\Large {Critical thermodynamics of the two-dimensional systems \\ in five-loop renormalization-group approximation}
E. V. Orlov
~A. I. Sokolov
{The RG functions of the $2D$ $n$-vector $\lambda \phi^4$ model are calculated in the five-loop approximation. Perturbative series for the $\beta$-function and critical exponents are resummed by the Pade-Borel and Pade-Borel-Leroy techniques, resummation procedures are optimized and an accuracy of the numerical results is estimated. In the Ising case $n = 1$ as well as in the others ($n = 0$, $n = -1$, $n = 2, 3,...32$) an account for the five-loop term is found to shift the Wilson fixed point location only briefly, leaving it outside the segment formed by the results of the corresponding lattice calculations; even error bars of the RG and lattice estimates do not overlap in the most cases studied. This is argued to reflect the influence of the singular (non-analytical) contribution to the $\beta$-function that can not be found perturbatively. The evaluation of the critical exponents for $n = 1$, $n = 0$ and $n = -1$ in the five-loop approximation and comparison of the numbers obtained with their known exact counterparts confirm the conclusion that non-analytical contributions are visible in two dimensions. For the $2D$ Ising model, the estimate $\omega = 1.31(3)$ for the correction-to-scaling exponent is found that is close to the value 4/3 resulting from the conformal invariance.}
hep-th0003140.tex
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\begin{document}
\baselineskip = 1.0 \baselineskip

\title
{\Large {Critical thermodynamics of the two-dimensional systems \\ 
in five-loop renormalization-group approximation}}

\author{E. V. Orlov, ~A. I. Sokolov}

\address
{Saint Petersburg Electrotechnical University, 
\\ 197376, Saint Petersburg, Russia}

\maketitle

\begin{abstract}
{The RG functions of the $2D$ $n$-vector $\lambda \phi^4$ model are 
calculated in the five-loop approximation. Perturbative series for 
the $\beta$-function and critical exponents are resummed by the Pade-Borel 
and Pade-Borel-Leroy techniques, resummation procedures are optimized and 
an accuracy of the numerical results is estimated. In the Ising case 
$n = 1$ as well as in the others ($n = 0$, $n = -1$, $n = 2, 3,...32$) an 
account for the five-loop term is found to shift the Wilson fixed point 
location only briefly, leaving it outside the segment formed by the results 
of the corresponding lattice calculations; even error bars of the RG and 
lattice estimates do not overlap in the most cases studied. This is argued 
to reflect the influence of the singular (non-analytical) contribution to 
the $\beta$-function that can not be found perturbatively. The evaluation 
of the critical exponents for $n = 1$, $n = 0$ and $n = -1$ in the five-loop 
approximation and comparison of the numbers obtained with their known exact 
counterparts confirm the conclusion that non-analytical contributions are 
visible in two dimensions. For the $2D$ Ising model, the estimate 
$\omega = 1.31(3)$ for the correction-to-scaling exponent is found that 
is close to the value 4/3 resulting from the conformal invariance.}    

\end{abstract}
\bigskip
\bigskip
\bigskip

{\bf What follows is the radically shortened version of the paper 
written in Russian and published in "Fizika Tverdogo Tela" 42, No 11, 
2087 (2000). It contains all the formulas, tables and complete list 
of references of the original paper available now via 
http://www.ioffe.rssi.ru/journals/ftt.} 

\bigskip
\bigskip
\bigskip

The Hamiltonian of the model describing the critical behavior of various 
two-dimensional systems reads:
\begin{equation}
H = 
\int d^2x \Biggl[{1 \over 2}( m_0^2 \varphi_{\alpha}^2
 + (\nabla \varphi_{\alpha})^2) 
+ {\lambda \over 24} (\varphi_{\alpha}^2)^2 \Biggr] ,
\label{eq:1} \\
\end{equation}
where $\varphi_{\alpha}$ is a real $n$-vector field, $m_0^2$ is proportional 
to $T - T_c^{(0)}$, $T_c^{(0)}$ being the mean-field transition temperature. 

