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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
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
Outline
Galilean Holography
Superfluid
Fermi surface
Generalized B-F theory and R-G critical points
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized 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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
I. Galilean Holography
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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
I. Galilean Holography
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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
I. Galilean Holography
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
I. Galilean Holography
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
I. Galilean Holography
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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 + . . . .
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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 + . . . .
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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 + . . . .
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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 + . . . .
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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 + . . . .
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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 + . . . .
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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)
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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*
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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)
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids
Galilean HolographySuperfluid
Fermi surfaceGeneralized B-F theory and R-G critical points
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.
Juven Wang (MIT) Non-Relativistic Holographic Quantum Liquids