chaos in general holographic space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf ·...
TRANSCRIPT
![Page 1: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/1.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Chaos in General Holographic Space-times
Nilanjan Sircar
WITS
Joburg Workshop on Black holes and Entanglement, March,2017
arXiv:1602.07307 [hep-th] (JHEP 1605 (2016) 091) with Jacob
Sonnenschein & Walter Tangarife
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 2: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/2.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Outline
1 Motivation & Introduction
2 Chaos, Butterfly-effect & Holography
3 Chaos in various systems
4 Summary & Future Directions
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 3: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/3.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Outline of talk
1 Motivation & Introduction
2 Chaos, Butterfly-effect & Holography
3 Chaos in various systems
4 Summary & Future Directions
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 4: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/4.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -1
Chaos refers to sensitive dependence on initial conditions,i.e. initially very similar states can evolve to be quitedifferent.
Chaos in context of thermalization.
In Quantum Information Theory and Black holes this isalso known as scrambling.
Black Holes are fastest scramblers in nature: t∗ ∼ β log S .Hayden & Preskill ’07, Sekino & Susskind ’08
Largest Lyapunov exponent is bounded by Black holeresult: λL ≤ 2πkB T
~ . Theories where the bound issaturated should have a gravity dual.Maldacena, Shenker, & Stanford ’15
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 5: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/5.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -1
Chaos refers to sensitive dependence on initial conditions,i.e. initially very similar states can evolve to be quitedifferent.
Chaos in context of thermalization.
In Quantum Information Theory and Black holes this isalso known as scrambling.
Black Holes are fastest scramblers in nature: t∗ ∼ β log S .Hayden & Preskill ’07, Sekino & Susskind ’08
Largest Lyapunov exponent is bounded by Black holeresult: λL ≤ 2πkB T
~ . Theories where the bound issaturated should have a gravity dual.Maldacena, Shenker, & Stanford ’15
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 6: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/6.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -1
Chaos refers to sensitive dependence on initial conditions,i.e. initially very similar states can evolve to be quitedifferent.
Chaos in context of thermalization.
In Quantum Information Theory and Black holes this isalso known as scrambling.
Black Holes are fastest scramblers in nature: t∗ ∼ β log S .Hayden & Preskill ’07, Sekino & Susskind ’08
Largest Lyapunov exponent is bounded by Black holeresult: λL ≤ 2πkB T
~ . Theories where the bound issaturated should have a gravity dual.Maldacena, Shenker, & Stanford ’15
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 7: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/7.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -1
Chaos refers to sensitive dependence on initial conditions,i.e. initially very similar states can evolve to be quitedifferent.
Chaos in context of thermalization.
In Quantum Information Theory and Black holes this isalso known as scrambling.
Black Holes are fastest scramblers in nature: t∗ ∼ β log S .Hayden & Preskill ’07, Sekino & Susskind ’08
Largest Lyapunov exponent is bounded by Black holeresult: λL ≤ 2πkB T
~ . Theories where the bound issaturated should have a gravity dual.Maldacena, Shenker, & Stanford ’15
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 8: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/8.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -1
Chaos refers to sensitive dependence on initial conditions,i.e. initially very similar states can evolve to be quitedifferent.
Chaos in context of thermalization.
In Quantum Information Theory and Black holes this isalso known as scrambling.
Black Holes are fastest scramblers in nature: t∗ ∼ β log S .Hayden & Preskill ’07, Sekino & Susskind ’08
Largest Lyapunov exponent is bounded by Black holeresult: λL ≤ 2πkB T
~ . Theories where the bound issaturated should have a gravity dual.Maldacena, Shenker, & Stanford ’15
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 9: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/9.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Bound on Chaos-1
Diagnostic of Chaos: C (t) = −〈[W (t),V (0)]2〉β ∼ O(1)
In semi classical limit, V = p and W = q(t) :
C (t) = ~2(∂q(t)∂q(0)
)2∼ ~2e2λLt .
C (t) ∼ O(1) at t∗ ∼ 1λL
log 1~
Dissipation time td : 〈V (0)V (t)〉 ∼ e− t
td .
Typically in thermal systems: td ∼ β
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 10: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/10.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Bound on Chaos-1
Diagnostic of Chaos: C (t) = −〈[W (t),V (0)]2〉β ∼ O(1)
In semi classical limit, V = p and W = q(t) :
C (t) = ~2(∂q(t)∂q(0)
)2∼ ~2e2λLt .
C (t) ∼ O(1) at t∗ ∼ 1λL
log 1~
Dissipation time td : 〈V (0)V (t)〉 ∼ e− t
td .
Typically in thermal systems: td ∼ β
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 11: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/11.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Bound on Chaos-1
Diagnostic of Chaos: C (t) = −〈[W (t),V (0)]2〉β ∼ O(1)
In semi classical limit, V = p and W = q(t) :
C (t) = ~2(∂q(t)∂q(0)
)2∼ ~2e2λLt .
C (t) ∼ O(1) at t∗ ∼ 1λL
log 1~
Dissipation time td : 〈V (0)V (t)〉 ∼ e− t
td .
Typically in thermal systems: td ∼ β
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 12: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/12.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Bound on Chaos-1
Diagnostic of Chaos: C (t) = −〈[W (t),V (0)]2〉β ∼ O(1)
In semi classical limit, V = p and W = q(t) :
C (t) = ~2(∂q(t)∂q(0)
)2∼ ~2e2λLt .
C (t) ∼ O(1) at t∗ ∼ 1λL
log 1~
Dissipation time td : 〈V (0)V (t)〉 ∼ e− t
td .
Typically in thermal systems: td ∼ β
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 13: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/13.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Bound on Chaos-1
Diagnostic of Chaos: C (t) = −〈[W (t),V (0)]2〉β ∼ O(1)
In semi classical limit, V = p and W = q(t) :
C (t) = ~2(∂q(t)∂q(0)
)2∼ ~2e2λLt .
C (t) ∼ O(1) at t∗ ∼ 1λL
log 1~
Dissipation time td : 〈V (0)V (t)〉 ∼ e− t
td .
