Holographic perspectives on the Kibble-Zurek...
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Tuesday, November 27, 12
Holographic perspectives on theKibble-Zurek mechanism
What is the Kibble-Zurek mechanism?
unbroken
broken
QFT with 2nd order phase transition:
• Example: superfluid
• Symmetry group U(1) broken for T < Tc.
• Order parameter 6= 0 for T < Tc.
• What happens when T is dynamic?
✏(t) ⌘ 1� T (t)
Tc.
• Density of defects after quench:
n ⇠ ⇠�(d�D).
What is the Kibble-Zurek mechanism?
unbroken
broken
Box of
superfluid
at T > Tc
QFT with 2nd order phase transition:
• Example: superfluid
• Symmetry group U(1) broken for T < Tc.
• Order parameter 6= 0 for T < Tc.
• What happens when T is dynamic?
✏(t) ⌘ 1� T (t)
Tc.
• Density of defects after quench:
n ⇠ ⇠�(d�D).
What is the Kibble-Zurek mechanism?
unbroken
broken
Box of
superfluid
at T > Tc
}
⇠
QFT with 2nd order phase transition:
• Example: superfluid
• Symmetry group U(1) broken for T < Tc.
• Order parameter 6= 0 for T < Tc.
• What happens when T is dynamic?
✏(t) ⌘ 1� T (t)
Tc.
• Density of defects after quench:
n ⇠ ⇠�(d�D).
What is the Kibble-Zurek mechanism?
unbroken
broken
Box of
superfluid
at T > Tc
}
⇠
QFT with 2nd order phase transition:
• Example: superfluid
• Symmetry group U(1) broken for T < Tc.
• Order parameter 6= 0 for T < Tc.
• What happens when T is dynamic?
✏(t) ⌘ 1� T (t)
Tc.
• Density of defects after quench:
n ⇠ ⇠�(d�D).
unbroken
broken
�tfreeze +tfreeze
Critical slowing down
• Critical exponents: ⇠eq = ⇠o
|✏|�⌫
and ⌧eq = ⌧o
|✏|�z⌫ .
• Inevitable 9 tfreeze such that
@⌧eq
@t
��t=tfreeze
⇠ 1.
• Characteristic scale: ⇠freeze ⌘ ⇠eq(t = tfreeze).
Zurek’s estimate of correction length
• Assume linear quench: ✏(t) = t/⌧Q.
)tfreeze ⇠ ⌧⌫z/(1+⌫z)Q , ⇠freeze ⇠ ⌧⌫/(1+⌫z)
Q .
• Density of topological defects when condensate first forms:
nKZ ⇠ 1
⇠d�Dfreeze
⇠ ⌧�(d�D)⌫/(1+⌫z)Q
�tfreeze +tfreeze
t
(t) ⇠ ✏(t)�
frozen
adiabatic
The Kibble-Zurek scaling
Motivational claims
1. Dynamics after +tfreeze need not be adiabatic.
• Adiabatic evolution only after teq � tfreeze.
2. No well-defined condensate until teq.
3. Dynamics after T < Tc responsible for KZ scaling.
4. ⇠(teq) � ⇠freeze ) far fewer defects formed
Motivational claims
1. Dynamics after +tfreeze need not be adiabatic.
• Adiabatic evolution only after teq � tfreeze.
