart from a velocity anisotropy ($v_0 \ne u_2$),
the fermionic fields $\xi^0$ and $\zeta$ plays the role of the
charge degrees of freedom in the corresponding Hubbard model.
We shall hence
refer to the gapless Majorana fermions $\xi^0$ and $\zeta$ as
``pseudo-charge'' excitations which account for
the central charge $c=1$ at the IR fixed point.
The spin and pseudo-charge separation is already
manifest at the Toulouse point in the Hamiltonian
(\ref{hfintot}) since
these fermions fields are decoupled from the magnetic excitations
described by the bosonic field ${\bar \Phi}_1$.
This separation of the different modes will also be clearly seen in
the expression of the
leading asymptotics of correlations functions
that we now estimate.

\subsection{Spin-spin correlation functions at the chiral fixed point}

We are in position to compute the leading
asymptotics of the spin-spin correlation functions of
the 3-chain cylinder model
at the chiral fixed point. Let us begin
our analysis by considering the spin-spin correlation functions
between the central spins.

\subsubsection{Correlation functions between the central spins}

Since the SU(2) symmetry is broken to an U(1) symmetry in
our Toulouse point approach ($g_{\parallel} \ne g_{\perp}$),
we shall discriminate between the z and perpendicular
components of the correlation function:
\begin{eqnarray}
\langle S_0^z\left(x,\tau\rightarrow) S_0^z\left(0,0\rightarrow) \rangle =
\langle J_{0R}^z\left(x,\tau\rightarrow) J_{0R}^z\left(0,0\rightarrow) \rangle +
\left(R\rightarrow L\rightarrow) \nonumber \\
+
\left(-1\rightarrow)^{x/a_0}
\langle n_0^z\left(x,\tau\rightarrow) n_0^z\left(0,0\rightarrow) \rangle,
\label{spincorzmiddle}
\end{eqnarray}
\begin{eqnarray}
\langle S_0^{+}\left(x,\tau\rightarrow) S_0^{-}\left(0,0\rightarrow) \rangle =
\langle J_{0R}^{+}\left(x,\tau\rightarrow) J_{0R}^{-}\left(0,0\rightarrow) \rangle +
\left(R\rightarrow L\rightarrow) \nonumber \\
+
\left(-1\rightarrow)^{x/a_0} \langle n_0^{+}\left(x,\tau\rightarrow)
n_0^{-}\left(0,0\rightarrow) \rangle.
\label{spincorpmmiddle}
\end{eqnarray}
Using the representation (\ref{spincurrrepfinal}, \ref{stagrepfinal}) of
the uniform and staggered parts of the spin density of
the central chain at the Toulouse point,
we get the following leading behaviours:
\begin{eqnarray}
\langle S_0^z\left(x,\tau\rightarrow) S_0^z\left(0,0\rightarrow) \rangle \sim
\langle {\cal J}_{R}^z\left(x,\tau\rightarrow)
{\cal J}_{R}^z\left(0,0\rightarrow) \rangle +
\left(R\rightarrow L\rightarrow) \nonumber \\
+
\left(-1\rightarrow)^{x/a_0} \pi^2 a_0^2
\langle {\cal N}^z\left(x,\tau\rightarrow) {\cal N}^z\left(0,0\rightarrow) \rangle
\langle \zeta_R\left(x,\tau\rightarrow) \zeta_R\left(0,0\rightarrow) \rangle
\langle \zeta_L\left(x,\tau\rightarrow) \zeta_L\left(0,0\rightarrow) \rangle
\label{spincorzmiddlenext}
\end{eqnarray}
and
\begin{eqnarray}
\langle S_0^{+}\left(x,\tau\rightarrow) S_0^{-}\left(0,0\rightarrow) \rangle \sim
\gamma_m^2 \langle {\cal J}_{R}^{+}\left(x,\tau\rightarrow)
{\cal J}_{R}^{-}\left(0,0\rightarrow) \rangle +
\left(R\rightarrow L\rightarrow) \nonumber \\
+
\left(-1\rightarrow)^{x/a_0}
\pi^2 a_0^2
\langle {\cal N}^+\left(x,\tau\rightarrow) {\cal N}^-\left(0,0\rightarrow) \rangle
\langle \zeta_R\left(x,\tau\rightarrow) \zeta_R\left(0,0\rightarrow) \rangle
\langle \zeta_L\left(x,\tau\rightarrow) \zeta_L\left(0,0\rightarrow) \rangle.
\label{spincorpmmiddlenext}
\end{eqnarray}
Accordingly, the estimation of these correlations simply reduces to
the computation of two-point functions of a massless bosonic field
${\bar \Phi}_1$ and a massless Majorana fermion $\zeta$.
The final expressions
for the leading
asymptotics of the correlation functions
between spins in the central chain at the chiral fixed point are thus
\begin{eqnarray}
\langle S_0^z\left(x,\tau\rightarrow) S_0^z\left(0,0\rightarrow) \rangle &\sim&
- \frac{1}{8\pi^2}
\left(\frac{1}{\left(x + \mbox{i} u_1 \tau\rightarrow)^2} +
\frac{1}{\left(x - \mbox{i} u_1 \tau\rightarrow)^2} \rightarrow) \nonumber\\
&+& \left(-1\rightarrow)^{x/a_0} \; \frac{\lambda^2 a_0}{8\pi^2}\;
\frac{1}{\left(x^2+u_2^2\tau^2\rightarrow)}
\frac{1}{\left(x^2+u_1^2\tau^2\rightarrow)^{1/2}},
\label{spincorzfinmiddle}
\end{eqnarray}
\begin{eqnarray}
\langle S_0^{+}\left(x,\tau\rightarrow) S_0^{-}\left(0,0\rightarrow) \rangle &\sim&
- \frac{\gamma_{m}^2 }{4\pi^2}  \;
\left(\frac{1}{\left(x + \mbox{i} u_1 \tau\rightarrow)^2} +
\frac{1}{\left(x - \mbox{i} u_1 \tau\rightarrow)^2} \rightarrow) \nonumber \\
&+& \left(-1\rightarrow)^{x/a_0} \; \frac{\lambda^2 a_0}{4\pi^2}\;
\frac{1}{\left(x^2+u_2^2\tau^2\rightarrow)}
\frac{1}{\left(x^2+u_1^2\tau^2\rightarrow)^{1/2}}.
\label{spincorpmfinmiddle}
\end{eqnarray}

