Holographic View on non-relativistic Superfluids, Fermi...

Transcript of Holographic View on non-relativistic Superfluids, Fermi...

IntroductionBoson Operators in Schr/NRCFT

Fermion Operators in Schr/NRCFTB-F theory formalism of RG critical points

Conclusion

Holographic View onnon-relativistic Superfluids,

Fermi Surfaces and RG fixed points

Juven Wang (MIT)

Jan 13, 2012 @ Natl Taiwan Univ

Juven Wang (MIT) Holographic View on non-relativistic Superfluids, Fermi Surfaces and RG fixed points

Conclusion

IntroductionEx 1: HologramEx 2: AdS/CFTEx 3: Schr/NRCFTAsymptotic Schrodinger SpacetimePartition Function

Boson Operators in Schr/NRCFTSetup & DictionarySuperfluids from Schr BHSuperfluids from Schr soliton

Fermion Operators in Schr/NRCFTSetup & DictionaryFermi SurfaceLandau Fermi Liquid & Senthil’s ansatzQuantum Phase Transition & Fermi Surface disappearance

B-F theory formalism of RG critical pointsKnown SolutionB-F theory and New SolutionNew Ideas

ConclusionJuven Wang (MIT) Holographic View on non-relativistic Superfluids, Fermi Surfaces and RG fixed points

Conclusion

Work based on:

(1) non-relativistic Superfluids- arXiv: 1103.3472, New J. Phys. 13, 115008 (2011),Allan Adams, JW.

(2) non-relativistic Fermi surface- to appear, Allan Adams, Raghu Mahajan, JW.

(3) B-F theory on RG critical points- JW, . . . .

Conclusion

Why do we use gravity-dual(string theory) to studyquantum many-body systems? . . .

Gravity Quantum

(1). Strongly coupled many-body quantum systems are hardto study by QFT, which gravity dual system is weak coupledand often classical gravity - an easier approach.

(2). Enhance understandings on both sidesgravity, string theory ⇔ quantum many-body systems QFT

Conclusion

Gravity Quantum

Conclusion

Gravity Quantum

Conclusion

Gravity Quantum

Conclusion

Four Take-Home Messages:

(1). Gravity dual of non-realtivistic conformal fieldtheory(NRCFT) can be useful description for strongly coupledmany-body quantum systems.

(2). Gravity dual’s Bosonic operators under NRCFTbackground shows superfluid, metal or insulator low energystates.

(3). Gravity dual’s Fermionic operators under NRCFTbackground shows Fermi surfaces(metalic), or Fermi surfacescollapses(insulator) low energy states.

(4). Use Gravitational B-F theory to formulate gravity duals ofCFT, NRCFTand Lifshitz field theory.

Conclusion

Ex 1: HologramEx 2: AdS/CFTEx 3: Schr/NRCFTAsymptotic Schrodinger SpacetimePartition Function

What is HOLOGRAPHY ?

Ex 1: Hologram

3D ⇔Fourier Trans⇔2D

Ex: Bulk side Dictionary Boundary side

Hologram 3D object Fourier Trans 2D image

Conclusion

Ex 2: AdS/CFT (Anti-de Sitter space/Conformal Field Theory)Ex: Bulk side Dictionary Boundary side

AdS/CFT (D+1)-dim gravity AdS/CFT D-dim Relativistic field theory

1997 Maldacena conjecture, Gubser, Klebanov&Polyakov and Witten

Conclusion

What is HOLOGRAPHY ? Aside from Ex2 AdS/CFT :

The hidden but profound connections between (a)Gravity,(b)Thermodynamics(Statistical Mech.) and (c)Quantum(Information).

(a) (b) (c)

Conclusion

(a) (b) (c)

Conclusion

(a) ↔ (b):(i) black hole thermodynamics: Hawking (TH = κ/2π), Bekenstein(SBH = A/4), Unruh (T = a/2π). (ii) Thermodynamics of Spacetime-The Einstein Equation of State(δQ = TdS): Jacobson, Verlinde.(iii) analogue model: acoustic black hole, He3(Volovik), Bose-EinsteinCondensation.

(b) ↔ (c):(i) partition function Z :Classical state mech e−βH (d-dim space) ⇔ Euclidean QFT (d-dim spacetime).Quantum state mech e−βH (D-dim space) ⇔ Euclidean QFT (D+1-dim spacetime).(ii) Nelson: derivation of Schrodinger eq from stochastic Brownian motion w/ friction.

(c) ↔ (a):(i) string theory, loop quantum gravity, (ii) emergent graviton,(iii) Levin-Wen string-net and Quantum Graphity.

Conclusion

AdS:(i) Poincare patches: ds2 = L2

(− r−2dt2 + r−2(dr 2 + d~x2)

)(ii) Hyperboloid submanifold: ds2 = −dt2 +

∑d~x2 with

−t2 +∑~x2 = −α2 constraint.

CFT:(i) ex: Massless Klein-Gordon QFT:

∫ddxdt 1

2∂µφ∂µφ

(ii) conformal symmetry: Lorentz group Mµν , time translation H, spacetranslation Pµ, scaling(dilatation) D and special conformal Kµ.

Conclusion

(continue)

Ex 2: AdS/CFT (Anti-de Sitter space/Conformal Field Theory)

Q: Why there is holography and duality relation between AdS/CFT?The best-undertood story is:AdS5 × S5 gravity and N = 4 SU(Nc) supersymmetric Yang-Mills(SYM).

hint 1 : matching of symmetries.

N = 4 SYM is invariance under conf (1, 3)× SO(6).

conformal group conf (1, 3) = Poincare sym group+scaling(or dilatation)operator(D)+special conformal transf.(Kµ);

Poincare sym group=Lorentz group+translations operator(Pµ)(Pµ: this includes time translation Hamiltonian P0 = H, plus spatial translationmomentum Pi );Lorentz group=Rotation(Mij )+Lorentz boost(M0j ), together Mµν .

SO(6) is from R symmetry.

Conclusion

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Conclusion

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Conclusion

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Conclusion

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AdS5 × S5 has diffeomorphism isometry group SO(2, 4)× SO(6).

conf (1, 3)× SO(6) isomorphic to SO(2, 4)× SO(6).

