Two-Dimensional Heisenberg Model with Nonlinear Interactions
[-0.5cm] {Sergio Caracciolo}
[-0.2cm] {Dipartimento di Fisica dell'Universit\`a di Milano, I-20100 Milano}
[-0.2cm] {INFN, Sez. di Pisa, and INFM-NEST, Pisa, ITALIA}
[-0.2cm]
[-0.35cm]
{Andrea Pelissetto }
[-0.2cm] {Dipartimento di Fisica and INFN -- Sezione di Roma I}
[-0.2cm] {Universit\`a degli Studi di Roma ``La Sapienza''}
[-0.2cm] {I-00185 Roma, ITALIA}
[-0.2cm] {\protect\makebox[5in]{\quad}} % To force authors' names to be written % vertically
one above another. % (\author seems to put them side-by-side % if there is room.)
We investigate a two-dimensional classical $N$-vector model with a nonlinear interaction $(1 + \bsigma_i\cdot \bsigma_j)^p$ in the large-$N$ limit. As observed for $N=3$ by Bl\"ote {\em et al.} [Phys. Rev. Lett. {88}, 047203 (2002)], we find a first-order transition for $p>p_c$ and no finite-temperature phase transitions for $p < p_c$. For $p>p_c$, both phases have short-range order, the correlation length showing a finite discontinuity at the transition. For $p=p_c$, there is a peculiar transition, where the spin-spin correlation length is finite while the energy-energy correlation length diverges. \bigskip PACS: 75.10.Hk, 05.50.+q, 64.60.Cn, 64.60.Fr
largeN.tex


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\date{February 26, 2002}


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\title{Two-Dimensional Heisenberg Model with Nonlinear Interactions}
\author{
  \\[-0.5cm]
  {\small Sergio Caracciolo}              \\[-0.2cm]
  {\small\it Dipartimento di Fisica dell'Universit\`a di Milano, I-20100 Milano}
                     \\[-0.2cm]
  {\small\it INFN, Sez. di Pisa, and INFM-NEST, Pisa, ITALIA}    \\[-0.2cm]
  {\small {\tt Sergio.Caracciolo@sns.it}}     \\[-0.2cm]
  \\[-0.35cm] \and
  {\small Andrea Pelissetto }        \\[-0.2cm]
  {\small\it Dipartimento di Fisica and INFN -- Sezione di Roma I}  \\[-0.2cm]
  {\small\it Universit\`a degli Studi di Roma ``La Sapienza''}      \\[-0.2cm]
  {\small\it I-00185 Roma, ITALIA}        \\[-0.2cm]
  {\small {\tt Andrea.Pelissetto@roma1.infn.it}}  \\[-0.2cm]
  {\protect\makebox[5in]{\quad}}  % To force authors' names to be written
                                  %   vertically, one above another.
                                  % (\author seems to put them side-by-side
                                  %   if there is room.)
   \\
}

\vspace{0.2cm}
\maketitle
\thispagestyle{empty}   % Suppress page number on front page.

\vspace{-0.4cm}

\begin{abstract}
We investigate a two-dimensional classical $N$-vector model 
with a nonlinear interaction $(1 + \bsigma_i\cdot \bsigma_j)^p$
in the large-$N$ limit. As observed for $N=3$ by 
Bl\"ote {\em et al.} [Phys. Rev. Lett. {\bf 88}, 047203 (2002)],
we find a first-order transition for $p>p_c$ and no finite-temperature
phase transitions for $p < p_c$.
For $p>p_c$, both phases have short-range order, the correlation length
showing a finite discontinuity at the transition.
For $p=p_c$, there is a peculiar transition, where the spin-spin
correlation length is finite while the energy-energy correlation length
diverges.

\bigskip 

PACS: 75.10.Hk, 05.50.+q, 64.60.Cn, 64.60.Fr

\end{abstract}

\clearpage



%%%% \doublespace

The two-dimensional Heisenberg model has been the object of 
extensive studies which mainly focused on the $O(N)$-symmetric
Hamiltonian
\be
H = - N \beta \sum_{\<ij\>} \bsigma_i\cdot \bsigma_j,
\ee
where $\bsigma_i$ is an $N$-dimensional unit spin and 
the sum is extended over all lattice nearest neighbors.
The behavior of this system in two dimensions is well
understood. It is disordered for all finite $\beta$
\cite{MW-66} and it is described for $\beta\to\infty$ 
by the perturbative renormalization group 
\cite{Polyakov-75,BZ-76,BLS-76}. The square-lattice
model has been extensively studied numerically
\cite{Wolff_89_90,EFGS-92,Kim,CEPS-95,CEMPS-96,MPS-96},
checking the perturbative predictions 
\cite{FT-86,CP-94,CP-95,ACPP-99,ABC-97}
and the nonperturbative constants 
\cite{HMN-90,HN-90,CPRV-97}.

