Mathematica precision on extensive eigenvalues calculations

Im trying to evaluate some dot products with eigenvectors and matrices using Mathematica. My problem is the following:

To an given matrix 6×6:

h[kx_, ky_, t1_, B1_, B2_, 
  B3_] := -4 t1 (tx (Cos[kx]) + ty (Cos[ky])) - 
  B1*O1- B2*O2 - 
  B3*O3

Where the variables tx, ty, O1, O2, O3, T are given on the code down on this post.

I need to get the 2 lowest eigenvalues and their associated eigenvectors at kx=0, ky=0, to do so, i use

NLowestEiv[expr_, n_] := 
 Sort[Transpose[Eigensystem[expr]], Re[#1[[1]]] < Re[#2[[1]]] &][[1 ;; n]]

NLowestEiv[h[0,0,1,B1,0,0],2]

For B1 between -1 and 1

And then compute this w matrix:

w=Table[Conjugate[NLowestEiv[h[0,0,1,B1,0,0],2]][[i]].Conjugate[NLowestEiv[h[0,0,1, B1,0,0],2]][[j]],{i,1,2},{j,1,2}]

For different values of B1.

The problem is, the result that i need depends on the signal of the elements of w and mathematica, due to the computational precision return really small values and oscilates very much between positive and negative values. This oscilation can be seen by using manipulate or plot:

Manipulate[Conjugate[NLowestEiv[h[0,0,1,B1,0,0],2]][[1]].Conjugate[NLowestEiv[h[0,0,1, B1,0,0],2]][[2]]], {B1,0,0}

Plot[Conjugate[NLowestEiv[h[0,0,1,B1,0,0],2]][[1]].Conjugate[NLowestEiv[h[0,0,1, B1,0,0],2]][[2]]], {B1,0,0}

How can i eliminate this oscilation?? I tried to use If as conditionals but can´t make it work.

Here is the definition of the matrices used on the top

O1={{10, 0, 0, 0, 0, 0}, {0, -2, 0, 0, 0, 0}, {0, 0, -8, 0, 0, 0}, {0, 0,
   0, -8, 0, 0}, {0, 0, 0, 0, -2, 0}, {0, 0, 0, 0, 0, 10}}

O2={{60, 0, 0, 0, 0, 0}, {0, -180, 0, 0, 0, 0}, {0, 0, 120, 0, 0, 0}, {0,
   0, 0, 120, 0, 0}, {0, 0, 0, 0, -180, 0}, {0, 0, 0, 0, 0, 60}}

O3={{0, 0, 0, 0, 12 Sqrt[5], 0}, {0, 0, 0, 0, 0, 12 Sqrt[5]}, {0, 0, 0, 
  0, 0, 0}, {0, 0, 0, 0, 0, 0}, {12 Sqrt[5], 0, 0, 0, 0, 0}, {0, 
  12 Sqrt[5], 0, 0, 0, 0}}

tx={{15/56, 0, -((3 Sqrt[5/2])/28), 0, (3 Sqrt[5])/56, 0}, {0, 3/56, 
  0, -(3/(28 Sqrt[2])), 0, (3 Sqrt[5])/56}, {-((3 Sqrt[5/2])/28), 0, 
  3/28, 0, -(3/(28 Sqrt[2])), 0}, {0, -(3/(28 Sqrt[2])), 0, 3/28, 
  0, -((3 Sqrt[5/2])/28)}, {(3 Sqrt[5])/56, 0, -(3/(28 Sqrt[2])), 0, 
  3/56, 0}, {0, (3 Sqrt[5])/56, 0, -((3 Sqrt[5/2])/28), 0, 15/56}}

ty={{15/56, 0, -((3 Sqrt[5/2])/28), 0, (3 Sqrt[5])/56, 0}, {0, 3/56, 
  0, -(3/(28 Sqrt[2])), 0, (3 Sqrt[5])/56}, {-((3 Sqrt[5/2])/28), 0, 
  3/28, 0, -(3/(28 Sqrt[2])), 0}, {0, -(3/(28 Sqrt[2])), 0, 3/28, 
  0, -((3 Sqrt[5/2])/28)}, {(3 Sqrt[5])/56, 0, -(3/(28 Sqrt[2])), 0, 
  3/56, 0}, {0, (3 Sqrt[5])/56, 0, -((3 Sqrt[5/2])/28), 0, 15/56}}

T={{0, 0, 0, 0, 0, -1}, {0, 0, 0, 0, 1, 0}, {0, 0, 0, -1, 0, 0}, {0, 0, 
  1, 0, 0, 0}, {0, -1, 0, 0, 0, 0}, {1, 0, 0, 0, 0, 0}}