Conclusion

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Conclusion

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Conclusion

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Q: Why there is holography and duality relation between AdS/CFT?The best-undertood story is:AdS5 × S5 gravity and N = 4 SU(Nc) supersymmetric Yang-Mills(SYM)

hint 2 : Maldacena conjecture AdS/CFT correspondence

AdS5 × S5 type IIB string theory ⇔ Nc stacks of D3 branes.gravity and string theory ⇔ gauge theory, QFT, CFT.

⇔Juven Wang (MIT) Holographic View on non-relativistic Superfluids, Fermi Surfaces and RG fixed points

Conclusion

(continue)

Q: Why there is holography and duality relation between AdS/CFT?The best-undertood story is:AdS5 × S5 gravity and N = 4 supersymmetric Yang-Mills(SYM) theory.

hint 3 : matching of parameters strong-weak couplings duality

R2

α′ ∼√

gsNc ∼√λ, gs ∼ g 2

YM ∼ λNc

, R4

`4p∼ R4√

G∼ Nc

hint 4 : Partition function and field operator correspondence

ZCFT [φ] = Zstring [ Φ|∂AdS ] ' e−Ssupergravity .S → S +

∫d4x φ(x) · O(x) , (source · response)

φ = Φ|∂AdS (operator-field)

Conclusion

(continue)

Q: Why there is holography and duality relation between AdS/CFT?The best-undertood story is:AdS5 × S5 gravity and N = 4 supersymmetric Yang-Mills(SYM) theory.

R2

α′ ∼√

gsNc ∼√λ, gs ∼ g 2

YM ∼ λNc

, R4

`4p∼ R4√

G∼ Nc

ZCFT [φ] = Zstring [ Φ|∂AdS ] ' e−Ssupergravity .S → S +

∫d4x φ(x) · O(x) , (source · response)

φ = Φ|∂AdS (operator-field)

Conclusion

hint 1 : matching of symmetries. conformal group ' isometry (Isomorphism)

Conclusion

AdS/CFT (D+1)-dim gravity AdS/CFT D-dim Rela field theory

hint 1 : matching of symmetries conformal group ' isometry (Isomorphism)

Conclusion

Ex 3: Schr/NRCFT (Schrodinger space(Schr)/NRCFT)

hint : matching of symmetries

NRCFT ′s Schr group isomorphic to isometry of Schr space .

LHS: Schrodinger group=Galilean group + translation +scaling(dilatation) operator+special conformal operator.

Galilean group: Rotation(Mi,j )+Galilean boost(Ki )translation(Pµ: includes Hamiltonian P0 = H and momentum Pi )scaling(dilatation) (D)special conformal operator (C)

Conclusion

RHS: Embed Schr group in a higher dimensional Conformal group.Free Schrodinger eq inside Free Klein-Gordon eq .Light-cone coordinate t, ξ. Compactify ξ to give discrete mass tower.

answer ds2 = −r−2zdt2 + r−2(2dtdξ + d~x2 + dr 2)(Son, Balasubramanian&McGreevy)

Schr/NRCFT (D+2)-dim gravity Schr/NRCFT D-dim NR field theory

Conclusion

Ex 1,2,3:

Conclusion

Schr/NRCFT correspondence

Asymptotic Schrodinger Spacetimepure Schr spacetime

ds2 = −r−2zdt2 + r−2(2dtdξ + d~x2 + dr 2)

The pure Schr spacetime provides no temperature for boundary fieldtheory, however asymptotic Schr with black hole(BH) does.

Neutral Schr BH with finite density (0 < r < rH):By TsssT (Null Melvin twist) on neutral black D3 branes of type IIB string.ds2

Ein

= K1/3“`− f + (f−1)2

4(K−1)

´dt2

Kr4 + 1+fr2K

dt dξ + K−1K

dξ2 + d~x2

r2 + dr2

f r2

”.

(arXiv: 0807.1099, 0807.1100, 0807.1111)

Charged Schr BH with finite density (rH < r <∞):By TsssT (Null Melvin twist) on charge black D3 branes of type IIB string.ds2

Ein

= K−1/3

Kr2

R2

„“1−f4β2 −r2f

”dt2+β2(1−f )dξ2+(1+f )dtdξ

«+ r2

R2 (dx21 +dx2

2 )+ R2

r2dr2

f

!(arXiv: 0907.1892, 0907.1920)

Conclusion

Ein

= K1/3“`− f + (f−1)2

4(K−1)

´dt2

Kr4 + 1+fr2K

dt dξ + K−1K

dξ2 + d~x2

r2 + dr2

f r2

”.

(arXiv: 0807.1099, 0807.1100, 0807.1111)

Ein

= K−1/3

Kr2

R2

„“1−f4β2 −r2f

«+ r2

R2 (dx21 +dx2

2 )+ R2

r2dr2

f

!(arXiv: 0907.1892, 0907.1920)

Conclusion

Ein

= K1/3“`− f + (f−1)2

4(K−1)

´dt2

Kr4 + 1+fr2K

dt dξ + K−1K

dξ2 + d~x2

r2 + dr2

f r2

”.

(arXiv: 0807.1099, 0807.1100, 0807.1111)

Ein

= K−1/3

Kr2

R2

„“1−f4β2 −r2f

«+ r2

R2 (dx21 +dx2

2 )+ R2

r2dr2

f

!(arXiv: 0907.1892, 0907.1920)

Conclusion

Ein

= K1/3“`− f + (f−1)2

4(K−1)

´dt2

Kr4 + 1+fr2K

dt dξ + K−1K

dξ2 + d~x2

r2 + dr2

f r2

”.

(arXiv: 0807.1099, 0807.1100, 0807.1111)

Ein

= K−1/3

Kr2

R2

„“1−f4β2 −r2f

«+ r2

R2 (dx21 +dx2

2 )+ R2

r2dr2

f

!(arXiv: 0907.1892, 0907.1920)

Conclusion

Neutral Schr BH with finite density:By TsssT (Null Melvin twist) on neutral black D3 branes of type IIB string.

Charged Schr BH with finite density:By TsssT (Null Melvin twist) on charge black D3 branes of type IIB string.

Conclusion

Partition FunctionPartition function and field operator correspondence

ZCFT [φ] = Zstring [ Φ|∂AdS ] ' e−Ssupergravity .