In this paper we study a more general Hamiltonian 
on the square lattice; more precisely, we consider
\be
H = - N\beta \sum_{x\mu} W(1 + \bsigma_x\cdot \bsigma_{x+\mu}),
\label{Hgenerica}
\ee
where $W(x)$ is a generic function such that 
$W(2) > W(x)$ for all $0\le x < 2$, in order to
guarantee that the system orders ferromagnetically 
for $\beta\to\infty$. A particular case of 
the Hamiltonian \reff{Hgenerica} has been extensively
studied in the years, the case in which $W(x)$ is 
a second-order polynomial. Such a choice of $W(x)$
gives rise to the so-called mixed
$O(N)$-$\R P^{N-1}$ model 
\cite{HM-82,MR-87,Ohno_90,KZ,%
Butera_92,CEPS_RP,Hasenbusch_96,%
Niedermayer_96,Catterall_98,SS-01}, which is relevant 
for liquid crystals \cite{Lasher-72,LL-72,KS-87,Lin-87,LL-87,BZCP-91}
and for some orientational transitions \cite{KJ-54}.

In a recent Letter \cite{BGH-02}, the authors analyzed 
a model with $W(x)= a x^p + b$ and found an additional
first-order transition for $p$ large enough. 
Here, we will study the same model, finding an 
analogous result: for $p>p_c\approx 4.537857$ a first-order
transition appears, the correlation length---and 
in general, all thermodynamic quantities---showing 
a finite discontinuity. Note that the appearance of a 
first-order transition in nonlinear models is not a new 
phenomenon. Indeed, for $N=\infty$ it was already
shown in Ref. \cite{MR-87} that a first-order transition
appears in mixed $O(N)$-$\R P^{N-1}$ models for some 
values of the couplings. It is of interest to understand the 
behavior for $p=p_c$. For such value of $p$,
Ref. \cite{BGH-02} found a peculiar phase transition:
while the spin-spin correlation length remains finite, the 
energy-energy correlation length diverges. 
Here, we will show that the same phenomenon occurs for $N=\infty$.
However, at variance with what observed in Ref. \cite{BGH-02},
the critical theory shows mean-field---not Ising---behavior.

Let us consider the Hamiltonian \reff{Hgenerica}
on a hypercubic $d$-dimensional lattice. We normalize 
$W(x)$ by requiring $W'(2)=1$ so that in the spin-wave limit 
\be
H = {N\beta\over 2} \int dx\, \partial_\mu\bsigma\cdot \partial_\mu\bsigma.
\ee 
We also fix 
$W(1) = 0$ so that $H=0$ for a random configuration.
Then, we introduce two new fields 
$\lambda_{x\mu}$ and $\rho_{x\mu}$ in order to linearize the 
dependence of the Hamiltonian on the spin coupling. We write
\be
\exp\left[ N\beta\, W(1 + \bsigma_x\cdot\bsigma_{x+\mu})\right] \sim
\int d\rho_{x\mu} d \lambda_{x\mu} \exp\left[
  {N\beta\over2} \lambda_{x\mu} 
      \left(1 + \bsigma_x\cdot\bsigma_{x+\mu} - \rho_{x\mu}\right)
  + N \beta W(\rho_{x\mu})\right].
\ee
As usual in the large-$N$ expansion, we also introduce a field $\mu_x$
in order to eliminate the constraint $\bsigma_x^2 = 1$. Thus, we write
\be
\delta\left(\bsigma_x^2 - 1\right) \sim 
\int d\mu_x \exp\left[
  - {N\beta\over2} \mu_x\left(\bsigma_x^2 - 1\right)\right].
\ee
With these transformations we can rewrite the partition function as 
\be
Z = \int \prod_{x\mu} [d\rho_{x\mu} d \lambda_{x\mu}]
         \prod_x [d\mu_x d\bsigma_x]\,  e^{NA}
\ee
where 
\be
A = {\beta\over2} \sum_{x\mu} 
   \left[\lambda_{x\mu} + \lambda_{x\mu}\bsigma_x\cdot\bsigma_{x+\mu} - 
         \lambda_{x\mu} \rho_{x\mu} + 2 W(\rho_{x\mu})\right] - 
   {\beta\over2} \sum_{x} 
     \left(\mu_x\bsigma_x^2 - \mu_x\right).
\ee
We perform a saddle-point integration by writing
\begin{eqnarray}
\lambda_{x\mu} &=& \alpha + \widehat{\lambda}_{x\mu}, \nonumber \\
\rho_{x\mu} &=& \tau + \widehat{\rho}_{x\mu}, \nonumber \\
\mu_{x} &=& \gamma + \widehat{\mu}_{x}. 
\end{eqnarray}
A standard calculation gives the following saddle-point equations
\cite{foot1}:
\begin{eqnarray}
&& d\beta(1-\tau) + {1\over \alpha} \left[(2 d + m_0^2) I(m_0^2) - 1\right] = 0,
\nonumber \\
&& \alpha - 2 W'(\tau) = 0, 
\nonumber \\
&& {\beta\over2} - {1\over \alpha} I(m_0^2) = 0,
\label{puntisella}
\end{eqnarray}
where we set $\gamma = \alpha(2d + m_0^2)/2$,
\be
I(m_0^2) = \int{d^dp\over (2\pi)^d}\, {1\over \hat{p}^2 + m_0^2},
\ee
and $\hat{p}^2 = 4 \sum_\mu \sin^2 p_\mu/2$. The variable $m_0$
has a simple interpretation: it is related to the 
spin-spin correlation length by $\xi_\sigma = 1/m_0$. 
From Eq. \reff{puntisella} we obtain finally
\be
\beta = {I(m_0^2)\over W'(\tau)},
\label{puntosella2}
\ee
where 
\be
\tau = \tau(m_0) \equiv 2 + {m_0^2\over 2d} - {1\over 2 d I(m_0^2)}.
\ee
The corresponding free energy can be written as 
\be
F = - \beta d W(\tau) + {1\over 2} \log I(m_0^2) + {1\over2} L(m_0^2),
\ee
where 
\be
L(m_0^2) = \int{d^dp\over (2\pi)^d}\,\log (\hat{p}^2 + m_0^2).
\ee
Focusing now on the two-dimensional case, let us show that, for any
$W(x)$, the spin-spin correlation length is always finite, i.e.  
$\xi_\sigma=\infty$, i.e. $m_0=0$, only for $\beta=\infty$.
Note first that $\tau = 2$ (resp. $\tau = 1$) for 
$m_0 = 0$ (resp. $m_0 = \infty$) and that $\tau(m_0)$ is 
a strictly decreasing function of $m_0$. Thus, 
$W'(\tau)$ is finite for all $m_0$. 
Then, since $I(0) = +\infty$, we find that $\xi_\sigma=\infty$  
only if $\beta=\infty$,
i.e. $\xi_\sigma$ is finite for all
finite $\beta$. 

