non-relativistic holographic quantum...

Galilean HolographySuperfluid

Fermi surfaceGeneralized B-F theory and R-G critical points

Non-Relativistic HolographicQuantum Liquids

Juven Wang (MIT)

Feb 29, 2012 @ APS March 2012

Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids



Outline

Galilean Holography

Superfluid

Fermi surface

Generalized B-F theory and R-G critical points




Work based on:

(1) non-relativistic Superfluids

- arXiv: 1103.3472, New J. Phys. 13, 115008 (2011),A Adams, JW.

(2) non-relativistic Fermi surface- to appear, JW, et al.

(3) Gravitational B-F theory on RG critical points

- to appear, JW.




I. Galilean Holography

1. Maldacena’s conjecture on AdS/CFT may be a tip of theiceberg - gauge-gravity duality.

Hint: (i) matching of symmetries(ii) matching of parameters(strong-weak couplings duality) (iii) Partition function andfield-operator correspondence

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

(a) (b) (c)Hawking, Bekenstein, Unruh, . . . . Thermodynamics of blackhole.





1. Maldacena’s conjecture on AdS/CFT may be a tip of theiceberg - gauge-gravity duality.

Hint: (i) matching of symmetries(ii) matching of parameters(strong-weak couplings duality) (iii) Partition function andfield-operator correspondence

Holography

Ex: Bulk side Dictionary Boundary side

Hologram 3D object Fourier Trans 2D image

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

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





2. Non-Relativistic Conformal Field Theory (Boundary - Field Theory)

NRCFT satisfies Schrodinger group(algebra of transformation invariance of freeSchrodinger equations). Operators N, D, Mij , Ki , Pi , C , H(Number, dilation,rotation, Galilean boost, translation, special conformal, Hamiltonian)

[A,B] Pj Kj D C HPi 0 −iδijN −iPi −iKi 0Ki iδijN 0 i(z − 1)Ki 0 iPi

D iPj (1− z)iKj 0 −2iC 2iHC iKj 0 2iC 0 iDH 0 −iPj −2iH −iD 0

[Mij , Mkl ] = i(δikMjk − δjkMil + δilMkj − δjlMki ),

[Mij , Kk ] = i(δikKj − δjkKi ), [Mij , Pk ] = i(δikPj − δjkPi ),

[Mij , C ] = [Mij , D] = [Mij , H] = 0,

[D,N] = i(2− z)N. Nishida&Son 2007

We focus on z = 2, in 2+1-dim boundary field theory.





3. Schrodinger spacetime(Schr) (Bulk - Gravity Theory)

Gravity dual of NRCFT. Son, Balasubramanian&McGreevy 2008

4. Galilean Holography(Schr/NRCFT), realizing Schrodinger group

by isometry.

−∂2τ + ~∂2 → −2∂t∂ξ + ~∂2 → −2i`∂t + ~∂2 , with Φ = φe i`ξ .

Field Theory Gravity Dual

Free (d+1)-dim Schrodinger eq Schrd+3 metric

Free (d+2)-dim Klein-Gordon eq AdSd+3 metric

Compactifying a light cone direction ξ. Embed Schr into K-G’s CFT group.

Pure AdS metric: ds2 = −r−2dτ 2 + r−2(dy 2 + d~x2 + dr 2)

Pure Schr metric: ds2 = −r−2zdt2 + r−2(2dtdξ + d~x2 + dr 2)

5. Dictionary: Partition function and field-operator correspondence

ZCFT [φ] = Zstring

ˆΦ|∂AdS

˜' e−Ssupergravity





3. Schrodinger spacetime(Schr) (Bulk - Gravity Theory)

Gravity dual of NRCFT. Son, Balasubramanian&McGreevy 2008

4. Galilean Holography(Schr/NRCFT), realizing Schrodinger group

by isometry. −∂2τ + ~∂2 → −2∂t∂ξ + ~∂2 → −2i`∂t + ~∂2 , with Φ = φe i`ξ .

Field Theory Gravity Dual

Free (d+1)-dim Schrodinger eq Schrd+3 metric

Free (d+2)-dim Klein-Gordon eq AdSd+3 metric

Compactifying a light cone direction ξ. Embed Schr into K-G’s CFT group.

Pure AdS metric: ds2 = −r−2dτ 2 + r−2(dy 2 + d~x2 + dr 2)

Pure Schr metric: ds2 = −r−2zdt2 + r−2(2dtdξ + d~x2 + dr 2)

5. Dictionary: Partition function and field-operator correspondence

ZCFT [φ] = Zstring

ˆΦ|∂AdS

˜' e−Ssupergravity




Superfluid from Gravity Dual of Boson Operators in NRCFT

Probe limit: Abelian Higgs model under Finite Temperature Schr metrics.

Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2)

,

Number(Mass) Operator : N = `− qMo , Φ = φe i`ξ

In Gravity Dual , gauge invariant momentum of compact(extra-)dimension ξ as dual to Number operator.φ = φ1r

∆− + φ2r∆+ + . . . ,

with conformal dimension ∆± = 2±√

4 + m2 + N2.At = µQ + ρQ r2 + . . . , Aξ = Mo + ρM r2 + . . . ,

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

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

ω〈Ax〉 = −i A2

ωA0

Ax = A0 + A2r2

2 + . . . .






Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2)

,


In Gravity Dual , gauge invariant momentum of compact(extra-)dimension ξ as dual to Number operator.

φ = φ1r∆− + φ2r

∆+ + . . . ,with conformal dimension ∆± = 2±

√4 + m2 + N2.

At = µQ + ρQ r2 + . . . , Aξ = Mo + ρM r2 + . . . ,



ω〈Ax〉 = −i A2

ωA0

Ax = A0 + A2r2

2 + . . . .






Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2)

,



∆− + φ2r∆+ + . . . ,





ω〈Ax〉 = −i A2

ωA0

Ax = A0 + A2r2

2 + . . . .






Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2)

,



∆− + φ2r∆+ + . . . ,



Dictionary:

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


ω〈Ax〉 = −i A2

ωA0

Ax = A0 + A2r2

2 + . . . .






Sprobe,AH =∫

d5x√−gEin

1e2

(− 1

4F 2 − |DΦ|2 −m2|Φ|2)

,



∆− + φ2r∆+ + . . . ,





ω〈Ax〉 = −i A2

ωA0

Ax = A0 + A2r2

2 + . . . .




II. Superfluids

Order Parameter 〈O〉 and Conductivity σ(ω) under Temp T , density Ω,

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




II. Superfluids

Order Parameter 〈O〉 under Temp T , density Ω,

〈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(near T=0) by tuning Ω




II. Superfluids

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

0.192

0.191

316

18

0

ΜQ

XO1HTcL\2nd order phase transition

XO1HT*L\ 1st order phase transition

Low T and High T condensates - compare free energy:

FC −FN = −T∫ C

NδSE

VD= −T (∆1 −∆2)

∫ C

Nφ2 dφ1.

Near the multicritical point shows Mean-Field theory behavior.Landau-Ginzburg free energy can be:

F (ϕ) = 12c2(T − Tc(µQ))ϕ2 + 1

4c4(µ∗ − µQ)ϕ4 + 16c6ϕ

6.




II. Superfluids

Order Parameter 〈O〉 under Temp T , density Ω, chemical potential µQ

〈O〉 v.s. µQ :

ΜQ1st order PT2nd order PT

Μ*

Multicritical Point from 2nd to 1st order phase transitions.

〈O〉 v.s. T :

TMetalSuperfluid

Tc

Finite T mean-field phase transition(w/ βMF = 1/2)

〈O〉 v.s. Ω:

WMetalSuperfluid

W*

Quantum phase transition(near T=0)




Fermi surface from Gravity Dual of Fermion Operators in NRCFT

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

∫d5x√−gEini ψ(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[∫

dd+1x(χO + Oχ)]〉QFT

χ ∝ ψ as source, O ∝ Π as response .

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




III. Fermi 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

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

2.0

k

Ω

quasi-particle like peak, Particle-Hole asymmetry ,

compare to Landau Fermi Liquid & Senthil’s ansatz.Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids



Quantum Phase Transition & Fermi Surface disappearance:

〈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




Fermi surface from Schr BH

〈O〉 v.s. β:Β

MetalInsulatorΒ*

Superfluids from Schr BH

〈O〉 v.s. T :T

MetalSuperfluidTc

〈O〉 v.s. Ω:W

MetalSuperfluidW*

〈O〉 v.s. µQ :ΜQ

1st order PT2nd order PTΜ*




IV. Gravitational 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) let Fd+3−p ≡ ∗dφp−1 + . . . ,− 1

2 |F |2 + λ

∫B ∧ F = − 1

2 |λB − dφ|2,. . . makes the gauge transf valid, and gauge choice fixes dφp−1 = 0.

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

Massive Field Theory

S =

∫dd+3x

√−g(R − 2Λ− 1

2|Hp+1|2 −

1

2λ2|Bp|2)




IV. Gravitational B-F theory

finite T finite density Schr BH spacetime for(a) ∀ d-dim, z = 2 and (Kovtun&Nickel, PRL 2008)

S =

∫dd+3x

√−g(R − a

2(∂µφ)(∂µφ)− 1

4e−aφ|Fµν |2 −

m2

2AµA

µ − V (φ))

(b) d = 2z − 4-dim, ∀ z (to appear - JW)

S =

∫dd+3x

√−ge−2ϕ(R − 2Λ− 1

2|Hz |2)− 1

2|Fz |2 + λ

∫Bz−1 ∧ Fz

Ex: Bulk side Dictionary Boundary side

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

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

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




Conclusion:

I. Galilean Holography can be useful to studynon-relativistic strong interacting systems (fieldtheories or condensed matter).

II. Supfluids and its phase transition as gravity dualof Boson operators in NRCFT.

III. Fermi surface and its disappearance as gravitydual of Fermion operators in NRCFT.

IV. Gravitational B-F theory extends to more andnew solutions with AdS/CFT, Schr/NRCFT andLif/LFT.




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

THANK YOU FOR YOUR ATTENTION.


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