UNIVERSALITY AND DYNAMIC
LOCALIZATION IN KIBBLE-ZUREK
SCALING OF THE QUANTUM ISING CHAIN
Michael Kolodrubetz
Boston University
In collaboration with:
B.K. Clark, D. Huse (Princeton)
A. Polkovnikov, A. Katz (BU)
OUTLINE


Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Part II: Transverse-field Ising chain
with a dynamic field
TRANSVERSE-FIELD ISING CHAIN
One-dimensional transverse-field Ising chain
TRANSVERSE-FIELD ISING CHAIN
One-dimensional transverse-field Ising chain
TRANSVERSE-FIELD ISING CHAIN
One-dimensional transverse-field Ising chain
Paramagnet (PM)
TRANSVERSE-FIELD ISING CHAIN
One-dimensional transverse-field Ising chain
Paramagnet (PM)
Ferromagnet (FM)
TRANSVERSE-FIELD ISING CHAIN
One-dimensional transverse-field Ising chain
Paramagnet (PM)
Ferromagnet (FM)
Quantum phase transition
QUANTUM PHASE TRANSITION
[Smirnov, php.math.unifi.it/users/paf/
LaPietra/files/Chelkak01.ppt]
QUANTUM PHASE TRANSITION
[Smirnov, php.math.unifi.it/users/paf/
LaPietra/files/Chelkak01.ppt]
QUANTUM PHASE TRANSITION
[Smirnov, php.math.unifi.it/users/paf/
LaPietra/files/Chelkak01.ppt]
QUANTUM PHASE TRANSITION
[Smirnov, php.math.unifi.it/users/paf/
LaPietra/files/Chelkak01.ppt]
QUANTUM PHASE TRANSITION
,
[Smirnov, php.math.unifi.it/users/paf/
LaPietra/files/Chelkak01.ppt]
QUANTUM PHASE TRANSITION
,
[Smirnov, php.math.unifi.it/users/paf/
LaPietra/files/Chelkak01.ppt]
QUANTUM PHASE TRANSITION
,
Correlation length
critical exponent
Dynamic
critical exponent
[Smirnov, php.math.unifi.it/users/paf/
LaPietra/files/Chelkak01.ppt]
QUANTUM PHASE TRANSITION
,
Correlation length
critical exponent
Dynamic
critical exponent
Ising:
QUANTUM PHASE TRANSITION
Can these results be extended
to non-equilbrium dynamics?
,
Correlation length
critical exponent
Dynamic
critical exponent
Ising:
KIBBLE-ZUREK RAMPS
Ramp rate
Kibble-Zurek
Ramp through the critical point
at a constant, finite rate
KIBBLE-ZUREK RAMPS
Ramp rate
KIBBLE-ZUREK RAMPS
Ramp rate
KIBBLE-ZUREK RAMPS
Ramp rate
Fall out of
equilibrium
KIBBLE-ZUREK RAMPS
Ramp rate
Fall out of
equilibrium
KIBBLE-ZUREK RAMPS
Ramp rate
Fall out of
equilibrium
KIBBLE-ZUREK RAMPS
Ramp rate
Fall out of
equilibrium
KIBBLE-ZUREK RAMPS
Ramp rate
KIBBLE-ZUREK RAMPS
Ramp rate
KIBBLE-ZUREK RAMPS
Ramp rate
KIBBLE-ZUREK RAMPS
Ramp rate
KIBBLE-ZUREK RAMPS
Ramp rate
KIBBLE-ZUREK RAMPS

Kibble-Zurek ramps show
non-equilibrium scaling
[Chandran et. al., Deng et. al., etc.]
KIBBLE-ZUREK RAMPS

Kibble-Zurek ramps show
non-equilibrium scaling

(in the limit of slow ramps)
[Chandran et. al., Deng et. al., etc.]
KIBBLE-ZUREK RAMPS

Kibble-Zurek ramps show
non-equilibrium scaling
(in the limit of slow ramps)
 More than a theory of defect production!