\begin{figure}[tb]
\hspace{-1cm}
\vspace{0cm}
\centerline{\psfig{width=12truecm,angle=-90,file=beta.eps}}
\vspace{0cm}
\caption{Function $\beta(m_0) \equiv I(m_0)/W'(\tau)$ vs $m_0$, for 
$p=4$, 4.5, 5, and 5.5. 
For any $p$, $\beta(m_0)\to\infty$ for $m_0\to 0$.}
\label{fig-beta}
\end{figure}

We want now to discuss the behavior for $\beta\to\infty$. 
From Eq. \reff{puntosella2}, we see that $\beta\to\infty$ for 
$m_0\to 0$ and possibly for $m_0\to \bar{m}_i$, where 
$W'(\tau(\bar{m}_i))=0$. If there is more than one solution, 
the relevant one corresponds to the lowest free energy. 
Now, for $\beta\to\infty$, we can simply write \cite{foot2}
$F \approx - 2 \beta W(\tau)$. Since $\tau(0)=2$ and 
$W(2) > W(\tau)$ for all $0\le \tau < 2$ because of the 
ferromagnetic condition, the relevant solution is the 
one with $m_0\to 0$. Then, using 
\be
I(m_0) = -{1\over2\pi} \log{m_0^2\over32} + O(m_0^2 \log m_0^2)
\ee
for $m_0\to 0$, we obtain 
\be
m_0^2 = 32 e^{-2\pi\beta + \pi W''(2)/2}[1 + O(\beta^{-1})],
\ee
in agreement with the standard perturbative renormalization-group 
predictions \cite{foot3}.

\begin{figure}[tb]
\hspace{-1cm}
\vspace{0cm}
\centerline{\psfig{width=12truecm,angle=-90,file=Free-energy.eps}}
\vspace{0cm}
\caption{The free energy $F(\beta)$ for $p=5$.
There is a critical point $C$ for $\beta_c \approx 1.543$.}
\label{fig-F}
\end{figure}


