F theory Some Solution

finite T finite density Schr BH spacetime for

(a) ∀ d -dim, z = 2 and (Kovtun&Nickel, 0809.2020, PRL) S =

Z

d^{d +3}x√

−g (R − a

2(∂_µφ)(∂^µφ) −1

4e^−aφ|F_µν|²−m²

2 A_µA^µ− V (φ) V (φ) = (Λ + Λ⁰)e^aφ+ (Λ − Λ⁰)e^bφ

ds²= r²K^{− dd +1}“

[_{4(K −1)}^{(f −1)2} − f ]r²dt²− (1 + f )dtdξ +^{K −1} r 2 d ξ²”

+ K 1 d +1“

r²dx²+^{dr 2} r 2 f

”

Free energy: F = −TlogZ = −TSE

F = −T_16πG⁻¹

d +3R d^{d +2}x 2r_H^−(d+2)= −T_16πG⁻¹

d +3

1

T∆ξV 2r_H^−(d+2)

=_G²

d +3∆ξπ^{d +1}2^3d2−1(d + 2)^−(d+2)T^{d +22} (^T_µ)^{d +22}

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

Known Solution

B-F theory and New Solution New Ideas

B-F theory Some Solution

finite T finite density Schr BH spacetime for

(a) ∀ d -dim, z = 2 and (Kovtun&Nickel, 0809.2020, PRL) S =

Z

d^{d +3}x√

−g (R − a

2(∂_µφ)(∂^µφ) −1

4e^−aφ|F_µν|²−m²

2 A_µA^µ− V (φ) V (φ) = (Λ + Λ⁰)e^aφ+ (Λ − Λ⁰)e^bφ

ds²= r²K^{− dd +1}“

[_{4(K −1)}^{(f −1)2} − f ]r²dt²− (1 + f )dtdξ +^{K −1} r 2 d ξ²”

+ K 1 d +1“

r²dx²+^{dr 2} r 2 f

”

Free energy: F = −TlogZ = −TSE

F = −T_16πG⁻¹

d +3R d^{d +2}x 2r_H^−(d+2)= −T_16πG⁻¹

d +3

1

T∆ξV 2r_H^−(d+2)

=_G²

d +3∆ξπ^{d +1}2^3d2−1(d + 2)^−(d+2)T^{d +22} (^T_µ)^{d +22}

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

Known Solution

B-F theory and New Solution New Ideas

B-F theory New Solution

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

S = Z

d^{d +3}x√

−g e^−2ϕ(R − 2Λ −1

2|Hz|²) −1

2|Fz|²+ λ Z

Bz−1∧ Fz ds_str² = ¹

Kr 2z(−f +_{4(K −1)}^{(1−f )2})dt²+^1+f

Kr 2dtd ξ +^{K −1}_K r^2(z−2)d ξ²+ ¹

Kr 2d~x(1,...,z−2)² + ¹

r 2d~x(z−1,...,2z−4)² +^{dr 2} r 2 f

Free energy: F = −TlogZ = −TS_E, with Ω_s≡ ^β2 rd +2

H

= ^β2 r2(z−1)

H

= (4πT )(d +2) (d +2)(d +2) (2|µ|)

d +4 2

F = ^2T

16πGd+3R d^{d +2}xr_H^{−(d+2)1−(z−2)Ωs} (1+Ωs )

z−1 2

= ²

Gd+3∆ξπ^{d +1}2^3d2−1

(d + 2)^−(d+2)T^{d +2}2 (^T_µ)^{d +2}2 1−(z−2)Ωs (1+Ωs )

z−1 2

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

Known Solution

B-F theory and New Solution New Ideas

B-F theory New Solution

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

S = Z

d^{d +3}x√

−g e^−2ϕ(R − 2Λ −1

2|Hz|²) −1

2|Fz|²+ λ Z

Bz−1∧ Fz ds_str² = ¹

Kr 2z(−f +_{4(K −1)}^{(1−f )2})dt²+^1+f

Kr 2dtd ξ +^{K −1}_K r^2(z−2)d ξ²+ ¹

Kr 2d~x(1,...,z−2)² + ¹

r 2d~x(z−1,...,2z−4)² +^{dr 2} r 2 f

Free energy: F = −TlogZ = −TS_E, with Ω_s≡ ^β2 rd +2

H

= ^β2 r2(z−1)

