Clear[d1, d2, d3, d4, L, [CapitalDelta]s, s1, s2, sp, sm, H, HT, IM,
v, smin, smax, f, g1, g2]
d1 = 1; d2 = 2; d3 = 4; d4 = 3;
smin = -1; smax = 1; [CapitalDelta]s = 0.1;
L = 5;
sp = {{0, 1}, {0, 0}};
sm = {{0, 0}, {1, 0}};
n = {{0, 0}, {0, 1}};
v = {{1, 0}, {0, 0}};
IM = IdentityMatrix[2];
H = MatrixForm[{{0, E^-s1 d1, E^s1 d3, 0}, {0, -d1 - d2 - s2/2, 0,
     0}, {0, 0, -d3 - d4 - s2/2, 0}, {0, E^-s1 d2, E^s1 d4, -s2}}];
HT = KroneckerProduct[H, IM, IM, IM] +
   KroneckerProduct[IM, H, IM, IM] +
   KroneckerProduct[IM, IM, H, IM] +
   KroneckerProduct[IM, IM, IM, H] +
   (d1*E^-s1) KroneckerProduct[sp, IM, IM, IM,
     v] + (d3*E^s1) KroneckerProduct[v, IM, IM, IM,
     sp] + (d2*E^-s1) KroneckerProduct[n, IM, IM, IM,
     sm] + (d4*E^s1) KroneckerProduct[sm, IM, IM, IM,
 n] - (d1) KroneckerProduct[n, IM, IM, IM,
 v] - (d3) KroneckerProduct[v, IM, IM, IM,
 n] - (d2) KroneckerProduct[n, IM, IM, IM,
 v] - (d4) KroneckerProduct[v, IM, IM, IM, n] - (s2/
 2) KroneckerProduct[n, IM, IM, IM, v] - (s2/2) KroneckerProduct[
 v, IM, IM, IM, n] - (s2) KroneckerProduct[n, IM, IM, IM, n];

Table[Max[Re[Eigenvalues[HT]]], {s1,smin,smax,[CapitalDelta]s},
{s2,smin,smax,[CapitalDelta]s}];
 f = ListInterpolation[%, {{smin, smax}, {smin, smax}}];
g1[s1_] = -D[f[s1, s2], {s1, 1}] /. s2 -> 0;
g2[s1_] = -D[f[s1, s2], {s2, 1}] /. s2 -> 0;
 ParametricPlot3D[{g1[s1], g2[s1]}, {s1, smin, smax}]