Let us now discuss the possibility of first-order phase transitions, which 
may arise from the presence of multiple solutions to 
Eq. \reff{puntosella2}. As in Ref. \cite{BGH-02}, we consider 
\be
W(x) = {2\over p}\left( {x\over2}\right)^p - {2^{1-p}\over p}.
\ee
In Fig. \ref{fig-beta} we report the function 
$\beta(m_0) \equiv I(m_0^2)/W'(\tau)$, for $p = 4,4.5,5,5.5$.
For $p=4,4.5$, for each $\beta$ there is a unique solution
$m_0$ and thus there are no phase transitions. On the other hand,
for $p=5,5.5$ there is the possibility of multiple solutions, in which case
the most relevant is the one that gives the lowest free energy. 
For $p=5$, we report the free energy in Fig. \ref{fig-F}.
We observe a first-order transition for 
$\beta \approx 1.543$ with a finite discontinuity of the correlation length,
$\Delta \xi_\sigma \approx 16.2$, and of all thermodynamic quantities. 
A numerical analysis of the gap equation
\reff{puntosella2} shows that a first-order transition exists 
for all $p > p_c \approx 4.537857$. For $p=p_c$, the thermodynamic functions
are nonanalytic for $\beta = \beta_c\approx 1.33472$.
In this case, 
\be
\beta - \beta_c \approx -0.035726 (m_0 - m_{0c})^3 + O[(m_0 - m_{0c})^4],
\ee
where $m_{0c} \approx 0.387537$. Consequently, repeating the discussion of 
Ref. \cite{MR-87},
\begin{eqnarray}
\xi_\sigma(\beta) &\approx& 2.5804 + 7.8682 (\beta - \beta_c)^{1/3} + 
      \cdots , \\
E (\beta) &\approx& 0.162274  + 0.314385 (\beta - \beta_c)^{1/3} + \cdots , \\
C (\beta) &\approx& 0.104795 (\beta - \beta_c)^{-2/3} + \cdots,
\end{eqnarray}
where $E$ and $C$ are respectively the energy and the specific heat per site.
Note that $C (\beta)$ diverges at the critical point, indicating that,
although spin-spin correlations are not critical, criticality is 
observed for energy-energy correlations. Indeed, consider
\be
D_Q(k) = \sum_{x\mu\nu} e^{ik\cdot(x-y)} 
   \langle Q(1 + \bsigma_x\cdot \bsigma_{x+\mu});
           Q(1 + \bsigma_y\cdot \bsigma_{y+\nu})\rangle,
\ee
where $Q(x)$ is an arbitrary regular function. For $N\to \infty$, 
\be
D_Q(k) = [Q'(\tau)]^2 \sum_{\mu\nu} 
     \langle \widehat{\rho}_\mu(-k); \widehat{\rho}_\nu(k)\rangle,
\ee 
so that
\be
N D_Q(0) = \left({Q'(\tau)\over W'(\tau)}\right)^2 C(\beta).
\ee
It follows $D_Q(0) \sim (\beta-\beta_c)^{-2/3}$ for any function $Q(x)$.
Thus, all correlation functions of the energy show a critical behavior.
In order to compute the associated correlation length, we determine 
$D_Q(k)$ for arbitrary $k$. We obtain
\be
N D_Q(k) = {2 [Q'(\tau)]^2[ A_2(k) A_0(k) - A_1(k)^2] \over 
   \beta^2 [W'(\tau)]^2 A_0(k) - \beta W''(\tau)[ A_2(k) A_0(k) - A_1(k)^2]},
\ee
where
\be
A_n(k) = \left.\int {d^2q\over (2\pi)^2} \, 
    { \left(\sum_\mu \cos q_\mu\right)^n \over 
    \left( \widehat{(q + k/2)}^2 + m_0^2\right)
    \left(\widehat{(q - k/2)}^2 + m_0^2\right) }\right. \; .
\ee
For $\beta\to \beta_c$ and $k\to 0$, we have 
\be
D_Q(k)^{-1} = a (\beta - \beta_c)^{2/3} + b k^2 + O(k^4),
\ee
with $a,b\not=0$. Thus, the energy-energy correlation length $\xi_E(\beta)$
behaves as 
\be
\xi_E(\beta) \sim (\beta - \beta_c)^{-1/3},
\ee
i.e. $\nu_E = 1/3$. We thus confirm the results of Ref. \cite{BGH-02}
on the existence of the critical theory for $p=p_c$, although we 
disagree on the nature of the critical behavior. Indeed,
Ref. \cite{BGH-02} suggested $\alpha = 1 - 1/\delta$, with $\delta$
assuming the Ising value $\delta=15$. 
Instead, we find the mean-field value $\delta = 3$.
It is unclear how our large-$N$ result is compatible with what observed 
for $N=3$. Indeed, the universality argument of Ref. \cite{BGH-02} 
would predict Ising behavior for any value of $N$. This issue deserves 
further investigations.

\bigskip

We thank Henk Bl\"ote and Henk Hilhorst for many useful comments.


       
\begin{thebibliography}{199}


\bibitem{MW-66}
N. D. Mermin and H. Wagner, Phys. Rev. Lett. {\bf 17}, 1133
(1966).
%%CITATION = PRLTA,17,1133;%%

\bibitem{Polyakov-75}  
A. M. Polyakov, Phys. Lett. B {\bf 59}, 79 (1975).
%%CITATION = PHLTA,B59,79;%%

\bibitem{BZ-76}  
E. Br\'ezin and J. Zinn-Justin, Phys. Rev. B {\bf 14}, 3110 (1976).
%%CITATION = PHRVA,B14,3110;%%

\bibitem{BLS-76}  
W. A. Bardeen, B. W. Lee, and R. E. Shrock, 
Phys. Rev. D {\bf 14}, 985 (1976).
%%CITATION = PHRVA,D14,985;%%

\bibitem{Wolff_89_90}  
U. Wolff, 
Phys. Rev. Lett. {\bf 62}, 361 (1989);
Nucl. Phys. B {\bf 334}, 581 (1990);
Phys. Lett. B {\bf 248}, 335 (1990).
%%CITATION = PRLTA,62,361;%%
%%CITATION = NUPHA,B334,581;%%
%%CITATION = PHLTA,B248,335;%%

\bibitem{EFGS-92}  
R. G. Edwards, S. J. Ferreira, J. Goodman, and A.D. Sokal,
Nucl. Phys. B {\bf 380}, 621 (1992).
%%CITATION = NUPHA,B380,621;%%

\bibitem{Kim}
J.-K. Kim, Phys. Rev. Lett. {\bf 70}. 1735 (1993);
Phys. Rev. D {\bf 50}, 4663 (1994).
%%CITATION = PRLTA,70,1735;%%
%%CITATION = PHRVA,D50,4663;%%

\bibitem{CEPS-95}
S.~Caracciolo, R.~G.~Edwards, A.~Pelissetto, and A.~D.~Sokal,
%``Asymptotic scaling in the two-dimensional 0(3) sigma model at
%correlation length 10**5,'' 
Phys.\ Rev.\ Lett.\  {\bf 75} (1995) 1891
[hep-lat/9411009];
Nucl. Phys. B (Proc. Suppl.) {\bf 42}, 752 (1995) [hep-lat/9411064].
%%CITATION = HEP-LAT 9411009;%%
%%CITATION = HEP-LAT 9411064;%%