\subsubsection{Correlation functions between spins of the surface chains}

Let us now compute the spin-spin correlation functions between
the surface chains:
\begin{eqnarray}
\langle S^z_a\left(x,\tau\rightarrow) S^z_b\left(0,0\rightarrow) \rangle =
\langle J_{aR}^z\left(x,\tau\rightarrow) J_{bR}^z\left(0,0\rightarrow) \rangle +
\left(R\rightarrow L\rightarrow) +
\left(-1\rightarrow)^{x/a_0}
\langle n^z_a\left(x,\tau\rightarrow) n^z_b\left(0,0\rightarrow) \rangle,
\label{spincorzout}
\end{eqnarray}
\begin{eqnarray}
\langle S^{+}_a\left(x,\tau\rightarrow) S^{-}_b\left(0,0\rightarrow) \rangle =
\langle J_{aR}^{+}\left(x,\tau\rightarrow) J_{bR}^{-}\left(0,0\rightarrow) \rangle +
\left(R\rightarrow L\rightarrow) +
\left(-1\rightarrow)^{x/a_0} \langle n^{+}_a\left(x,\tau\rightarrow)
n^{-}_b\left(0,0\rightarrow) \rangle.
\label{spincorpmout}
\end{eqnarray}
where $a,b=(1,2)$ is the chain index.
We shall follow the same route as
for the central chain
by using the expression of the uniform and staggered
spin fields (\ref{spincurrrepfinal}, \ref{stagrepfinal})
at the Toulouse point:
\begin{eqnarray}
\langle S_a^z\left(x,\tau\rightarrow) S_b^z\left(0,0\rightarrow) \rangle \sim
\langle {\cal J}_{R}^z\left(x,\tau\rightarrow)
{\cal J}_{R}^z\left(0,0\rightarrow) \rangle +
\left(R\rightarrow L\rightarrow) \nonumber \\
+
\left(-1\rightarrow)^{x/a_0} \mu^2
\langle {\cal N}^z\left(x,\tau\rightarrow) {\cal N}^z\left(0,0\rightarrow) \rangle
\nonumber \\
\left(
\langle \mu_4\left(x,\tau\rightarrow) \mu_4\left(0,0\rightarrow) \rangle
\langle \mu_0\left(x,\tau\rightarrow) \mu_0\left(0,0\rightarrow) \rangle
+ \tau_a \tau_b
\langle \sigma_4\left(x,\tau\rightarrow) \sigma_4\left(0,0\rightarrow) \rangle
\langle \sigma_0\left(x,\tau\rightarrow) \sigma_0\left(0,0\rightarrow) \rangle
\rightarrow)
\label{spincorzoutnext}
\end{eqnarray}
where we recall that $\tau_1=1,\tau_2 =-1$
and the perpendicular part reads
\begin{eqnarray}
\langle S_a^{+}\left(x,\tau\rightarrow) S_b^{-}\left(0,0\rightarrow) \rangle \sim
\langle {\cal J}_{R}^{+}\left(x,\tau\rightarrow)
{\cal J}_{R}^{-}\left(0,0\rightarrow) \rangle
\left({\bar \gamma}_m^2 + \tau_a \tau_b a_0^2
\langle \zeta_L\left(x,\tau\rightarrow) \zeta_L\left(0,0\rightarrow) \rangle
\langle \xi^0_R\left(x,\tau\rightarrow) \xi^0_R\left(0,0\rightarrow) \rangle
\rightarrow)
\nonumber \\
+\left(R,L\rightarrow)\rightarrow \left(L,R\rightarrow)
+
\left(-1\rightarrow)^{x/a_0}
\mu^2
\langle {\cal N}^+\left(x,\tau\rightarrow) {\cal N}^-\left(0,0\rightarrow) \rangle
\nonumber \\
\left(
\langle \mu_4\left(x,\tau\rightarrow) \mu_4\left(0,0\rightarrow) \rangle
\langle \mu_0\left(x,\tau\rightarrow) \mu_0\left(0,0\rightarrow) \rangle
+ \tau_a \tau_b
\langle \sigma_4\left(x,\tau\rightarrow) \sigma_4\left(0,0\rightarrow) \rangle
\langle \sigma_0\left(x,\tau\rightarrow) \sigma_0\left(0,0\rightarrow) \rangle
\rightarrow).
\label{spincorpmoutnext}