[Chandran et. al., Deng et. al., etc.]
KIBBLE-ZUREK RAMPS

Kibble-Zurek ramps show
non-equilibrium scaling
(in the limit of slow ramps)
 More than a theory of defect production!

[Chandran et. al., Deng et. al., etc.]
KIBBLE-ZUREK SCALING
Excess heat
KIBBLE-ZUREK OBSERVABLES
KIBBLE-ZUREK OBSERVABLES
KIBBLE-ZUREK OBSERVABLES
KIBBLE-ZUREK OBSERVABLES
KIBBLE-ZUREK OBSERVABLES
KIBBLE-ZUREK SCALING
KIBBLE-ZUREK SCALING
KIBBLE-ZUREK SCALING
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
 phase
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
 phase
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
 phase

Quadratic  Integrable
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
 phase


Quadratic  Integrable
Work in subspace where each mode
(k,-k) is either occupied or unoccupied
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
 phase


Quadratic  Integrable
Work in subspace where each mode
(k,-k) is either occupied or unoccupied
EQUILIBRIUM SCALING
“Spin up”  (k,-k) unoccupied
“Spin down”  (k,-k) occupied
EQUILIBRIUM SCALING
“Spin up”  (k,-k) unoccupied
“Spin down”  (k,-k) occupied
EQUILIBRIUM SCALING
“Spin up”  (k,-k) unoccupied
“Spin down”  (k,-k) occupied
Low energy, long wavelength theory?
EQUILIBRIUM SCALING
“Spin up”  (k,-k) unoccupied
“Spin down”  (k,-k) occupied
Low energy, long wavelength theory
KIBBLE-ZUREK SCALING
KIBBLE-ZUREK SCALING
Low energy, long wavelength theory?
KIBBLE-ZUREK SCALING
Low energy, long wavelength theory?
KIBBLE-ZUREK SCALING
Low energy, long wavelength theory
KIBBLE-ZUREK SCALING
Schrödinger
Equation
OR
Observable
KIBBLE-ZUREK SCALING
Schrödinger
Equation
OR
Observable
Fixed
KIBBLE-ZUREK SCALING
Schrödinger
Equation
OR
Observable
Fixed
KIBBLE-ZUREK SCALING
KIBBLE-ZUREK SCALING
KIBBLE-ZUREK SCALING
KIBBLE-ZUREK SCALING
KIBBLE-ZUREK SCALING
KIBBLE-ZUREK SCALING
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain


Dynamics near QCP gives
non-equilibrium critical scaling theory
Part II: Transverse-field Ising chain
with a dynamic field
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Are the results universal?


Part II: Transverse-field Ising chain
with a dynamic field
UNIVERSALITY
Theory
Sachdev et al. (2002)
Experiment
Greiner group (Harvard)
Nagerl group (Innsbruck)
UNIVERSALITY
Theory
Sachdev et al. (2002)
Experiment
Greiner group (Harvard)
Nagerl group (Innsbruck)
UNIVERSALITY
or
Theory
Sachdev et al. (2002)
Experiment
Greiner group (Harvard)
Nagerl group (Innsbruck)
UNIVERSALITY
Ramp the tilt
linearly in time
or
Theory
Sachdev et al. (2002)
Experiment
Greiner group (Harvard)
Nagerl group (Innsbruck)
UNIVERSALITY
Ramp the tilt
linearly in time:
Solve numerically
with DMRG
or
Theory
Sachdev et al. (2002)
Experiment
Greiner group (Harvard)
Nagerl group (Innsbruck)
UNIVERSALITY
UNIVERSALITY
UNIVERSALITY
UNIVERSALITY
UNIVERSALITY
UNIVERSALITY
Matches to analytical
solution of the Ising chain!
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Dynamics are universal to Ising-like QPTs


Part II: Transverse-field Ising chain
with a dynamic field
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Dynamics are universal to Ising-like QPTs
 What are some properties of the scaling functions?