\bibitem{CEMPS-96}
S. Caracciolo, R. G. Edwards, T. Mendes, A. Pelissetto, and A. D. Sokal,
Nucl. Phys. B (Proc. Suppl.) {\bf 47}, 763 (1996) [hep-lat/9509033].
%%CITATION = HEP-LAT 9509033;%%

\bibitem{MPS-96}  
T. Mendes, A. Pelissetto, and A. D. Sokal,
%% Multi-Grid Monte Carlo via $XY$ embedding.  I. General theory and
%% two-dimensional $O(N)$-symmetric nonlinear $\sigma$-models,
Nucl. Phys. B {\bf 477}, 203 (1996) [hep-lat/9604015].
%%CITATION = NUPHA,B477,203;%%

\bibitem{FT-86}  
M. Falcioni and A. Treves, 
Nucl. Phys. B {\bf 265}, 671 (1986).
%%CITATION = NUPHA,B265,671;%%

\bibitem{CP-94}
S.~Caracciolo and A.~Pelissetto,
%% Lattice Perturbation Theory for $O(N)$-Symmetric $\sigma$-Models
%% with General Nearest-Neighbour Action. I. Conventional Perturbation Theory,
Nucl. Phys. B {\bf 420}, 141 (1994) [hep-lat/9401015].
%%CITATION = NUPHA,B420,141;%%

\bibitem{CP-95}
S.~Caracciolo and A.~Pelissetto,
%% Four-Loop Perturbative Expansion for the Lattice $N$-Vector Model,
Nucl. Phys. B {\bf 455}, 619 (1995) [hep-lat/9510015]. 
The results presented here contained
a small numerical error. The correct results are given in Ref. \cite{ACPP-99}.
%%CITATION = NUPHA,B455,619;%%

\bibitem{ACPP-99}
B. All\'es, S. Caracciolo, A. Pelissetto, and M. Pepe,
Nucl. Phys. B {\bf 562}, 581 (1999) [hep-lat/9906014].
%%CITATION = NUPHA,B562,581;%%

\bibitem{ABC-97} 
B. All\'es, A. Buonanno, and G. Cella,
Nucl. Phys. B {\bf 500}, 513 (1997) [hep-lat/9701001];
Nucl. Phys. B (Proc. Suppl.) {\bf 53}, 677 (1997) [hep-lat/9608002].
%%CITATION = NUPHA,B500,513;%%
%%CITATION = NUPHZ,53,677;%%

\bibitem{HMN-90}  
P. Hasenfratz, M. Maggiore, and F. Niedermayer, 
Phys. Lett. B {\bf 245}, 522 (1990).
%%CITATION = PHLTA,B245,522;%%

\bibitem{HN-90} 
P. Hasenfratz and F. Niedermayer, 
Phys. Lett. B {\bf 245}, 529 (1990).
%%CITATION = PHLTA,B245,529;%%

\bibitem{CPRV-97}
M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, 
Phys. Lett. B {\bf 402}, 141 (1997) [hep-lat/9702010].
%%CITATION = PHLTA,B402,141;%%

\bibitem{HM-82}  
S. Hikami and T. Maskawa, Prog. Theor. Phys. {\bf 67}, 1038 (1982).
%%CITATION = PTPKA,67,1038;%%
 
\bibitem{MR-87}  N. Magnoli and F. Ravanini,
Z. Phys. C {\bf 34}, 43 (1987).   % 1/N expansion for mixed S^N/RP^N models
%%CITATION = ZEPYA,C34,43;%%
 
\bibitem{Ohno_90}  
K. Ohno, H.-O. Carmesin, H. Kawamura, and Y. Okabe,
Phys. Rev. B {\bf 42}, 10360 (1990).
%%CITATION = PHRVA,B42,10360;%%
 
\bibitem{KZ} 
H.~Kunz and G.~Zumbach, 
J. Phys. A {\bf 22}, L1043 (1989); % 1/N expansion
Phys. Lett. B {\bf 257}, 299 (1991); % Monte Carlo
Phys. Rev. B {\bf 46}, 662 (1992); % Monte Carlo
J. Phys. A {\bf 25}, 6155 (1992).
%% Phase transition in a nematic $N$-vector model: the large $N$ limit         
%%CITATION = JPAGB,A22,1043;%%
%%CITATION = PHLTA,B257,299;%%
%%CITATION = PHRVA,B46,662;%%
%%CITATION = JPAGB,A25,6155;%%

\bibitem{Butera_92}  
P. Butera and M. Comi, Phys. 
Rev. B {\bf 46}, 11141 (1992) [hep-lat/9205027].
%%CITATION = PHRVA,B46,11141;%%
 
\bibitem{CEPS_RP}  
S. Caracciolo, R.G. Edwards, A. Pelissetto, and A.D. Sokal, 
Nucl. Phys. B (Proc. Suppl.) {\bf 26}, 595 (1992) [hep-lat/9201003];
Nucl. Phys. B (Proc. Suppl.) {\bf 30}, 815 (1993) [hep-lat/9212012];
Phys. Rev. Lett. {\bf 71}, 3906 (1993) [hep-lat/9307022].
%%CITATION = NUPHZ,26,595;%%
%%CITATION = NUPHZ,30,815;%%
%%CITATION = PRLTA,71,3906;%%
 
