I think the problem is that the Manipulate argument
Plot[Evaluate[
Flatten[Table[
ni[1/(10^(LT))], {a, -1, 0}, {\[Epsilon]i, 0, 1}]]], {LT, -2, 5}]
does not contain any explicit reference to \[Mu]
. Try with
z[\[Beta]_, \[Mu]_] := Exp[\[Beta] \[Mu]];
ni[\[Beta]_, \[Mu]_] :=
1/((z[\[Beta], \[Mu]]^-1) Exp[\[Beta] \[Epsilon]i] + a)
Manipulate[
Plot[Evaluate[
Flatten[Table[
ni[1/(10^(LT)), \[Mu]], {a, -1, 0}, {\[Epsilon]i, 0,
1}]]], {LT, -2, 5}], {\[Mu], 0, 3}]