Typically in thermal systems: td ∼ β
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 14: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/14.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Bound on Chaos-2
V(0)
V(0)
W(t)
W(t)
F (t) = Tr [yVyW (t)yVyW (t)] wherey4 = 1
Z e−βH .
Conjecture (t � td , t � t∗):ddt (Fd − F (t)) ≤ 2π
β (Fd − F (t))
Chaotic system: (Fd − F (t)) ∼ εeλLt
→ λL ≤ 2πβ
Holographically (2 + 1 dim bulk):
F (t) = f0 − f1N2 e
2πβ
tShenker & Stanford ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 15: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/15.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Bound on Chaos-2
V(0)
V(0)
W(t)
W(t)
F (t) = Tr [yVyW (t)yVyW (t)] wherey4 = 1
Z e−βH .
Conjecture (t � td , t � t∗):ddt (Fd − F (t)) ≤ 2π
β (Fd − F (t))
Chaotic system: (Fd − F (t)) ∼ εeλLt
→ λL ≤ 2πβ
Holographically (2 + 1 dim bulk):
F (t) = f0 − f1N2 e
2πβ
tShenker & Stanford ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 16: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/16.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Bound on Chaos-2
V(0)
V(0)
W(t)
W(t)
F (t) = Tr [yVyW (t)yVyW (t)] wherey4 = 1
Z e−βH .
Conjecture (t � td , t � t∗):ddt (Fd − F (t)) ≤ 2π
β (Fd − F (t))
Chaotic system: (Fd − F (t)) ∼ εeλLt
→ λL ≤ 2πβ
Holographically (2 + 1 dim bulk):
F (t) = f0 − f1N2 e
2πβ
tShenker & Stanford ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 17: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/17.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Bound on Chaos-2
V(0)
V(0)
W(t)
W(t)
F (t) = Tr [yVyW (t)yVyW (t)] wherey4 = 1
Z e−βH .
Conjecture (t � td , t � t∗):ddt (Fd − F (t)) ≤ 2π
β (Fd − F (t))
Chaotic system: (Fd − F (t)) ∼ εeλLt
→ λL ≤ 2πβ
Holographically (2 + 1 dim bulk):
F (t) = f0 − f1N2 e
2πβ
tShenker & Stanford ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 18: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/18.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -2
For such chaotic/thermal systems, a small perturbationcan change the pattern of correlation drastically :Disruption of entanglement/ Butterfly effect.
The diagnostic used in this case is Thermo-MutualInformation Morrison & Roberts ’12.
It is recently studied in context of AdS/CFT by variouspeople including Shenker, Stanford, Susskind, Roberts,Leichenauer, ...
We have extended their work in context of:
Black Dp branes.Lifshitz Black brane.Higher Derivative Black brane.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 19: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/19.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -2
For such chaotic/thermal systems, a small perturbationcan change the pattern of correlation drastically :Disruption of entanglement/ Butterfly effect.
The diagnostic used in this case is Thermo-MutualInformation Morrison & Roberts ’12.
It is recently studied in context of AdS/CFT by variouspeople including Shenker, Stanford, Susskind, Roberts,Leichenauer, ...
We have extended their work in context of:
Black Dp branes.Lifshitz Black brane.Higher Derivative Black brane.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 20: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/20.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -2
For such chaotic/thermal systems, a small perturbationcan change the pattern of correlation drastically :Disruption of entanglement/ Butterfly effect.
The diagnostic used in this case is Thermo-MutualInformation Morrison & Roberts ’12.
It is recently studied in context of AdS/CFT by variouspeople including Shenker, Stanford, Susskind, Roberts,Leichenauer, ...
We have extended their work in context of:
Black Dp branes.Lifshitz Black brane.Higher Derivative Black brane.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 21: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/21.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -2
For such chaotic/thermal systems, a small perturbationcan change the pattern of correlation drastically :Disruption of entanglement/ Butterfly effect.
The diagnostic used in this case is Thermo-MutualInformation Morrison & Roberts ’12.
It is recently studied in context of AdS/CFT by variouspeople including Shenker, Stanford, Susskind, Roberts,Leichenauer, ...
We have extended their work in context of:
Black Dp branes.Lifshitz Black brane.Higher Derivative Black brane.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 22: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/22.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -2
For such chaotic/thermal systems, a small perturbationcan change the pattern of correlation drastically :Disruption of entanglement/ Butterfly effect.
The diagnostic used in this case is Thermo-MutualInformation Morrison & Roberts ’12.
It is recently studied in context of AdS/CFT by variouspeople including Shenker, Stanford, Susskind, Roberts,Leichenauer, ...
We have extended their work in context of:
Black Dp branes.Lifshitz Black brane.Higher Derivative Black brane.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 23: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/23.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -2
For such chaotic/thermal systems, a small perturbationcan change the pattern of correlation drastically :Disruption of entanglement/ Butterfly effect.
The diagnostic used in this case is Thermo-MutualInformation Morrison & Roberts ’12.
It is recently studied in context of AdS/CFT by variouspeople including Shenker, Stanford, Susskind, Roberts,Leichenauer, ...
We have extended their work in context of:
Black Dp branes.Lifshitz Black brane.Higher Derivative Black brane.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 24: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/24.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Motivation -2
For such chaotic/thermal systems, a small perturbationcan change the pattern of correlation drastically :Disruption of entanglement/ Butterfly effect.
The diagnostic used in this case is Thermo-MutualInformation Morrison & Roberts ’12.
It is recently studied in context of AdS/CFT by variouspeople including Shenker, Stanford, Susskind, Roberts,Leichenauer, ...
We have extended their work in context of:
Black Dp branes.Lifshitz Black brane.Higher Derivative Black brane.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 25: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/25.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
“AdS/CFT Correspondence” or Holography
QFT(d + 1) Gravity
((d−1)+1)
r
Oφ
Duality between N = 4 SU(N)Super-Yang Mills’ theory in (3 + 1)-dimand type IIB Super-strings inAdS5 × S5. Maldacena ’97; Gubser, Klebanov, Polyakov
’98 ; Witten ’98
Simplifies in the limit of large ’t Hooftcoupling (λ = g2
YMN � 1) and largeN � 1 to a duality between classicaltype IIB super-gravity and full quantumSuper-Yang Mills’ theory at leadingorder in λ and N.
