JuvenWang(MIT)Jan13,2012@NatlTaiwanUniv HolographicViewonnon-relativisticSuperfluids,FermiSurfacesandRGfixedpoints

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Holographic View on non-relativistic Superfluids, Fermi Surfaces and RG fixed points

Juven Wang (MIT)

Jan 13, 2012 @ Natl Taiwan Univ

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Introduction Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Boson Operators in Schr/NRCFT Setup & Dictionary

Superfluids from Schr BH Superfluids from Schr soliton Fermion Operators in Schr/NRCFT

Setup & Dictionary Fermi Surface

Landau Fermi Liquid & Senthil’s ansatz

Quantum Phase Transition & Fermi Surface disappearance B-F theory formalism of RG critical points

Known Solution

B-F theory and New Solution New Ideas

Conclusion

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical 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, . . . .

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

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

Gravity Quantum

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

(2). Enhance understandings on both sides

gravity, string theory ⇔ quantum many-body systems QFT

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

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

Gravity Quantum

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

(2). Enhance understandings on both sides

gravity, string theory ⇔ quantum many-body systems QFT

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

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

Gravity Quantum

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

(2). Enhance understandings on both sides

gravity, string theory ⇔ quantum many-body systems QFT

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

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

Gravity Quantum

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

(2). Enhance understandings on both sides

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Four Take-Home Messages:

(1). Gravity dual of non-realtivistic conformal field

theory(NRCFT) can be useful description for strongly coupled many-body quantum systems.

(2). Gravity dual’s Bosonic operators under NRCFT

background shows superfluid, metal or insulator low energy states.

(3). Gravity dual’s Fermionic operators under NRCFT

background shows Fermi surfaces(metalic), or Fermi surfaces collapses(insulator) low energy states.

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

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ? Ex 1: Hologram

3D ⇔Fourier Trans⇔2D

Ex: Bulk side Dictionary Boundary side Hologram 3D object Fourier Trans 2D image

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 2: AdS/CFT

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

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

(b)Thermodynamics(Statistical Mech.) and(c)Quantum(Information).

(a) (b) (c)

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

(b)Thermodynamics(Statistical Mech.) and(c)Quantum(Information).

(a) (b) (c)

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

(b)Thermodynamics(Statistical Mech.) and(c)Quantum(Information).

(a) ↔ (b):

(i)black hole thermodynamics: Hawking (T_H= 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-Einstein Condensation.

(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 Schr¨odinger eq from stochastic Brownian motion w/ friction.

(c) ↔ (a):

(i)stringtheory,loop quantum gravity, (ii)emergent graviton, (iii) Levin-Wenstring-netandQuantum Graphity.

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

(b)Thermodynamics(Statistical Mech.) and(c)Quantum(Information).

(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-Einstein Condensation.

(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 Schr¨odinger eq from stochastic Brownian motion w/ friction.

(c) ↔ (a):

(i)stringtheory,loop quantum gravity, (ii)emergent graviton, (iii) Levin-Wenstring-netandQuantum Graphity.

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

(b)Thermodynamics(Statistical Mech.) and(c)Quantum(Information).

(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-Einstein Condensation.

(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 Schr¨odinger eq from stochastic Brownian motion w/ friction.

(c) ↔ (a):

(i)stringtheory,loop quantum gravity, (ii)emergent graviton, (iii) Levin-Wenstring-netandQuantum Graphity.

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

(b)Thermodynamics(Statistical Mech.) and(c)Quantum(Information).

(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-Einstein Condensation.

(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 Schr¨odinger eq from stochastic Brownian motion w/ friction.

(c) ↔ (a):

(i)stringtheory,loop quantum gravity, (ii)emergent graviton, (iii) Levin-Wenstring-netandQuantum Graphity.

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 2: AdS/CFT

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

AdS:

(i) Poincare patches: ds²= L² − r⁻²dt²+ r⁻²(dr²+ d~x²)
(ii) Hyperboloid submanifold: ds²= −dt²+P d~x²with

−t²+P ~x²= −α²constraint.