S → S +∫

d4x φ(x) · O(x) , (source · response)φ = Φ|∂AdS (operator-field)

S → S +∫

d4x A(x) · J (x) , (source · response)Aµ = Aµ|∂AdS (operator-field)

Boson Operators in NRCFT:scalar field in Schr

Fermion Operators in NRCFTDirac spinor field in Schr

Conclusion

S → S +∫

Conclusion

S → S +∫

Conclusion

S → S +∫

Conclusion

S → S +∫

Conclusion

S → S +∫

Conclusion

Summary So Far:

Bulk side Dictionary Boundary side

Boson scalar field field-operator Boson operator

Fermion Dirac spinor field field-operator Fermion operator

Conclusion

Setup & DictionarySuperfluids from Schr BHSuperfluids from Schr soliton

(I) Boson Operators in Schr/NRCFT

Focus on 5-dim Schr and 3-dim NRCFT correspondence at z = 2.

Setup: Probe limit: Abelian Higgs model

Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2

),

φ = φ1r ∆− + φ2r ∆+ + . . . ,with conformal dimension ∆± = 2±

√4 + m2 + q2M2

o .At = µQ + ρQ r 2 + . . . ,Aξ = Mo + ρM r 2 + . . . ,

Ax = A0 + A2r2

2 + . . . .

Dictionary:φ = Φ|∂AdS (operator-field correpondence).Spontaneous Symmetry Breaking: turn off source, only left with response.

φ1 = 0, φ2 = 〈O2〉 , or φ2 = 0, φ1 = 〈O1〉 for superfluid.

Conductivity: σ(ω) = 〈Jx〉〈Ex〉 = −i 〈Jx〉

ω〈Ax〉 = −i A2

ωA0

Conclusion

Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2

),

√4 + m2 + q2M2

Ax = A0 + A2r2

2 + . . . .

ω〈Ax〉 = −i A2

ωA0

Conclusion

Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2

),

√4 + m2 + q2M2

Ax = A0 + A2r2

2 + . . . .

Dictionary:

φ = Φ|∂AdS (operator-field correpondence).Spontaneous Symmetry Breaking: turn off source, only left with response.

ω〈Ax〉 = −i A2

ωA0

Conclusion

Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2

),

√4 + m2 + q2M2

Ax = A0 + A2r2

2 + . . . .

ω〈Ax〉 = −i A2

ωA0

Conclusion

Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2

),

√4 + m2 + q2M2

Ax = A0 + A2r2

2 + . . . .

ω〈Ax〉 = −i A2

ωA0

Conclusion

Before using Gravity Dual description . . .

Recall: What is Superfluid?

Spontaneous U(1) Symmetry Breaking of coordinate θ,which conjugate field is number N strongly fluctuating.

(i)QFT description L = iφ†∂0φ− 12m∂iφ†∂iφ− g2(φ†φ− ρ)2, with

φ(x , t) =pρ+ h(x , t)e iθ(x,t), integrate out h, thus L = 1

4g2 (∂0θ)2 − ρ2m

(∂iθ)2.

(ii)Lattice Hamiltonian H = −tP〈i,j〉(ψ

†i ψj + ψ†j ψi ) + U

Pi (Ni − N)2,

By the 2nd quantization, ψj =p

Njeiθj , commutators [ψi , ψ

†j ] = δi,j , thus

[θi , Nj ] = −iδi,j . Continuum field limit is free Klein-Gordon equation.

Above all have linear dispersion ω ∝ k .

(iii) In our Gravity Dual system, we have gauge invariant momentum ofcompact (extra-)dimension ξ as dual to Number operator.

Conclusion

4g2 (∂0θ)2 − ρ2m

(∂iθ)2.

Pi (Ni − N)2,

Conclusion

4g2 (∂0θ)2 − ρ2m

(∂iθ)2.

Pi (Ni − N)2,

Conclusion

4g2 (∂0θ)2 − ρ2m

(∂iθ)2.

Pi (Ni − N)2,

Conclusion

Superfluids from Schr BH

0.0 0.2 0.4 0.6 0.8 1.0 1.20.00

0.02

0.04

0.06

0.08

0.10

TTc

XO1\

0 2 4 6 8 100.00

0.05

0.10

0.15

ΩTc

Re@ΣHΩLD

1.261.051.0.960.880.650.370.290.240.190.160.080.050.01

TTc

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

ΩTc

Im@ΣHΩLD

1.261.051.0.960.880.650.370.290.240.190.160.080.050.01

TTc

0.0 0.2 0.4 0.6 0.8 1.0 1.20.00

0.02

0.04

0.06

0.08

TTc

XO1\

0 2 4 6 8 100.0

0.5

1.0

1.5

ΩTc

Re@ΣHΩLD

1.281.010.950.60.330.20.060.020.010.

TTc

0 2 4 6 8 10

-2

-1

0

1

2

3

ΩTc

Im@ΣHΩLD

1.28

1.01

0.95

0.6

0.33

0.2

0.06

0.02

0.01

0.TTc

0.0 0.2 0.4 0.6 0.8 1.0 1.20.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

TTc

XO1\

0 2 4 6 8 100.0

0.5

1.0

1.5

ΩTc

Re@ΣHΩLD

1.28

1.

0.95

0.72

0.36

0.24

0.15

0.1

0.07

0.06

0.01TTc

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0

ΩTc

Im@ΣHΩLD

1.28

1.

0.95

0.72

0.36

0.24

0.15

0.1

0.07

0.06

0.01TTc

Conclusion

〈O〉 v.s. T :

TMetalSuperfluid

Tc

Finite T mean-field Phase Transition(w/ βMF = 1/2) by tuning T

〈O〉 v.s. Ω:

WMetalSuperfluid

W*

Quantum Phase Transition(at T=0) by tuning Ω

Conclusion

Low T and High T condensates:

0 1 2 3 4 50.00

0.02

0.04

0.06

0.08

0.10

0.12

T

XO1\

Low T and High T condensates - compare free energy:

FC −FN = −TR CNδSEVD

= −T (∆1 −∆2)R CN φ2 dφ1

Conclusion

0 1 2 3 4 50.00

0.05

0.10

0.15

0.20

T

XO1\

TH IΜQM

2nd Tc IΜQM

1st T* IΜQM

716

38

516

14

732

1364

0.195

0.192

0.191

316

18

0

ΜQ

XO1HTcL\2nd order phase transition

XO1HT*L\ 1st order phase transition

= −T (∆1 −∆2)R CN φ2 dφ1

Conclusion

2nd order phase transitionT=Tc

1st order phase transitionT=T*

0.0 0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

Μt

XO1\

= −T (∆1 −∆2)R CN φ2 dφ1

Conclusion

2nd order phase transitionT=Tc

1st order phase transitionT=T*

0.0 0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

Μt

XO1\

ΜQ1st order PT2nd order PT

Μ*

Near the multicritical point shows Mean-Field theory behavior. Landau-Ginzburg freeenergy can be: F (ϕ) = 1

2c2(T − Tc (µQ))ϕ2 + 1

4c4(µ∗ − µQ)ϕ4 + 1

6c6ϕ

6.