Presently refers to more general classof dualities.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 26: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/26.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
“AdS/CFT Correspondence” or Holography
QFT(d + 1) Gravity
((d−1)+1)
r
Oφ
Duality between N = 4 SU(N)Super-Yang Mills’ theory in (3 + 1)-dimand type IIB Super-strings inAdS5 × S5. Maldacena ’97; Gubser, Klebanov, Polyakov
’98 ; Witten ’98
Simplifies in the limit of large ’t Hooftcoupling (λ = g2
YMN � 1) and largeN � 1 to a duality between classicaltype IIB super-gravity and full quantumSuper-Yang Mills’ theory at leadingorder in λ and N.
Presently refers to more general classof dualities.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 27: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/27.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
“AdS/CFT Correspondence” or Holography
QFT(d + 1) Gravity
((d−1)+1)
r
Oφ
Duality between N = 4 SU(N)Super-Yang Mills’ theory in (3 + 1)-dimand type IIB Super-strings inAdS5 × S5. Maldacena ’97; Gubser, Klebanov, Polyakov
’98 ; Witten ’98
Simplifies in the limit of large ’t Hooftcoupling (λ = g2
YMN � 1) and largeN � 1 to a duality between classicaltype IIB super-gravity and full quantumSuper-Yang Mills’ theory at leadingorder in λ and N.
Presently refers to more general classof dualities.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 28: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/28.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Entanglement Entropy
B
A
Σ
t=const.
Consider an entangling surface Σ whichdivides the space in to two separatesub-systems.
Integrate out the the degrees offreedom living “outside” (region B).
The reduced system is now describedby a density matrix ρA.
“Entanglement entropy” or vonNeuman entropy:SEE = −Tr (ρA log ρA).
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 29: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/29.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Entanglement Entropy
B
A
Σ
t=const.
Consider an entangling surface Σ whichdivides the space in to two separatesub-systems.
Integrate out the the degrees offreedom living “outside” (region B).
The reduced system is now describedby a density matrix ρA.
“Entanglement entropy” or vonNeuman entropy:SEE = −Tr (ρA log ρA).
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 30: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/30.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Entanglement Entropy
B
A
Σ
t=const.
Consider an entangling surface Σ whichdivides the space in to two separatesub-systems.
Integrate out the the degrees offreedom living “outside” (region B).
The reduced system is now describedby a density matrix ρA.
“Entanglement entropy” or vonNeuman entropy:SEE = −Tr (ρA log ρA).
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 31: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/31.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Entanglement Entropy
B
A
Σ
t=const.
Consider an entangling surface Σ whichdivides the space in to two separatesub-systems.
Integrate out the the degrees offreedom living “outside” (region B).
The reduced system is now describedby a density matrix ρA.
“Entanglement entropy” or vonNeuman entropy:SEE = −Tr (ρA log ρA).
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 32: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/32.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Holographic Entanglement Entropy
B Σ
γA
Bulk
r
t=const.A
SEE = min∂γA=Σ
(Area(γA)
4GN
)Ryu &
Takayanagi ’06
Generalized to time dependentsituations (Covariantprescription): min→extremum.Hubeny, Rangamani, & Takayanagi ’07
In presence of dilaton: Area →Area in Einstein frame Ryu &
Takayanagi ’06, Klebanov et al.. ’07.
For higher derivative gravity:Area →
∫γA
f [RγA] de Boer et al.. ’11,
Hung et al.. ’11, Dong ’13, Camps ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 33: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/33.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Holographic Entanglement Entropy
B Σ
γA
Bulk
r
t=const.A
SEE = min∂γA=Σ
(Area(γA)
4GN
)Ryu &
Takayanagi ’06
Generalized to time dependentsituations (Covariantprescription): min→extremum.Hubeny, Rangamani, & Takayanagi ’07
In presence of dilaton: Area →Area in Einstein frame Ryu &
Takayanagi ’06, Klebanov et al.. ’07.
For higher derivative gravity:Area →
∫γA
f [RγA] de Boer et al.. ’11,
Hung et al.. ’11, Dong ’13, Camps ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 34: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/34.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Holographic Entanglement Entropy
B Σ
γA
Bulk
r
t=const.A
SEE = min∂γA=Σ
(Area(γA)
4GN
)Ryu &
Takayanagi ’06
Generalized to time dependentsituations (Covariantprescription): min→extremum.Hubeny, Rangamani, & Takayanagi ’07
In presence of dilaton: Area →Area in Einstein frame Ryu &
Takayanagi ’06, Klebanov et al.. ’07.
For higher derivative gravity:Area →
∫γA
f [RγA] de Boer et al.. ’11,
Hung et al.. ’11, Dong ’13, Camps ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 35: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/35.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Holographic Entanglement Entropy
B Σ
γA
Bulk
r
t=const.A
SEE = min∂γA=Σ
(Area(γA)
4GN
)Ryu &
Takayanagi ’06
Generalized to time dependentsituations (Covariantprescription): min→extremum.Hubeny, Rangamani, & Takayanagi ’07
In presence of dilaton: Area →Area in Einstein frame Ryu &
Takayanagi ’06, Klebanov et al.. ’07.
For higher derivative gravity:Area →
∫γA
f [RγA] de Boer et al.. ’11,
Hung et al.. ’11, Dong ’13, Camps ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 36: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/36.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Mutual Information
I (A;B) = S(A) + S(B)− S(A ∪ B)
Finite quantity, UV divergence in EE cancels.
Strong sub-additivity: I (A;B) ≥ 0
Measures total classical and quantum correlation betweentwo regions.
I (A;B) ≥ 12
(〈OAOB〉−〈OA〉〈OB〉||OA||||OB ||
)2Wolf et al.. ’08i.e.