H

= (4πT )(d +2) (d +2)(d +2) (2|µ|)

d +4 2

F = ^2T

16πGd+3R d^{d +2}xr_H^{−(d+2)1−(z−2)Ωs} (1+Ωs )

z−1 2

= ²

Gd+3∆ξπ^{d +1}2^3d2−1

(d + 2)^−(d+2)T^{d +2}2 (^T_µ)^{d +2}2 1−(z−2)Ωs (1+Ωs )

z−1 2

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

Known Solution

B-F theory and New Solution New Ideas

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 T solutions for ∀z.

2. Free energy F (T , µ) has the unphysical form T^#(^T_µ)^#, instead of physical result T^#µ^#.

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

Known Solution

B-F theory and New Solution New Ideas

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 T solutions for ∀z.

2. Free energy F (T , µ) has the unphysical form T^#(^T_µ)^#, instead of physical result T^#µ^#.

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

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

2. A proposal on gravity dual solution realizingSuperfluid in NRCFT (not the probe limit): The UV theory isNRCFT(z = 2)but the IR theory isLifshitz field theory(z = 2).

Meanings:

(1) Short range behavior is NRCFT, sym of free Schr eq - gravity dual is Schr. 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 Lif_{d +2} UV Schr_{d +3}

(3)Break Number U(1) symmetry(assuperfluid)geometricallyby shrinking U(1) ξ circle, where −i ∂ξ corresponds to Number operator.

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

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

2. A proposal on gravity dual solution realizingSuperfluid in NRCFT (not the probe limit): The UV theory isNRCFT(z = 2)but the IR theory isLifshitz field theory(z = 2).

Meanings:

(1) Short range behavior is NRCFT, sym of free Schr eq - gravity dual is Schr. 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 Lif_{d +2} UV Schr_{d +3}

(3)Break Number U(1) symmetry(assuperfluid)geometricallyby shrinking U(1) ξ circle, where −i ∂ξ corresponds to Number 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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

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

2. A proposal on gravity dual solution realizingSuperfluid in NRCFT (not the probe limit): The UV theory isNRCFT(z = 2)but the IR theory isLifshitz field theory(z = 2).

Meanings:

(1) Short range behavior is NRCFT, sym of free Schr eq - gravity dual is Schr. 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 Lif_{d +2} UV Schr_{d +3}

(3)Break Number U(1) symmetry(assuperfluid)geometricallyby shrinking U(1) ξ circle, where −i ∂ξ corresponds to Number operator.

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

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

2. A proposal on gravity dual solution realizingSuperfluid in NRCFT (not the probe limit): The UV theory isNRCFT(z = 2)but the IR theory isLifshitz field theory(z = 2).

Meanings:

(1) Short range behavior is NRCFT, sym of free Schr eq - gravity dual is Schr. 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 Lif_{d +2} UV Schr_{d +3}

(3)Break Number U(1) symmetry(assuperfluid)geometricallyby shrinking U(1) ξ circle, where −i ∂ξ corresponds to Number 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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

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

2. A proposal on gravity dual solution realizingSuperfluid in NRCFT (not the probe limit): The UV theory isNRCFT(z = 2)but the IR theory isLifshitz field theory(z = 2).

Meanings:

(1) Short range behavior is NRCFT, sym of free Schr eq - gravity dual is Schr. 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 Lif_{d +2} UV Schr_{d +3}

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

IR Lifd +2 UV Schrd +3

(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), i τb∼ i τ_b+ N/T , yb∼ y_b+ MLξ

i τs∼ i τs+ MLξ, ys∼ ys− N/T .

