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首页二维狄拉克费米子的纠缠熵与背景规范场:解析与物理特性
本文主要探讨的是二维环上的狄拉克费米子在背景规范场中的纠缠熵、Rényi熵以及互信息的研究。作者Bom Soo Kim来自美国肯塔基大学物理与天文学系,他利用扭曲边界条件与背景规范场之间的等价关系,推导出了这些量子信息量的通用公式。这项工作源于arXiv论文1705.01859,其中提出的物理结果既新颖又引人入胜。 论文的核心内容集中在对低温度、大半径和高温极限下纠缠熵和互信息的精确计算。在低温极限下,系统的性质通常比较清晰,可以进行常规分析。然而,到了高温极限,情况则呈现出显著的不同,显示出两种非平凡的极限,这些极限依赖于一个模量参数,并且两者之间的转换并非平滑,揭示了高温环境下新的物理特性。 这个研究对于理解量子场论,特别是弦理论和统计力学中的量子纠缠行为具有重要意义,因为它涉及到了二维场论和低维度量子场论的基本概念。通过这些计算,研究者能够探索不同物理环境对量子系统纠缠现象的影响,这对于量子信息科学和量子计算的发展具有潜在的应用价值。 此外,这篇论文作为Open Access资源,意味着它可供全球范围内的研究人员免费访问和利用,促进了学术交流和知识共享。论文的发表日期和流程表明了作者对科研严谨性的重视,从接收、接受到最终发表,每个阶段都遵循了严格的学术标准。 总结来说,本研究不仅深化了对狄拉克费米子在背景规范场中的量子纠缠现象的理解,而且提供了一个数学工具,用于分析不同物理条件下纠缠度量的变化,为量子信息理论和凝聚态物理领域的前沿研究做出了贡献。
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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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JHEP08(2017)041
T . The corresponding eigenvalues of the T are e
i
k
n
2π
, where k = −
(n−1)
2
, −
(n−1)
2
+ 1,
. . . ,
(n−1)
2
. This reveals that the space is simply connected but the field ψ is not singled
valued. By diagonalizing the T matrix, the problem is reduced to n decoupled fields ψ
k
living on a single plane. These fields are multivalued, since when encircling u
˜a
or v
˜a
they
are multiplied by e
i
k
n
2π
or e
−i
k
n
2π
, respectively.
The multivaluedness can be removed by introducing an external constant gauge
field coupled with the fields ψ
k
, which is single-valued. This can be described by the
Lagrangian density
L
k
= i
¯
ψ
k
γ
µ
∂
µ
+ i
˜
A
k
µ
ψ
k
. (3.4)
The singular gauge transformation, ψ
k
(x) → exp
−i
R
x
x
0
dx
0
µ
˜
A
k
µ
(x
0
)
ψ
k
(x), would get rid
of the gauge field
˜
A
k
µ
, leaving the fields to be multivalued. Thus the proper boundary
conditions for the ˜a-th cut on the fields ψ
k
are
I
C
u
˜a
dx
µ
˜
A
k
µ
(x) = −
2πk
n
,
I
C
v
˜a
dx
µ
˜
A
k
µ
(x) =
2πk
n
. (3.5)
This holds for any two circles going around u
˜a
and v
˜a
. Putting together, we get
µν
∂
ν
˜
A
k
µ
(x) = 2π
k
n
p
X
˜a=1
[δ(x − u
˜a
) − δ(x − v
˜a
)] . (3.6)
The R´enyi entropy can be evaluated with Z[n] =
Q
n−1
2
k=−
(n−1)
2
Z
k
, where Z
k
can be
obtained by the vacuum expectation value in the free Dirac theory
Z
k
= he
i
R
˜
A
k
µ
J
µ
k
d
2
x
i , (3.7)
where J
µ
k
and
˜
A
k
µ
are is the Dirac current and background gauge field. Z
k
can be evaluated
using the bosonization technique. Then the current is
J
µ
k
→
1
2π
µν
∂
ν
φ , (3.8)
where φ is a real scalar field. For a free massless Dirac field, L
φ
=
1
2
∂
µ
φ∂
µ
φ. Thus
Z
k
= he
i
R
˜
A
k
µ
1
2π
µν
∂
ν
φd
2
x
i =
D
e
−i
k
n
P
p
˜a=1
(φ(u
˜a
)−φ(v
˜a
))
E
=
p
Y
˜a=1
hσ
k
(u
˜a
)σ
−k
(v
˜a
)i , (3.9)
where the vacuum expectation values correspond to those of the massless Dirac theory,
and σ
k
(u
˜a
) and σ
−k
(v
˜a
) are the twist operators with conformal dimension
k
2
2n
2
in the Z
n
orbifold theory for free Dirac fermion [24].
Since L
φ
is quadratic in φ, one can perform the path integral for the field φ.
he
−i
R
f(x)φ(x)d
2
x
i = e
−
1
2
R
f(x)G(x−y)f (y)d
2
xd
2
y
, (3.10)
– 7 –
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