With ϕ ∼ 〈O〉 and with coefficients c2, c4, c6 > 0.

Mean-Field exponent 1/2: 〈O〉 ∼ (µQ/µ∗ − 1)1/2

Conclusion

〈O〉 v.s. T :

TMetalSuperfluid

Tc

〈O〉 v.s. Ω:

WMetalSuperfluid

W*

〈O〉 v.s. µQ :

Μ*

Conclusion

Superfluids from Schr soliton

Recall AdS soliton, studied in Hawking-Page transition and Witten.

Analogue to the AdS soliton, take Schrodinger black hole solution to do adouble Wick rotation. i.e.(τ, y)→ i(y , τ), (t, ξ)→ (t, ξ), (β,Ω)→ −i(β,Ω),

we get Schrodinger soliton:

ds2soliton,Ein = K

1/3s

((−fs+ (fs−1)2

4(Ks−1)

)dt2

Ks r4 + 1+fsr2Ks

dt dξ+ Ks−1Ks

dξ2+ d~x2

r2 + dr2

fs r2

).

Conclusion

we get Schrodinger soliton.

0.00 0.05 0.10 0.150.0

0.1

0.2

0.3

0.4

0.5

W

XO1\

0 10 20 30 40 50-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Ω

Im@ΣHΩLD

1.09

0.997

0.701

0.319

0.106

WWc

〈O〉 v.s. Ω:W

InsulatorSuperfluidWc

Quantum Phase Transition(at T=0) by tuning Ω

Conclusion

we get Schrodinger soliton.

0 1 2 3 40.0

0.1

0.2

0.3

0.4

ΜQ

XO1\

0 5 10 15 20

-0.5

0.0

0.5

Ω

Im@ΣHΩLD

0 5 10 15 20

-0.2

0.0

0.2

0.4

Ω

Im@ΣHΩLD

〈O〉 v.s. µQ :ΜQ

SuperfluidInsulatorΜc

Quantum Phase Transition(at T=0) by tuning µQ

Conclusion

〈O〉 v.s. T :T

MetalSuperfluidTc

〈O〉 v.s. Ω:W

MetalSuperfluidW*

1st order PT2nd order PTΜ*

〈O〉 v.s. Ω:W

SuperfluidInsulatorΜc

Conclusion

Setup & DictionaryFermi SurfaceLandau Fermi Liquid & Senthil’s ansatzQuantum Phase Transition & Fermi Surface disappearance

(II) Fermion Operators in Schr/NRCFT

Setup:

Probe limit:Dirac fermions coupled to gauge field in charged Schr BH spacetimeSprobe,Dirac =

∫d5x√−gEiniψ(eµa ΓaDµ −m)ψ

Dictionary:

S∂ =∫∂M d3xdξ

√−gg rr ψψ

Π+ = −√−gg rr ψ−, Π− =

√−gg rr ψ+

exp[−Sgrav [ψ, ψ](r →∞)] = 〈exp[R

dd+1x(χO + Oχ)]〉QFT

χ ∝ ψ as source, O ∝ Π as response .

Green′s function ≡ G = response(R)/source(S) ∝ O/χ

Conclusion

(II) Fermion Operators in Schr/NRCFT

Setup:

Probe limit:Dirac fermions coupled to gauge field in charged Schr BH spacetimeSprobe,Dirac =

∫d5x√−gEiniψ(eµa ΓaDµ −m)ψ

Dictionary:

S∂ =∫∂M d3xdξ

√−gg rr ψψ

Π+ = −√−gg rr ψ−, Π− =

√−gg rr ψ+

exp[−Sgrav [ψ, ψ](r →∞)] = 〈exp[R

dd+1x(χO + Oχ)]〉QFT

χ ∝ ψ as source, O ∝ Π as response .

Green′s function ≡ G = response(R)/source(S) ∝ O/χ

Conclusion

Fermion Surface

0.0 0.5 1.0 1.5 2.0 2.50

1000

2000

3000

4000

k

ImG1

1.51.41.31.21.11.0.950.90.80.70.60.50.40.30.20.10-0.1Ω

0.0 0.5 1.0 1.5 2.0 2.5

-4000

-2000

0

2000

4000

k

ReG1

1.4

1.3

1.2

1.1

1.

0.95

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

Ω

we find near the quasi-particle like peak has scalings (k⊥ ≡ |k − kF |),ω∗(k⊥) ∼ kz

⊥, z ' 1.14Im[G (ω∗(k⊥), k⊥)] ∼ k−α⊥ , α ' 1.00

Conclusion

Fermion Surface

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

k

Ω

we find near the quasi-particle like peak has scalings (k⊥ ≡ |k − kF |),ω∗(k⊥) ∼ kz

⊥, z ' 1.14Im[G (ω∗(k⊥), k⊥)] ∼ k−α⊥ , α ' 1.00.Particle-Hole asymmetry .

Conclusion

Compare to Landau Fermi Liquid(LFL) and Senthil’s ansatz.

LFLG(k, ω) = 1

ω−ξk−Σ(ω,k)= 1

ω−(ξk +ReΣ(k,ω))−iImΣ(k,ω)= Z

(ω−ωF )−ξk + i2τk

with γ(ω) = κ(ω − ωF )n.