\end{eqnarray}
The Ising models labelled by ``0'' and ``4''
associated with respectively the Majorana fermion $\xi^0$
and $\zeta$
are decoupled and
critical at the chiral fixed point.
The correlation functions of the order and disorder operators
$\sigma_{0,4}, \mu_{0,4}$
are known exactly. In the
long time and long distance limit they are given by (see Appendix B):
\begin{eqnarray}
\langle \mu_0\left(x,\tau\rightarrow) \mu_0\left(0,0\rightarrow)\rangle &=&
\langle \sigma_0\left(x,\tau\rightarrow)
\sigma_0\left(0,0\rightarrow)\rangle  \nonumber \\
 &\sim&\frac{a_0^{1/4}}{(x^2 + v_1^2
\tau^2)^{1/8}},
\label{sigmucor0}
\end{eqnarray}
\begin{eqnarray}
\langle \mu_4\left(x,\tau\rightarrow) \mu_4\left(0,0\rightarrow)\rangle &=&
\langle \sigma_4\left(x,\tau\rightarrow) \sigma_4\left(0,0\rightarrow)\rangle
\nonumber \\
 &\sim&\frac{a_0^{1/4}}{(x^2 + u_2^2
\tau^2)^{1/8}}.
\label{sigmucor4}
\end{eqnarray}
This enables us to obtain
the result for the asymptotics of
the correlation functions between spins
of the surface chains:
\begin{eqnarray}
\langle S^z_a\left(x,\tau\rightarrow) S^z_b\left(0,0\rightarrow) \rangle \sim
 -\frac{1}{8\pi^2}
\left(\frac{1}{\left(x + \mbox{i} u_1 \tau\rightarrow)^2} +
\frac{1}{\left(x - \mbox{i} u_1 \tau\rightarrow)^2} \rightarrow) \nonumber \\
+ (-1)^{x/a_0}\delta_{ab}
\frac{\mu^2\lambda^2}{\pi^2 \sqrt a_0} \frac{1}{\left(x^2 +
u_1^2\tau^2\rightarrow)^{1/2}} \frac{1}{\left(x^2 + v_1^2
\tau^2\rightarrow)^{1/8}}\frac{1}{\left(x^2 + u_2^2
\tau^2\rightarrow)^{1/8}},
\label{spincorzoutfin}
\end{eqnarray}
\begin{eqnarray}
\langle S^{+}_a\left(x,\tau\rightarrow) S^{-}_b\left(0,0\rightarrow) \rangle \sim
-\frac{{\bar \gamma}_{m}^2}{4\pi^2}
\left(\frac{1}{\left(x + \mbox{i} u_1 \tau\rightarrow)^2}
+
\frac{1}{\left(x - \mbox{i} u_1 \tau\rightarrow)^2}\rightarrow)
\nonumber \\
- \frac{a_0^2 \tau_a \tau_b}{16 \pi^4}
\left[
\frac{1}{\left(x - \mbox{i} u_1 \tau\rightarrow)^2}
\frac{1}{\left(x - \mbox{i} v_1 \tau\rightarrow)}
\frac{1}{\left(x + \mbox{i} u_2 \tau\rightarrow)}
+
\frac{1}{\left(x + \mbox{i} u_1 \tau\rightarrow)^2}
\frac{1}{\left(x + \mbox{i} v_1 \tau\rightarrow)}
\frac{1}{\left(x - \mbox{i} u_2 \tau\rightarrow)}
\rightarrow] \nonumber \\
+ (-1)^{x/a_0} \delta_{ab}
\frac{2 \mu^2 \lambda^2}{\pi^2 \sqrt a_0} \frac{1}{\left(x^2 +
u_1^2\tau^2\rightarrow)^{1/2} } \frac{1}{\left(x^2 + v_1^2
\tau^2\rightarrow)^{1/8}}\frac{1}{\left(x^2 + u_2^2
\tau^2\rightarrow)^{1/8}}.
\label{spincorpmoutfin}
\end{eqnarray}
Here again, the leading contribution of the uniform
part of the correlation functions is that of a single
S=1/2 Heisenberg spin chain. The other critical modes
manifest themselves in subleading contributions in the uniform part
and directly in the staggered part.
In particular, the contributions of the Majorana fermions
$\xi^0$ and $\zeta$ in the staggered part show up through
their corresponding order and disorder Ising operators.