I (A;B) = 0 =⇒ 〈OAOB〉 = 〈OA〉〈OB〉
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 37: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/37.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Mutual Information
I (A;B) = S(A) + S(B)− S(A ∪ B)
Finite quantity, UV divergence in EE cancels.
Strong sub-additivity: I (A;B) ≥ 0
Measures total classical and quantum correlation betweentwo regions.
I (A;B) ≥ 12
(〈OAOB〉−〈OA〉〈OB〉||OA||||OB ||
)2Wolf et al.. ’08i.e.
I (A;B) = 0 =⇒ 〈OAOB〉 = 〈OA〉〈OB〉
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 38: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/38.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Mutual Information
I (A;B) = S(A) + S(B)− S(A ∪ B)
Finite quantity, UV divergence in EE cancels.
Strong sub-additivity: I (A;B) ≥ 0
Measures total classical and quantum correlation betweentwo regions.
I (A;B) ≥ 12
(〈OAOB〉−〈OA〉〈OB〉||OA||||OB ||
)2Wolf et al.. ’08i.e.
I (A;B) = 0 =⇒ 〈OAOB〉 = 〈OA〉〈OB〉
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 39: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/39.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Mutual Information
I (A;B) = S(A) + S(B)− S(A ∪ B)
Finite quantity, UV divergence in EE cancels.
Strong sub-additivity: I (A;B) ≥ 0
Measures total classical and quantum correlation betweentwo regions.
I (A;B) ≥ 12
(〈OAOB〉−〈OA〉〈OB〉||OA||||OB ||
)2Wolf et al.. ’08i.e.
I (A;B) = 0 =⇒ 〈OAOB〉 = 〈OA〉〈OB〉
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 40: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/40.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Mutual Information
I (A;B) = S(A) + S(B)− S(A ∪ B)
Finite quantity, UV divergence in EE cancels.
Strong sub-additivity: I (A;B) ≥ 0
Measures total classical and quantum correlation betweentwo regions.
I (A;B) ≥ 12
(〈OAOB〉−〈OA〉〈OB〉||OA||||OB ||
)2Wolf et al.. ’08i.e.
I (A;B) = 0 =⇒ 〈OAOB〉 = 〈OA〉〈OB〉
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 41: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/41.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Holographic Mutual Information
A B
γA γB
γA∪B
min(Area(γA∪B)) = Area(γA∪B)
if Area(γA∪B) < Area(γA) + Area(γB)
= Area(γA) + Area(γB) otherwise
For later case,
I (A;B) = S(A) + S(B)− S(A ∪ B) = 0
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 42: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/42.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Holes and Finite Temperature QFT
QFT at Temperature T ≡ Black Holes with Hawkingtemperature T .
Black Hole metric:
ds2 = −f (r)dt2 +dr2
f (r)+ dΣ2
⊥
f (rH) = 0 ; T =1
β=|f ′(rH)|
4π
For asymptotically AdS black holes: f (r)→ r2
L2 as r →∞
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 43: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/43.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Holes and Finite Temperature QFT
QFT at Temperature T ≡ Black Holes with Hawkingtemperature T .
Black Hole metric:
ds2 = −f (r)dt2 +dr2
f (r)+ dΣ2
⊥
f (rH) = 0 ; T =1
β=|f ′(rH)|
4π
For asymptotically AdS black holes: f (r)→ r2
L2 as r →∞
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 44: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/44.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Holes and Finite Temperature QFT
QFT at Temperature T ≡ Black Holes with Hawkingtemperature T .
Black Hole metric:
ds2 = −f (r)dt2 +dr2
f (r)+ dΣ2
⊥
f (rH) = 0 ; T =1
β=|f ′(rH)|
4π
For asymptotically AdS black holes: f (r)→ r2
L2 as r →∞
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 45: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/45.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Holes: Kruskal-Szekeres Coordinates
Define Kruskal-Szekeres Coordinates (u, v):
uv = −e4πTr∗(r), u/v = e−4πTt
ds2 = − 4f (r)
16π2T 2e−4πTr∗(r)dudv + dΣ2
⊥
dr∗ = drf (r) .
We can further re-define coordinates (“Penrose Diagram“)region by: U = tan−1(u), V = tan−1(v).
AdS black holes: AdS boundary at uv = −1 andSingularity at uv = 1.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 46: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/46.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Holes: Kruskal-Szekeres Coordinates
Define Kruskal-Szekeres Coordinates (u, v):
uv = −e4πTr∗(r), u/v = e−4πTt
ds2 = − 4f (r)
16π2T 2e−4πTr∗(r)dudv + dΣ2
⊥
dr∗ = drf (r) .
We can further re-define coordinates (“Penrose Diagram“)region by: U = tan−1(u), V = tan−1(v).
AdS black holes: AdS boundary at uv = −1 andSingularity at uv = 1.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 47: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/47.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Holes: Kruskal-Szekeres Coordinates
Define Kruskal-Szekeres Coordinates (u, v):
uv = −e4πTr∗(r), u/v = e−4πTt
ds2 = − 4f (r)
16π2T 2e−4πTr∗(r)dudv + dΣ2
⊥
dr∗ = drf (r) .
We can further re-define coordinates (“Penrose Diagram“)region by: U = tan−1(u), V = tan−1(v).
AdS black holes: AdS boundary at uv = −1 andSingularity at uv = 1.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 48: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/48.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Eternal AdS Black Holes
uv
tt
tt = const.
r = const.