We calculate the $\beta$-function and the critical exponents for the model 
(1) within the massive theory. The Green function, the four-point vertex and 
the $\phi^2$ insertion are normalized in a conventional way:
\begin{eqnarray}
G_R^{-1} (0, m, g_4) = m^2 , \qquad \quad 
{{\partial G_R^{-1} (p, m, g_4)} \over {\partial p^2}}
\bigg\arrowvert_{p^2 = 0} = 1 , \\
\nonumber
\Gamma_R (0, 0, 0, m, g) = m^2 g_4, \qquad \quad  
\Gamma_R^{1,2} (0, 0, m, g_4) = 1. 
\label{eq:2}
\end{eqnarray}

Since the four-loop RG expansions at $n = 1$ are known [1] we are in a 
position to find corresponding series for arbitrary $n$ and to calculate
the five-loop terms. The results of our calculations are as follows:    
\begin{eqnarray}
{\beta (g) \over 2} &=& - g + g^2 - {g^3 \over (n + 8)^2} 
\biggl( 10.33501055~n + 47.67505273 \biggr) \ \
\nonumber \\
&+& {g^4 \over (n + 8)^3} \biggl( 5.000275928~n^2 
+ 149.1518586~n + 524.3766023 \biggr) \ \
\nonumber \\
&-& {g^5 \over (n + 8)^4} \biggl( 0.088842906~n^3 + 179.6975910~n^2   
+ 2611.154798~n + 7591.108694 \biggr) \ \
\nonumber \\
&+& {g^6 \over (n + 8)^5} 
\biggl( -0.00407946~n^4 + 80.3096~n^3   
+ 5253.56~n^2 + 53218.6~n + 133972 \biggr) .  \ \ 
\label{eq:3}
\end{eqnarray}
  
\begin{eqnarray}
\gamma^{-1} &=& 1 - {{n + 2} \over {n + 8}}~g 
+ {g^2 \over (n + 8)^2}~(n + 2)~3.375628955 \ \
\nonumber \\
&-& {g^3 \over (n + 8)^3} \biggl( 4.661884772~n^2 + 34.41848329~n 
+ 50.18942749 \biggr) \ \
\nonumber \\
&+& {g^4 \over (n + 8)^4} \biggl( 0.318993036~n^3 + 71.70330240~n^2 
+ 429.4244948~n + 574.5877236 \biggr) \ \
\nonumber \\
&-& {g^5 \over (n + 8)^5} \biggl( 0.0938051~n^4 + 85.4975~n^3 
+ 1812.19~n^2 + 8453.70~n + 10341.1 \biggr) . \ \
\label{eq.4}
\end{eqnarray}
\begin{eqnarray}
\eta &=& {g^2 \over (n + 8)^2}~(n + 2)~0.9170859698 
- {g^3 \over (n + 8)^2}~(n + 2)~0.05460897758 \ \
\nonumber \\
&+& {g^4 \over (n + 8)^4} \biggl( - 0.0926844583~n^3 + 4.05641051~n^2 
+ 29.2511668~n + 41.5352155 \biggr) \ \
\nonumber \\
&-& {g^5 \over (n + 8)^5} \biggl( 0.0709196~n^4 + 1.05240~n^3 
+ 57.7615~n^2 + 325.329~n + 426.896 \biggr) . \
\label{eq.5}
\end{eqnarray}
Instead of the renormalized coupling constant $g_4$, a rescaled coupling  
\begin{equation}
g = {n + 8 \over {24 \pi}} g_4,
\label{eq:6}
\end{equation}
is used as an argument in above RG series. This variable is more convenient 
since it does not go to zero under $n \to {\infty}$ but approaches the 
finite value equal to unity.