CFT:

(i) ex: Massless Klein-Gordon QFT:R d^dxdt ¹₂∂_µφ∂^µφ

(ii) conformal symmetry: Lorentz group M_µν, time translation H, space

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS₅× S⁵ gravity andN = 4SU(N_c) supersymmetric Yang-Mills(SYM).

hint 1 : matching of symmetries.

N = 4 SYMis invariance underconf (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 translation momentum Pi);

Lorentz group=Rotation(Mij)+Lorentz boost(M0j), together Mµν.

SO(6) is from R symmetry.

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS₅× S⁵ gravity andN = 4SU(N_c) supersymmetric Yang-Mills(SYM).

hint 1 : matching of symmetries.

N = 4 SYMis invariance underconf (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 translation momentum Pi);

Lorentz group=Rotation(Mij)+Lorentz boost(M0j), together Mµν.

SO(6) is from R symmetry.

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS₅× S⁵ gravity andN = 4SU(N_c) supersymmetric Yang-Mills(SYM).

hint 1 : matching of symmetries.

N = 4 SYMis invariance underconf (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 translation momentum Pi);

Lorentz group=Rotation(Mij)+Lorentz boost(M0j), together Mµν.

SO(6) is from R symmetry.

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS₅× S⁵ gravity andN = 4SU(N_c) supersymmetric Yang-Mills(SYM).

hint 1 : matching of symmetries.

N = 4 SYMis invariance underconf (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 translation momentum Pi);

Lorentz group=Rotation(Mij)+Lorentz boost(M0j), together Mµν.

SO(6) is from R symmetry.

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS₅× S⁵ gravity andN = 4SU(N_c) supersymmetric Yang-Mills(SYM).

hint 1 : matching of symmetries.

N = 4 SYMis invariance underconf (1, 3) × SO(6).

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

AdS₅× S⁵has diffeomorphism isometry groupSO(2, 4) × SO(6). conf (1, 3) × SO(6)isomorphic toSO(2, 4) × SO(6).

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS₅× S⁵ gravity andN = 4SU(N_c) supersymmetric Yang-Mills(SYM).

hint 1 : matching of symmetries.

N = 4 SYMis invariance underconf (1, 3) × SO(6).

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

AdS₅× S⁵has diffeomorphism isometry groupSO(2, 4) × SO(6).

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS₅× S⁵ gravity andN = 4SU(N_c) supersymmetric Yang-Mills(SYM).

hint 1 : matching of symmetries.

N = 4 SYMis invariance underconf (1, 3) × SO(6).

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

AdS₅× S⁵has diffeomorphism isometry groupSO(2, 4) × SO(6).

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

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS5× S⁵gravity andN = 4SU(Nc) supersymmetricYang-Mills(SYM) hint 2 : Maldacena conjecture AdS/CFT correspondence

AdS₅× S⁵type IIB string theory

⇔

gravity and string theory

⇔

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS₅× S⁵gravity andN = 4supersymmetricYang-Mills(SYM)theory.

hint 3 : matching of parameters strong-weak couplings duality

R² α⁰ ∼√

g_sN_c ∼√

λ, g_s ∼ g_YM² ∼_N^λ

c, ^R_`4⁴

p ∼ ^√R4

G ∼ Nc

hint 4 : Partition function and field operator correspondence ZCFT[φ] = Zstring[ Φ|_∂AdS] ' e^−Ssupergravity.

S → S +R d⁴x φ(x ) · O(x ) , (source · response) φ = Φ|_∂AdS (operator-field)

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

(continue)

Ex 2: AdS/CFT

The best-undertood story is:

AdS₅× S⁵gravity andN = 4supersymmetricYang-Mills(SYM)theory.

hint 3 : matching of parameters strong-weak couplings duality

R² α⁰ ∼√

g_sN_c ∼√

λ, g_s ∼ g_YM² ∼_N^λ

c, ^R_`4⁴

p ∼ ^√R4

G ∼ Nc

hint 4 : Partition function and field operator correspondence ZCFT[φ] = Zstring[ Φ|_∂AdS] ' e^−Ssupergravity.