Kruskal-Szekeres Diagram
I
III
II
IV
RL
Penrose Diagram
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 49: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/49.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Thermo Field Double (TFD)
Consider two QFTs with isomorphic Hilbert Spaces HL
and HR . Thermofield double is an particular entangledstate in HL ⊗HR :
|TFD〉 =1√Z (β)
∑n
e−β2
En |n〉L|n〉R
Z (β) =∑
n e−βEn
ρL = TrR |TFD〉 〈TFD| = 1Z(β)
∑n e−βEn |n〉L 〈n|L
← Thermal Density Matrix
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 50: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/50.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Thermo Field Double (TFD)
Consider two QFTs with isomorphic Hilbert Spaces HL
and HR . Thermofield double is an particular entangledstate in HL ⊗HR :
|TFD〉 =1√Z (β)
∑n
e−β2
En |n〉L|n〉R
Z (β) =∑
n e−βEn
ρL = TrR |TFD〉 〈TFD| = 1Z(β)
∑n e−βEn |n〉L 〈n|L
← Thermal Density Matrix
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 51: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/51.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
TFD ≡ Maximally extended Black Holes
I
III
II
IV
QFTL QFTR
TFD is dual to the maximalextension of the eternal blackhole. The pair of QFTs living onthe two boundaries correspond tothe two QFTs in the definition ofTFD. Israel ’76, Maldacena ’01
Entanglement between the twoQFTs is given by the thermalentropy of the black hole.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 52: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/52.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
TFD ≡ Maximally extended Black Holes
I
III
II
IV
QFTL QFTR
TFD is dual to the maximalextension of the eternal blackhole. The pair of QFTs living onthe two boundaries correspond tothe two QFTs in the definition ofTFD. Israel ’76, Maldacena ’01
Entanglement between the twoQFTs is given by the thermalentropy of the black hole.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 53: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/53.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Hole and (Thermo-) Mutual Information
X
AB
X
AB
Consider the mutual information for strips of size L:
IAB = SA + SB − SA∪B
For BTZ Black Hole (2 + 1-dimension) :
IAB = max(LAdS
GNlog sinh(πLT ), 0)
Shenker & Stanford ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 54: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/54.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Hole and (Thermo-) Mutual Information:Plot
IAB
L TLcT
There exists a critical strip size L = Lc beyond which theMutual Information is non zero.Shenker & Stanford ’13, Leichenauer ’14, Sonnenschein-NS-Tangarife ’16
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 55: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/55.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Outline
1 Motivation & Introduction
2 Chaos, Butterfly-effect & Holography
3 Chaos in various systems
4 Summary & Future Directions
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 56: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/56.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Perturbing TFD: Shock Wave geometry
δM
tW
M + δM
M
α
Consider the Kruskal coordinates (u, v), (u, v) to the left andright of the perturbation respectively. In the limit of smallperturbation δM
M � 1 and large tw � 1: v = v + α, u = uwith,
α =c2
c1
δMβ
SBHe−
2πβ
(r∗(∞)−tw )
Using Bekenstein-Hawking Formula and First Law ofThermodynamics. Shenker & Stanford ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 57: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/57.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Heuristic Calculation of Scrambling time
α
tW should correspond to scrambling time t∗ when the effect ofthe perturbation is order one.
α ∼ 1
which gives for perturbation δM ∼ T ,
t∗ =β
2πlog SBH +
β
2πlog
(c1
c2er∗(∞)
)SBH ∼ N2 � 1.Shenker & Stanford ’13
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 58: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/58.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Diagnostics for Chaos -I
We are interested in looking for signature of disruption ofentanglement/ correlation due to a small perturbation.
So in the Shockwave geometry corresponding to a smallperturbation of the Thermofield double we can calculatetwo sided mutual information (thermo-mutual information(TMI)).
As seen before the TMI is non zero only for beyond somecritical strip size L > Lc .
Now with L > Lc we can calculate the TMI in theShockwave geometry.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 59: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/59.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Diagnostics for Chaos -I
We are interested in looking for signature of disruption ofentanglement/ correlation due to a small perturbation.
So in the Shockwave geometry corresponding to a smallperturbation of the Thermofield double we can calculatetwo sided mutual information (thermo-mutual information(TMI)).
As seen before the TMI is non zero only for beyond somecritical strip size L > Lc .
Now with L > Lc we can calculate the TMI in theShockwave geometry.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 60: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/60.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Diagnostics for Chaos -I
We are interested in looking for signature of disruption ofentanglement/ correlation due to a small perturbation.
So in the Shockwave geometry corresponding to a smallperturbation of the Thermofield double we can calculatetwo sided mutual information (thermo-mutual information(TMI)).
As seen before the TMI is non zero only for beyond somecritical strip size L > Lc .
Now with L > Lc we can calculate the TMI in theShockwave geometry.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 61: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/61.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Diagnostics for Chaos -I
We are interested in looking for signature of disruption ofentanglement/ correlation due to a small perturbation.
So in the Shockwave geometry corresponding to a smallperturbation of the Thermofield double we can calculatetwo sided mutual information (thermo-mutual information(TMI)).
As seen before the TMI is non zero only for beyond somecritical strip size L > Lc .
Now with L > Lc we can calculate the TMI in theShockwave geometry.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 62: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/62.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Diagnostics for Chaos-II
X
AB
X
AB
In this geometry TMI is function ofIAB(L, α(tW )) = SA(L) + SB(L)− SA∪B(L, α(tw )).
We define scrambling time (t∗) as whenIAB(L, α(tW = t∗)) = 0 for a given L > Lc .
t∗ = β2π log SBH +O(N0), same conclusion as α ∼ 1
analysis.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 63: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/63.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Diagnostics for Chaos-II
X
AB
X
AB
In this geometry TMI is function ofIAB(L, α(tW )) = SA(L) + SB(L)− SA∪B(L, α(tw )).
We define scrambling time (t∗) as whenIAB(L, α(tW = t∗)) = 0 for a given L > Lc .
t∗ = β2π log SBH +O(N0), same conclusion as α ∼ 1
analysis.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 64: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/64.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Diagnostics for Chaos-II
X
AB
X
AB
In this geometry TMI is function ofIAB(L, α(tW )) = SA(L) + SB(L)− SA∪B(L, α(tw )).