2. No well-defined condensate until teq.
3. Dynamics after T < Tc responsible for KZ scaling.
4. ⇠(teq) � ⇠freeze ) far fewer defects formed
November 14, 2013 1:19 WSPC - Proceedings Trim Size: 9in x 6in reviewKZMv2
5
of ε within the interval ε ∈ [−ε̂, ε̂], where
ε̂ ≡ |ε(t̂)| ∼(
τ0τQ
)1
1+zν
. (8)
Spontaneous symmetry breaking entails degeneracy of the ground state.In an extended system, causally disconnected regions will make independentchoices of the vacuum in the new phase. A summary of the topologicalclassification of the resulting defects using homotopy theory is presented inthe Appendix A. The KZM sets the average size of these domains by thevalue of the equilibrium correlation length at ε̂,19
ξ̂ ≡ ξ[ε̂] = ξ0
(
τQτ0
)ν
1+zν
. (9)
This is the main prediction of the KZM.This simple form of a power law of t̂ (and, consequently, of ξ̂) arises
only when the relaxation time of the system scales as a power law of ε.This need not always be the case. For example, in the Kosterlitz-Thoulessphase transition universality class, of relevance to 2D Bose gases, the criticalslowing down is described by a more complicated (exponential) dependenceon ε. A more complex dependence of t̂ and ξ̂ on τQ (rather than a simplepower law) would be then predicted as a result.30
The above estimate of the ξ̂ is often recast as an estimate for the result-ing density of topological defects,
n ∼ ξ̂d
ξ̂D=
1
ξ0D−d
(
τ0τQ
)(D−d) ν1+zν
, (10)
where D and d are the dimensions of the space and of the defects (e.g.,D = 3 and d = 1 for vortex lines in a 3D superfluid). This order-of-magnitude prediction usually overestimates the real density of defects ob-served in numerics. A better estimate is obtained by using a factor f , tomultiply ξ̂ in the above equations, where f ≈ 5−10 depends on the specificmodel.29,31–35 Thus, while KZM provides an order-of-magnitude estimateof the density of defects, it does not provide a precise prediction of theirnumber. However, if one were able to check the power law above, one couldclaim that the KZM holds and show that the non-equilibrium dynamicsacross the phase transition is also universal. This requires the ability tomeasure the average number of excitations after driving the system at agiven quench rate, and repeating this measurement for different quenchrates.
excerpt from [Del Campo & Zurek]
Motivational claims
Employ holographic duality
1. Dynamics after +tfreeze need not be adiabatic.
• Adiabatic evolution only after teq � tfreeze.
2. No well-defined condensate until teq.
3. Dynamics after T < Tc responsible for KZ scaling.
4. ⇠(teq) � ⇠freeze ) far fewer defects formed
November 14, 2013 1:19 WSPC - Proceedings Trim Size: 9in x 6in reviewKZMv2
5
of ε within the interval ε ∈ [−ε̂, ε̂], where
ε̂ ≡ |ε(t̂)| ∼(
τ0τQ
)1
1+zν
. (8)
Spontaneous symmetry breaking entails degeneracy of the ground state.In an extended system, causally disconnected regions will make independentchoices of the vacuum in the new phase. A summary of the topologicalclassification of the resulting defects using homotopy theory is presented inthe Appendix A. The KZM sets the average size of these domains by thevalue of the equilibrium correlation length at ε̂,19
ξ̂ ≡ ξ[ε̂] = ξ0
(
τQτ0
)ν
1+zν
. (9)
This is the main prediction of the KZM.This simple form of a power law of t̂ (and, consequently, of ξ̂) arises
only when the relaxation time of the system scales as a power law of ε.This need not always be the case. For example, in the Kosterlitz-Thoulessphase transition universality class, of relevance to 2D Bose gases, the criticalslowing down is described by a more complicated (exponential) dependenceon ε. A more complex dependence of t̂ and ξ̂ on τQ (rather than a simplepower law) would be then predicted as a result.30
The above estimate of the ξ̂ is often recast as an estimate for the result-ing density of topological defects,
n ∼ ξ̂d
ξ̂D=
1
ξ0D−d
(
τ0τQ
)(D−d) ν1+zν
, (10)
where D and d are the dimensions of the space and of the defects (e.g.,D = 3 and d = 1 for vortex lines in a 3D superfluid). This order-of-magnitude prediction usually overestimates the real density of defects ob-served in numerics. A better estimate is obtained by using a factor f , tomultiply ξ̂ in the above equations, where f ≈ 5−10 depends on the specificmodel.29,31–35 Thus, while KZM provides an order-of-magnitude estimateof the density of defects, it does not provide a precise prediction of theirnumber. However, if one were able to check the power law above, one couldclaim that the KZM holds and show that the non-equilibrium dynamicsacross the phase transition is also universal. This requires the ability tomeasure the average number of excitations after driving the system at agiven quench rate, and repeating this measurement for different quenchrates.
excerpt from [Del Campo & Zurek]
Motivational claims
Employ holographic duality
First holographic study:
[Sonner, del Campo, Zurek: two weeks ago]
1. Dynamics after +tfreeze need not be adiabatic.
• Adiabatic evolution only after teq � tfreeze.
