Dephases

Report
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



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