universality and dynamic localization in kibble- zurek scaling of the quantum ising chain
DESCRIPTION
Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain. Michael Kolodrubetz Boston University In collaboration with: B.K. Clark, D. Huse (Princeton) A. Polkovnikov , A. Katz (BU). Outline. - PowerPoint PPT PresentationTRANSCRIPT
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UNIVERSALITY AND DYNAMIC LOCALIZATION IN KIBBLE-ZUREK SCALING OF THE QUANTUM ISING CHAIN
Michael Kolodrubetz
Boston University
In collaboration with: B.K. Clark, D. Huse (Princeton)A. Polkovnikov, A. Katz (BU)
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OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Part II: Transverse-field Ising chain with a dynamic field
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TRANSVERSE-FIELD ISING CHAIN
One-dimensional transverse-field Ising chain
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One-dimensional transverse-field Ising chain
TRANSVERSE-FIELD ISING CHAIN
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One-dimensional transverse-field Ising chain
Paramagnet (PM)
TRANSVERSE-FIELD ISING CHAIN
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One-dimensional transverse-field Ising chain
Paramagnet (PM) Ferromagnet (FM)
TRANSVERSE-FIELD ISING CHAIN
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TRANSVERSE-FIELD ISING CHAIN
One-dimensional transverse-field Ising chain
Paramagnet (PM) Ferromagnet (FM)
Quantum phase transition
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QUANTUM PHASE TRANSITION
[Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]
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QUANTUM PHASE TRANSITION
[Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]
![Page 10: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/10.jpg)
QUANTUM PHASE TRANSITION
[Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]
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QUANTUM PHASE TRANSITION
[Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]
![Page 12: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/12.jpg)
QUANTUM PHASE TRANSITION
,
[Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]
![Page 13: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/13.jpg)
QUANTUM PHASE TRANSITION
,
[Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]
![Page 14: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/14.jpg)
QUANTUM PHASE TRANSITION
Correlation lengthcritical exponent
Dynamiccritical exponent
,
[Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]
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QUANTUM PHASE TRANSITION
Correlation lengthcritical exponent
Dynamiccritical exponent
,
Ising:
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QUANTUM PHASE TRANSITION
Correlation lengthcritical exponent
Dynamiccritical exponent
,
Ising:
Can these results be extendedto non-equilbrium dynamics?
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KIBBLE-ZUREK RAMPS
Ramp rate
Kibble-ZurekRamp through the critical
point at a constant, finite rate
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KIBBLE-ZUREK RAMPS
Ramp rate
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KIBBLE-ZUREK RAMPS
Ramp rate
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KIBBLE-ZUREK RAMPS
Ramp rate
Fall out ofequilibrium
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KIBBLE-ZUREK RAMPS
Ramp rate
Fall out ofequilibrium
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KIBBLE-ZUREK RAMPS
Ramp rate
Fall out ofequilibrium
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KIBBLE-ZUREK RAMPS
Ramp rate
Fall out ofequilibrium
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KIBBLE-ZUREK RAMPS
Ramp rate
Slower
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KIBBLE-ZUREK RAMPS
Ramp rate
Slower
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KIBBLE-ZUREK RAMPS
Ramp rate
Slower
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KIBBLE-ZUREK RAMPS
Ramp rate
Slower
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KIBBLE-ZUREK RAMPS
Ramp rate
Slower
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KIBBLE-ZUREK RAMPS
Kibble-Zurek ramps shownon-equilibrium scaling
[Chandran et. al., Deng et. al., etc.]
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KIBBLE-ZUREK RAMPS
Kibble-Zurek ramps shownon-equilibrium scaling (in the limit of slow ramps)
[Chandran et. al., Deng et. al., etc.]
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KIBBLE-ZUREK RAMPS
Kibble-Zurek ramps shownon-equilibrium scaling (in the limit of slow ramps) More than a theory of defect production!
[Chandran et. al., Deng et. al., etc.]
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KIBBLE-ZUREK RAMPS
Kibble-Zurek ramps shownon-equilibrium scaling (in the limit of slow ramps) More than a theory of defect production!
[Chandran et. al., Deng et. al., etc.]