\bibitem{Hasenbusch_96}  
M. Hasenbusch, Phys. Rev. D {\bf 53}, 3445 (1996) [hep-lat/9507008].
%%CITATION = PHRVA,D53,3445;%%
 
\bibitem{Niedermayer_96}  
F. Niedermayer, P. Weisz, and D.-S. Shin,
Phys. Rev. D {\bf 53}, 5918 (1996) [hep-lat/9507005].
%%CITATION = PHRVA,D53,5918;%%

\bibitem{Catterall_98}  
S. M. Catterall, M. Hasenbusch, R. R. Horgan and R. Renken, 
Phys. Rev. D {\bf 58}, 074510 (1998) [hep-lat/9801032].
%%CITATION = PHRVA,D58,074510;%%

\bibitem{SS-01}
A. D. Sokal and A. O. Starinets,
Nucl. Phys. B {\bf 601}, 425 (2001) [hep-lat/0011043].
%%CITATION = NUPHA,B601,425;%%

%%%%%%% d=3 RP^2 MODEL %%%%%%%%%
 
\bibitem{Lasher-72} 
G. Lasher, Phys. Rev. A {\bf 5}, 1350 (1972).
%%CITATION = PHRVA,A5,1350;%%
 
\bibitem{LL-72}  
P. A. Lebwohl and G. Lasher, 
Phys. Rev. A {\bf 6}, 426 (1972).
%%CITATION = PHRVA,A6,426;%%
 
\bibitem{KS-87}  
G. Kohring and R. E. Shrock, Nucl. Phys. B {\bf 285}, 504 (1987). 
%%CITATION = NUPHA,B285,504;%%

%%%%%%% d=3 MIXED ISOVECTOR/ISOTENSOR MODEL %%%%%%%%%
 
\bibitem{Lin-87}  
Lin Lei, Mol. Cryst. Liq. Cryst. {\bf 146}, 41 (1987).
%%CITATION = MCLCA,146,41;%%
 
\bibitem{LL-87}  
K. M. Leung and Lin Lei, 
Mol. Cryst. Liq. Cryst. {\bf 146}, 71 (1987).
%%CITATION = MCLCA,146,71;%%
 
\bibitem{BZCP-91}  
F. Biscarini, C. Zannoni, C. Chiccoli, and P. Pasini,
Molec. Phys. {\bf 73}, 439 (1991).
%%CITATION = MOPHA,73,439;%%

\bibitem{KJ-54}  
T. J. Krieger and H. M. James, 
J. Chem. Phys. {\bf 22}, 796 (1954).
%%CITATION = JCPSA,22,796;%%

\bibitem{BGH-02}
H. W. J. Bl\"ote, W. Guo, and H. J. Hilhorst,
Phys. Rev. Lett. {\bf 88}, 047203 (2002) 
[cond-mat/0201174].
%%CITATION = PRLTA,88,047203;%%

\bibitem{foot1}
If $W'(1) = 0$, there is an additional solution with 
$\tau = 1$, $\alpha = 0$, $\gamma = \beta^{-1}$.

\bibitem{foot2}
This is obvious for $m_0\to\bar{m}_i$, since $I(\bar{m}_i)$
and $L(\bar{m}_i)$ are finite. For $m_0\to 0$, it is enough to 
observe that $L(0)$ is finite, while $\log I(m_0^2)\approx \log \beta$
because of the gap equations.

\bibitem{foot3}
The large-$\beta$ behavior of $\xi_\sigma$ for any $N\ge 3$ and any 
potential $W(x)$ is given in Ref.~\cite{CP-94}.

%% \bibitem{CP_1overN} S.~Caracciolo and A.~Pelissetto,
%``Corrections to finite-size scaling in the lattice N-vector model
%   for  N = infinity,''
%% Phys.\ Rev.\ D {\bf 58}, 105007 (1998) 
%% [hep-lat/9804001].





\end{thebibliography}


\end{document}

cond-mat0202506.extracted_bib
\begin{thebibliography}{199}


\bibitem{MW-66}
N. D. Mermin and H. Wagner, Phys. Rev. Lett. {\bf 17}, 1133
(1966).
%%CITATION = PRLTA,17,1133;%%

\bibitem{Polyakov-75}  
A. M. Polyakov, Phys. Lett. B {\bf 59}, 79 (1975).
%%CITATION = PHLTA,B59,79;%%

\bibitem{BZ-76}  
E. Br\'ezin and J. Zinn-Justin, Phys. Rev. B {\bf 14}, 3110 (1976).
%%CITATION = PHRVA,B14,3110;%%

\bibitem{BLS-76}  
W. A. Bardeen, B. W. Lee, and R. E. Shrock, 
Phys. Rev. D {\bf 14}, 985 (1976).
%%CITATION = PHRVA,D14,985;%%

\bibitem{Wolff_89_90}  
U. Wolff, 
Phys. Rev. Lett. {\bf 62}, 361 (1989);
Nucl. Phys. B {\bf 334}, 581 (1990);
Phys. Lett. B {\bf 248}, 335 (1990).
%%CITATION = PRLTA,62,361;%%
%%CITATION = NUPHA,B334,581;%%
%%CITATION = PHLTA,B248,335;%%