2. No well-defined condensate until teq.
3. Dynamics after T < Tc responsible for KZ scaling.
4. ⇠(teq) � ⇠freeze ) far fewer defects formed
November 14, 2013 1:19 WSPC - Proceedings Trim Size: 9in x 6in reviewKZMv2
5
of ε within the interval ε ∈ [−ε̂, ε̂], where
ε̂ ≡ |ε(t̂)| ∼(
τ0τQ
)1
1+zν
. (8)
Spontaneous symmetry breaking entails degeneracy of the ground state.In an extended system, causally disconnected regions will make independentchoices of the vacuum in the new phase. A summary of the topologicalclassification of the resulting defects using homotopy theory is presented inthe Appendix A. The KZM sets the average size of these domains by thevalue of the equilibrium correlation length at ε̂,19
ξ̂ ≡ ξ[ε̂] = ξ0
(
τQτ0
)ν
1+zν
. (9)
This is the main prediction of the KZM.This simple form of a power law of t̂ (and, consequently, of ξ̂) arises
only when the relaxation time of the system scales as a power law of ε.This need not always be the case. For example, in the Kosterlitz-Thoulessphase transition universality class, of relevance to 2D Bose gases, the criticalslowing down is described by a more complicated (exponential) dependenceon ε. A more complex dependence of t̂ and ξ̂ on τQ (rather than a simplepower law) would be then predicted as a result.30
The above estimate of the ξ̂ is often recast as an estimate for the result-ing density of topological defects,
n ∼ ξ̂d
ξ̂D=
1
ξ0D−d
(
τ0τQ
)(D−d) ν1+zν
, (10)
where D and d are the dimensions of the space and of the defects (e.g.,D = 3 and d = 1 for vortex lines in a 3D superfluid). This order-of-magnitude prediction usually overestimates the real density of defects ob-served in numerics. A better estimate is obtained by using a factor f , tomultiply ξ̂ in the above equations, where f ≈ 5−10 depends on the specificmodel.29,31–35 Thus, while KZM provides an order-of-magnitude estimateof the density of defects, it does not provide a precise prediction of theirnumber. However, if one were able to check the power law above, one couldclaim that the KZM holds and show that the non-equilibrium dynamicsacross the phase transition is also universal. This requires the ability tomeasure the average number of excitations after driving the system at agiven quench rate, and repeating this measurement for different quenchrates.
excerpt from [Del Campo & Zurek]
A holographic model of a charged superfluid
Action: [Hartnoll, Herzog & Horowitz: 0803.3295]
Sgrav =
1
16⇡GN
Zd
4x
p�G
R+ ⇤+
1
q
2
��F
2 � |D�|2 �m
2|�|2��
,
where ⇤ = �3 and m
2= �2.
• Near-boundary asymptotics of � encodes QFT condensate h i.
• Spontaneous symmetry breaking:
– Black-brane solutions with T > Tc have � = 0.
– Black-brane solutions with T < Tc have � 6= 0.
A holographic model of a charged superfluid
Action: [Hartnoll, Herzog & Horowitz: 0803.3295]
Sgrav =
1
16⇡GN
Zd
4x
p�G
R+ ⇤+
1
q
2
��F
2 � |D�|2 �m
2|�|2��
,
where ⇤ = �3 and m
2= �2.
• Near-boundary asymptotics of � encodes QFT condensate h i.
• Spontaneous symmetry breaking:
– Black-brane solutions with T > Tc have � = 0.
– Black-brane solutions with T < Tc have � 6= 0.
Game plan:
• Start at T > Tc in distant past.
• Cool black brane through Tc.
• Watch � and h i form.
A holographic model of a charged superfluid
Action: [Hartnoll, Herzog & Horowitz: 0803.3295]
Sgrav =
1
16⇡GN
Zd
4x
p�G
R+ ⇤+
1
q
2
��F
2 � |D�|2 �m
2|�|2��
,
where ⇤ = �3 and m
2= �2.
• Near-boundary asymptotics of � encodes QFT condensate h i.
• Spontaneous symmetry breaking:
– Black-brane solutions with T > Tc have � = 0.