We define scrambling time (t∗) as whenIAB(L, α(tW = t∗)) = 0 for a given L > Lc .
t∗ = β2π log SBH +O(N0), same conclusion as α ∼ 1
analysis.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 65: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/65.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Diagnostics for Chaos-Plot
IAB
α(tW )α(t∗)
General behavior of TMI in Shockwave geometries except forsome parameter region of higher derivative gravity.Shenker & Stanford ’13, Leichenauer ’14, Sonnenschein-NS-Tangarife ’16
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 66: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/66.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Analytic Results in BTZ black hole
In 2 + 1-dimensional BTZ black hole calculation can beperformed analytically. Shenker & Stanford ’13
For L > Lc : I (A,B) = LAdSGN
(log sinh πL
β − log(1 + α
2
))α = E β
2SBHe2π tw/β
For high temperature, LT � 1: t∗ = L2 + β
2π log 2SBHβE .
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 67: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/67.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Analytic Results in BTZ black hole
In 2 + 1-dimensional BTZ black hole calculation can beperformed analytically. Shenker & Stanford ’13
For L > Lc : I (A,B) = LAdSGN
(log sinh πL
β − log(1 + α
2
))α = E β
2SBHe2π tw/β
For high temperature, LT � 1: t∗ = L2 + β
2π log 2SBHβE .
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 68: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/68.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Analytic Results in BTZ black hole
In 2 + 1-dimensional BTZ black hole calculation can beperformed analytically. Shenker & Stanford ’13
For L > Lc : I (A,B) = LAdSGN
(log sinh πL
β − log(1 + α
2
))α = E β
2SBHe2π tw/β
For high temperature, LT � 1: t∗ = L2 + β
2π log 2SBHβE .
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 69: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/69.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Analytic Results in BTZ black hole
In 2 + 1-dimensional BTZ black hole calculation can beperformed analytically. Shenker & Stanford ’13
For L > Lc : I (A,B) = LAdSGN
(log sinh πL
β − log(1 + α
2
))α = E β
2SBHe2π tw/β
For high temperature, LT � 1: t∗ = L2 + β
2π log 2SBHβE .
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 70: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/70.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
F (t) in BTZ Black Hole
In geodesic approximation Cornalba et al ’06, Shenker & Stanford ’13,
F (t) ∼(
1
1 + α2
)2ml
where α = E β2SBH
e2π tw/β.F (t) is initially order 1 but starts decaying exponentialy at timet ∼ t∗.Note in BTZ case, the calculation of F (t) and MutualInformation are essentially same and given by geodesics.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 71: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/71.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Outline
1 Motivation & Introduction
2 Chaos, Butterfly-effect & Holography
3 Chaos in various systems
4 Summary & Future Directions
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 72: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/72.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Dp Branes
Black Dp Branes correspond to p + 1-dimensional SuperYang Mills’ theories with 16 supercharges. p = 3corresponds to usual AdS/CFT. Itzhaki, Maldacena, Sonnenschein, &
Yankielowicz
p 6= 3 corresponds to non-conformal field theories. TheYang-Mills’ coupling is dimension full for p 6= 3.
p 6= 3 has a non-trivial dilaton which couples to metric. Sowe need modified Ryu-Takayanagi principle to calculateentanglement entropy. Ryu & Takayanagi ’06, Klebanov et al.. ’07.
Validity of super-gravity solution requires
1� g2eff � N
47−p where, g2
eff = g2YMN rp−3
l2(p−3)s
.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 73: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/73.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Dp Branes
Black Dp Branes correspond to p + 1-dimensional SuperYang Mills’ theories with 16 supercharges. p = 3corresponds to usual AdS/CFT. Itzhaki, Maldacena, Sonnenschein, &
Yankielowicz
p 6= 3 corresponds to non-conformal field theories. TheYang-Mills’ coupling is dimension full for p 6= 3.
p 6= 3 has a non-trivial dilaton which couples to metric. Sowe need modified Ryu-Takayanagi principle to calculateentanglement entropy. Ryu & Takayanagi ’06, Klebanov et al.. ’07.
Validity of super-gravity solution requires
1� g2eff � N
47−p where, g2
eff = g2YMN rp−3
l2(p−3)s
.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 74: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/74.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Dp Branes
Black Dp Branes correspond to p + 1-dimensional SuperYang Mills’ theories with 16 supercharges. p = 3corresponds to usual AdS/CFT. Itzhaki, Maldacena, Sonnenschein, &
Yankielowicz
p 6= 3 corresponds to non-conformal field theories. TheYang-Mills’ coupling is dimension full for p 6= 3.
p 6= 3 has a non-trivial dilaton which couples to metric. Sowe need modified Ryu-Takayanagi principle to calculateentanglement entropy. Ryu & Takayanagi ’06, Klebanov et al.. ’07.
Validity of super-gravity solution requires
1� g2eff � N
47−p where, g2
eff = g2YMN rp−3
l2(p−3)s
.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 75: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/75.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Dp Branes
Black Dp Branes correspond to p + 1-dimensional SuperYang Mills’ theories with 16 supercharges. p = 3corresponds to usual AdS/CFT. Itzhaki, Maldacena, Sonnenschein, &
Yankielowicz
p 6= 3 corresponds to non-conformal field theories. TheYang-Mills’ coupling is dimension full for p 6= 3.
p 6= 3 has a non-trivial dilaton which couples to metric. Sowe need modified Ryu-Takayanagi principle to calculateentanglement entropy. Ryu & Takayanagi ’06, Klebanov et al.. ’07.
Validity of super-gravity solution requires
1� g2eff � N
47−p where, g2
eff = g2YMN rp−3
l2(p−3)s
.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 76: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/76.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Dp Branes contd.
So we need to put an UV-cutoff Λ and the validity at theother end can be achieved by choosing a appropriatetemperature.
Temperature and Entropy is given as,
sBH = c(p)(g2YMN)
p−35−p N2T
9−p5−p
Holography is not well defined for p ≥ 5.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 77: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/77.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Dp Branes contd.
So we need to put an UV-cutoff Λ and the validity at theother end can be achieved by choosing a appropriatetemperature.
Temperature and Entropy is given as,
sBH = c(p)(g2YMN)
p−35−p N2T
9−p5−p
Holography is not well defined for p ≥ 5.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 78: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/78.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Dp Branes contd.