\subsubsection{Correlation functions between spins of
the surface chains and of the central chain}

The last spin-spin correlation functions to estimate
at the chiral fixed point
consists in correlation between a spin of the surface
chain ${\bf S}_a$ ($a=1,2$) and a spin of the central chain ${\bf S}_0$.
Using the representation (\ref{spincurrrepfinal}, \ref{stagrepfinal})
of the different
spin fields at the Toulouse point,
we obtain following the same route as for the previous
correlation functions:
\begin{eqnarray}
\langle S_a^z\left(x,\tau\rightarrow) S_0^z\left(0,0\rightarrow) \rangle \sim
 \frac{1}{8\pi^2}
\left(\frac{1}{\left(x + \mbox{i} u_1 \tau\rightarrow)^2} +
\frac{1}{\left(x - \mbox{i} u_1 \tau\rightarrow)^2} \rightarrow),
\label{spincorzmiddlesurf}
\end{eqnarray}
\begin{eqnarray}
\langle S_a^{+}\left(x,\tau\rightarrow) S_0^{-}
\left(0,0\rightarrow) \rangle \sim
-\frac{{\bar \gamma}_{m}\gamma_m}{4\pi^2}
\left(\frac{1}{\left(x + \mbox{i} u_1 \tau\rightarrow)^2}
+
\frac{1}{\left(x - \mbox{i} u_1 \tau\rightarrow)^2}\rightarrow).
\label{spincorpmmiddlesurf}
\end{eqnarray}
It is interesting to notice that
the staggered parts of these correlation functions
are short-ranged at the chiral fixed point.
This result stems from the very special structure of the spectrum
of the Hamiltonian at the Toulouse point discussed above.

We end this section by estimating the NMR
relaxation rate $1/ T_1$ at low temperature.
The slowest correlation
functions of the model at the chiral fixed point correspond to
the staggered correlations between the spins of the
surface chains (see Eqs. (\ref{spincorzoutfin},
\ref{spincorpmoutfin})).
We emphasize that the exponents occuring in these
correlation functions
are {\it universal} and
characterize a new universality class
in spin ladders. We deduce from these correlation
function (\ref{spincorzoutfin},
\ref{spincorpmoutfin}) the low-temperature dependence of the NMR
relaxation rate:
\begin{equation}
\frac{1}{T_1} \sim \sqrt{T}
\label{nmr}
\end{equation}
in contrast with the S=1/2 Heisenberg
spin chain where $1/ T_1 \sim \;  \rm const$\cite{book,book1}.
Finally, we remark that the result (\ref{nmr})
corresponds to the leading behaviour of the
NMR relaxation rate of the one-dimensional
Hubbard model away from half-filling at $U=\infty$\cite{book,book1}.
The main reason for this stems from the presence of the pseudo-charge
degrees of freedom $\xi^0$, $\zeta$ in the correlation function
(\ref{spincorzoutfin},
\ref{spincorpmoutfin})
that enter through their corresponding
Ising order-disorder operators ($\sigma_{0,4},
\mu_{0,4}$) with scaling dimension $1/8$.

\section{Stability of the chiral fixed point of the 3-chain cylinder model}

In Sections IV and V, we have carefully analysed the
Hamiltonian consisting in only current-current interaction
describing two marginally coupled SU(2)$_2$ and
SU(2)$_1$ WZNW models.
The model with
only current-current interaction has been solved
exactly using a Toulouse point
approach.
At the strong coupling fixed point,
this Hamiltonian belongs to the chiral stabilized liquids universality
class with central charge $c=2$. The critical fields at
the chiral fixed point consist of two decoupled gapless
modes: a magnetic sector described by an effective S=1/2
Heisenberg chain associated with the bosonic field
${\bar \Phi}_1$ and the pseudo-charge degrees of freedom
stemming from the two Majorana fermions $\xi^0$ and $\zeta$.
Moreover, the Toulouse point solution captures
all universal properties of the chiral fixed point
as shown in Appendix D.

The next step of our approach is to analyse the stability
of this chiral fixed point.
The non-perturbative basis provided by the Toulouse
point approach enables us to investigate the stability
of the chiral fixed point under several operators
such as backscattering contributions.
In particular, we shall  first investigate the effect of the interchain
backscattering operator to
deduce the nature of the
low energy physics of the 3-chain cylinder model in the vinicity of
the line ${\bar g} = 0$.

\subsection{Effect of the interchain
backscattering perturbation at the chiral fixed point}