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

IR Lifd +2 UV Schrd +3

(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 (τ, y ), or (τ, y ),

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

IR Lifd +2 UV Schrd +3

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

(i)Witten’s AdS soliton- Do NOT work. Hawking-Page transition.

(ii)Schr¨odinger soliton- shrink ξ cigar?

Do NOT work. Double Wick rotate: (τb, yb) → i (ys, τs), (tb, ξb) → (ts, ξs), (βb, Ωb) → −i (βs, Ωs) Periodic identification for (τ_b, ξ_b), (ys, ξs) shows different ensemble system: Schr BH: it_b= it_b+ N/T , ξ_b= ξ_b+ N(µM/T ) + MLξ

Schr soliton: its= its− i N/Ts, ξs= ξs+ i N(µMs/Ts) + MLξ

We hadsuperfluid in Schr soliton in the probe limitby introducing boson hair. However, in Schr case, so far we cannot consider Hawking-Page or Witten’s confined-deconfined picture as in AdS.AdS5with a compact y direction for 2+1-dim CMThas been studied by Nishioka, Ryu, Takayanagi, arXiv 0911.0962.

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

IR Lifd +2 UV Schrd +3

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

(i)Witten’s AdS soliton- Do NOT work. Hawking-Page transition.

(ii)Schr¨odinger soliton- shrink ξ cigar? Do NOT work. Double Wick rotate: (τb, yb) → i (ys, τs), (tb, ξb) → (ts, ξs), (βb, Ωb) → −i (βs, Ωs) Periodic identification for (τ_b, ξ_b), (ys, ξs) shows different ensemble system:

Schr BH: it_b= it_b+ N/T , ξ_b= ξ_b+ N(µM/T ) + MLξ

Schr soliton: its= its− i N/Ts, ξs= ξs+ i N(µMs/Ts) + MLξ

We hadsuperfluid in Schr soliton in the probe limitby introducing boson hair. However, in Schr case, so far we cannot consider Hawking-Page or Witten’s confined-deconfined picture as in AdS.AdS5with a compact y direction for 2+1-dim CMThas been studied by Nishioka, Ryu, Takayanagi, arXiv 0911.0962.

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

IR Lifd +2 UV Schrd +3

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

(i)Witten’s AdS soliton- Do NOT work. Hawking-Page transition.

(ii)Schr¨odinger soliton- shrink ξ cigar? Do NOT work. Double Wick rotate: (τb, yb) → i (ys, τs), (tb, ξb) → (ts, ξs), (βb, Ωb) → −i (βs, Ωs) Periodic identification for (τ_b, ξ_b), (ys, ξs) shows different ensemble system:

Schr BH: it_b= it_b+ N/T , ξ_b= ξ_b+ N(µM/T ) + MLξ

Schr soliton: its= its− i N/Ts, ξs= ξs+ i N(µMs/Ts) + MLξ

We hadsuperfluid in Schr soliton in the probe limitby introducing boson hair.

However, in Schr case, so far we cannot consider Hawking-Page or Witten’s confined-deconfined picture as in AdS.AdS5with a compact y direction for 2+1-dim CMThas been studied by Nishioka, Ryu, Takayanagi, arXiv 0911.0962.

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

IR Lif_{d +2} UV Schr_{d +3}

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

(i)Witten’s AdS soliton- Do NOT work. Hawking-Page transition.

(ii)Schr¨odinger 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 edge of the bubble). (b) Klebanov-Strassler, hep-th/0007191 (shrinks S^p with p > 1 and a remaining S^q).

(iv) some preliminary results by numerical method. How about analytic solution?

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

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

IR Lif_{d +2} UV Schr_{d +3}

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

(i)Witten’s AdS soliton- Do NOT work. Hawking-Page transition.

(ii)Schr¨odinger 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 edge of the bubble). (b) Klebanov-Strassler, hep-th/0007191 (shrinks S^p with p > 1 and a remaining S^q).