Senthil’s ansatzG(k, ω) = c0(k − kF )−αF0( c1(ω−ωF )

(k−kF )z)

Conclusion

LFLG(k, ω) = 1

Senthil’s ansatz for k < kF

G(k, ω) = c0(k − kF )−αF0( c1(ω−ωF )(k−kF )z

) = c0(k−kF )−α

log(−(ω−ωF )

c1(k−kF )z)+iγ0

0.0 0.5 1.0 1.50

500

1000

1500

2000

2500

Ω

ImG1

k<kF

0.0 0.5 1.0 1.5

-1000

-500

0

500

1000

1500

2000

Ω

ReG1

k<kF

Conclusion

LFLG(k, ω) = 1

Senthil’s ansatz for k > kF

G(k, ω) = c0(k − kF )−αF0( c1(ω−ωF )(k−kF )z

) = c0(k−kF )−α

log(−(ω−ωF )

c1(k−kF )z)−iγ0

0.0 0.5 1.0 1.5 2.00

500

1000

1500

2000

Ω

ImG1

k>kF

0.0 0.5 1.0 1.5 2.0

-1000

-500

0

500

1000

Ω

ReG1

k>kF

Conclusion

Quantum Phase Transition & Fermi Surface disappearance

β > β∗

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

50 000

100 000

150 000

200 000

k

ImG1

1.51.41.31.21.11.0.90.80.70.60.50.40.30.20.10-0.1Ω

0.0 0.5 1.0 1.5 2.0 2.50

10 000

20 000

30 000

40 000

k

ImG1

1.5

1.4

1.3

1.2

1.1

1.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1Ω

β ' β∗

0.0 0.5 1.0 1.5 2.0 2.50

1000

2000

3000

4000

k

ImG1

1.51.41.31.21.11.0.950.90.80.70.60.50.40.30.20.10-0.1Ω

β < β∗

0.0 0.5 1.0 1.5 2.0 2.50

1000

2000

3000

4000

5000

k

ImG1

1.51.41.31.21.11.0.90.80.70.60.50.40.30.20.10-0.1Ω

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

250

k

ImG1

3.2.52.12.052.042.032.022.012.1.91.81.71.61.51.41.31.21.11.0.90.80.70.60.50.40.30.20.10-0.1-0.5-1.Ω

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

2.5

k

ImG1

1.5

1.4

1.3

1.2

1.1

1.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1Ω

Conclusion

〈O〉 v.s. β:Β

MetalInsulatorΒ*

β > β∗

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50

50 000

100 000

150 000

200 000

k

ImG1

1.51.41.31.21.11.0.90.80.70.60.50.40.30.20.10-0.1Ω

0.0 0.5 1.0 1.5 2.0 2.50

10 000

20 000

30 000

40 000

k

ImG1

1.5

1.4

1.3

1.2

1.1

1.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1Ω

β ' β∗

0.0 0.5 1.0 1.5 2.0 2.50

1000

2000

3000

4000

k

ImG1

1.51.41.31.21.11.0.950.90.80.70.60.50.40.30.20.10-0.1Ω

β < β∗

0.0 0.5 1.0 1.5 2.0 2.50

1000

2000

3000

4000

5000

k

ImG1

1.51.41.31.21.11.0.90.80.70.60.50.40.30.20.10-0.1Ω

0.0 0.5 1.0 1.5 2.0 2.50

50

100

150

200

250

k

ImG1

3.2.52.12.052.042.032.022.012.1.91.81.71.61.51.41.31.21.11.0.90.80.70.60.50.40.30.20.10-0.1-0.5-1.Ω

0.0 0.5 1.0 1.5 2.0 2.50.0

0.5

1.0

1.5

2.0

2.5

k

ImG1

1.5

1.4

1.3

1.2

1.1

1.

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1Ω

Conclusion

Fermi surface from Schr BH

〈O〉 v.s. β:

ΒMetalInsulator

Β*

〈O〉 v.s. T :

TMetalSuperfluid

Tc

〈O〉 v.s. Ω:

WMetalSuperfluid

W*

〈O〉 v.s. µQ :

Μ*

〈O〉 v.s. Ω:

W

〈O〉 v.s. µQ :

ΜQSuperfluidInsulator

Μc

Conclusion

Fermions in charged Schr BH

The parameters of phase space: ∆,T , µQ ,M, β,conformal dimension, temperature, charge density, Number(Mass),background density.

Bosons in Schr BHThe parameters of phase space: ∆,T , µQ ,M,Ωconformal dimension, temperature, charge density, Number(Mass),background density.

Bosons in Schr solitonThe parameters of phase space: ∆,mG , µQ ,M,Ωconformal dimension, mass gap(∼ 1/Lξ), charge density, Number(Mass),background density.

asymptotics AdSd+2 Schrd+3

scalar conformal dim ∆± = d+12±q

( d+12

)2 + m2 ∆± = d+22±q

( d+22

)2 + m2 + (`− qMo )2

spinor conformal dim ∆± = d+12± m ∆± = d+2

2±q

((m ± 12

)2 + (`− qMo )2

Conclusion

Known SolutionB-F theory and New SolutionNew Ideas

(III) Gravitational B-F theory formalism of RG critical points

(i) AdS/CFT (D+1)-dim gravity AdS/CFT D-dim Rela FT

(ii) Lif/Lifshitz FT (D+1)-dim gravity Lif/LFT D-dim Rela FT

Sch r/NRCFT (D+2)-dim gravity Schr/NRCFT D-dim NR FT

Conclusion

Known Solution

AdS/CFT (D+1)-dim gravity AdS/CFT D-dim Rela FT

Lif/Lifshitz FT (D+1)-dim gravity Lif/LFT D-dim Rela FT

CFT (z=1):Boundary: Conformal group.

Lifshitz FT (∀z):Boundary: RotationMi,j without Lorentz boost+translation(Pµ: includes Hamiltonian P0 = H and momentum Pi )+‘dynamical’ scaling(D, which sends t → λz t, x → λx)

AdS and Lif Bulk solutionds2 = L2

(− r−2zdt2 + r−2(dr 2 + d~x2)

)solved by field strength H = dB and massive 1-form B.

S =

∫dD+1x

√−g(R − 2Λ− 1

2|H2|2 −

1

2λ2|B1|2)

Conclusion

Known Solution

(− r−2zdt2 + r−2(dr 2 + d~x2)

S =

∫dD+1x

√−g(R − 2Λ− 1

2|H2|2 −

1

2λ2|B1|2)

Conclusion

Known Solution

(− r−2zdt2 + r−2(dr 2 + d~x2)

S =

∫dD+1x

√−g(R − 2Λ− 1

2|H2|2 −

1

2λ2|B1|2)

Conclusion

Known Solution

Schr/NRCFT (D+2)-dim gravity Schr/NRCFT D-dim NR FT

NRCFT :Boundary: Schrodinger group.

Schr Bulk solutionds2 = L2

(− r−2zdt2 + r−2(2dtdξ + d~x2 + dr 2)

),

solved by flux H = dB and massive p-form B.

