U NIVERSALITY AND D YNAMIC L OCALIZATION IN K IBBLE -Z UREK Michael Kolodrubetz Boston University In collaboration with: B.K. Clark, D. Huse (Princeton) A. Polkovnikov, A. Katz (BU)

K IBBLE -Z UREK S CALING Disordered Ordered

K IBBLE -Z UREK SCALING Ramp rate Kibble-Zurek Ramp through the critical point at a constant, finite rate

K IBBLE -Z UREK SCALING Ramp rate

K IBBLE -Z UREK SCALING Ramp rate

K IBBLE -Z UREK SCALING Ramp rate

K IBBLE -Z UREK SCALING Ramp rate Fall out of equilibrium

K IBBLE -Z UREK SCALING Ramp rate Fall out of equilibrium

K IBBLE -Z UREK SCALING Ramp rate Slower

K IBBLE -Z UREK SCALING Ramp rate Slower

K IBBLE -Z UREK SCALING Ramp rate Slower

K IBBLE -Z UREK SCALING Ramp rate Slower

K IBBLE -Z UREK SCALING Ramp rate Slower

K IBBLE -Z UREK SCALING Recent work: Kibble-Zurek ramps show non-equilibrium scaling (in the limit of slow ramps) [Chandran et. al., Deng et. al., etc.]

K IBBLE -Z UREK SCALING Recent work: Kibble-Zurek ramps show non-equilibrium scaling (in the limit of slow ramps) More predictions than just defect production! [Chandran et. al., Deng et. al., etc.]

K IBBLE -Z UREK SCALING Excess heat

K IBBLE -Z UREK SCALING

Schrödinger Equation OR Observable

K IBBLE -Z UREK SCALING Schrödinger Equation OR Observable Fixed

K IBBLE -Z UREK SCALING Schrödinger Equation OR Observable Fixed Universal dynamics! =

T RANSVERSE - FIELD I SING CHAIN

Paramagnet (PM) Ferromagnet (FM)

T RANSVERSE - FIELD I SING CHAIN Paramagnet (PM) Ferromagnet (FM)

K IBBLE -Z UREK SCALING Ramp rate Slower

K IBBLE -Z UREK SCALING Ramp rate Slower

K IBBLE -Z UREK SCALING Excess heat

K IBBLE -Z UREK SCALING

Dynamics does not depend on ramp rate!

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Part II: Kibble-Zurek with a dynamic field

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? Part II: Kibble-Zurek with a dynamic field

U NIVERSALITY Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck)

U NIVERSALITY Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck)

U NIVERSALITY or Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck)

U NIVERSALITY or Ramp the tilt linearly in time Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck)

U NIVERSALITY or Ramp the tilt linearly in time: Solve numerically with DMRG Theory Sachdev et al. (2002) Experiment Greiner group (Harvard) Nagerl group (Innsbruck)

U NIVERSALITY

Dynamics are universal!

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Dynamics are universal to Ising-like QPTs Part II: Kibble-Zurek with a dynamic field

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Dynamics are universal to Ising-like QPTs Non-trivial scaling functions Part II: Kibble-Zurek with a dynamic field

N ON - EQUILIBRIUM PROPERTIES Spin-spin correlation function

N ON - EQUILIBRIUM PROPERTIES Thermal

N ON - EQUILIBRIUM PROPERTIES Kibble-Zurek Thermal

N ON - EQUILIBRIUM PROPERTIES Kibble-Zurek Thermal Antiferromagnetic

N ON - EQUILIBRIUM PROPERTIES Antiferromagnetic

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal Part II: Kibble-Zurek with a dynamic field

O UTLINE Part I: Universality of Kibble-Zurek scaling 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: Kibble-Zurek with a dynamic field

O UTLINE Part I: Universality of Kibble-Zurek scaling 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: Kibble-Zurek with a dynamic field Motivating example:  4 theory

D YNAMIC -F IELD K IBBLE -Z UREK

“Inflaton” “Higgs field”

D YNAMIC -F IELD K IBBLE -Z UREK “Inflaton” “Higgs field”

D YNAMIC -F IELD K IBBLE -Z UREK

When does field get trapped?