To evaluate the Wilson fixed point location $g^*$ and numerical values of 
the critical exponents, the resummation procedure based on the Borel-Leroy
transformation 
\begin{equation}
f(x) = \sum_{i = 0}^{\infty} c_i x^i = \int\limits_0^{\infty} 
e^{-t} t^b F(xt) dt, \ \ \ \ \   
\nonumber\\
F(y) = \sum_{i = 0}^{\infty} {c_i \over (i + b)!} y^i \ \ , 
\label{eq:7}
\end{equation}
is used. The analytical extension of the Borel transforms is performed
by exploiting relevant Pad\'e approximants [L/M]. In particular, four 
subsequent diagonal and near-diagonal approximants $[1/1]$, $[2/1]$, 
$[2/2]$, and $[3/2]$ turn out to lead to numerical estimates for $g^*$ 
which rapidly converge, via damped oscillations, to the asymptotic 
values (see Table I). As is seen from Table II, these asymptotic values, 
however, differ appreciably from numerical estimates for $g^*$ given by 
the lattice and Monte Carlo calculations; such estimates are usually 
extracted from the data obtained for the linear ($\chi$) and non-linear 
($\chi_4$) susceptibilities related to each another via $g_4$: 
\begin{equation}
\chi_4 = {\partial^3M \over{\partial H^3}} \Bigg\arrowvert_{H = 0} 
= - \chi^2 m^{-2} g_4, \qquad \quad 
\label{eq:8} \\
\end{equation}
An account for higher-order (six-loop, seven-loop, etc.) terms in the RG 
expansion (3) will not avoid this discrepancy which is thus believed 
to reflect the influence of the singular (non-analytical) contribution 
to the $\beta$-function. 

The critical exponents for the Ising model ($n = 1$) and for those with 
$n = 0$ and $n = -1$ are estimated by the Pad\'e-Borel summation of the 
five-loop expansions (4), (5) for $\gamma^{-1}$ � $\eta$. Both the 
five-loop RG (Table I) and the lattice (Table II) estimates for $g^*$ are 
used in the course of the critical exponent evaluation. To get an idea 
about an accuracy of the numerical results obtained the exponents are 
estimated using different Pad\'e approximants, under various values of 
the shift parameter $b$, etc. In particular, the exponent $\eta$ is 
estimates in two principally different ways: by direct summation of the 
series (5) and via the resummation of RG expansions for exponents
\begin{equation}
\eta^{(2)} = {1 \over \nu} + \eta - 2, 
\qquad \qquad \eta^{(4)} = {1 \over \nu} - 2,  
\label{eq:9}
\end{equation}
which possess a regular structure favoring the rapid convergence of the 
iteration procedure. The typical error bar thus found is about 0.05. 

The results obtained are collected in Table III. As is seen, for small 
exponent $\eta$ and in some other cases the differences between the 
five-loop RG estimates and known exact values of the critical exponents 
exceed the error bar mentioned. Moreover, in the five-loop approximation
the correction-to-scaling exponent $\omega$ of the 2D Ising model is found
to be close to the value 4/3 predicted by the conformal theory but differs 
markedly from the exact value $\omega = 1$ [33]. This confirms the 
conclusion that non-analytical contributions are visible in two dimensions.  

We thank B. N. Shalaev for numerous useful discussions of the 
critical thermodynamics of 2D systems. The work was supported by the 
Ministry of Education of Russian Federation (Grant 97-14.2-16), by the 
International Science Foundation (A. I. S., Grant �99-943), and by Saint 
Petersburg Administration (E. V. O., grant ASP 298496).

