documentclass[a4paper,11pt,oneside,onecolumn]{article}
usepackage{pgfplots}
usetikzlibrary{pgfplots.groupplots}
begin{document}
begin{center}
begin{tikzpicture}[baseline]
begin{axis}[
axis y line=left,
axis x line=bottom,
grid=none,
xmin=0,xmax=0.8,
ymin=0,ymax=1.2,
xlabel=$frac{PX}{E^2}$,ylabel=$frac{QX}{E^2}$,
width=0.9columnwidth,
height=0.45columnwidth,
xtick={0,0.2,...,0.8},
ytick={0,0.2,...,1.2},
anchor=center,
y tick label style={
    /pgf/number format/.cd,
    fixed,
    fixed zerofill,
    precision=1,
    /tikz/.cd
},
x tick label style={
    /pgf/number format/.cd,
    fixed,
    fixed zerofill,
    precision=1,
    /tikz/.cd
}
]
foreach tf in {
    -0.2,
    0,
    0.2%
} {%
    addplot[domain=0:0.8,samples=2000
    ] {sqrt(1/2-x*tf+sqrt(1/4-x^2-x*tf))} node[right] {$tanphi=$tf} ;
    addplot[domain=0:0.8,samples=2000
    ] {sqrt(1/2-x*tf-sqrt(1/4-x^2-x*tf))} ;
}
end{axis}
end{tikzpicture}
end{center}
end{document}

Undefined control sequence. end{axis}
Undefined control sequence. end{axis}
Undefined control sequence. end{axis}