Part II: Transverse-field Ising chain
with a dynamic field
NON-EQUILIBRIUM PROPERTIES
Spin-spin correlation function
NON-EQUILIBRIUM PROPERTIES
Spin-spin correlation function
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
Ground state
NON-EQUILIBRIUM PROPERTIES
Ground state
NON-EQUILIBRIUM PROPERTIES
Ground state
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
Inverted
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
NON-EQUILIBRIUM PROPERTIES
Antiferromagnetic
NON-EQUILIBRIUM PROPERTIES
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Dynamics are universal to Ising-like QPTs
 Long-time dynamics are athermal


Part II: Transverse-field Ising chain
with a dynamic field
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Dynamics are universal to Ising-like QPTs
 Long-time dynamics are athermal
 Finite size scaling, dephasing, experiments…


Part II: Transverse-field Ising chain
with a dynamic field
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Dynamics are universal to Ising-like QPTs
 Long-time dynamics are athermal
 Finite size scaling, dephasing, experiments…


Part II: Transverse-field Ising chain
with a dynamic field
DYNAMIC-FIELD ISING CHAIN

Basic idea: Add (classical) dynamics
to the transverse field
DYNAMIC-FIELD ISING CHAIN

Basic idea: Add (classical) dynamics
to the transverse field
DYNAMIC-FIELD ISING CHAIN


Basic idea: Add (classical) dynamics
to the transverse field
“Friction” = back-action of spins on field
DYNAMIC-FIELD ISING CHAIN

Basic idea: Add (classical) dynamics
to the transverse field
“Friction” = back-action of spins on field
 Mass is extensive (
)
 Mean-field coupling between field and spins

DYNAMIC-FIELD ISING CHAIN

Basic idea: Add (classical) dynamics
to the transverse field
“Friction” = back-action of spins on field
 Mass is extensive (
)
 Mean-field coupling between field and spins

What happens when field tries to
pass through the critical point?
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
as
DYNAMIC-FIELD ISING CHAIN
as

Field motion
arrested by
QCP!
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN

Dynamics dominated
by critical behavior
DYNAMIC-FIELD ISING CHAIN
Dynamics dominated
by critical behavior
 Linearize the
Hamiltonian

DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
What happens for
other models?
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN

Crossover tunable via…

…dimensionality
DYNAMIC-FIELD ISING CHAIN

Crossover tunable via…
…dimensionality
 …critical exponents

DYNAMIC-FIELD ISING CHAIN

Crossover tunable via…
…dimensionality
 …critical exponents


Possibility of
as
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Are the results universal?
 What are some properties of the scaling functions?


Part II: Transverse-field Ising chain
with a dynamic field

Field is trapped at QCP by critical absorption
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Are the results universal?
 What are some properties of the scaling functions?


Part II: Transverse-field Ising chain
with a dynamic field


Field is trapped at QCP by critical absorption
Dynamics of field during trapping?
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
Overdamped/underdamped?
DYNAMIC-FIELD ISING CHAIN
Measure velocity at QCP
Overdamped/underdamped?
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
Hypothesis
Initial momentum is the
relevant scale for dynamics
DYNAMIC-FIELD ISING CHAIN
Hypothesis
Initial momentum is the
relevant scale for dynamics
DYNAMIC-FIELD ISING CHAIN
Hypothesis
Initial momentum is the
relevant scale for dynamics
DYNAMIC-FIELD ISING CHAIN
Hypothesis
Initial momentum is the
relevant scale for dynamics
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Are the results universal?
 What are some properties of the scaling functions?


Part II: Transverse-field Ising chain
with a dynamic field


System is trapped at QCP by critical absorption
Trapping dynamics show scaling collapse
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Are the results universal?
 What are some properties of the scaling functions?