\bigskip
\bigskip
 
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\bigskip

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\newpage
\begin{table}
\caption{The Wilson fixed point coordinate for models with $n = 1$, $n = 0$ 
and $n = -1$ in four subsequent RG approximations and the final five-loop 
estimates for $g^*(n)$.}
\begin{tabular}{|c|c|c|c|c|c|}
~~~$n$~~~& [1/1]~~~& [2/1]~~~& [2/2]~~~& [3/2]~~~& $g^*$, 5-loop~~\\
\hline
\hline
1  & 2.4246~~~~& 1.7508~~~~& 1.8453~~~~& 1.8286~~~~& 1.837 $\pm$ 0.03~~~~\\
\hline
0  & 2.5431~~~~& 1.7587~~~~& 1.8743~~~~& 1.8402~~~~& 1.86 $\pm$ 0.04~~~~\\
\hline
-1 & 2.6178~~~~& 1.7353~~~~& 1.8758~~~~& 1.8278~~~~& 1.85 $\pm$ 0.05~~~~\\
\end{tabular}
\end{table}

\begin{table}
\caption{The Wilson fixed point coordinate $g^*$ and critical exponent  
$\omega$ for $-1 \le n \le 32$ obtained in the five-loop RG approximation.  
The values of $g^*$ extracted from high-temperature (HT) and strong 
coupling (SC) expansions, found by Monte Carlo simulations (MC), obtained 
by the constrained resummation of the $\epsilon$-expansion for $g^*$ 
($\epsilon$-exp.), and given by corresponding $1/n$-expansion ($1/n$-exp.) 
are also presented for comparison.}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
$n$ & -1 & 0 & 1 & 2 & 3 & 4 & 8 & 16 & 32 \\
\hline
\multicolumn{10}{|c|}{$g^*$} \\
\hline
RG, 5-loop & 1.85(5) & 1.86(4) & 1.837(30) & 1.80(3) & 1.75(2) 
& 1.70(2) & 1.52(1) & 1.313(3) & 1.170(2) \\
&  &  &  &  &  & ($b=1$) & ($b=1$) & ([4/1], & ([4/1], \\
&  &  &  &  &  &         &         &  [3/1]) &  [3/1]) \\
\hline
HT exp. [22,24] &  & 1.679(3) & 1.754(1) & 1.81(1) & 1.724(9) 
& 1.655(16) &    &    &  \\
\hline
MC [25,29] &  &  & 1.71(12) & 1.76(3) & 1.73(3) &  &  &  &  \\  
\hline
SC [23] & 1.473(8) & 1.673(8) & 1.746(8) & 1.81(2) 
& 1.73(4) &  &  &  & \\
\hline
$\epsilon$-exp. [24] &  & 1.69(7) & 1.75(5) & 1.79(3) & 1.72(2) 
& 1.64(2) & 1.45(2) & 1.28(1) & 1.16(1) \\
\hline
1/n-exp. [24] &  &  &  &  & 1.758 & 1.698 & 1.479 & 1.283 & 1.154 \\
\hline
\hline
\multicolumn{10}{|c|}{$\omega$} \\
\hline
RG, 5-loop & 1.32(4) & 1.31(3) & 1.31(3) & 1.32(3) & 1.33(2) 
& 1.37(3) & 1.50(2) & 1.70(1) & 1.85(2) \\
\end{tabular}
\end{table}

\begin{table}
\caption{Critical exponents for $n = 1$, $n = 0$, and $n = -1$ obtained 
via the Pad\'e-Borel summation of the five-loop RG expansions for 
$\gamma^{-1}$ and $\eta$. The known exact values of these exponents are 
presented for comparison.}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
~~~$n$~~~&  & $g^*$ & $\gamma$ & $\eta$ & $\nu$ & $\alpha$ & $\beta$ \\
\hline
\hline
1  & RG    & 1.837        & 1.779 & 0.146 & 0.960~~~& 0.081~~~& 0.070 \\
   &       & 1.754 (HT)~~~& 1.739 & 0.131 & 0.931~~~& 0.139~~~& 0.061 \\
\hline   
   & exact~~&              &  7/4   &  1/4   &  1     &  0       & 1/8 \\   
   &       &              & (1.75) & (0.25) &        &          & (0.125) \\
\hline   
\hline
0  & RG    & 1.86         & 1.449 & 0.128 & 0.774 & 0.452 & 0.049 \\
   &       & 1.679 (HT)~~~& 1.402 & 0.101 & 0.738 & 0.524 & 0.037 \\
\hline
   & exact~~& & 43/32        & 5/24         & 3/4~~  & 1/2~~ & 5/64 \\
   &       & & (1.34375)~~~~& (0.20833)~~~~& (0.75) & (0.5) & (0.078125)~~~~\\
\hline
\hline
-1 & RG    & 1.85         & 1.184 & 0.082 & 0.617 & 0.765 & 0.025 \\
   &       & 1.473 (SC)~~~& 1.155 & 0.049 & 0.592 & 0.816 & 0.014 \\
\hline   
   & exact~~& & 37/32        &  3/20  &   5/8   &  3/4   & 3/64 \\
   &       & & (1.15625)~~~~& (0.15) & (0.625) & (0.75) & (0.046875)~~~~\\
\end{tabular}
\end{table}
\end{document}