– Black-brane solutions with T < Tc have � 6= 0.
T > Tc
Game plan:
• Start at T > Tc in distant past.
• Cool black brane through Tc.
• Watch � and h i form.
A holographic model of a charged superfluid
Action: [Hartnoll, Herzog & Horowitz: 0803.3295]
Sgrav =
1
16⇡GN
Zd
4x
p�G
R+ ⇤+
1
q
2
��F
2 � |D�|2 �m
2|�|2��
,
where ⇤ = �3 and m
2= �2.
• Near-boundary asymptotics of � encodes QFT condensate h i.
• Spontaneous symmetry breaking:
– Black-brane solutions with T > Tc have � = 0.
– Black-brane solutions with T < Tc have � 6= 0.
T < Tc
T > Tc
Game plan:
• Start at T > Tc in distant past.
• Cool black brane through Tc.
• Watch � and h i form.
Stochastic driving
1. Stochastic processes choose di↵erent vacua at di↵erent x.
2. Boundary conditions limu!0 A⌫ = µ�⌫0, limu!0 @u� = '.
3. Statistics h'⇤(t,x)'(t0,x0
)i = ⇣�(t� t0)�2(x� x0).
4. Mimics backreaction of GN suppressed Hawking radiation.
'
⇣ ⇠ 1/N2
Stochastic driving
1. Stochastic processes choose di↵erent vacua at di↵erent x.
2. Boundary conditions limu!0 A⌫ = µ�⌫0, limu!0 @u� = '.
3. Statistics h'⇤(t,x)'(t0,x0
)i = ⇣�(t� t0)�2(x� x0).
4. Mimics backreaction of GN suppressed Hawking radiation.
'
⇣ ⇠ 1/N2
Movies show |h (t,x)i|2
Slow quenchFast quench
Movies show |h (t,x)i|2
Slow quenchFast quench
Movies show |h (t,x)i|2
Slow quenchFast quench
0 50 100 150 200 2500
0.25
0.5
0.75
1
t
tfreeze
Condensate growth
Adiabatic growth
|h i|2 ⇠ ✏(t)2�
avg� |h
i|2�
0 50 100 150 200 2500
0.25
0.5
0.75
1
t
tfreezeteq
Condensate growth
Adiabatic growth
|h i|2 ⇠ ✏(t)2�
avg� |h
i|2�
0 50 100 150 200 2500
0.25
0.5
0.75
1
t
tfreezeteq
Condensate growth
Adiabatic growth
|h i|2 ⇠ ✏(t)2�
avg� |h
i|2�
non-adiabatic
growth
0 50 100 150 200 2500
0.25
0.5
0.75
1
t
tfreezeteq
Condensate growth
Adiabatic growth
|h i|2 ⇠ ✏(t)2�
101 102 103100
101
102
t freezeteq√τQ
⌧Q
avg� |h
i|2�
non-adiabatic
growth
Non-adiabatic condensate growth
• Correlation function C(t, r) ⌘ h ⇤(t,x+ r) (t,x)i.
• Linear response
C(t, q) = ⇣
Zdt |G
R
(t, t0, q)|2.
• Relation to black brane quasinormal modes
GR
(t, t0, q) = ✓(t� t0)H(q)e�i
R 0tt
dt
00!
o
(✏(t
00),q)
where !o
is ✏ < 0 quasinormal mode analytically continued to ✏ > 0
• Instability for ✏ > 0
Im!o
= b✏z⌫ � a✏(z�2)⌫q2 +O(q4) > 0.
• Modes with q < qmax
with qmax
⇠ ✏(t)⌫ form condensate.
Non-adiabatic condensate growth (II)
At t > tfreeze
,
C(t, r) ⇠ C0
(t)e� r2
`co
(t)2 ,
where
C0
(t) ⇠ ⇣tfreeze
`co
(t)�dexp
(✓t
tfreeze
◆1+⌫z
).
and
`co
(t) = ⇠freeze
✓t
tfreeze
◆ 1+(z�2)⌫2
.
Linear response breaks down when C0
(t) ⇠ ✏(t)2�
teq
⇠ [logR]
1
1+⌫z tfreeze
, R ⇠ ⇣�1⌧(d�z)⌫�2�
1+⌫z
Q .