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KIBBLE-ZUREK SCALING
Excess heat
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KIBBLE-ZUREK OBSERVABLES
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KIBBLE-ZUREK OBSERVABLES
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KIBBLE-ZUREK OBSERVABLES
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KIBBLE-ZUREK OBSERVABLES
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KIBBLE-ZUREK OBSERVABLES
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KIBBLE-ZUREK SCALING
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KIBBLE-ZUREK SCALING
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KIBBLE-ZUREK SCALING
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TRANSVERSE-FIELD ISING CHAIN
Sachdev: “Quantum Phase Transitions”
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TRANSVERSE-FIELD ISING CHAIN
Wigner fermionizeSachdev: “Quantum Phase Transitions”
phase
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TRANSVERSE-FIELD ISING CHAIN
Wigner fermionizeSachdev: “Quantum Phase Transitions”
phase
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TRANSVERSE-FIELD ISING CHAIN
Wigner fermionize
Quadratic Integrable
Sachdev: “Quantum Phase Transitions”
phase
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TRANSVERSE-FIELD ISING CHAIN
Wigner fermionize
Quadratic Integrable Work in subspace where each mode
(k,-k) is either occupied or unoccupied
Sachdev: “Quantum Phase Transitions”
phase
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TRANSVERSE-FIELD ISING CHAIN
Wigner fermionize
Quadratic Integrable Work in subspace where each mode
(k,-k) is either occupied or unoccupied
Sachdev: “Quantum Phase Transitions”
phase
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EQUILIBRIUM SCALING
“Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied
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EQUILIBRIUM SCALING
“Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied
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EQUILIBRIUM SCALING
Low energy, long wavelength theory?
“Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied
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EQUILIBRIUM SCALING
Low energy, long wavelength theory
“Spin up” (k,-k) unoccupied “Spin down” (k,-k) occupied
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KIBBLE-ZUREK SCALING
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KIBBLE-ZUREK SCALING
Low energy, long wavelength theory?
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KIBBLE-ZUREK SCALING
Low energy, long wavelength theory?
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KIBBLE-ZUREK SCALING
Low energy, long wavelength theory
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KIBBLE-ZUREK SCALING
SchrödingerEquation
ORObservable
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KIBBLE-ZUREK SCALING
SchrödingerEquation
ORObservable
Fixed
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KIBBLE-ZUREK SCALING
SchrödingerEquation
ORObservable
Fixed
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KIBBLE-ZUREK SCALING
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KIBBLE-ZUREK SCALING
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KIBBLE-ZUREK SCALING
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KIBBLE-ZUREK SCALING
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KIBBLE-ZUREK SCALING
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KIBBLE-ZUREK SCALING
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OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Part II: Transverse-field Ising chain with a dynamic field
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OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Are the results universal?
Part II: Transverse-field Ising chain with a dynamic field
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UNIVERSALITY
TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)
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UNIVERSALITY
TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)
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UNIVERSALITY
or
TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)
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UNIVERSALITY
or
Ramp the tilt linearly in time
TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)
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UNIVERSALITY
or
Ramp the tilt linearly in time: Solve numerically
with DMRG
TheorySachdev et al. (2002)ExperimentGreiner group (Harvard)Nagerl group (Innsbruck)
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UNIVERSALITY
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UNIVERSALITY
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UNIVERSALITY
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UNIVERSALITY
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UNIVERSALITY
![Page 77: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/77.jpg)
UNIVERSALITY
Matches to analytical solution of the Ising
chain!
![Page 78: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/78.jpg)
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Dynamics are universal to Ising-like QPTs
Part II: Transverse-field Ising chain with a dynamic field
![Page 79: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/79.jpg)
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Dynamics are universal to Ising-like QPTs What are some properties of the scaling
functions?
Part II: Transverse-field Ising chain with a dynamic field
![Page 80: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/80.jpg)
NON-EQUILIBRIUM PROPERTIES
Spin-spin correlation function
![Page 81: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/81.jpg)
NON-EQUILIBRIUM PROPERTIES
Spin-spin correlation function
![Page 82: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/82.jpg)
NON-EQUILIBRIUM PROPERTIES
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NON-EQUILIBRIUM PROPERTIES
Ground state
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NON-EQUILIBRIUM PROPERTIES
Ground state
![Page 85: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/85.jpg)
NON-EQUILIBRIUM PROPERTIES
Ground state
![Page 86: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/86.jpg)
NON-EQUILIBRIUM PROPERTIES
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NON-EQUILIBRIUM PROPERTIES
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NON-EQUILIBRIUM PROPERTIES
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NON-EQUILIBRIUM PROPERTIES
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NON-EQUILIBRIUM PROPERTIES
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NON-EQUILIBRIUM PROPERTIES
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NON-EQUILIBRIUM PROPERTIES
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NON-EQUILIBRIUM PROPERTIES
Inverted
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NON-EQUILIBRIUM PROPERTIES
Thermal
![Page 95: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/95.jpg)
NON-EQUILIBRIUM PROPERTIES
Kibble-Z
urek
Thermal
![Page 96: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/96.jpg)
NON-EQUILIBRIUM PROPERTIES
Kibble-Z
urek
Thermal
Antiferromagnetic
![Page 97: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/97.jpg)
NON-EQUILIBRIUM PROPERTIES
![Page 98: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/98.jpg)
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal
Part II: Transverse-field Ising chain with a dynamic field
![Page 99: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/99.jpg)
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal Finite size scaling, dephasing, experiments…
Part II: Transverse-field Ising chain with a dynamic field
![Page 100: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/100.jpg)
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Dynamics are universal to Ising-like QPTs Long-time dynamics are athermal Finite size scaling, dephasing, experiments…
Part II: Transverse-field Ising chain with a dynamic field
![Page 101: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/101.jpg)
DYNAMIC-FIELD ISING CHAIN
Basic idea: Add (classical) dynamics to the transverse field
![Page 102: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/102.jpg)
DYNAMIC-FIELD ISING CHAIN
Basic idea: Add (classical) dynamics to the transverse field
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DYNAMIC-FIELD ISING CHAIN
Basic idea: Add (classical) dynamics to the transverse field
“Friction” = back-action of spins on field
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DYNAMIC-FIELD ISING 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
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DYNAMIC-FIELD ISING 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?