S =

∫dd+3x

√−g(R − 2Λ− 1

2|Hp+1|2 −

1

2λ2|Bp|2)

Conclusion

B-F theory

Consider two fluxes H = dB and F = dC with a topological term B ∧ F :

S =

∫dd+3x

√−g(R − 2Λ− 1

2|Hp+1|2 −

1

2|Fd+3−p|2) + λ

∫Bp ∧ Fd+3−p ,

(i) B ∧ F is topological, ∵ λ∫

Bp ∧ Fd+3−p does not depend on metric g .(ii) EOMs:Einstein δg : no B ∧ F term contribution.Maxwell EOMs: d(∗dC ) ∝ λdB and d(∗dB) ∝ λdC(iii) λ is constrained by EOMs.

Conclusion

B-F theory

S =

∫dd+3x

√−g(R − 2Λ− 1

2|Hp+1|2 −

1

2|Fd+3−p|2) + λ

∫Bp ∧ Fd+3−p ,

(iv) let Fd+3−p ≡ ∗dφp−1,− 1

2 |Fd+3−p|2 + λ∫

Bp ∧ Fd+3−p = − 12 | ∗ dφp−1|2 + λ

∫Bp ∧ ∗dφp−1

− 12 |Fd+3−p|2 + λ

∫Bp ∧ Fd+3−p = − 1

2 |λBp − dφp−1|2 + . . . ,. . . are extra term to make the gauge transf valid,gauge choice fix dφp−1.

(v) Alternatively, consider integrating out F field to get massive B field.

Massive Field

S =

∫dd+3x

√−g(R − 2Λ− 1

2|Hp+1|2 −

1

2λ2|Bp|2)

Conclusion

B-F theory

S =

∫dd+3x

√−g(R − 2Λ− 1

2|Hp+1|2 −

1

2|Fd+3−p|2) + λ

∫Bp ∧ Fd+3−p ,

(iv) let Fd+3−p ≡ ∗dφp−1,− 1

2 |Fd+3−p|2 + λ∫

Bp ∧ Fd+3−p = − 12 | ∗ dφp−1|2 + λ

∫Bp ∧ ∗dφp−1

− 12 |Fd+3−p|2 + λ

∫Bp ∧ Fd+3−p = − 1

2 |λBp − dφp−1|2 + . . . ,. . . are extra term to make the gauge transf valid,gauge choice fix dφp−1.

(v) Alternatively, consider integrating out F field to get massive B field.

Massive Field

S =

∫dd+3x

√−g(R − 2Λ− 1

2|Hp+1|2 −

1

2λ2|Bp|2)

Conclusion

B-F theory

New Solutionfinite T finite density Schr BH spacetime for(a) ∀ d-dim, z = 2 and (b) d = 2z − 4-dim, ∀ z (Papers to appear - JW)

Conclusion

B-F theory

Some Solutionfinite T finite density Schr BH spacetime for(a) ∀ d-dim, z = 2 and (Kovtun&Nickel, 0809.2020, PRL)

S =

∫dd+3x

√−g(R − a

2(∂µφ)(∂µφ)− 1

4e−aφ|Fµν |2 −

m2

2AµAµ − V (φ)

V (φ) = (Λ + Λ′)eaφ + (Λ− Λ′)ebφ

ds2 = r2K− d

d+1“

[(f−1)2

4(K−1)− f ]r2dt2 − (1 + f )dtdξ + K−1

r2 dξ2”

+ K1

d+1“r2dx2 + dr2

r2 f

”

Free energy: F = −TlogZ = −TSE

F = −T −116πGd+3

∫dd+2x2r

−(d+2)H = −T −1

16πGd+3

1T ∆ξV 2r

−(d+2)H

= 2Gd+3

∆ξπd+123d2 −1(d + 2)−(d+2)T

d+22 (T

µ )d+2

2

Conclusion

B-F theory

Some Solutionfinite T finite density Schr BH spacetime for(a) ∀ d-dim, z = 2 and (Kovtun&Nickel, 0809.2020, PRL)

S =

∫dd+3x

√−g(R − a

2(∂µφ)(∂µφ)− 1

4e−aφ|Fµν |2 −

m2

2AµAµ − V (φ)

V (φ) = (Λ + Λ′)eaφ + (Λ− Λ′)ebφ

ds2 = r2K− d

d+1“

[(f−1)2

4(K−1)− f ]r2dt2 − (1 + f )dtdξ + K−1

r2 dξ2”

+ K1

d+1“r2dx2 + dr2

r2 f

”Free energy: F = −TlogZ = −TSE

F = −T −116πGd+3

∫dd+2x2r

−(d+2)H = −T −1

16πGd+3

1T ∆ξV 2r

−(d+2)H

= 2Gd+3

∆ξπd+123d2 −1(d + 2)−(d+2)T

d+22 (T

µ )d+2

2

Conclusion

B-F theory

New Solutionfinite T finite density Schr BH spacetime for(b) d = 2z − 4-dim, ∀ z (Papers to appear - JW)

S =

∫dd+3x

√−ge−2ϕ(R − 2Λ− 1

2|Hz |2)− 1

2|Fz |2 + λ

∫Bz−1 ∧ Fz

ds2str = 1

Kr2z (−f +(1−f )2

4(K−1))dt2 + 1+f

Kr2 dtdξ + K−1K

r2(z−2)dξ2 + 1Kr2 d~x2

(1,...,z−2) + 1r2 d~x2

(z−1,...,2z−4) + dr2

r2 f

Free energy: F = −TlogZ = −TSE , with Ωs ≡ β2

rd+2H

= β2

r2(z−1)H

=(4πT )(d+2)

(d+2)(d+2)(2|µ|)d+4

2

F = 2T16πGd+3

Rdd+2xr

−(d+2)H

1−(z−2)Ωs

(1+Ωs )z−1

2

= 2Gd+3

∆ξπd+123d2−1

(d + 2)−(d+2)Td+2

2 ( Tµ

)d+2

21−(z−2)Ωs

(1+Ωs )z−1

2

Conclusion

B-F theory

New Solutionfinite T finite density Schr BH spacetime for(b) d = 2z − 4-dim, ∀ z (Papers to appear - JW)