Let us begin by studying the effect of the interchain coupling
in the 3-chain cylinder model in the vinicity of the chiral fixed point:
\begin{equation}
{\cal H}_{b} = {\tilde g}\; {\bf n}_0 \cdot {\bf n}_+ .
\label{intercback}
\end{equation}
From Eq. (\ref{barecoupmod1}), we see that
the coupling constant ${\tilde g}$ is
independent from the coupling $g$ of the current-current interaction.
One can thus always suppose that  ${\tilde g}$ is small to investigate
the backscattering term (\ref{intercback}) as a weak perturbation
at the chiral fixed point: $|{\tilde g}| \ll 1$.
Using the bosonized form of the staggered spin fields
given by Eqs. (\ref{stagbosovarphi}, \ref{stagbosophia}),
we have:
\begin{eqnarray}
{\cal H}_{b} = \frac{2 {\tilde g} \lambda^2}{\pi^2 a_0^2}
\left(\cos\left(\sqrt{\pi} \vartheta_{-}\rightarrow)
\cos\left(\sqrt{2\pi} \vartheta - \sqrt{\pi} \Theta\rightarrow)
+ \cos\left(\sqrt{\pi} \varphi_{-}\rightarrow)
\sin\left(\sqrt{2\pi} \varphi\rightarrow)
\sin\left(\sqrt{\pi} \Phi\rightarrow)\rightarrow)
\label{interbos}
\end{eqnarray}
so that in terms of the different fields at the Toulouse
point, the backscattering term (\ref{intercback})  reads
as follows:
\begin{eqnarray}
{\cal H}_{b} = \frac{2 {\tilde g} \lambda^2}{\pi^2 a_0^2}
\left(\cos\left(\sqrt{\pi} \vartheta_{-}\rightarrow)
\cos\left(\sqrt{\pi} {\bar \Theta}_2\rightarrow)
-\frac{1}{2} \cos\left(\sqrt{\pi} \varphi_{-}\rightarrow)
\cos\left(\sqrt{\pi} {\bar \Phi}_2\rightarrow) \rightarrow. \nonumber \\
\left.
+ \frac{1}{2} \cos\left(\sqrt{\pi} \varphi_{-}\rightarrow)
\cos\left(3\sqrt{\pi} {\bar \Phi}_2 -\sqrt{8\pi}
{\bar \Phi}_1\rightarrow) \rightarrow).
\label{interbostous}
\end{eqnarray}
Using  the Ising dictionnary (see Appendix B),
it can be shown that the contribution in
the magnetic sector of Eq. (\ref{interbostous})
is irrelevant.
The backscattering term affects mostly the
pseudo-charge sector. Indeed, using the results (\ref{proffappendfin})
from the structure of the massive modes at the Toulouse point,
one finds the following estimate at the chiral fixed point:
\begin{equation}
{\cal H}_{b} \sim \frac{\lambda^2
\mu {\tilde g}}{\pi^2 a_0^2} \mu_0 \mu_4 .
\label{interfin}
\end{equation}
We thus conclude that
the backscattering operator opens a gap, $\Delta_c$, in the
pseudo-charge sector but has no
effect on the magnetic (spinons) excitations.
Standard scaling arguments give an
estimate of the mass gap since
the operator (\ref{interfin}) has scaling dimension
$1/4$ and thus $\Delta_c \sim {\tilde g}^{4/7}$.
The chiral fixed point is thus unstable in the far IR limit,
and one expects that the system will flow to
the $c=1$ fixed point
of the standard three-leg spin ladder\cite{arri}.
Of course, the very applicability of the perturbative approach
to the chiral fixed point requires
that $\Delta_c \ll m$,
the condition which can always be satisfied for
sufficiently small ${\tilde g}$.
Under this condition, there exists an
intermediate but still low-energy region
$\Delta_c \ll E \ll m$ where
the $c=2$ behaviour caused by frustration is dominant.
The physics in this region is
universal and cannot be
understood without having recourse to the chiral
fixed point.
At lower energies, $E \ll \Delta_c$, the system will eventually
cross over to the conventional
critical $c=1$ behaviour.

\subsection{Effect of a weak interaction between
surface chains at the chiral fixed point}

In this subsection, we investigate the stability of the
chiral fixed point against a weak interaction between
the spins of the surface chains:
\begin{equation}
{\cal O}_{12} = g_{12} {\bf S}_1 \cdot {\bf S}_2
\simeq g_{12} \left({\bf n}_1 \cdot {\bf n}_2 +
{\bf J}_1 \cdot {\bf J}_2 \rightarrow).
\label{boundaryint}
\end{equation}
Such a term accounts for periodic transverse boundary
condition in the 3-chain cylinder model and may
change drastically the physics of the chiral fixed point.
In Eq. (\ref{boundaryint}), the coupling constant $g_{12}$
is independent of the coupling $g$ of the current-current
interaction of the model so that on can always consider
${\cal O}_{12}$ as a weak perturbation at the chiral fixed
point: $|g_{12}| \ll 1$.

Let us first analyse the ${\bf n}_1 \cdot {\bf n}_2$ contribution
at the chiral fixed point.
Using the bosonized form (\ref{stagbosophia}) of the staggered magnetizations
of the surface chains and the Toulouse basis (\ref{transtouPhi}), one
finds:
\begin{equation}
{\bf n}_1 \cdot {\bf n}_2 = \frac{\lambda^2}{2 \pi^2 a_0^2}
\left[ \cos\left(\sqrt{4\pi} \varphi_-\rightarrow) +
2 \cos\left(\sqrt{4\pi} \vartheta_-\rightarrow)
- \cos\left(\sqrt{8\pi}{\bar \Phi}_1 - \sqrt{4\pi}{\bar \Phi}_2 \rightarrow)
\rightarrow].
\end{equation}
Using the refermionization
formula (\ref{cosphim}, \ref{sinphim}, \ref{referformphi2}),
this operator can be expressed in terms of the different fields
occuring in the Toulouse point solution:
\begin{eqnarray}
{\bf n}_1 \cdot {\bf n}_2 = \frac{\mbox{i} \lambda^2}{2 \pi a_0}
\left[- \xi^3_R \xi^3_L + 3 \xi^0_R \xi^0_L -\left(\eta_R \eta_L
+ \zeta_R \zeta_L\rightarrow) \cos\left(\sqrt{8\pi}{\bar \Phi}_1\rightarrow)
\rightarrow. \nonumber \\
\left.
- \left(\zeta_R \eta_L
- \eta_R \zeta_L\rightarrow) \sin\left(\sqrt{8\pi}{\bar \Phi}_1\rightarrow)
\rightarrow].
\end{eqnarray}
Since the Majorana fermions $\xi^3$ and $\eta$ are short-ranged
fields , we can drop their contribution so that:
\begin{equation}
{\bf n}_1 \cdot {\bf n}_2 \simeq \frac{\mbox{i} \lambda^2}{2 \pi a_0}
\left[3 \xi^0_R \xi^0_L - \zeta_R \zeta_L
\cos\left(\sqrt{8\pi}{\bar \Phi}_1\rightarrow) \rightarrow].
\label{n1n2fin}
\end{equation}
The two terms in this equation are of different nature.
One the one hand, the first contribution is a relevant perturbation
(with scaling dimension 1) which identifies with the energy
density operator of the Ising model corresponding to the Majorana
fermion $\xi^0$ and drives the latter model out-of criticality.
On the other hand, the second operator in Eq. (\ref{n1n2fin})
is a naively irrelevant contribution with scaling dimension 3
but, as we shall see below, it is crucial to keep it since it couples
degrees of freedom of different nature: Magnetic ones described
by the bosonic field ${\bar \Phi}_1$ and a half of the pseudo-charge
degrees of freedom stemming from the Majorana fermion $\zeta_{R,L}$.

The same analysis can be made for the current-current contribution
in Eq. (\ref{boundaryint}).
In particular, using the results (\ref{outercurz}, \ref{outercurpm}),
the leading part of this perturbation at the chiral fixed point
affects mostly the magnetic degrees of freedom:
\begin{equation}
{\bf J}_1 \cdot {\bf J}_2 \simeq 2 {\cal J}^z_R {\cal J}^z_L
-\frac{{\bar \gamma}^2_m}{2\pi a_0} \cos\left(\sqrt{8\pi}{\bar \Phi}_1\rightarrow)
\label{j1j2nfin}
\end{equation}
which can be rewritten at the SU(2) point (${\bar \gamma}^2_m =1$,
see Appendix A) in a rotationally invariant form:
\begin{equation}
{\bf J}_1 \cdot {\bf J}_2 \simeq 2 {\vec {\cal J}}_R \cdot
{\vec {\cal J}}_L .
\label{j1j2fin}
\end{equation}

With the two results (\ref{n1n2fin}, \ref{j1j2fin}) at hand, we are
now in position
to investigate the stability of the chiral fixed point upon
switching on a weak interaction between the surface chains.
The effective interaction (\ref{boundaryint}) at the chiral fixed point
writes:
\begin{equation}
{\cal O}_{12} \simeq 2 g_{12} {\vec {\cal J}}_R \cdot
{\vec {\cal J}}_L  +
\frac{g_{12}\mbox{i} \lambda^2}{2 \pi a_0}
\left[3 \xi^0_R \xi^0_L - \zeta_R \zeta_L
\cos\left(\sqrt{8\pi}{\bar \Phi}_1\rightarrow) \rightarrow].
\label{boundaryinteff}
\end{equation}
Due to the first term, the effect of the operator (${\cal O}_{12}$)
on the chiral fixed point strongly depends on the sign of
the coupling $g_{12}$.

For an antiferromagnetic interaction (i.e. $g_{12} >0$),
we expect the succession of two transitions in that case.
A first one (Ising transition), as soon as the coupling
$g_{12}$ is switched on,
the model is still critical but with a smaller central charge $c=3/2$
since, as already emphasized, the Majorana fermion $\xi^0$ acquires
a mass. As a consequence, the Z$_2$ symmetry with respect to the interchange
of the two surface chains is now broken.
When $g_{12}$ increases, there is a critical value when
$2 g_{12}$ exceeds the coupling constant $\gamma <0$ of the marginal irrelevant
current-current interaction of the effective S=1/2 Heisenberg
chain associated with the bosonic field ${\bar \Phi}_1$.
In that case, the magnetic sector becomes massive and enters in
a dimerized phase as in the J$_1$-J$_2$ problem\cite{haldane1}.
Moreover, in this massive phase, the bosonic field ${\bar \Phi}_1$
becomes locked and the operator $\cos\left(\sqrt{8\pi}{\bar \Phi}_1\rightarrow)$
acquires a non-zero expectation value:
$\langle \cos\left(\sqrt{8\pi}{\bar \Phi}_1\rightarrow)\rangle \ne 0$.
As a consequence,
the magnetic excitation
affects indirectly the pseudo-charge
degrees of freedom by giving a mass term for the Majorana fermion
$\zeta$ due to the third term in Eq. (\ref{boundaryinteff}):
$ -g_{12}\mbox{i} \lambda^2
\langle \cos\left(\sqrt{8\pi}{\bar \Phi}_1\rightarrow)\rangle
\zeta_R \zeta_L
/(2 \pi a_0)$.
Therefore, in the dimerized phase, all degrees of freedom are massive.
In summary, the chiral fixed point is unstable against the presence
of an antiferromagnetic interaction between the surface chains.
There are two distinct Ising transitions in the pseudo-charge sector
whereas the spin part spontaneously dimerized for a sufficiently
strong value of the interaction $g_{12}$.

