is defined as pf[i ˜H(k)] = √det{i ˜H(k)}40. As the EGap
vanishes exactly at k = π, and k = 0, for J′ = 2J + 2hz,
and J′ = −2J + 2hz, respectively, Pfaffians, pf[i ˜H(k =
π)], and pf[i ˜H(k = 0)] are evaluated. Those values are
given by
pf[i ˜H(π)] =
1
2 (
−J −
J′
2
+ hz),
pf[i ˜H(0)] =
1
2 (
J −
J′
2
+ hz).
Hence the sign of pf[i ˜H(π)] and pf[i ˜H(0)] obey the rela-
tions:
sign(pf[i ˜H(π)]) =
−ve,
J′ > 2J + 2hz,
−ve, −2J + 2hz < J′ < 2J + 2hz,
+ve,
J′ < −2J + 2hz,
sign(pf[i ˜H(0)]) =
−ve,
J′ > 2J + 2hz,
+ve, −2J + 2hz < J′ < 2J + 2hz,
+ve,
J′ < −2J + 2hz.
Now, the Pfaffian invariant, Q = sign(pf[i ˜H(π)]) ×
sign(pf[i ˜H(0)]), is given by40
Q =
+ve,
J′ > 2J + 2hz,
−ve, −2J + 2hz < J′ < 2J + 2hz,
+ve,
J′ < −2J + 2hz.
This result indicates that Q is negative in the annular
region enclosed by the boundary lines J′ = 2J +2hz, and
J′ = −2J + 2hz, while Q is positive elsewhere. In order
to characterize the topology, value of bulk topological
invariant, i. e., winding number (ν) has been determined,
which is defined as
ν =
1
2π ∮C (ˆg(k) ×
d
dk
ˆg(k))dk,
where ˆg(k) = g(k)/|g(k)|, and C is a closed curve in the
gz-gy plane. Winding number enumerates the number of
winding around the origin, and at the same time, it will
be accounted as positive when the curve C is traversed
along the counter clockwise direction. For γ = 1, three
different sets of contours, {(a), (b), (c)}, {(d), (e), (f)},
and {(g), (h), (i)}, as shown in Fig 11 are drawn for
hz = −1, 0, 1, respectively. In each triplet set, three
different contours are consecutively drawn for J′/J =
−3, 0, +3. However, in each diagram direction of the
contour is counter clockwise. So, the value of winding
number, ν = 1 for the diagrams (b), (e) and (h), since
in each case it encloses the origin once. Whereas, for the
remaining diagrams it is zero as they do not enclose the
origin. On the other hand, for γ = −1, the shapes of the
contours would be same but winding around clockwise
direction. Hence the nontrivial topological phase is define
by ν = −1, in this case. When γ > 0 (γ < 0), value of
winding number for H in the parameter space is given by
ν = {
1 (−1), −2J + 2hz < J′ < 2J + 2hz,
0,
J′ > 2J + 2hz and J′ < −2J + 2hz.
So, the points, γ = 0, and J′ = 2(hz ± J), are actually
the multicritical points.
In order to check the bulk-boundary correspondence
rule24, variation of eigen energies for H under open
boundary condition is shown in Fig. 12, with respect
to J′/J, for hz = −1 (a), hz = 0 (b), and hz = 1
(c) when J = 1, and γ = 1. Pair of zero energy edge
states are present in the topologically nontrivial region,
2(hz − J) < J′ < 2(J + hz). Finally a combined phase
diagrams for magnetic and topological phases is shown in