JHEP08(2017)041
3 Entropies in the presence of background gauge fields
In this section, we first review the construction of the entanglement and R´enyi entropies
using the Replica trick [2–4], followed by their generalization in the presence of the twisted
boundary conditions and also background gauge fields. We then perform detailed compu-
tations of the entropies in the various limits.
3.1 Basics
Let us consider a reduced density matrix for a free Dirac fermion ψ on an Euclidean
plane by tracing its vacuum state over the degrees of freedom lying outside a given set of
disjoint intervals (u
˜a
, v
˜a
), ˜a = 1, . . . , p. The density matrix ρ(ψ
in
, ψ
out
) can be written as a
functional integral on an Euclidean plane with appropriate boundary conditions ψ = ψ
in
,
and ψ = ψ
out
along each side of the cuts (u
˜a
, v
˜a
)
ρ(ψ
in
, ψ
out
) =
1
Z[1]
Z
Dψe
−S[ψ]
, (3.1)
where Z[1] is a normalization factor and gives the density matrix T r(ρ) = 1. To evaluate
the R´enyi entropy, we consider T r(ρ
n
) with n copies of the cut plane by sewing together a
cut (u
˜a
, v
˜a
)
k
o
with the next cut (u
˜a
, v
˜a
)
k+1
i
, for all cuts ˜a = 1, . . . , p, and the replica copies
k = 1, . . . , n. The copy n + 1 coincides with the first one. The trace of ρ
n
is Z[n] for the
fields, and thus
T r(ρ
n
) = Z[n]/Z[1]
n
. (3.2)
Following [22], we consider n-copies of fermions on a single Riemann surface instead
of a fermion on n-copies of Riemann surfaces.
ψ =
ψ
1
(x)
.
.
.
ψ
n
(x)
, (3.3)
where ψ
k
(x) represents the field on the k-th copy. For a fermion, we need to introduce a −
sign for a trace, which is required when we connect the last copy to the first one. Further-
more, each fermion copy requires another − sign due to a 2π rotation before connecting
to the next copy [23]. Putting them together for a single cut (u
˜a
, v
˜a
) with the n-replica
copies, from the first copy (u
˜a
, v
˜a
)
1
in
to the last one (u
˜a
, v
˜a
)
n+1
out
, requires (−1)
n+1
factor. In
this way, performing the trace for each cut can be described by multiplying the matrix
T =
0 1
0 1
. .
0 1
(−1)
(n+1)
0
for u
˜a
located on one end of a cut and T
−1
for v
˜a
located on the other end of the cut.
By changing a basis by a unitary transformation in the replica space, one can diagonalize
– 6 –
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