Transcript Dephases
NON-EQUILIBRIUM DYNAMIC
CRITICAL SCALING OF THE
QUANTUM ISING CHAIN
Michael 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-thermal
Non-equilibrium
Dephasing resistant
Experimentally viable
CLASSICAL PHASE TRANSITIONS
“Magnetization”
Landau-Ginzburg
functional
CLASSICAL PHASE TRANSITIONS
“Magnetization”
CLASSICAL PHASE TRANSITIONS
“Magnetization”
Thermal fluctuations
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 length
critical exponent
Dynamic
critical exponent
[Smirnov, php.math.unifi.it/users/paf/
LaPietra/files/Chelkak01.ppt]
CRITICAL SCALING
Ising:
,
Correlation length
critical exponent
Dynamic
critical exponent
CRITICAL SCALING
Ising:
,
Order parameter
critical exponent
Correlation length
critical exponent
Dynamic
critical exponent
CRITICAL SCALING
Ising:
,
Order parameter
critical exponent
Correlation length
critical exponent
Dynamic
critical exponent
KIBBLE-ZUREK RAMPS
Ramp rate
KIBBLE-ZUREK RAMPS
Ramp rate
KIBBLE-ZUREK RAMPS
Ramp rate
Adiabatic
KIBBLE-ZUREK RAMPS
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
METHOD
IDEA
WHEN IT WORKS
“Old-school”
Kibble-Zurek
and
set the
“interesting” time
and length scales
-Ramp to the QCP
-Ramp to deep in
the FM phase
[Kibble 1976,
Zurek 1985]
KIBBLE-ZUREK RAMPS
METHOD
IDEA
WHEN IT WORKS
“Old-school”
Kibble-Zurek
and
set the
“interesting” time
and length scales
-Ramp to the QCP
-Ramp to deep in
the FM phase
Most quantities
show scaling
collapse when
scaled by and
Throughout the
ramp
[Kibble 1976,
Zurek 1985]
Kibble-Zurek scaling
[Deng et. al. 2008,
Erez et. al., in prep.,
Polkovnikov, …]
KIBBLE-ZUREK RAMPS
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
Ramp rate
Impulse
Adiabatic
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
phase
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
phase
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
phase
Quadratic Integrable
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
phase
Quadratic Integrable
Hamiltonian conserves parity for each mode k
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
phase
Quadratic Integrable
Hamiltonian conserves parity for each mode k
Work in subspace where parity is even
TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum
Phase Transitions”
Wigner fermionize
phase
Quadratic Integrable
Hamiltonian conserves parity for each mode k
Work in subspace where parity is even
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
Low energy,
long wavelength theory?
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
Low energy,
long wavelength theory
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK RAMPS
Low energy,
long wavelength theory
Ramp rate
Impulse
Adiabatic
KIBBLE-ZUREK SCALING LIMIT
Schrödinger
Equation
OR
Observable
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 effects
can be ignored
FINITE-SIZE SCALING
FINITE-SIZE SCALING
EQUILIBRIUM VIA DYNAMICS
KZ scaling
function
If dynamic scaling functions exist, they must
have the equilibrium critical exponents
Equilibrium
scaling
function
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
ATHERMAL PROPERTIES
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
…
…
Protocol
Ramp to create excitations
Freeze the Hamiltonian
Wait
Dephasing in
integrable model:
Generalized Gibb’s
ensemble (GGE)
DEPHASING
…
…
Dephasing in
integrable model:
Generalized Gibb’s
ensemble (GGE)
Does dephasing occur
during the KibbleZurek ramp?
DEPHASING
…
…
Dephasing in
integrable model:
Generalized Gibb’s
ensemble (GGE)
DEPHASING
…
…
as
Dephasing in
integrable model:
Generalized Gibb’s
ensemble (GGE)
DEPHASING
…
…
as
Dephasing in
integrable model:
Generalized Gibb’s
ensemble (GGE)
DEPHASING
Cubic ramp:
…
…
DEPHASING
Cubic ramp:
…
…
as
DEPHASING
Cubic ramp:
…
…
as
UNIVERSALITY
UNIVERSALITY
UNIVERSALITY
Additional terms change (renormalize) the nonuniversal 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 against
open 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 on
neighboring sites