\bibitem{EFGS-92}  
R. G. Edwards, S. J. Ferreira, J. Goodman, and A.D. Sokal,
Nucl. Phys. B {\bf 380}, 621 (1992).
%%CITATION = NUPHA,B380,621;%%

\bibitem{Kim}
J.-K. Kim, Phys. Rev. Lett. {\bf 70}. 1735 (1993);
Phys. Rev. D {\bf 50}, 4663 (1994).
%%CITATION = PRLTA,70,1735;%%
%%CITATION = PHRVA,D50,4663;%%

\bibitem{CEPS-95}
S.~Caracciolo, R.~G.~Edwards, A.~Pelissetto, and A.~D.~Sokal,
%``Asymptotic scaling in the two-dimensional 0(3) sigma model at
%correlation length 10**5,'' 
Phys.\ Rev.\ Lett.\  {\bf 75} (1995) 1891
[hep-lat/9411009];
Nucl. Phys. B (Proc. Suppl.) {\bf 42}, 752 (1995) [hep-lat/9411064].
%%CITATION = HEP-LAT 9411009;%%
%%CITATION = HEP-LAT 9411064;%%

\bibitem{CEMPS-96}
S. Caracciolo, R. G. Edwards, T. Mendes, A. Pelissetto, and A. D. Sokal,
Nucl. Phys. B (Proc. Suppl.) {\bf 47}, 763 (1996) [hep-lat/9509033].
%%CITATION = HEP-LAT 9509033;%%

\bibitem{MPS-96}  
T. Mendes, A. Pelissetto, and A. D. Sokal,
%% Multi-Grid Monte Carlo via $XY$ embedding.  I. General theory and
%% two-dimensional $O(N)$-symmetric nonlinear $\sigma$-models,
Nucl. Phys. B {\bf 477}, 203 (1996) [hep-lat/9604015].
%%CITATION = NUPHA,B477,203;%%

\bibitem{FT-86}  
M. Falcioni and A. Treves, 
Nucl. Phys. B {\bf 265}, 671 (1986).
%%CITATION = NUPHA,B265,671;%%

\bibitem{CP-94}
S.~Caracciolo and A.~Pelissetto,
%% Lattice Perturbation Theory for $O(N)$-Symmetric $\sigma$-Models
%% with General Nearest-Neighbour Action. I. Conventional Perturbation Theory,
Nucl. Phys. B {\bf 420}, 141 (1994) [hep-lat/9401015].
%%CITATION = NUPHA,B420,141;%%

\bibitem{CP-95}
S.~Caracciolo and A.~Pelissetto,
%% Four-Loop Perturbative Expansion for the Lattice $N$-Vector Model,
Nucl. Phys. B {\bf 455}, 619 (1995) [hep-lat/9510015]. 
The results presented here contained
a small numerical error. The correct results are given in Ref. \cite{ACPP-99}.
%%CITATION = NUPHA,B455,619;%%

\bibitem{ACPP-99}
B. All\'es, S. Caracciolo, A. Pelissetto, and M. Pepe,
Nucl. Phys. B {\bf 562}, 581 (1999) [hep-lat/9906014].
%%CITATION = NUPHA,B562,581;%%

\bibitem{ABC-97} 
B. All\'es, A. Buonanno, and G. Cella,
Nucl. Phys. B {\bf 500}, 513 (1997) [hep-lat/9701001];
Nucl. Phys. B (Proc. Suppl.) {\bf 53}, 677 (1997) [hep-lat/9608002].
%%CITATION = NUPHA,B500,513;%%
%%CITATION = NUPHZ,53,677;%%

\bibitem{HMN-90}  
P. Hasenfratz, M. Maggiore, and F. Niedermayer, 
Phys. Lett. B {\bf 245}, 522 (1990).
%%CITATION = PHLTA,B245,522;%%

\bibitem{HN-90} 
P. Hasenfratz and F. Niedermayer, 
Phys. Lett. B {\bf 245}, 529 (1990).
%%CITATION = PHLTA,B245,529;%%

\bibitem{CPRV-97}
M. Campostrini, A. Pelissetto, P. Rossi, and E. Vicari, 
Phys. Lett. B {\bf 402}, 141 (1997) [hep-lat/9702010].
%%CITATION = PHLTA,B402,141;%%

\bibitem{HM-82}  
S. Hikami and T. Maskawa, Prog. Theor. Phys. {\bf 67}, 1038 (1982).
%%CITATION = PTPKA,67,1038;%%
 
\bibitem{MR-87}  N. Magnoli and F. Ravanini,
Z. Phys. C {\bf 34}, 43 (1987).   % 1/N expansion for mixed S^N/RP^N models
%%CITATION = ZEPYA,C34,43;%%
 
\bibitem{Ohno_90}  
K. Ohno, H.-O. Carmesin, H. Kawamura, and Y. Okabe,
Phys. Rev. B {\bf 42}, 10360 (1990).
%%CITATION = PHRVA,B42,10360;%%
 