S → S +R d⁴x φ(x ) · O(x ) , (source · response) φ = Φ|_∂AdS (operator-field)

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 2: AdS/CFT

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

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

hint 2 : Maldacena conjecture AdS/CFT correspondence hint 3 : matching of parameters strong-weak couplings duality hint 4 : Partition function and field operator correspondence

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 2: AdS/CFT

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 1997 Maldacena conjecture, Gubser, Klebanov&Polyakov and Witten

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

hint 2 : Maldacena conjecture AdS/CFT correspondence hint 3 : matching of parameters strong-weak couplings duality hint 4 : Partition function and field operator correspondence

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 3: Schr/NRCFT

NRCFT⁰s Schr group isomorphic to isometry of Schr space . LHS:Schr¨odinger 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 P_i) scaling(dilatation) (D)

special conformal operator (C )

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 3: Schr/NRCFT

NRCFT⁰s Schr group isomorphic to isometry of Schr space .

LHS:Schr¨odinger 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 P_i) scaling(dilatation) (D)

special conformal operator (C )

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 3: Schr/NRCFT

NRCFT⁰s Schr group isomorphic to isometry of Schr space . LHS:Schr¨odinger 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 P_i) scaling(dilatation) (D)

special conformal operator (C )

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 3: Schr/NRCFT

NRCFT⁰s Schr group isomorphic to isometry of Schr space . LHS:Schr¨odinger 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 P_i) scaling(dilatation) (D)

special conformal operator (C )

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 3: Schr/NRCFT

NRCFT⁰s Schr group isomorphic to isometry of Schr space . LHS:Schr¨odinger group=Galilean group + translation + scaling(dilatation) operator+special conformal operator.

RHS: Embed Schr group in a higher dimensional Conformal group. FreeSchr¨odingereq inside FreeKlein-Gordoneq .

Light-cone coordinate t, ξ. Compactify ξ to give discrete mass tower. answer ds²= −r^−2zdt²+ r⁻²(2dtd ξ + d~x²+ dr²)

(Son, Balasubramanian&McGreevy)

Ex: Bulk side Dictionary Boundary side

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

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 3: Schr/NRCFT

NRCFT⁰s Schr group isomorphic to isometry of Schr space . LHS:Schr¨odinger group=Galilean group + translation + scaling(dilatation) operator+special conformal operator.

RHS: Embed Schr group in a higher dimensional Conformal group.

FreeSchr¨odingereq inside FreeKlein-Gordoneq .

Light-cone coordinate t, ξ. Compactify ξ to give discrete mass tower.

answer ds²= −r^−2zdt²+ r⁻²(2dtd ξ + d~x²+ dr²)

(Son, Balasubramanian&McGreevy)

Ex: Bulk side Dictionary Boundary side

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 3: Schr/NRCFT

NRCFT⁰s Schr group isomorphic to isometry of Schr space . LHS:Schr¨odinger group=Galilean group + translation + scaling(dilatation) operator+special conformal operator.

RHS: Embed Schr group in a higher dimensional Conformal group.

FreeSchr¨odingereq inside FreeKlein-Gordoneq .

Light-cone coordinate t, ξ. Compactify ξ to give discrete mass tower.

answer ds²= −r^−2zdt²+ r⁻²(2dtd ξ + d~x²+ dr²)

(Son, Balasubramanian&McGreevy)

Ex: Bulk side Dictionary Boundary side

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

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What is HOLOGRAPHY ?

Ex 3: Schr/NRCFT

NRCFT⁰s Schr group isomorphic to isometry of Schr space . LHS:Schr¨odinger group=Galilean group + translation + scaling(dilatation) operator+special conformal operator.