D YNAMIC -F IELD K IBBLE -Z UREK

Mass density

D YNAMIC -F IELD K IBBLE -Z UREK Mass density

D YNAMIC -F IELD K IBBLE -Z UREK Mass density Trapped

D YNAMIC -F IELD K IBBLE -Z UREK Mass density Ising: Higgs: Trapped

D YNAMIC -F IELD K IBBLE -Z UREK Mass density Ising: Higgs: Trapped

D YNAMIC -F IELD K IBBLE -Z UREK Mass density Ising: Higgs: Trapped

D YNAMIC -F IELD K IBBLE -Z UREK Mass density Ising: Higgs: Trapped

D YNAMIC -F IELD K IBBLE -Z UREK Mass density Ising: Higgs: Trapped

D YNAMIC -F IELD K IBBLE -Z UREK Mass density Ising: Higgs: Trapped

D YNAMIC -F IELD K IBBLE -Z UREK

Dynamics dominated by critical behavior

D YNAMIC -F IELD K IBBLE -Z UREK

Fluctuations around QCP

D YNAMIC -F IELD K IBBLE -Z UREK

Fluctuations around QCP

D YNAMIC -F IELD K IBBLE -Z UREK Fluctuations around QCP

D YNAMIC -F IELD K IBBLE -Z UREK Fluctuations around QCP

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Finite size scaling, dephasing, experiments… Part II: Kibble-Zurek with a dynamic field System is trapped at QCP by critical absorption

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Finite size scaling, dephasing, experiments… Part II: Kibble-Zurek with a dynamic field System is trapped at QCP by critical absorption Should work equally well for Higgs, etc.

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Finite size scaling, dephasing, experiments… Part II: Kibble-Zurek with a dynamic field System is trapped at QCP by critical absorption Should work equally well for Higgs, etc. Do dynamics show scaling collapse?

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Finite size scaling, dephasing, experiments… Part II: Kibble-Zurek with a dynamic field System is trapped at QCP by critical absorption Should work equally well for Higgs, etc. Do dynamics show scaling collapse? Expect scaling for

D YNAMIC -F IELD K IBBLE -Z UREK Scaling hypothesis Initial momentum is the relevant scale for dynamics

D YNAMIC -F IELD K IBBLE -Z UREK Scaling hypothesis Initial momentum is the relevant scale for dynamics

D YNAMIC -F IELD K IBBLE -Z UREK Scaling hypothesis Initial momentum is the relevant scale for dynamics

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Finite size scaling, dephasing, experiments… Part II: Kibble-Zurek with a dynamic field System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Finite size scaling, dephasing, experiments… Part II: Kibble-Zurek with a dynamic field System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Effect of ground state potential?

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Finite size scaling, dephasing, experiments… Part II: Kibble-Zurek with a dynamic field System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Effect of ground state potential Is RG relevant

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Finite size scaling, dephasing, experiments… Part II: Kibble-Zurek with a dynamic field System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Effect of ground state potential Is RG relevant Trapping in certain regimes

D YNAMIC -F IELD K IBBLE -Z UREK

O UTLINE Part I: Universality of Kibble-Zurek scaling Dynamics near QCP gives non-equilibrium critical scaling theory Are the results universal? What are some properties of the scaling functions? Finite size scaling, dephasing, experiments… Part II: Kibble-Zurek with a dynamic field System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Trapping can occur with ground state potential In progress: scaling with potential, emergent mass,  4 theory, inflationary models…

S UMMARY Part I: Universality of Kibble-Zurek scaling Part II: Kibble-Zurek with a dynamic field

T RANSVERSE - FIELD I SING CHAIN

 phase

T RANSVERSE - FIELD I SING CHAIN  phase

E QUILIBRIUM SCALING “Spin up”  (k,-k) unoccupied “Spin down”  (k,-k) occupied

E QUILIBRIUM SCALING Low energy, long wavelength theory? “Spin up”  (k,-k) unoccupied “Spin down”  (k,-k) occupied

E QUILIBRIUM SCALING Low energy, long wavelength theory “Spin up”  (k,-k) unoccupied “Spin down”  (k,-k) occupied

K IBBLE -Z UREK SCALING

Low energy, long wavelength theory?

K IBBLE -Z UREK SCALING Low energy, long wavelength theory?

K IBBLE -Z UREK SCALING Low energy, long wavelength theory

N ON - EQUILIBRIUM PROPERTIES

Inverted

D YNAMIC -F IELD I SING CHAIN Basic idea: Add (classical) dynamics to the transverse field

D YNAMIC -F IELD I SING CHAIN Basic idea: Add (classical) dynamics to the transverse field

D YNAMIC -F IELD I SING CHAIN Basic idea: Add (classical) dynamics to the transverse field “Friction” = back-action of spins on field

D YNAMIC -F IELD I SING 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

D YNAMIC -F IELD I SING 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?