NON-EQUILIBRIUM DYNAMIC CRITICAL SCALING OF THE QUANTUM ISING CHAINMichael Kolodrubetz

Princeton University

In collaboration with: Bryan Clark, David Huse

David Pekker

Krishnendu Sengupta

QUANTUM STATE OF TRANSVERSE-FIELD ISING MODEL DURING SLOW RAMP IS…

Universal

Non-equilibrium

Experimentally viable

Non-thermal

Dephasing resistant

CLASSICAL PHASE TRANSITIONS

“Magnetization”

Landau-Ginzburgfunctional

CLASSICAL PHASE TRANSITIONS

“Magnetization”

“Magnetization”

Thermal fluctuations

CLASSICAL PHASE TRANSITIONS

QUANTUM PHASE TRANSITIONS

One-dimensional transverse-field Ising chain

QUANTUM PHASE TRANSITIONS

One-dimensional transverse-field Ising chain

Paramagnet (PM) Ferromagnet (FM)

QUANTUM PHASE TRANSITIONS

One-dimensional transverse-field Ising chain

Paramagnet (PM) Ferromagnet (FM)

Quantum fluctuations

CRITICAL SCALING

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

CRITICAL SCALING

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

CRITICAL SCALING

Correlation lengthcritical exponent

Dynamiccritical exponent

,

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

CRITICAL SCALING

,

Ising:

Correlation lengthcritical exponent

Dynamiccritical exponent

CRITICAL SCALING

,

Ising:

Correlation lengthcritical exponent

Dynamiccritical exponent

Order parametercritical exponent

CRITICAL SCALING

,

Ising:

Correlation lengthcritical exponent

Dynamiccritical exponent

Order parametercritical exponent

KIBBLE-ZUREK RAMPS

Ramp rate

KIBBLE-ZUREK RAMPS

Ramp rate

KIBBLE-ZUREK RAMPS

Adiabatic

Ramp rate

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

KIBBLE-ZUREK RAMPS

METHOD IDEA WHEN IT WORKS

“Old-school”Kibble-Zurek[Kibble 1976, Zurek 1985]

and set the “interesting” time and length scales

-Ramp to the QCP-Ramp to deep in the FM phase

KIBBLE-ZUREK RAMPS

METHOD IDEA WHEN IT WORKS

“Old-school”Kibble-Zurek[Kibble 1976, Zurek 1985]

and set the “interesting” time and length scales

-Ramp to the QCP-Ramp to deep in the FM phase

Kibble-Zurek scaling [Deng et. al. 2008, Erez et. al., in prep., Polkovnikov, …]

Most quantities show scaling collapse when scaled by and

Throughout the ramp

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

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 Hamiltonian conserves parity for each mode k

Sachdev: “Quantum Phase Transitions”

phase

TRANSVERSE-FIELD ISING CHAIN

Wigner fermionize

Quadratic Integrable Hamiltonian conserves parity for each mode k Work in subspace where parity is even

Sachdev: “Quantum Phase Transitions”

phase

TRANSVERSE-FIELD ISING CHAIN

Wigner fermionize

Quadratic Integrable Hamiltonian conserves parity for each mode k Work in subspace where parity is even

Sachdev: “Quantum Phase Transitions”

phase

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

Low energy, long wavelength theory

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

Low energy, long wavelength theory?