Part II: Transverse-field Ising chain
with a dynamic field



System is trapped at QCP by critical absorption
Trapping dynamics show scaling collapse
Analytical understanding of late-time dynamics?
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
Dephasing
DYNAMIC-FIELD ISING CHAIN
Dephasing
Are long-time dynamics welldescribed by the dephased ensemble?
(generalized Gibbs ensemble / GGE)
DYNAMIC-FIELD ISING CHAIN
Manually dephase
Are long-time dynamics welldescribed by the dephased ensemble?
(generalized Gibbs ensemble / GGE)
DYNAMIC-FIELD ISING CHAIN
Manually dephase
Are long-time dynamics welldescribed by the dephased ensemble?
(generalized Gibbs ensemble / GGE)
DYNAMIC-FIELD ISING CHAIN
Manually dephase
Are long-time dynamics welldescribed by the dephased ensemble?
YES!
DYNAMIC-FIELD ISING CHAIN

Approximate dynamics by adiabatic PT

Based on unpublished work by Anatoli Polkovnikov
and Luca D’Alessio
DYNAMIC-FIELD ISING CHAIN

Approximate dynamics by adiabatic PT
Based on unpublished work by Anatoli Polkovnikov
and Luca D’Alessio
 Start from stationary state of

DYNAMIC-FIELD ISING CHAIN

Approximate dynamics by adiabatic PT
Based on unpublished work by Anatoli Polkovnikov
and Luca D’Alessio
 Start from stationary state of
 Go to the “moving frame” of


Frame that locally diagonalizes
DYNAMIC-FIELD ISING CHAIN

Approximate dynamics by adiabatic PT
Based on unpublished work by Anatoli Polkovnikov
and Luca D’Alessio
 Start from stationary state of
 Go to the “moving frame” of

Frame that locally diagonalizes



DYNAMIC-FIELD ISING CHAIN

Approximate dynamics by adiabatic PT
Based on unpublished work by Anatoli Polkovnikov
and Luca D’Alessio
 Start from stationary state of
 Go to the “moving frame” of

Frame that locally diagonalizes




Treat
term via 2nd order time-dependent PT
DYNAMIC-FIELD ISING CHAIN

Approximate dynamics by adiabatic PT
DYNAMIC-FIELD ISING CHAIN

Approximate dynamics by adiabatic PT

Need to know…
Initial condition on ,
 Mode occupation numbers

DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Are the results universal?
 What are some properties of the scaling functions?


Part II: Transverse-field Ising chain
with a dynamic field



System is trapped at QCP by critical absorption
Trapping dynamics show scaling collapse
Late-time dynamics are given by dephasing
OUTLINE

Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Dynamics near QCP gives
non-equilibrium critical scaling theory
 Are the results universal?
 What are some properties of the scaling functions?


Part II: Transverse-field Ising chain
with a dynamic field



System is trapped at QCP by critical absorption
Trapping dynamics show scaling collapse
Late-time dynamics are given by dephasing
FUTURE DIRECTIONS

Analytically understand the dynamics via APT
FUTURE DIRECTIONS
Analytically understand the dynamics via APT
 Remove the offset potential


Is it RG relevant?
FUTURE DIRECTIONS
Analytically understand the dynamics via APT
 Remove the offset potential



Is it RG relevant?
Tune the scaling of excess heat

FUTURE DIRECTIONS
Analytically understand the dynamics via APT
 Remove the offset potential



Is it RG relevant?
Tune the scaling of excess heat

Generalize Ising model to higher dimensions
 Use models with other critical exponents

FUTURE DIRECTIONS
Analytically understand the dynamics via APT
 Remove the offset potential



Is it RG relevant?
Tune the scaling of excess heat

Generalize Ising model to higher dimensions
 Use models with other critical exponents
 What happens if
as

FUTURE DIRECTIONS
Analytically understand the dynamics via APT
 Remove the offset potential



Is it RG relevant?
Tune the scaling of excess heat

Generalize Ising model to higher dimensions
 Use models with other critical exponents
 What happens if
as


Relationship to the Higgs boson?
SUMMARY


Part I: Kibble-Zurek scaling of the
transverse-field Ising chain
Part II: Transverse-field Ising chain
with a dynamic field
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
TRANSVERSE-FIELD ISING CHAIN
TRANSVERSE-FIELD ISING CHAIN
TRANSVERSE-FIELD ISING CHAIN
TRANSVERSE-FIELD ISING CHAIN
TRANSVERSE-FIELD ISING CHAIN
TRANSVERSE-FIELD ISING CHAIN
TRANSVERSE-FIELD ISING CHAIN