RHS: Embed Schr group in a higher dimensional Conformal group.

FreeSchr¨odingereq inside FreeKlein-Gordoneq .

Light-cone coordinate t, ξ. Compactify ξ to give discrete mass tower.

answer ds²= −r^−2zdt²+ r⁻²(2dtd ξ + d~x²+ dr²)

(Son, Balasubramanian&McGreevy)

Ex: Bulk side Dictionary Boundary side

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

What isHOLOGRAPHY ? Ex 1,2,3:

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) Holographic View on non-relativistic Superfluids, Fermi Surfaces and RG fixed points

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Schr/NRCFT correspondence

Asymptotic Schr¨ odinger Spacetime

pure Schr spacetime

ds²= −r^−2zdt²+ r⁻²(2dtd ξ + d~x²+ dr²)

Thepure Schr spacetimeprovides no temperature for boundary field theory, howeverasymptotic Schrwith black hole(BH) does.

Neutral Schr BHwith finite density (0 < r < r_H):

By TsssT (Null Melvin twist) on neutral black D3branes of type IIB string. ds_Ein²

= K^1/3

“` − f +_{4(K −1)}^{(f −1)2}´_dt²

Kr⁴ +^1+f_r2Kdt d ξ +^{K −1}_K d ξ²+^d~_r^x2²+_{f r}^dr22

” . (arXiv: 0807.1099, 0807.1100, 0807.1111)

Charged Schr BHwith finite density (rH < r < ∞):

By TsssT (Null Melvin twist) on charge black D3branes of type IIB string. ds_Ein²

= K^{−1/3 Kr}_R2²

„“

1−f 4β²−r²f”

dt²+β²(1−f )d ξ²+(1+f )dtd ξ

«

+_R^r2₂(dx₁²+dx₂²)+^R_r2^2dr_f²

!

(arXiv: 0907.1892, 0907.1920)

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Schr/NRCFT correspondence

Asymptotic Schr¨ odinger Spacetime

pure Schr spacetime

ds²= −r^−2zdt²+ r⁻²(2dtd ξ + d~x²+ dr²)

Thepure Schr spacetimeprovides no temperature for boundary field theory, howeverasymptotic Schrwith black hole(BH) does.

Neutral Schr BHwith finite density (0 < r < r_H):

By TsssT (Null Melvin twist) on neutral black D3branes of type IIB string. ds_Ein²

= K^1/3

“` − f +_{4(K −1)}^{(f −1)2}´_dt²

Kr⁴ +^1+f_r2Kdt d ξ +^{K −1}_K d ξ²+^d~_r^x2²+_{f r}^dr22

” . (arXiv: 0807.1099, 0807.1100, 0807.1111)

Charged Schr BHwith finite density (rH < r < ∞):

By TsssT (Null Melvin twist) on charge black D3branes of type IIB string. ds_Ein²

= K^{−1/3 Kr}_R2²

„“

1−f 4β²−r²f”

dt²+β²(1−f )d ξ²+(1+f )dtd ξ

«

+_R^r2₂(dx₁²+dx₂²)+^R_r2^2dr_f²

!

(arXiv: 0907.1892, 0907.1920)

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Schr/NRCFT correspondence

Asymptotic Schr¨ odinger Spacetime

pure Schr spacetime

ds²= −r^−2zdt²+ r⁻²(2dtd ξ + d~x²+ dr²)

Thepure Schr spacetimeprovides no temperature for boundary field theory, howeverasymptotic Schrwith black hole(BH) does.

Neutral Schr BHwith finite density (0 < r < r_H):

By TsssT (Null Melvin twist) on neutral black D3branes of type IIB string.

ds_Ein²

= K^1/3

“` − f +_{4(K −1)}^{(f −1)2}´_dt²

Kr⁴ +^1+f_r2Kdt d ξ +^{K −1}_K d ξ²+^d~_r^x2²+^dr_{f r}²2

” . (arXiv: 0807.1099, 0807.1100, 0807.1111)

Charged Schr BHwith finite density (rH < r < ∞):

By TsssT (Null Melvin twist) on charge black D3branes of type IIB string. ds_Ein²

= K^{−1/3 Kr}_R2²

„“

1−f 4β²−r²f”

dt²+β²(1−f )d ξ²+(1+f )dtd ξ

«

+_R^r2₂(dx₁²+dx₂²)+^R_r2^2dr_f²

!