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

Low energy, long wavelength theory

KIBBLE-ZUREK RAMPS

Adiabatic

Impulse

Ramp rate

Low energy, long wavelength theory

KIBBLE-ZUREK SCALING LIMIT

SchrödingerEquation

ORObservable

Fixed

KIBBLE-ZUREK SCALING LIMIT

KIBBLE-ZUREK SCALING LIMIT

KIBBLE-ZUREK SCALING LIMIT

KIBBLE-ZUREK SCALING LIMIT

KIBBLE-ZUREK OBSERVABLES

Excess heat

Spin-spin correlation function

KIBBLE-ZUREK OBSERVABLES

Excess heat

Spin-spin correlation function

KIBBLE-ZUREK OBSERVABLES

KIBBLE-ZUREK OBSERVABLES

KIBBLE-ZUREK OBSERVABLES

KIBBLE-ZUREK OBSERVABLES

KIBBLE-ZUREK OBSERVABLES

KIBBLE-ZUREK OBSERVABLES

KIBBLE-ZUREK OBSERVABLES

FINITE-SIZE SCALING

FINITE-SIZE SCALING

FINITE-SIZE SCALING

Finite size effectscan be ignored

FINITE-SIZE SCALING

FINITE-SIZE SCALING

EQUILIBRIUM VIA DYNAMICS

KZ scalingfunction

Equilibriumscalingfunction

If dynamic scaling functions exist, they must have the equilibrium critical

exponents

FINITE-SIZE SCALING

FINITE-SIZE SCALING

LANDAU-ZENER DYNAMICS

LANDAU-ZENER DYNAMICS

LANDAU-ZENER DYNAMICS

FINITE-SIZE SCALING

LANDAU-ZENER DYNAMICS

ATHERMAL PROPERTIES

ATHERMAL PROPERTIES

ATHERMAL PROPERTIES

ATHERMAL PROPERTIES

ATHERMAL PROPERTIES

Inverted

ATHERMAL PROPERTIES

Kibble-Z

urek

ATHERMAL PROPERTIES

Kibble-Z

urek

Thermal

DEPHASING

Protocol Ramp to create excitations Freeze the Hamiltonian Wait

DEPHASING

Protocol Ramp to create excitations Freeze the Hamiltonian Wait

… …

DEPHASING

… …

Protocol Ramp to create excitations Freeze the Hamiltonian Wait

DEPHASING

… …

Protocol Ramp to create excitations Freeze the Hamiltonian Wait

DEPHASING

… …

Protocol Ramp to create excitations Freeze the Hamiltonian Wait

DEPHASING

… …

Protocol Ramp to create excitations Freeze the Hamiltonian Wait

DEPHASING

Dephasing inintegrable model:

Generalized Gibb’s ensemble

(GGE)… …

Protocol Ramp to create excitations Freeze the Hamiltonian Wait

DEPHASING

Dephasing inintegrable model:

Generalized Gibb’s ensemble

(GGE)… …

Does dephasing occur during the Kibble-

Zurek ramp?

DEPHASING

Dephasing inintegrable model:

Generalized Gibb’s ensemble

(GGE)… …

DEPHASING

Dephasing inintegrable model:

Generalized Gibb’s ensemble

(GGE)

as

… …

DEPHASING

Dephasing inintegrable model:

Generalized Gibb’s ensemble

(GGE)

as

Dephases

… …

DEPHASING

Cubic ramp:

… …

DEPHASING

Cubic ramp:

as

… …

DEPHASING

Cubic ramp:

as

Does Not Dephase

… …

UNIVERSALITY

UNIVERSALITY

UNIVERSALITY

Additional terms change (renormalize) the non-universal aspects of the critical point

They do not change critical scaling Critical exponents Scaling functions

Debated for non-integrable system dynamics

UNIVERSALITY

UNIVERSALITY

UNIVERSALITY

Paramagnet

Antiferromagnet

UNIVERSALITY

Paramagnet

Antiferromagnet

Ramp the tilt ( ) linearly in time

UNIVERSALITY

UNIVERSALITY

UNIVERSALITY

UNIVERSALITY

CONCLUSIONS

Solved dynamic critical scaling behavior of the TFI chain Athermal negative correlations Phase-locked high order ramps

Strong numerical evidence for universality Tilted boson model has

same scaling functions Experimentally accessible

Athermal features robust againstopen boundary conditions

Open b.c. simplifies measurement Time scales already available

[Simon et. al., 2007]

DEPHASING VIA QUASIPARTICLES

DEPHASING VIA QUASIPARTICLES

OPEN BOUNDARY CONDITIONS

OPEN BOUNDARY CONDITIONS

OPEN BOUNDARY CONDITIONS

UNIVERSALITY

Remove spin ups onneighboring sites