For a ferromagnetic interaction ($g_{12} <0$),
the Majorana fermion $\xi^0$ still acquires a mass but now
the bosonic field ${\bar \Phi}_1$ and the Majorana fermion $\zeta$
remain massless excitations so that
the resulting model has central charge $c=3/2$.
One should note that a {\sl pseudo-charge fractionalization} occurs
in this $c=3/2$ phase: A half of the degrees of freedom, described
by the Majorana fermion $\xi^0$, in
the pseudo-charge sector becomes massive by an Ising transition
whereas the other (i.e. the Majorana fermion $\zeta$) is still critical.
A similar phenomenon for the charge degrees of freedom of a one
dimensional model has been reported in Ref.\cite{fabgoners}.
The fractionalization manifests itself in the spin-spin correlation
functions in this phase.
The asymptotics of the correlation between spins of the central
chain are still given
by Eqs. (\ref{spincorzfinmiddle}, \ref{spincorpmfinmiddle}).
The only modifications occur in the calculation of the correlation
functions for the surface chains.
For a ferromagnetic interaction ($g_{12} >0$),
one observes from Eq. (\ref{boundaryinteff})
that the mass $m_0$ of the Majorana fermion $\xi^0$ is positive
($m_0 = -3\lambda^2 g_{12}/(2\pi a_0)$) so that the corresponding
Ising model is in its disordered phase: $\langle \sigma_0 \rangle =0$
and $\langle \mu_0 \rangle \ne 0$.
Using Eqs. (\ref{spincorzoutnext}, \ref{spincorpmoutnext}), one
deduces the following estimate
for the leading asymptotics of the correlation functions:
\begin{eqnarray}
\langle S^z_a\left(x,\tau\rightarrow) S^z_b\left(0,0\rightarrow) \rangle \sim
 -\frac{1}{8\pi^2}
\left(\frac{1}{\left(x + \mbox{i} u_1 \tau\rightarrow)^2} +
\frac{1}{\left(x - \mbox{i} u_1 \tau\rightarrow)^2} \rightarrow) \nonumber \\
+ (-1)^{x/a_0} A
\frac{1}{\left(x^2 +
u_1^2\tau^2\rightarrow)^{1/2}}
\frac{1}{\left(x^2 + u_2^2
\tau^2\rightarrow)^{1/8}},
\label{newspincorzoutfin}
\end{eqnarray}
\begin{eqnarray}
\langle S^{+}_a\left(x,\tau\rightarrow) S^{-}_b\left(0,0\rightarrow) \rangle \sim
-\frac{{\bar \gamma}_{m}^2}{4\pi^2}
\left(\frac{1}{\left(x + \mbox{i} u_1 \tau\rightarrow)^2}
+
\frac{1}{\left(x - \mbox{i} u_1 \tau\rightarrow)^2}\rightarrow)
\nonumber \\
+ 2 (-1)^{x/a_0} A
\frac{1}{\left(x^2 +
u_1^2\tau^2\rightarrow)^{1/2} }
\frac{1}{\left(x^2 + u_2^2
\tau^2\rightarrow)^{1/8}}
\label{newspincorpmoutfin}
\end{eqnarray}
where $A$ is a non-universal constant.
In this $c=3/2$ phase, the slowest correlations still
correspond to the correlator of the staggered magnetization
of the surface chains and the low-temperature dependence of
the NMR relaxation rate is now given by:
\begin{equation}
\frac{1}{T_1} \sim T^{1/4}
\end{equation}
in contrast with the estimate (\ref{nmr}) obtained at the chiral
fixed point.

\section{Concluding remarks}

In this work, we have studied some specific models which
display in the long distance limit some CSL physics and
belongs to a new non-Fermi-Liquid class of fixed points
describing the IR behaviour of one-dimensional interacting
chiral fermions.
The possible realizations of this CSL state, proposed in
this paper, belong to several topics of the one-dimensional
Quantum Magnetism: the (N+1)-chain cylinder as
a special frustrated spin ladder, its asymmetric doped version
with a doped central chain as an example of a Luttinger
liquid in an active environment, and finally the Kondo-Heisenberg
model with N channels away from half-filling as a generalized
multichannel Kondo lattice.

The common feature of all these models stems from
the fact that their Hamiltonians in the continuum limit
have the same structure consisting of two marginally
coupled SU(2)$_N$ and SU(2)$_1$ WZNW models with only
current-current interactions. In particular,
this Hamiltonian decomposes into two commuting {\sl chirally
asymmetric} parts which transform into each other under the
time-reversal symmetry. Each part flows to an intermediate
fixed point  where the system as a whole displays critical
properties characterized by a non-trivial symmetry group
belonging to the universality class of the CSL state\cite{andrei}.
All the possible lattice realizations are expected to exhibit
critical properties in the IR limit with, in general, non-integer
central charge.

