x2

x1

z

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