(arXiv: 0907.1892, 0907.1920)

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Schr/NRCFT correspondence

Asymptotic Schr¨ odinger Spacetime

pure Schr spacetime

ds²= −r^−2zdt²+ r⁻²(2dtd ξ + d~x²+ dr²)

Thepure Schr spacetimeprovides no temperature for boundary field theory, howeverasymptotic Schrwith black hole(BH) does.

Neutral Schr BHwith finite density (0 < r < r_H):

By TsssT (Null Melvin twist) on neutral black D3branes of type IIB string.

ds_Ein²

= K^1/3

“` − f +_{4(K −1)}^{(f −1)2}´_dt²

Kr⁴ +^1+f_r2Kdt d ξ +^{K −1}_K d ξ²+^d~_r^x2²+^dr_{f r}²2

” . (arXiv: 0807.1099, 0807.1100, 0807.1111)

Charged Schr BHwith finite density (rH < r < ∞):

By TsssT (Null Melvin twist) on charge black D3branes of type IIB string.

ds_Ein²

= K^{−1/3 Kr}_R2²

„“

1−f 4β²−r²f”

dt²+β²(1−f )d ξ²+(1+f )dtd ξ

«

+_R^r2₂(dx₁²+dx₂²)+^R_r2^2dr_f²

!

(arXiv: 0907.1892, 0907.1920)

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Asymptotic Schr¨odinger Spacetime pure Schr spacetime

ds²= −r^−2zdt²+ r⁻²(2dtd ξ + d~x²+ dr²)

Thepure Schr spacetimeprovides no temperature for boundary field theory, howeverasymptotic Schrwith black hole(BH) does.

Neutral Schr BHwith finite density:

By TsssT (Null Melvin twist) on neutral black D3branes of type IIB string.

Charged Schr BHwith finite density:

By TsssT (Null Melvin twist) on charge black D3branes of type IIB string.

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Partition Function

Partition function and field operator correspondence

ZCFT[φ] = Zstring[ Φ|_∂AdS] ' e^−Ssupergravity. S → S +R d⁴x φ(x ) · O(x ) , (source · response) φ = Φ|_∂AdS (operator-field)

S → S +R d⁴x A(x ) · J (x ) , (source · response) Aµ= Aµ|_∂AdS (operator-field)

BosonOperators inNRCFT: scalar fieldin Schr

FermionOperators inNRCFT Dirac spinorfield inSchr

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Partition Function

Partition function and field operator correspondence ZCFT[φ] = Zstring[ Φ|_∂AdS] ' e^−Ssupergravity.

S → S +R d⁴x φ(x ) · O(x ) , (source · response) φ = Φ|_∂AdS (operator-field)

S → S +R d⁴x A(x ) · J (x ) , (source · response) Aµ= Aµ|_∂AdS (operator-field)

BosonOperators inNRCFT: scalar fieldin Schr

FermionOperators inNRCFT Dirac spinorfield inSchr

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Partition Function

Partition function and field operator correspondence ZCFT[φ] = Zstring[ Φ|_∂AdS] ' e^−Ssupergravity.

S → S +R d⁴x φ(x ) · O(x ) , (source · response) φ = Φ|_∂AdS (operator-field)

S → S +R d⁴x A(x ) · J (x ) , (source · response) Aµ= Aµ|_∂AdS (operator-field)

BosonOperators inNRCFT: scalar fieldin Schr

FermionOperators inNRCFT Dirac spinorfield inSchr

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Partition Function

Partition function and field operator correspondence ZCFT[φ] = Zstring[ Φ|_∂AdS] ' e^−Ssupergravity.

