universality and dynamic localization in kibble- zurek scaling of the quantum ising chain
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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
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
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 lengthcritical exponent
Dynamiccritical exponent
,
[Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]
QUANTUM PHASE TRANSITION
Correlation lengthcritical exponent
Dynamiccritical exponent
,
Ising:
QUANTUM PHASE TRANSITION
Correlation lengthcritical exponent
Dynamiccritical exponent
,
Ising:
Can these results be extendedto non-equilbrium dynamics?
KIBBLE-ZUREK RAMPS
Ramp rate
Kibble-ZurekRamp 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 ofequilibrium
KIBBLE-ZUREK RAMPS
Ramp rate
Fall out ofequilibrium
KIBBLE-ZUREK RAMPS
Ramp rate
Fall out ofequilibrium
KIBBLE-ZUREK RAMPS
Ramp rate
Fall out ofequilibrium
KIBBLE-ZUREK RAMPS
Ramp rate
Slower
KIBBLE-ZUREK RAMPS
Ramp rate
Slower
KIBBLE-ZUREK RAMPS
Ramp rate
Slower
KIBBLE-ZUREK RAMPS
Ramp rate
Slower
KIBBLE-ZUREK RAMPS
Ramp rate
Slower
KIBBLE-ZUREK RAMPS
Kibble-Zurek ramps shownon-equilibrium scaling
[Chandran et. al., Deng et. al., etc.]
KIBBLE-ZUREK RAMPS
Kibble-Zurek ramps shownon-equilibrium scaling (in the limit of slow ramps)
[Chandran et. al., Deng et. al., etc.]
KIBBLE-ZUREK RAMPS
Kibble-Zurek ramps shownon-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 shownon-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
Wigner fermionizeSachdev: “Quantum Phase Transitions”
phase
TRANSVERSE-FIELD ISING CHAIN
Wigner fermionizeSachdev: “Quantum Phase Transitions”
phase
TRANSVERSE-FIELD ISING CHAIN
Wigner fermionize
Quadratic Integrable
Sachdev: “Quantum Phase Transitions”
phase
TRANSVERSE-FIELD ISING CHAIN
Wigner fermionize
Quadratic Integrable Work in subspace where each mode
(k,-k) is either occupied or unoccupied
Sachdev: “Quantum Phase Transitions”
phase
TRANSVERSE-FIELD ISING CHAIN
Wigner fermionize
Quadratic Integrable Work in subspace where each mode
(k,-k) is either occupied or unoccupied
Sachdev: “Quantum Phase Transitions”
phase
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
Low energy, long wavelength theory?
“Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied
EQUILIBRIUM SCALING
Low energy, long wavelength theory
“Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied
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
SchrödingerEquation
ORObservable
KIBBLE-ZUREK SCALING
SchrödingerEquation
ORObservable
Fixed
KIBBLE-ZUREK SCALING
SchrödingerEquation
ORObservable
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 givesnon-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 givesnon-equilibrium critical scaling theory
Are the results universal?
Part II: Transverse-field Ising chain with a dynamic field
UNIVERSALITY
TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)
UNIVERSALITY
TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)
UNIVERSALITY
or
TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)
UNIVERSALITY
or
Ramp the tilt linearly in time
TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)
UNIVERSALITY
or
Ramp the tilt linearly in time: Solve numerically
with DMRG
TheorySachdev et al. (2002)ExperimentGreiner 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 givesnon-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 givesnon-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
Thermal
NON-EQUILIBRIUM PROPERTIES
Kibble-Z
urek
Thermal
NON-EQUILIBRIUM PROPERTIES
Kibble-Z
urek
Thermal
Antiferromagnetic
NON-EQUILIBRIUM PROPERTIES
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-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 givesnon-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 givesnon-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
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
DYNAMIC-FIELD ISING CHAIN
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-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 givesnon-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
Overdamped/underdamped?
DYNAMIC-FIELD ISING CHAIN
Overdamped/underdamped?
Measure velocity at QCP
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
DYNAMIC-FIELD ISING CHAIN
HypothesisInitial momentum is the
relevant scale for dynamics
DYNAMIC-FIELD ISING CHAIN
HypothesisInitial momentum is the
relevant scale for dynamics
DYNAMIC-FIELD ISING CHAIN
HypothesisInitial momentum is the
relevant scale for dynamics
DYNAMIC-FIELD ISING CHAIN
HypothesisInitial momentum is the
relevant scale for dynamics
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-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 givesnon-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
Are long-time dynamics well-described by the dephased
ensemble?(generalized Gibbs ensemble /
GGE)
Dephasing
DYNAMIC-FIELD ISING CHAIN
Manually dephase
Are long-time dynamics well-described by the dephased
ensemble?(generalized Gibbs ensemble /
GGE)
DYNAMIC-FIELD ISING CHAIN
Manually dephase
Are long-time dynamics well-described by the dephased
ensemble?(generalized Gibbs ensemble /
GGE)
DYNAMIC-FIELD ISING CHAIN
Manually dephase
Are long-time dynamics well-described 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 givesnon-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 givesnon-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
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