![Page 106: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/106.jpg)
DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
![Page 108: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/108.jpg)
DYNAMIC-FIELD ISING CHAIN
as
![Page 109: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/109.jpg)
DYNAMIC-FIELD ISING CHAIN
as
Field motion arrested by
QCP!
![Page 110: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/110.jpg)
DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
Dynamics dominated by critical behavior
![Page 117: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/117.jpg)
DYNAMIC-FIELD ISING CHAIN
Dynamics dominated by critical behavior
Linearize the Hamiltonian
![Page 118: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/118.jpg)
DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
![Page 131: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/131.jpg)
DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
![Page 134: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/134.jpg)
DYNAMIC-FIELD ISING CHAIN
![Page 135: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/135.jpg)
DYNAMIC-FIELD ISING CHAIN
What happens for other models?
![Page 136: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/136.jpg)
DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
![Page 138: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/138.jpg)
Crossover tunable via… …dimensionality
DYNAMIC-FIELD ISING CHAIN
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Crossover tunable via… …dimensionality …critical exponents
DYNAMIC-FIELD ISING CHAIN
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Crossover tunable via… …dimensionality …critical exponents
Possibility of as
DYNAMIC-FIELD ISING CHAIN
![Page 141: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/141.jpg)
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Are the results universal? What are some properties of the scaling
functions?
Part II: Transverse-field Ising chain with a dynamic field
Field is trapped at QCP by critical absorption
![Page 142: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/142.jpg)
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Are the results universal? What are some properties of the scaling
functions?
Part II: Transverse-field Ising chain with a dynamic field
Field is trapped at QCP by critical absorption Dynamics of field during trapping?
![Page 143: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/143.jpg)
DYNAMIC-FIELD ISING CHAIN
![Page 144: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/144.jpg)
Overdamped/underdamped?
DYNAMIC-FIELD ISING CHAIN
![Page 145: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/145.jpg)
Overdamped/underdamped?
Measure velocity at QCP
DYNAMIC-FIELD ISING CHAIN
![Page 146: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/146.jpg)
DYNAMIC-FIELD ISING CHAIN
![Page 147: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/147.jpg)
DYNAMIC-FIELD ISING CHAIN
![Page 148: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/148.jpg)
DYNAMIC-FIELD ISING CHAIN
HypothesisInitial momentum is the
relevant scale for dynamics
![Page 149: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/149.jpg)
DYNAMIC-FIELD ISING CHAIN
HypothesisInitial momentum is the
relevant scale for dynamics
![Page 150: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/150.jpg)
DYNAMIC-FIELD ISING CHAIN
HypothesisInitial momentum is the
relevant scale for dynamics
![Page 151: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/151.jpg)
DYNAMIC-FIELD ISING CHAIN
HypothesisInitial momentum is the
relevant scale for dynamics
![Page 152: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/152.jpg)
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Are the results universal? What are some properties of the scaling
functions?
Part II: Transverse-field Ising chain with a dynamic field
System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse
![Page 153: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/153.jpg)
OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Are the results universal? What are some properties of the scaling
functions?
Part II: Transverse-field Ising chain with a dynamic field
System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Analytical understanding of late-time dynamics?