S =

∫dd+3x

√−ge−2ϕ(R − 2Λ− 1

2|Hz |2)− 1

2|Fz |2 + λ

∫Bz−1 ∧ Fz

ds2str = 1

Kr2z (−f +(1−f )2

4(K−1))dt2 + 1+f

Kr2 dtdξ + K−1K

r2(z−2)dξ2 + 1Kr2 d~x2

(1,...,z−2) + 1r2 d~x2

(z−1,...,2z−4) + dr2

r2 f

Free energy: F = −TlogZ = −TSE , with Ωs ≡ β2

rd+2H

= β2

r2(z−1)H

=(4πT )(d+2)

(d+2)(d+2)(2|µ|)d+4

2

F = 2T16πGd+3

Rdd+2xr

−(d+2)H

1−(z−2)Ωs

(1+Ωs )z−1

2

= 2Gd+3

∆ξπd+123d2−1

(d + 2)−(d+2)Td+2

2 ( Tµ

)d+2

21−(z−2)Ωs

(1+Ωs )z−1

2

Conclusion

Comments:1. B-F theory as a gravitational effective action for AdS, Lif, Schr metrics- gravity dual of CFT, Lifshitz field theory, NRCFT. Find new finite Tsolutions for ∀z .

2. Free energy F (T , µ) has the unphysical form T #(Tµ )#, instead of

physical result T #µ#.

Conclusion

Other Ideas on B-F theory

1. Use B-F theory to interpolate different asymptotic types of metrics -between AdS, Lif, Schr.

2. A proposal on gravity dual solution realizing Superfluid in NRCFT(not the probe limit): The UV theory is NRCFT(z = 2) but the IR theoryis Lifshitz field theory(z = 2).Meanings:(1) Short range behavior is NRCFT, sym of free Schr eq - gravity dual isSchr. Long rang behavior is Lifshitz - gravity dual is Lif.(2) A gravity dual realizes shrinking extra-dim ξ, a smooth cigar,interpolating d + 3-dim UV Schr to d + 2-dim IR Lif.

IR Lifd+2 UV Schrd+3

(3) Break Number U(1) symmetry (as superfluid) geometrically byshrinking U(1) ξ circle, where −i∂ξ corresponds to Number operator.

Conclusion

2. A proposal on gravity dual solution realizing Superfluid in NRCFT(not the probe limit): The UV theory is NRCFT(z = 2) but the IR theoryis Lifshitz field theory(z = 2).

Meanings:(1) Short range behavior is NRCFT, sym of free Schr eq - gravity dual isSchr. Long rang behavior is Lifshitz - gravity dual is Lif.(2) A gravity dual realizes shrinking extra-dim ξ, a smooth cigar,interpolating d + 3-dim UV Schr to d + 2-dim IR Lif.

Conclusion

2. A proposal on gravity dual solution realizing Superfluid in NRCFT(not the probe limit): The UV theory is NRCFT(z = 2) but the IR theoryis Lifshitz field theory(z = 2).Meanings:(1) Short range behavior is NRCFT, sym of free Schr eq - gravity dual isSchr. Long rang behavior is Lifshitz - gravity dual is Lif.

(2) A gravity dual realizes shrinking extra-dim ξ, a smooth cigar,interpolating d + 3-dim UV Schr to d + 2-dim IR Lif.

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Conclusion

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Conclusion

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Conclusion

(4) How to shrink an extra-dim ξ circle?

(i) Witten’s AdS soliton picture - Do NOT work. Take d+3-dim gravity,compactify one dimension, dual to d+1-dim gauge theory. Euclideanized:

BH soliton

Double Wick rotation: (τ, y)→ i(y , τ), periodic identification (τb, yb), or (τs , ys),iτb ∼ iτb + N/T , yb ∼ yb + MLξ

iτs ∼ iτs + MLξ, ys ∼ ys − N/T .

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Conclusion

(4) How to shrink an extra-dim ξ circle?(i) Witten’s AdS soliton picture - Do NOT work. Take d+3-dim gravity,compactify one dimension, dual to d+1-dim gauge theory. Euclideanized:

BH soliton

Double Wick rotation: (τ, y)→ i(y , τ), periodic identification (τb, yb), or (τs , ys),iτb ∼ iτb + N/T , yb ∼ yb + MLξ

iτs ∼ iτs + MLξ, ys ∼ ys − N/T .

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Conclusion

(4) How to shrink an extra-dim ξ circle?(i) Witten’s AdS soliton - Do NOT work. Hawking-Page transition.(ii) Schrodinger soliton - shrink ξ cigar?

Do NOT work. Double Wickrotate: (τb, yb)→ i(ys , τs), (tb, ξb)→ (ts , ξs), (βb,Ωb)→ −i(βs ,Ωs)Periodic identification for (τb, ξb), (ys , ξs) shows different ensemble system:Schr BH: itb = itb + N/T , ξb = ξb + N(µM/T ) + MLξSchr soliton: its = its − iN/Ts , ξs = ξs + iN(µMs /Ts) + MLξWe had superfluid in Schr soliton in the probe limit by introducing boson hair.However, in Schr case, so far we cannot consider Hawking-Page or Witten’sconfined-deconfined picture as in AdS. AdS5 with a compact y direction for 2+1-dimCMT has been studied by Nishioka, Ryu, Takayanagi, arXiv 0911.0962.

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Conclusion

(4) How to shrink an extra-dim ξ circle?(i) Witten’s AdS soliton - Do NOT work. Hawking-Page transition.(ii) Schrodinger soliton - shrink ξ cigar? Do NOT work. Double Wickrotate: (τb, yb)→ i(ys , τs), (tb, ξb)→ (ts , ξs), (βb,Ωb)→ −i(βs ,Ωs)Periodic identification for (τb, ξb), (ys , ξs) shows different ensemble system:Schr BH: itb = itb + N/T , ξb = ξb + N(µM/T ) + MLξSchr soliton: its = its − iN/Ts , ξs = ξs + iN(µMs /Ts) + MLξ

We had superfluid in Schr soliton in the probe limit by introducing boson hair.However, in Schr case, so far we cannot consider Hawking-Page or Witten’sconfined-deconfined picture as in AdS. AdS5 with a compact y direction for 2+1-dimCMT has been studied by Nishioka, Ryu, Takayanagi, arXiv 0911.0962.