(iv) some preliminary results by numerical method. How about analytic solution?

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

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

IR Lif_{d +2} UV Schr_{d +3}

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

(i)Witten’s AdS soliton- Do NOT work. Hawking-Page transition.

(ii)Schr¨odinger 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 edge of the bubble). (b) Klebanov-Strassler, hep-th/0007191 (shrinks S^p with p > 1 and a remaining S^q).

(iv) some preliminary results by numerical method. How about analytic solution?

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

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

Known Solution

B-F theory and New Solution New Ideas

Other Ideas on B-F theory

IR Lif_{d +2} UV Schr_{d +3}

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

(i)Witten’s AdS soliton- Do NOT work. Hawking-Page transition.

(ii)Schr¨odinger 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 edge of the bubble). (b) Klebanov-Strassler, hep-th/0007191 (shrinks S^p with p > 1 and a remaining S^q).

(iv) some preliminary results by numerical method. How about analytic solution?

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

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

Known Solution

B-F theory and New Solution New Ideas

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 T solutions for ∀z.

2. Free energy F (T , µ) has the unphysical form T^#(^T_µ)^#, instead of physical result T^#µ^#.

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

IR Lifd +2 UV Schrd +3

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

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

Known Solution

B-F theory and New Solution New Ideas

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 T solutions for ∀z.

2. Free energy F (T , µ) has the unphysical form T^#(^T_µ)^#, instead of physical result T^#µ^#.

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

IR Lifd +2 UV Schrd +3

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

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

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.

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

hOi v.s. T :

T Metal Superfluid

Tc

hOi v.s. Ω:

W Metal Superfluid

W*

hOi v.s. µ_Q:

Μ_Q 1st order PT 2nd order PT

Μ*

Superfluids from Schr soliton

hOi v.s. Ω:

W Insulator Superfluid

Wc

hOi v.s. µ_Q:

ΜQ

Superfluid Insulator

Μc

Fermi surface from Schr BH

hOi v.s. β:

Β Metal Insulator

Β^*

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

Fermions in charged Schr BH

The parameters of phase space: ∆, T , µ_Q, M, β,

conformal dimension, temperature, charge density, Number(Mass), background density.

Bosons in Schr BH

The parameters of phase space: ∆, T , µ_Q, M, Ω

conformal dimension, temperature, charge density, Number(Mass), background density.

Bosons in Schr soliton

The parameters of phase space: ∆, mG, µQ, M, Ω

conformal dimension, mass gap(∼ 1/Lξ), charge density, Number(Mass), background density.

asymptotics AdS_{d +2} Schr_{d +3}

scalar conformal dim ∆_±= ^{d +1}₂ ±q

(^{d +1}₂ )²+ m² ∆_±= ^{d +2}₂ ±q

(^{d +2}₂ )²+ m²+ (` − qM_o)²

d +1 d +2 q

1 2 2

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

Ex: Bulk side Dictionary Boundary side

Hologram 3D object Fourier Trans 2D image 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 Schr/NRCFT (D+2)-dim gravity Schr/NRCFT D-dim NR FT

Gravity Thermo Quantum

Gravitational B-F theory formalism of RG critical points

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

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.00 0.02 0.04 0.06 0.08 0.10

TTc XO1\

0 2 4 6 8 10

0.00 0.05 0.10 0.15

ِTc Re@ΣHΩLD

1.26 1.05 1.

0.96 0.88 0.65 0.37 0.29 0.24 0.19 0.16 0.08 0.05 0.01 TTc

0 2 4 6 8 10

0.0 0.2 0.4 0.6 0.8 1.0

ِTc Im@ΣHΩLD

1.26 1.05 1.

0.96 0.88 0.65 0.37 0.29 0.24 0.19 0.16 0.08 0.05 0.01 TTc

在文檔中 JuvenWang(MIT)Jan13,2012@NatlTaiwanUniv HolographicViewonnon-relativisticSuperfluids,FermiSurfacesandRGfixedpoints (頁 91-117)