S → S +R d⁴x φ(x ) · O(x ) , (source · response) φ = Φ|_∂AdS (operator-field)

S → S +R d⁴x A(x ) · J (x ) , (source · response) Aµ = Aµ|_∂AdS (operator-field)

BosonOperators inNRCFT: scalar fieldin Schr

FermionOperators inNRCFT Dirac spinorfield inSchr

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Partition Function

Partition function and field operator correspondence ZCFT[φ] = Zstring[ Φ|_∂AdS] ' e^−Ssupergravity.

S → S +R d⁴x φ(x ) · O(x ) , (source · response) φ = Φ|_∂AdS (operator-field)

S → S +R d⁴x A(x ) · J (x ) , (source · response) Aµ = Aµ|_∂AdS (operator-field)

BosonOperators inNRCFT:

scalar fieldin Schr

FermionOperators inNRCFT Dirac spinorfield inSchr

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Partition Function

Partition function and field operator correspondence ZCFT[φ] = Zstring[ Φ|_∂AdS] ' e^−Ssupergravity.

S → S +R d⁴x φ(x ) · O(x ) , (source · response) φ = Φ|_∂AdS (operator-field)

S → S +R d⁴x A(x ) · J (x ) , (source · response) Aµ = Aµ|_∂AdS (operator-field)

BosonOperators inNRCFT:

scalar fieldin Schr

FermionOperators inNRCFT Dirac spinorfield inSchr

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Ex 1: Hologram Ex 2: AdS/CFT Ex 3: Schr/NRCFT

Asymptotic Schr¨odinger Spacetime Partition Function

Summary So Far:

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

Bulk side Dictionary Boundary side Boson scalar field field-operator Boson operator Fermion Dirac spinor field field-operator Fermion operator

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

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Setup & Dictionary Superfluids from Schr BH Superfluids from Schr soliton

(I) Boson Operators in Schr/NRCFT

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

Setup:

S

= R d

x √

−g

−

F

− |DΦ|

− m

|Φ|

,

φ = φ1r^∆−+ φ2r^∆++ . . . ,

with conformal dimension ∆_±= 2 ±p4 + m²+ q²M_o². At= µQ+ ρQr²+ . . . ,

Aξ= Mo+ ρMr²+ . . . , A_x= A₀+ A₂^r₂² + . . . . Dictionary:

φ = Φ|_∂AdS (operator-field correpondence).

Spontaneous Symmetry Breaking: turn off source, only left withresponse. φ₁= 0, φ₂= hO₂i , or φ₂= 0, φ₁= hO₁i forsuperfluid.

Conductivity: σ(ω) = ^hJ_hE^xi

xi = −i_ωhA^hJxi

xi= −i_ωA^A2

0

Introduction Boson Operators in Schr/NRCFT Fermion Operators in Schr/NRCFT B-F theory formalism of RG critical points Conclusion

Setup & Dictionary Superfluids from Schr BH Superfluids from Schr soliton

(I) Boson Operators in Schr/NRCFT

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

Setup:

S

= R d

x √

−g

−

F

− |DΦ|

− m

|Φ|

,

φ = φ1r^∆−+ φ2r^∆++ . . . ,

with conformal dimension ∆_±= 2 ±p4 + m²+ q²M_o². At = µQ+ ρQr²+ . . . ,

Aξ = Mo+ ρMr²+ . . . , A_x = A₀+ A₂^r₂² + . . . .

Dictionary:

φ = Φ|_∂AdS (operator-field correpondence).

Spontaneous Symmetry Breaking: turn off source, only left withresponse. φ₁= 0, φ₂= hO₂i , or φ₂= 0, φ₁= hO₁i forsuperfluid.

Conductivity: σ(ω) = ^hJ_hE^xi

xi = −i_ωhA^hJxi

xi= −i_ωA^A2

0

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