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
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
Paramagnet
Antiferromagnet
UNIVERSALITY
Paramagnet
Antiferromagnet
Ramp the tilt ( ) linearly in time
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