For the $N=2$ case, by allowing anisotropic interactions in
spin space, the model of two marginally
coupled WZNW models can be solved using a Toulouse point approach. It
provides a direct reading of the spectrum by identifying
the nature of the massive and massless degrees of freedom in
the IR limit. Moreover, the Toulouse limit solution captures
all universal properties of the model including the SU(2)
symmetric case. At the strong coupling fixed point,
the massless degrees of freedom with
central charge $c=2$ consist of an effective S=1/2
Heisenberg spin chain and two critical Ising models which
act as pseudo-charge degrees of freedom.
The special nature of the chiral fixed point identified by
the Toulouse point approach
reveals itself in two important facts for the 3-chain cylinder model.
On the one hand, the magnetic excitations consist of
a strongly correlated bound-state made by two spinons
of the surface chains and an antispinon of the central chain.
On the other hand, the slowest spin-spin correlation functions
corresponds to the staggered magnetizations
contribution of the surface chains with an unusual exponent which
characterizes a new universality class in spin ladders
and manifest itself in physical quantity such as the
low-temperature behaviour of the NMR relaxation rate:
$1/T_1 \sim \sqrt T$.

The Toulouse limit solution provides also a non-perturbative
basis to investigate the stability of the chiral fixed point
under the presence of several operators such as backscattering
contributions.
In particular, the interchain backscattering term of the 3-chain
cylinder model destabilizes the chiral fixed point and the
system will cross over to a fixed point with central charge $c=1$
which is expected to be the one for the non-frustrated 3-leg spin
ladder with open transverse boundary conditions.
However, there still exists an intermediate low-energy region governed
by the chiral fixed point.
This fixed point becomes also unstable upon
switching on a weak interaction between the surface chains.
For a ferromagnetic interaction, the system exhibits an interesting
critical phase with a smaller central charge $c=3/2$ where a
half of the pseudo charge degrees of freedom becomes massive by
an Ising transition and the low temperature
behaviour of the NMR relaxation rate is now: $1/T_1 \sim T^{1/4}$.
In the case of antiferromagnetic interaction (frustrated periodic
transverse boundary conditions), we expect the succession of two
transitions with an intermediate $c=3/2$ phase followed by a massive
phase where the magnetic sector is spontaneously dimerized.

Regarding perspectives, it is clearly of interest to further
explore the phase diagram of the (N+1)-chain cylinder model in
the vinicity of the chiral fixed point.
The experience gained by the Toulouse approach for $N=2$
leads us to expect, though it requires a proof, that the
intermediate fixed point will become unstable and
the system may cross over to a fixed point presumably
characterized by a central charge $c=1$.
A Toulouse limit solution for the general $N$ is thus clearly desirable
to shed light on this problem.
Using a parafermionic description of the SU(2)$_N$ spin current\cite{zamolo2},
one can readily find a Toulouse limit for every $N$
with the decoupling of a bosonic field corresponding to the
bosonic field ${\bar \Phi}_1$ in the $N=2$ case.
Unfortunately, the other critical and massive degrees of freedom
strongly interact with each other: The resulting effective theory
cannot mapped onto Majorana fermions as in $N=2$ case.
Otherwisely stated,
when N exceeds two, the Toulouse limit does not correspond to
a theory of free particles. The simplest case is the $N=4$ case
which is currently under study.
With this respect, the situation is in close parallel to
the multichannel Kondo problem since when the number of channels
exceeds two, the Toulouse limit is described by a non-trivial
field theory\cite{fabrizio,tsvelikto}.

This analogy leads us to expect that the simple model of a
Luttinger liquid in an antiferromagnetic environment considered in
this work and the multichannel Kondo-Heisenberg model away from
half-filling should exhibit a non-Fermi-liquid low-temperature
behaviour with enhanced composite pairing correlations (odd-frequency
pairing) as in the two channel Kondo problem\cite{emery} or
the one-dimensional Kondo lattice\cite{zachar,georges}.
This question in the
$N=2$ case will be addressed by the Toulouse point approach
in a forthcoming publication\cite{azarial}.
We hope that the chirally stabilized liquid state with all
its physical properties will be observed in numerical simulations
and in further experiments on spin ladder systems.

{\bf Acknowlegdments}

The authors are indepted to A. A. Nersesyan for
important contributions related to this work.
We would like also to thank
D. Allen and A. O. Gogolin
for very interesting discussions.
One of us (P. L.) would like to thank the
I.C.T.P. (Trieste) for hospitality during the completion
of this work.
This paper is dedicated to Meli.

{\bf Note added}

When this work was completed, we became aware of a work
by N. Andrei and E. Orignac\cite{orignac} who have also
determined the low energy theory of the one dimensional
multichannel Kondo-Heisenberg lattice away from half filling
and found that composite order parameters induce the dominant
instabilities for a number of channels less than 4.

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