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Conclusion

(4) How to shrink an extra-dim ξ circle?(i) Witten’s AdS soliton - Do NOT work. Hawking-Page transition.(ii) Schrodinger soliton - shrink ξ cigar? Do NOT work. Double Wickrotate: (τb, yb)→ i(ys , τs), (tb, ξb)→ (ts , ξs), (βb,Ωb)→ −i(βs ,Ωs)Periodic identification for (τb, ξb), (ys , ξs) shows different ensemble system:Schr BH: itb = itb + N/T , ξb = ξb + N(µM/T ) + MLξSchr soliton: its = its − iN/Ts , ξs = ξs + iN(µMs /Ts) + MLξWe had superfluid in Schr soliton in the probe limit by introducing boson hair.However, in Schr case, so far we cannot consider Hawking-Page or Witten’sconfined-deconfined picture as in AdS. AdS5 with a compact y direction for 2+1-dimCMT has been studied by Nishioka, Ryu, Takayanagi, arXiv 0911.0962.

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Conclusion

(4) How to shrink an extra-dim ξ circle?(i) Witten’s AdS soliton - Do NOT work. Hawking-Page transition.(ii) Schrodinger soliton - Do NOT work.

(iii) More fancier method: (a) Lin, Lunin, Juan Maldacena, arXiv hep-th:0409174 (one of two sphere can be shrink to zero smoothly on the edgeof the bubble). (b) Klebanov-Strassler, hep-th/0007191 (shrinks Sp withp > 1 and a remaining Sq).(iv) some preliminary results by numerical method. How about analyticsolution?(v) Future directions: (a) lift to 10-d string or 11-d M theory, (b) studyfree energy of analytic solution(Lif, Schr, Lif-Schr) and phases of vev. (c)Better RG picture between these critical points?

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Conclusion

(4) How to shrink an extra-dim ξ circle?(i) Witten’s AdS soliton - Do NOT work. Hawking-Page transition.(ii) Schrodinger soliton - Do NOT work.(iii) More fancier method: (a) Lin, Lunin, Juan Maldacena, arXiv hep-th:0409174 (one of two sphere can be shrink to zero smoothly on the edgeof the bubble). (b) Klebanov-Strassler, hep-th/0007191 (shrinks Sp withp > 1 and a remaining Sq).

(iv) some preliminary results by numerical method. How about analyticsolution?(v) Future directions: (a) lift to 10-d string or 11-d M theory, (b) studyfree energy of analytic solution(Lif, Schr, Lif-Schr) and phases of vev. (c)Better RG picture between these critical points?

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Conclusion

(4) How to shrink an extra-dim ξ circle?(i) Witten’s AdS soliton - Do NOT work. Hawking-Page transition.(ii) Schrodinger soliton - Do NOT work.(iii) More fancier method: (a) Lin, Lunin, Juan Maldacena, arXiv hep-th:0409174 (one of two sphere can be shrink to zero smoothly on the edgeof the bubble). (b) Klebanov-Strassler, hep-th/0007191 (shrinks Sp withp > 1 and a remaining Sq).(iv) some preliminary results by numerical method. How about analyticsolution?

(v) Future directions: (a) lift to 10-d string or 11-d M theory, (b) studyfree energy of analytic solution(Lif, Schr, Lif-Schr) and phases of vev. (c)Better RG picture between these critical points?

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Conclusion

(4) How to shrink an extra-dim ξ circle?(i) Witten’s AdS soliton - Do NOT work. Hawking-Page transition.(ii) Schrodinger soliton - Do NOT work.(iii) More fancier method: (a) Lin, Lunin, Juan Maldacena, arXiv hep-th:0409174 (one of two sphere can be shrink to zero smoothly on the edgeof the bubble). (b) Klebanov-Strassler, hep-th/0007191 (shrinks Sp withp > 1 and a remaining Sq).(iv) some preliminary results by numerical method. How about analyticsolution?(v) Future directions: (a) lift to 10-d string or 11-d M theory, (b) studyfree energy of analytic solution(Lif, Schr, Lif-Schr) and phases of vev. (c)Better RG picture between these critical points?

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Conclusion

3. Use B-F theory to interpolate different asymptotic - AdS, Lif, Schr.

4. IR Lifd+2 has near horizon geometry non-AdS. Possible resolution forTsssT - to get correct free energy form T #µ# - a physical realization ofNR superfluid.

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Conclusion

3. Use B-F theory to interpolate different asymptotic - AdS, Lif, Schr.

4. IR Lifd+2 has near horizon geometry non-AdS. Possible resolution forTsssT - to get correct free energy form T #µ# - a physical realization ofNR superfluid.

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Conclusion

Conclusion= Four Take-Home Messages:

(1). Gravity dual of non-realtivistic conformal fieldtheory(NRCFT) can be useful description for strongly coupledmany-body quantum systems.

(2). Gravity dual’s Bosonic operators under NRCFTbackground shows superfluid, metal or insulator low energystates.

(3). Gravity dual’s Fermionic operators under NRCFTbackground shows Fermi surfaces(metalic), or Fermi surfacescollapses(insulator) low energy states.

(4). Use Gravitational B-F theory to formulate gravity duals ofCFT, NRCFTand Lifshitz field theory.

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Conclusion

〈O〉 v.s. T :

TMetalSuperfluid

Tc

〈O〉 v.s. Ω:

WMetalSuperfluid

W*

〈O〉 v.s. µQ :

Μ*

〈O〉 v.s. Ω:

W

〈O〉 v.s. µQ :

ΜQSuperfluidInsulator

Μc

Fermi surface from Schr BH

〈O〉 v.s. β:

ΒMetalInsulator

Β*

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Conclusion

Fermions in charged Schr BH

The parameters of phase space: ∆,T , µQ ,M, β,conformal dimension, temperature, charge density, Number(Mass),background density.

Bosons in Schr BHThe parameters of phase space: ∆,T , µQ ,M,Ωconformal dimension, temperature, charge density, Number(Mass),background density.

Bosons in Schr solitonThe parameters of phase space: ∆,mG , µQ ,M,Ωconformal dimension, mass gap(∼ 1/Lξ), charge density, Number(Mass),background density.

asymptotics AdSd+2 Schrd+3

scalar conformal dim ∆± = d+12±q

( d+12

)2 + m2 ∆± = d+22±q

( d+22

)2 + m2 + (`− qMo )2

spinor conformal dim ∆± = d+12± m ∆± = d+2

2±q

((m ± 12

)2 + (`− qMo )2