So we need to put an UV-cutoff Λ and the validity at theother end can be achieved by choosing a appropriatetemperature.
Temperature and Entropy is given as,
sBH = c(p)(g2YMN)
p−35−p N2T
9−p5−p
Holography is not well defined for p ≥ 5.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 79: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/79.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Black Dp Branes: Results
t∗ = r∗(Λ) + β2π
(log sBH (β, p, λ,N) + log c1
c2(p) + logα∗( L
β , p))
r∗(Λ) =2`
n(p − 5)
(Λ
`
) n2 (p−5)
, p 6= 5
= ` log
(Λ
rh
), p = 5
n = 88+(7−p)(3−p)
> 0
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 80: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/80.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Lifshitz Black Branes
We are here interested in non-relativistic scale invarianttheories with:
t → λz t ; x → λt for z 6= 1.
The holographic dual to such field theories at finitetemperature is generically called Lifshitz Black Holes.Kachru,
Liu, & Mulligan ’08
sBH = 1
4G(4)N
(2π`
z
) 2z T
2z .
General lore is that the Ryu-Takayanagi principle is notmodified for this case.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 81: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/81.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Lifshitz Black Branes
We are here interested in non-relativistic scale invarianttheories with:
t → λz t ; x → λt for z 6= 1.
The holographic dual to such field theories at finitetemperature is generically called Lifshitz Black Holes.Kachru,
Liu, & Mulligan ’08
sBH = 1
4G(4)N
(2π`
z
) 2z T
2z .
General lore is that the Ryu-Takayanagi principle is notmodified for this case.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 82: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/82.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Lifshitz Black Branes
We are here interested in non-relativistic scale invarianttheories with:
t → λz t ; x → λt for z 6= 1.
The holographic dual to such field theories at finitetemperature is generically called Lifshitz Black Holes.Kachru,
Liu, & Mulligan ’08
sBH = 1
4G(4)N
(2π`
z
) 2z T
2z .
General lore is that the Ryu-Takayanagi principle is notmodified for this case.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 83: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/83.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Lifshitz Black Branes
We are here interested in non-relativistic scale invarianttheories with:
t → λz t ; x → λt for z 6= 1.
The holographic dual to such field theories at finitetemperature is generically called Lifshitz Black Holes.Kachru,
Liu, & Mulligan ’08
sBH = 1
4G(4)N
(2π`
z
) 2z T
2z .
General lore is that the Ryu-Takayanagi principle is notmodified for this case.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 84: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/84.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Lifshitz Black Branes: Results
t∗ = β2π log sBH − `2
z2Λ+ β
2π log( 12e−ψ( z
2)−γ) + β
2π logα∗(LT1z )
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 85: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/85.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Higher derivative Black Branes
We consider AdS black-brane solutions in case of Einsteingravity corrected with higher curvature terms.
It corresponds to finite T Conformal Theories with centralcharges a 6= c .
Most general higher curvature gravity theory with secondorder equation of motion is known as Lovelock theory.
We will consider 4 + 1-dimensional Lovelock theory, whichcorresponds to just addition of Gauss-Bonnet term alongwith usual Einstein-Hilbert term in the action.
Higher curvature corrections in bulk ⇐⇒ λ′tHooft
corrections in dual field theory.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 86: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/86.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Higher derivative Black Branes
We consider AdS black-brane solutions in case of Einsteingravity corrected with higher curvature terms.
It corresponds to finite T Conformal Theories with centralcharges a 6= c .
Most general higher curvature gravity theory with secondorder equation of motion is known as Lovelock theory.
We will consider 4 + 1-dimensional Lovelock theory, whichcorresponds to just addition of Gauss-Bonnet term alongwith usual Einstein-Hilbert term in the action.
Higher curvature corrections in bulk ⇐⇒ λ′tHooft
corrections in dual field theory.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 87: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/87.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Higher derivative Black Branes
We consider AdS black-brane solutions in case of Einsteingravity corrected with higher curvature terms.
It corresponds to finite T Conformal Theories with centralcharges a 6= c .
Most general higher curvature gravity theory with secondorder equation of motion is known as Lovelock theory.
We will consider 4 + 1-dimensional Lovelock theory, whichcorresponds to just addition of Gauss-Bonnet term alongwith usual Einstein-Hilbert term in the action.
Higher curvature corrections in bulk ⇐⇒ λ′tHooft
corrections in dual field theory.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 88: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/88.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Higher derivative Black Branes
We consider AdS black-brane solutions in case of Einsteingravity corrected with higher curvature terms.
It corresponds to finite T Conformal Theories with centralcharges a 6= c .
Most general higher curvature gravity theory with secondorder equation of motion is known as Lovelock theory.
We will consider 4 + 1-dimensional Lovelock theory, whichcorresponds to just addition of Gauss-Bonnet term alongwith usual Einstein-Hilbert term in the action.
Higher curvature corrections in bulk ⇐⇒ λ′tHooft
corrections in dual field theory.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 89: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/89.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Higher derivative Black Branes
We consider AdS black-brane solutions in case of Einsteingravity corrected with higher curvature terms.
It corresponds to finite T Conformal Theories with centralcharges a 6= c .
Most general higher curvature gravity theory with secondorder equation of motion is known as Lovelock theory.
We will consider 4 + 1-dimensional Lovelock theory, whichcorresponds to just addition of Gauss-Bonnet term alongwith usual Einstein-Hilbert term in the action.
Higher curvature corrections in bulk ⇐⇒ λ′tHooft
corrections in dual field theory.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 90: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/90.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Gauss Bonnet Action
Sgrav =1
16πGN
∫d5x√−g
(R +
12
L2+λGBL
2
2X4
)X4 = RµνρσR
µνρσ − 4RµνRµν + R2
The entropy density of Gauss-Bonnet black brane:Cai ’02
sBH =L3
4GNs(λGB )T 3
The value of λGB is bounded by causality constraints:Brigante et al.