\bibitem{KZ} 
H.~Kunz and G.~Zumbach, 
J. Phys. A {\bf 22}, L1043 (1989); % 1/N expansion
Phys. Lett. B {\bf 257}, 299 (1991); % Monte Carlo
Phys. Rev. B {\bf 46}, 662 (1992); % Monte Carlo
J. Phys. A {\bf 25}, 6155 (1992).
%% Phase transition in a nematic $N$-vector model: the large $N$ limit         
%%CITATION = JPAGB,A22,1043;%%
%%CITATION = PHLTA,B257,299;%%
%%CITATION = PHRVA,B46,662;%%
%%CITATION = JPAGB,A25,6155;%%

\bibitem{Butera_92}  
P. Butera and M. Comi, Phys. 
Rev. B {\bf 46}, 11141 (1992) [hep-lat/9205027].
%%CITATION = PHRVA,B46,11141;%%
 
\bibitem{CEPS_RP}  
S. Caracciolo, R.G. Edwards, A. Pelissetto, and A.D. Sokal, 
Nucl. Phys. B (Proc. Suppl.) {\bf 26}, 595 (1992) [hep-lat/9201003];
Nucl. Phys. B (Proc. Suppl.) {\bf 30}, 815 (1993) [hep-lat/9212012];
Phys. Rev. Lett. {\bf 71}, 3906 (1993) [hep-lat/9307022].
%%CITATION = NUPHZ,26,595;%%
%%CITATION = NUPHZ,30,815;%%
%%CITATION = PRLTA,71,3906;%%
 
\bibitem{Hasenbusch_96}  
M. Hasenbusch, Phys. Rev. D {\bf 53}, 3445 (1996) [hep-lat/9507008].
%%CITATION = PHRVA,D53,3445;%%
 
\bibitem{Niedermayer_96}  
F. Niedermayer, P. Weisz, and D.-S. Shin,
Phys. Rev. D {\bf 53}, 5918 (1996) [hep-lat/9507005].
%%CITATION = PHRVA,D53,5918;%%

\bibitem{Catterall_98}  
S. M. Catterall, M. Hasenbusch, R. R. Horgan and R. Renken, 
Phys. Rev. D {\bf 58}, 074510 (1998) [hep-lat/9801032].
%%CITATION = PHRVA,D58,074510;%%

\bibitem{SS-01}
A. D. Sokal and A. O. Starinets,
Nucl. Phys. B {\bf 601}, 425 (2001) [hep-lat/0011043].
%%CITATION = NUPHA,B601,425;%%

%%%%%%% d=3 RP^2 MODEL %%%%%%%%%
 
\bibitem{Lasher-72} 
G. Lasher, Phys. Rev. A {\bf 5}, 1350 (1972).
%%CITATION = PHRVA,A5,1350;%%
 
\bibitem{LL-72}  
P. A. Lebwohl and G. Lasher, 
Phys. Rev. A {\bf 6}, 426 (1972).
%%CITATION = PHRVA,A6,426;%%
 
\bibitem{KS-87}  
G. Kohring and R. E. Shrock, Nucl. Phys. B {\bf 285}, 504 (1987). 
%%CITATION = NUPHA,B285,504;%%

%%%%%%% d=3 MIXED ISOVECTOR/ISOTENSOR MODEL %%%%%%%%%
 
\bibitem{Lin-87}  
Lin Lei, Mol. Cryst. Liq. Cryst. {\bf 146}, 41 (1987).
%%CITATION = MCLCA,146,41;%%
 
\bibitem{LL-87}  
K. M. Leung and Lin Lei, 
Mol. Cryst. Liq. Cryst. {\bf 146}, 71 (1987).
%%CITATION = MCLCA,146,71;%%
 
\bibitem{BZCP-91}  
F. Biscarini, C. Zannoni, C. Chiccoli, and P. Pasini,
Molec. Phys. {\bf 73}, 439 (1991).
%%CITATION = MOPHA,73,439;%%

\bibitem{KJ-54}  
T. J. Krieger and H. M. James, 
J. Chem. Phys. {\bf 22}, 796 (1954).
%%CITATION = JCPSA,22,796;%%

\bibitem{BGH-02}
H. W. J. Bl\"ote, W. Guo, and H. J. Hilhorst,
Phys. Rev. Lett. {\bf 88}, 047203 (2002) 
[cond-mat/0201174].
%%CITATION = PRLTA,88,047203;%%

\bibitem{foot1}
If $W'(1) = 0$, there is an additional solution with 
$\tau = 1$, $\alpha = 0$, $\gamma = \beta^{-1}$.

\bibitem{foot2}
This is obvious for $m_0\to\bar{m}_i$, since $I(\bar{m}_i)$
and $L(\bar{m}_i)$ are finite. For $m_0\to 0$, it is enough to 
observe that $L(0)$ is finite, while $\log I(m_0^2)\approx \log \beta$
because of the gap equations.

\bibitem{foot3}
The large-$\beta$ behavior of $\xi_\sigma$ for any $N\ge 3$ and any 
potential $W(x)$ is given in Ref.~\cite{CP-94}.

%% \bibitem{CP_1overN} S.~Caracciolo and A.~Pelissetto,
%``Corrections to finite-size scaling in the lattice N-vector model
%   for  N = infinity,''
%% Phys.\ Rev.\ D {\bf 58}, 105007 (1998) 
%% [hep-lat/9804001].





\end{thebibliography}