Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain

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




Paramagnet (PM)



Paramagnet (PM) Ferromagnet (FM)




Paramagnet (PM) Ferromagnet (FM)

Quantum phase transition

QUANTUM PHASE TRANSITION

[Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]








,



,



Correlation lengthcritical exponent

Dynamiccritical exponent

,





,

Ising:




,

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


KIBBLE-ZUREK RAMPS

Ramp rate


KIBBLE-ZUREK RAMPS

Ramp rate


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)


KIBBLE-ZUREK RAMPS

Kibble-Zurek ramps shownon-equilibrium scaling (in the limit of slow ramps) More than a theory of defect production!


KIBBLE-ZUREK RAMPS

Kibble-Zurek ramps shownon-equilibrium scaling (in the limit of slow ramps) More than a theory of defect production!


KIBBLE-ZUREK SCALING

Excess heat

KIBBLE-ZUREK OBSERVABLES









Sachdev: “Quantum Phase Transitions”


Wigner fermionizeSachdev: “Quantum Phase Transitions”

phase


Wigner fermionizeSachdev: “Quantum Phase Transitions”

phase


Wigner fermionize

Quadratic Integrable


phase


Wigner fermionize

Quadratic Integrable Work in subspace where each mode

(k,-k) is either occupied or unoccupied


phase


Wigner fermionize

Quadratic Integrable Work in subspace where each mode

(k,-k) is either occupied or unoccupied


phase

EQUILIBRIUM SCALING

“Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied

EQUILIBRIUM SCALING


EQUILIBRIUM SCALING

Low energy, long wavelength theory?


EQUILIBRIUM SCALING

Low energy, long wavelength theory








Low energy, long wavelength theory


SchrödingerEquation

ORObservable



ORObservable

Fixed



ORObservable

Fixed







OUTLINE


Dynamics near QCP givesnon-equilibrium critical scaling theory


OUTLINE



Are the results universal?


UNIVERSALITY

TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)

UNIVERSALITY


UNIVERSALITY

or


UNIVERSALITY

or

Ramp the tilt linearly in time


UNIVERSALITY

or

Ramp the tilt linearly in time: Solve numerically

with DMRG


UNIVERSALITY

UNIVERSALITY

UNIVERSALITY

UNIVERSALITY

UNIVERSALITY

UNIVERSALITY

Matches to analytical solution of the Ising

chain!

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Dynamics are universal to Ising-like QPTs


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Dynamics are universal to Ising-like QPTs What are some properties of the scaling

functions?


NON-EQUILIBRIUM PROPERTIES

Spin-spin correlation function


Spin-spin correlation function



Ground state


Ground state


Ground state









Inverted


Thermal


Kibble-Z

urek

Thermal


Kibble-Z

urek

Thermal

Antiferromagnetic


OUTLINE



Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal


OUTLINE



Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal Finite size scaling, dephasing, experiments…


OUTLINE



Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal Finite size scaling, dephasing, experiments…


DYNAMIC-FIELD ISING CHAIN

Basic idea: Add (classical) dynamics to the transverse field





“Friction” = back-action of spins on field



“Friction” = back-action of spins on field Mass is extensive ( ) Mean-field coupling between field and spins



“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?




as


as

Field motion arrested by

QCP!








Dynamics dominated by critical behavior


Dynamics dominated by critical behavior

Linearize the Hamiltonian



















What happens for other models?



Crossover tunable via… …dimensionality


Crossover tunable via… …dimensionality …critical exponents


Crossover tunable via… …dimensionality …critical exponents

Possibility of as


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Are the results universal? What are some properties of the scaling

functions?


Field is trapped at QCP by critical absorption

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functions?


Field is trapped at QCP by critical absorption Dynamics of field during trapping?


Overdamped/underdamped?


Overdamped/underdamped?

Measure velocity at QCP





HypothesisInitial momentum is the

relevant scale for dynamics










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functions?


System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse

OUTLINE




functions?


System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Analytical understanding of late-time dynamics?



Dephasing


Are long-time dynamics well-described by the dephased

ensemble?(generalized Gibbs ensemble /

GGE)

Dephasing


Manually dephase



GGE)


Manually dephase



GGE)


Manually dephase


ensemble?

YES!


Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli

Polkovnikov and Luca D’Alessio



Polkovnikov and Luca D’Alessio Start from stationary state of



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


Approximate dynamics by adiabatic PT


Approximate dynamics by adiabatic PT

Need to know… Initial condition on , Mode occupation numbers



OUTLINE




functions?


System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Late-time dynamics are given by dephasing

OUTLINE




functions?


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


Is it RG relevant? Tune the scaling of excess heat

FUTURE DIRECTIONS



Generalize Ising model to higher dimensions Use models with other critical exponents

FUTURE DIRECTIONS



Generalize Ising model to higher dimensions Use models with other critical exponents What happens if as

FUTURE DIRECTIONS



Generalize Ising model to higher dimensions Use models with other critical exponents What happens if as

Relationship to the Higgs boson?

SUMMARY












Documents

Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain