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首页磁场增强的超杨米尔斯理论混沌特性:纠缠、扩散与内部作用加强
本文探讨了在恒定强度磁场B下的高温超杨米尔斯理论的混沌性质,利用全息方法作为研究工具。超杨米尔斯理论是一种量子场论,它在强相互作用物理学中占有重要地位,尤其是在描述弦理论和黑洞物理中。背景磁场的存在对理论的行为产生了显著影响。 在理论框架中,研究者注意到存在一个从紫外(UV)到红外(IR)的重规化流,这种流意味着理论在不同的能量尺度上展现出不同的行为。当温度T变化时,系统经历了一个能量交叉点,这个能量与温度的关系是非平凡的,表现为无量纲比T/T^2的单调增长。这种现象揭示了理论内在动态的复杂性和非线性效应。 论文重点研究了纠缠、扩散和扩展等混沌特性。在红外理论中,纠缠和蝴蝶速度(衡量混沌系统中信息传播的速度)通常表现出增强的趋势,这与先前的理论预测形成了对比。然而,这些增强并未超出光速的极限,体现了相对论约束的微妙平衡。 此外,作者还关注了扩散系数与蝴蝶速度之间的关系,特别是引用了Blake提出的下限。电扩散常数在磁场的作用下遵守这一限制,进一步证实了磁场强化了系统的内部相互作用。这种内部交互作用的加强不仅影响了基本粒子的行为,也对理论的整体稳定性及可能的物理现象有深远影响。 本研究通过全息方法深入分析了磁性黄铜(这里可能指的是带有磁场的超杨米尔斯系统)中的复杂现象,揭示了温度、磁场和能量尺度如何共同塑造了混沌性质,并且强调了磁场对内部相互作用的显著影响。这对于理解量子场论的热力学行为,尤其是在极端条件下,具有重要的理论价值和实际应用潜力。
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JHEP09(2018)131
u v
Future
Interior
Past
Interior
Left
Exterior
Right
Exterior
Figure 1. Penrose diagram for the two-sided black branes we consider. This geometry is dual to
a thermofield double state |T F Di made by entangling two copies of the boundary theory.
null trajectory very close to the past horizon, as we will now see. Let (u, v) = (u
0
, v
0
) be
the initial position of the perturbation in Kruskal coordinates. Under time evolution these
coordinates change as (u
0
, v
0
) → (e
−
2π
β
t
u
0
, e
2π
β
t
v
0
), and this means that, as time passes, the
perturbation gets more and more localized at u = 0, and stretched along the v-direction.
Besides that, from the point of view of the t = 0 frame, the energy of the perturbation
increases exponentially as t
0
,
7
moves farther into the past, i.e. E = E
0
e
2π
β
t
0
. As a result,
for t
0
far enough into the past, the energy-momentum tensor of the perturbation can be
very well approximated by
T
shock
uu
= E
0
e
2π
β
t
0
δ(u) a(x
i
) , (3.8)
where E
0
is the asymptotic energy of the perturbation and a(x
i
) is some function rep-
resenting the localization of the operator O
2
(t
0
). Note that the shock wave divides the
geometry into two halves: the causal future of the shock wave (the region u > 0), and its
causal past (the region u < 0).
The backreaction to this perturbation on the geometry is actually very simple and can
be described by a shift v → v + α(t, x
i
) in the causal future of the shock wave, while the
causal past is unaffected.
8
This is illustrated in figure 2.
The shock wave metric is simply given by [24, 51, 52]
ds
2
= 2A(u, v)dudv + G
ij
dx
i
dx
j
− 2A(u, v)α(t, x
i
)δ(u)du
2
, (3.9)
where the shock wave profile α(t, x
i
) has to obey
"
G
ii
∂
i
∂
i
−
2π
β
2
G
ii
(r
H
)G
0
ii
(r
H
)
G
0
tt
(r
H
)
#
α(t, x
i
) =
8πG
N
E
A(r
H
)
e
2πt/β
a(x
i
) , (3.10)
7
In our convention the Killing time coordinate t runs forward on the right boundary and backwards on
the left. Hence, a perturbation on the left boundary at the time t
0
> 0 is in the past of the t = 0 slice of
the geometry.
8
This was first done in [51] for Minkowski spacetime, and then generalized for generic curved spacetimes
in [52]. More details about the case of anisotropic metrics can be found, for instance, in [24].
– 7 –
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JHEP09(2018)131
O
2
(t
0
)|T F Di
α
Figure 2. Penrose diagram for the shock wave geometry. This geometry is dual to a thermofield
double state perturbed at a time t
0
in the far past.
and G
ij
has been considered to be diagonal. Taking a(x
i
) ∼ δ(x
i
) and assuming that
|~x| 1, the above equation has a solution of the form
α(t, x
i
) ∼ exp
2π
β
t − t
∗
−
|x
i
|
v
i
B
, (3.11)
where t
∗
∼
β
2π
log
1
G
N
∼
β
2π
log N
2
is the scrambling time and
v
i
B
=
1
p
G
ii
(r
H
)
p
G
0
tt
(r
H
)
q
G
jj
(r
H
)G
0
jj
(r
H
)
, (3.12)
is the butterfly velocity along the x
i
-direction. By comparing (3.11) and (1.2) we can also
extract the Lyapunov exponent of the system as λ
L
= 2π/β.
We now specialize our formula for the butterfly velocity of the magnetic brane solution
described in section 2.
Along the direction of the magnetic field, the butterfly velocity reads
v
2
B,k
=
U
0
W
2
V
0
V
+
W
0
W
r=r
H
, (3.13)
while the butterfly velocity along any direction perpendicular to the magnetic field is
given by
v
2
B,⊥
=
U
0
V
2
V
0
V
+
W
0
W
r=r
H
. (3.14)
Both v
2
B,k
and v
2
B,⊥
are functions of the ratio B/T
2
, which controls the strength of the
magnetic field on the system. Figure 3 shows how these velocities are affected by the
presence of the external magnetic field.
v
B
at the UV fixed point (B/T
2
→ 0). The gravitational dual to the UV fixed
point reads
9
U(r) = (r + r
H
/2)
2
1 −
(3r
H
/2)
2
(r + r
H
/2)
2
,
V (r) = W (r) = (r + r
H
/2)
2
, (3.15)
9
As explained in appendix A, the coordinate r in (3.15) has been shifted so that the Hawking temperature
is the same across the family of solutions.
– 8 –
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