’08, Buchel et al. ’09
−7/36 ≤ λGB ≤ 9/100
It is recently pointed out that Gauss-Bonnet as an exacttheory violates causality for any λGB .Camanho et al. ’14
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 91: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/91.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Holographic Entanglement Entropy in HigherDerivative theories
SEE =1
4GN
∫γ
d3σ√γ(1 + λGBL
2Rγ)
+ λGBL2 1
2GN
∫∂γ
d2σ√hK
de Boer et al. ’11, Hung et al. ’11
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 92: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/92.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Higher derivative Black Branes: Results
λ= 0.0
λ=-0.1
L /β = 0.8
Jump during
thermalization
Jump at the end
of thermalization
0.5 1.0 1.5 2.0 2.5 3.00
2
4
6
8
10
12
14
α
I(A,B)
For positive λGB the result is qualitatively similar to λGB = 0case.
t∗ =β
2πlog sBH +
β
2π
(π2− 4λGB
)Similar jumps was noticed in time evolution of Entanglemententropy with higher derivative correction Caceres et. al. ’15.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 93: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/93.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Outline
1 Motivation & Introduction
2 Chaos, Butterfly-effect & Holography
3 Chaos in various systems
4 Summary & Future Directions
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 94: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/94.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Summary
We discussed the concept of Chaos/ Scrambling/Butterfly Effect in context of Holography.
We have shown that the dual thermal field theoryscrambles information at time scales given by β
2π log SBH
for various geometries.
We have used vanishing of thermo-Mutual Information asa signature for scrambling.
The results are qualitatively similar in conformal (a = c),non-conformal and non-relativistic cases.
For conformal theories with (a 6= c), dual to Gauss-BonnetBlack hole the behavior can be very different depending onsign of coupling.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 95: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/95.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Summary
We discussed the concept of Chaos/ Scrambling/Butterfly Effect in context of Holography.
We have shown that the dual thermal field theoryscrambles information at time scales given by β
2π log SBH
for various geometries.
We have used vanishing of thermo-Mutual Information asa signature for scrambling.
The results are qualitatively similar in conformal (a = c),non-conformal and non-relativistic cases.
For conformal theories with (a 6= c), dual to Gauss-BonnetBlack hole the behavior can be very different depending onsign of coupling.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 96: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/96.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Summary
We discussed the concept of Chaos/ Scrambling/Butterfly Effect in context of Holography.
We have shown that the dual thermal field theoryscrambles information at time scales given by β
2π log SBH
for various geometries.
We have used vanishing of thermo-Mutual Information asa signature for scrambling.
The results are qualitatively similar in conformal (a = c),non-conformal and non-relativistic cases.
For conformal theories with (a 6= c), dual to Gauss-BonnetBlack hole the behavior can be very different depending onsign of coupling.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 97: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/97.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Summary
We discussed the concept of Chaos/ Scrambling/Butterfly Effect in context of Holography.
We have shown that the dual thermal field theoryscrambles information at time scales given by β
2π log SBH
for various geometries.
We have used vanishing of thermo-Mutual Information asa signature for scrambling.
The results are qualitatively similar in conformal (a = c),non-conformal and non-relativistic cases.
For conformal theories with (a 6= c), dual to Gauss-BonnetBlack hole the behavior can be very different depending onsign of coupling.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 98: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/98.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Summary
We discussed the concept of Chaos/ Scrambling/Butterfly Effect in context of Holography.
We have shown that the dual thermal field theoryscrambles information at time scales given by β
2π log SBH
for various geometries.
We have used vanishing of thermo-Mutual Information asa signature for scrambling.
The results are qualitatively similar in conformal (a = c),non-conformal and non-relativistic cases.
For conformal theories with (a 6= c), dual to Gauss-BonnetBlack hole the behavior can be very different depending onsign of coupling.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 99: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/99.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Future directions-1
Thermo-Mutual Information, used as a probe forscrambling involves regions in two different copies of thefield theory. It would be useful to express this as a probein a single field theory.
Exponential fall of out of time 4 point correlationfunction is also used as a probe of Scrambling. Connectionto Chaos is more transparent in this definition, alsodefinition of Lyapunov exponent is natural. In case of BTZblack hole, the two definitions can be shown to beequivalent, at least for some heavy operators. Preciseconnection in general is missing.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 100: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/100.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Future directions-1
Thermo-Mutual Information, used as a probe forscrambling involves regions in two different copies of thefield theory. It would be useful to express this as a probein a single field theory.
Exponential fall of out of time 4 point correlationfunction is also used as a probe of Scrambling. Connectionto Chaos is more transparent in this definition, alsodefinition of Lyapunov exponent is natural. In case of BTZblack hole, the two definitions can be shown to beequivalent, at least for some heavy operators. Preciseconnection in general is missing.
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 101: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/101.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Future directions-2
Whether the higher derivative correction beyond GB termcan smooth out the jumps? If not, significance of suchjumps?
Extension to non-commutative theories, General HigherDerivative gravity confining theories... Reynolds et al 1604.04099, Huang
et al 1609.08841, Alishahiha et al 1610.02890
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 102: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/102.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Future directions-2
Whether the higher derivative correction beyond GB termcan smooth out the jumps? If not, significance of suchjumps?
Extension to non-commutative theories, General HigherDerivative gravity confining theories... Reynolds et al 1604.04099, Huang
et al 1609.08841, Alishahiha et al 1610.02890
Nilanjan Sircar Chaos in General Holographic Space-times
![Page 103: Chaos in General Holographic Space-timesneo.phys.wits.ac.za/workshop_8/talks/sircar.pdf · Holography Chaos in various systems Summary & Future Directions Motivation -1 Chaosrefers](https://reader034.vdocuments.mx/reader034/viewer/2022043010/5fa069f7648deb5bcf5302a7/html5/thumbnails/103.jpg)
Chaos inGeneral
HolographicSpace-times
NilanjanSircar
Motivation &Introduction
Chaos,Butterfly-effect &Holography
Chaos invarioussystems
Summary &FutureDirections
Thank You
Nilanjan Sircar Chaos in General Holographic Space-times