![Page 154: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/154.jpg)
DYNAMIC-FIELD ISING CHAIN
![Page 155: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/155.jpg)
DYNAMIC-FIELD ISING CHAIN
Dephasing
![Page 156: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/156.jpg)
DYNAMIC-FIELD ISING CHAIN
Are long-time dynamics well-described by the dephased
ensemble?(generalized Gibbs ensemble /
GGE)
Dephasing
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DYNAMIC-FIELD ISING CHAIN
Manually dephase
Are long-time dynamics well-described by the dephased
ensemble?(generalized Gibbs ensemble /
GGE)
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DYNAMIC-FIELD ISING CHAIN
Manually dephase
Are long-time dynamics well-described by the dephased
ensemble?(generalized Gibbs ensemble /
GGE)
![Page 159: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/159.jpg)
DYNAMIC-FIELD ISING CHAIN
Manually dephase
Are long-time dynamics well-described by the dephased
ensemble?
YES!
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DYNAMIC-FIELD ISING CHAIN
Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli
Polkovnikov and Luca D’Alessio
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DYNAMIC-FIELD ISING CHAIN
Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli
Polkovnikov and Luca D’Alessio Start from stationary state of
![Page 162: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/162.jpg)
DYNAMIC-FIELD ISING CHAIN
Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli
Polkovnikov and Luca D’Alessio Start from stationary state of Go to the “moving frame” of
Frame that locally diagonalizes
![Page 163: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/163.jpg)
DYNAMIC-FIELD ISING CHAIN
Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli
Polkovnikov and Luca D’Alessio Start from stationary state of Go to the “moving frame” of
Frame that locally diagonalizes
![Page 164: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/164.jpg)
DYNAMIC-FIELD ISING CHAIN
Approximate dynamics by adiabatic PT Based on unpublished work by Anatoli
Polkovnikov and Luca D’Alessio Start from stationary state of Go to the “moving frame” of
Frame that locally diagonalizes
Treat term via 2nd order time-dependent PT
![Page 165: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/165.jpg)
DYNAMIC-FIELD ISING CHAIN
Approximate dynamics by adiabatic PT
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DYNAMIC-FIELD ISING CHAIN
Approximate dynamics by adiabatic PT
Need to know… Initial condition on , Mode occupation numbers
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Are the results universal? What are some properties of the scaling
functions?
Part II: Transverse-field Ising chain with a dynamic field
System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Late-time dynamics are given by dephasing
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OUTLINE
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Dynamics near QCP givesnon-equilibrium critical scaling theory
Are the results universal? What are some properties of the scaling
functions?
Part II: Transverse-field Ising chain with a dynamic field
System is trapped at QCP by critical absorption Trapping dynamics show scaling collapse Late-time dynamics are given by dephasing
![Page 171: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/171.jpg)
FUTURE DIRECTIONS
Analytically understand the dynamics via APT
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FUTURE DIRECTIONS
Analytically understand the dynamics via APT Remove the offset potential
Is it RG relevant?
![Page 173: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/173.jpg)
FUTURE DIRECTIONS
Analytically understand the dynamics via APT Remove the offset potential
Is it RG relevant? Tune the scaling of excess heat
![Page 174: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/174.jpg)
FUTURE DIRECTIONS
Analytically understand the dynamics via APT Remove the offset potential
Is it RG relevant? Tune the scaling of excess heat
Generalize Ising model to higher dimensions Use models with other critical exponents
![Page 175: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/175.jpg)
FUTURE DIRECTIONS
Analytically understand the dynamics via APT Remove the offset potential
Is it RG relevant? Tune the scaling of excess heat
Generalize Ising model to higher dimensions Use models with other critical exponents What happens if as
![Page 176: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/176.jpg)
FUTURE DIRECTIONS
Analytically understand the dynamics via APT Remove the offset potential
Is it RG relevant? Tune the scaling of excess heat
Generalize Ising model to higher dimensions Use models with other critical exponents What happens if as
Relationship to the Higgs boson?
![Page 177: Universality and Dynamic Localization in Kibble- Zurek Scaling of the Quantum Ising Chain](https://reader031.vdocument.in/reader031/viewer/2022012916/56814d0a550346895dba4074/html5/thumbnails/177.jpg)
SUMMARY
Part I: Kibble-Zurek scaling of the transverse-field Ising chain
Part II: Transverse-field Ising chain with a dynamic field
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DYNAMIC-FIELD ISING CHAIN
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DYNAMIC-FIELD ISING CHAIN
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TRANSVERSE-FIELD ISING CHAIN
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TRANSVERSE-FIELD ISING CHAIN
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TRANSVERSE-FIELD ISING CHAIN
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TRANSVERSE-FIELD ISING CHAIN
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TRANSVERSE-FIELD ISING CHAIN
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TRANSVERSE-FIELD ISING CHAIN
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TRANSVERSE-FIELD ISING CHAIN