23 Page 4 of 17 Eur. Phys. J. C (2018) 78 :23
Consider a d-dimensional CFT on Minkowski space and
choose a spherical entangling surface of radius R as a subsys-
tem. The computation of entanglement entropy of this sub-
system with the rest of the system can be performed by calcu-
lating the reduced density matrix ρ
v
. However, the authors in
[8], using the conformal structure of the theory, mapped this
problem of the entanglement entropy to the thermal entropy
on a hyperbolic space R×H
d−1
. They showed that the causal
development of the ball enclosed by the spherical entangling
surface can be mapped to a hyperbolic space R ×H
d−1
; with
curvature of H
d−1
space given by the radius of the spherical
entangling region R. An important point of this mapping was
that the vacuum of the original CFT mapped to a thermal bath
with temperature
T
0
=
1
2π R
(5)
on the hyperbolic space. Now relating the density matrix
ρ
therm
in the new spacetime R × H
d−1
to the old spacetime
ρ
v
by a unitary transformation, ρ
v
= U
−1
ρ
therm
U , we get
ρ
v
= U
−1
e
[−H / T
0
]
Z(T
0
)
U, (6)
where Z(T
0
) = Tr[e
−H / T
0
]. For the Rényi entropy we also
need the qth power of ρ
v
. From the above equation, we get
ρ
q
v
= U
−1
e
[−qH/T
0
]
Z(T
0
)
q
U. (7)
Taking the trace of both sides of Eq. (7), we get
Tr[ρ
q
v
]=
Z(T
0
/q)
Z(T
0
)
q
, (8)
as U and its inverse cancel each other upon taking trace. Now,
using the definition of Rényi entropy as in Eq. (2), we arrive
at
S
q
=
1
1 − q
ln Z(T
0
/q) − q ln Z(T
0
)
. (9)
The above expression for the Rényi entropy can also be writ-
ten in terms of the free energy F (T ) =−T ln Z (T ):
S
q
=
q
1 − q
1
T
0
F(T
0
) − F(T
0
/q)
, (10)
and further, using the thermodynamic relation S
therm
=
−∂ F/∂ T , we can rewrite the above expression as
S
q
=
q
q − 1
1
T
0
T
0
T
0
/q
dTS
therm
(T ); (11)
here, just to clarify again, S
therm
is the thermal entropy of a
d-dimensional CFT on R × H
d−1
, while S
q
is the desired
Rényi entropy. Equation (11) was the main result of [8,10],
which relates Rényi (and hence entanglement) entropy of
a spherical entangling region in d-dimensional CFT to the
thermal entropy on a hyperbolic space. As pointed out in
[8], the above analysis just mapped one difficult problem to
another equally difficult problem and is not particularly use-
ful for practical purposes. However, its true usefulness can be
realized via the AdS/CFT correspondence. In the AdS/CFT
correspondence, the thermal state of the boundary CFT cor-
responds to an appropriate non-extremal black hole in the
bulk AdS spacetime, with thermal entropy corresponding
to black hole entropy. Therefore, using the AdS/CFT cor-
respondence, we can relate S
therm
appearing in Eq. (11)to
that of the black hole entropy, which is relatively easy to com-
pute. Since on the boundary side our CFT is on R × H
d−1
,
its dual gravity theory will be described by a topological
black hole with hyperbolic event horizon. In any event, in
this AdS/CFT approach, the Rényi entropy is now given by
the horizon entropy of the corresponding hyperbolic black
hole, which can easily be computed using Ward’s standard
formula. However, since we are interested in calculating the
effects of the corrections of the entropy of the black hole on
the Rényi entropy, here we will use an approach based on
symmetry arguments on the horizon to calculate the black
hole entropy, instead of Wald’s formula. This is the topic of
the discussion of the next section.
3 Logarithmic corrections to the entropy of the black
hole from the Cardy formula
In this section, we will describe the necessary steps to calcu-
late the asymptotic form of density of states from a two-
dimensional conformal algebra.
4
This form of density of
states will be used in a later section to calculate the black hole
entropy and, further, to compute logarithmic corrections. In
this section, we will mostly follow the notations used in [35]
and refer the reader to [35] for a detailed discussion.
5
We start with a standard Virasoro algebra of the two con-
formal field theory with central charges c, ¯c:
L
m
, L
n
= (m − n)L
m+n
+
c
12
m(m
2
− 1)δ
m+n,0
¯
L
m
,
¯
L
n
= (m − n)
¯
L
m+n
+
¯c
12
m(m
2
− 1)δ
m+n,0
,
L
m
,
¯
L
n
= 0; (12)
here L
m
and
¯
L
m
are the generators of holomorphic and
antiholomorphic diffeomorphisms. If ρ(,
¯
) denotes the
degeneracy of states carrying L
0
= and
¯
L
0
=
¯
eigen-
values, then one can define the partition function on the two-
torus of modulus τ = τ
1
+iτ
2
as
4
See [42], for generalization of the Cardy formula in higher-
dimensional CFT.
5
Extension of the Cardy formula beyond the subleading order was
performed in [43].
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