Non-adiabatic condensate growth (II)
At t > tfreeze
,
C(t, r) ⇠ C0
(t)e� r2
`co
(t)2 ,
where
C0
(t) ⇠ ⇣tfreeze
`co
(t)�dexp
(✓t
tfreeze
◆1+⌫z
).
and
`co
(t) = ⇠freeze
✓t
tfreeze
◆ 1+(z�2)⌫2
.
Linear response breaks down when C0
(t) ⇠ ✏(t)2�
teq
⇠ [logR]
1
1+⌫z tfreeze
, R ⇠ ⇣�1⌧(d�z)⌫�2�
1+⌫z
Q .
0 50 100 150 200 2500
0.25
0.5
0.75
1
0 2 4 610−5
10−4
10−3
10−2
10−1
100
t (t/tfreeze)1+⌫z
avg� |h
i|2�
Comparing to unstable mode analysis (I)
e(t/tfreeze)1+⌫z
300 600 900 1200 15001
1.5
2
2.5
3
t freeze/√τQ
teq /√τQ
const.
c√
log c ′√τQ
101 102 103100
101
102
t freezeteq√τQ
⌧Q
Comparing to unstable mode analysis (II)
⌧Q
For holography (mean field exponents)
• R ⇠ 1⇣p⌧Q
and tfreeze ⇠p⌧Q.
• teq ⇠plogR tfreeze
Consequences of extended non-adiabatic growth
⇥10�4
t = tfreeze t = 0.7teq t = 0.85teq t = teq
If teq
� tfreeze
then
• No well-defined vortices form until t ⇠ teq
.
• `co
(teq
) = ⇠freeze
⇣teq
tfreeze
⌘ 1+(z�2)⌫2 � ⇠
freeze
.
• Far fewer defects formed than KZ predicts
n/nKZ ⇠ (teq
/tfreeze
)
� (d�D)(1�(z�2)⌫)2 .
• State at t = �tfreeze
is irrelevant.
How natural is teq � tfreeze?
1. All holographic theories have teq � tfreeze.
• GN suppressed Hawking ) ⇣ ⇠ 1/N2 and
teq ⇠ [logN ]1/(1+⌫z)tfreeze.
2. Universality classes (d� z)⌫ � 2� > 0 have teq � tfreeze.
• teq ⇠ [log ⌧Q]1/(1+⌫z)tfreeze.
• Example: superfluid 4He.
) Log correction to density of defects
n
nKZ⇠ [log ⌧Q]
� (d�D)(1+(z�2)⌫)2(1+z⌫ nKZ .
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
t
t/tfreezer
C(t, r)/C(t, r = 0)
• Smear over scales ⇠ ⇠freeze.
⇠ FW
HM/⇠
freeze
IR coarsening before condensate formation
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
t
t/tfreezer
C(t, r)/C(t, r = 0)
• Smear over scales ⇠ ⇠freeze.
⇠ FW
HM/⇠
freeze
factor of 5!
IR coarsening before condensate formation
0 1 2 3 4 50
0.2
0.4
0.6
0.8
1
t
t/tfreezer
C(t, r)/C(t, r = 0)
• Smear over scales ⇠ ⇠freeze.
⇠ FW
HM/⇠
freeze
factor of 5!
Due to explosive growth of IR modes after +tfreeze.
IR coarsening before condensate formation
Counting defects
⇥10�4
t = tfreeze t = 0.7teq t = 0.85teq t = teq
101 102 103101
102
Nvort ices(
1.92LξFWHM
)2
τ−1/2Q
⌧Q
O(25) fewer vortices
than KZ estimate
For holography,
n ⇠ 1plogN
⌧�1/2Q .
Summary
• For wide class of theories there exists new scale teq.
• Exposive growth of IR modes between tfreeze < t < teq.
• If teq � tfreeze
– Initial correlation ⇠freeze not imprinted on final state.
– Far fewer defects formed than KZ predicts.
– Log corrections to KZ scaling law.
Sudden quenches
0 0.05 0.10
20
40
60
80
Nvort icesϵf
✏final
`co
⇠ 1/qmax
⇠ ✏�⌫final