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Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Massive Quantum Liquids from holographic Angel’s Trumpets Matthias C. Wapler [email protected] Center for Quantum Spacetime (CQUeST), Sogang University Erwin Schrödinger Insitute, August 5th 2010 M.W.: JHEP 1005, 019 (2010) [arXiv:1002.0336 [hep-th]] MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization 1) Angel’s Trumpets (Brugmansia) MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization 1) Angel’s Trumpets (Brugmansia) What does that have to do with QCD? MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization 2) Low-temperature limit Look at the “low-temperature limit” of “top-down” holographic fundamental matter confined to d + 1 dimensions at: Quark mass Mq T > Tdeconf. d Baryon number density ρ T d > Tdeconf. ⇒ interesting scaling (see later...) MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization 2) Low-temperature limit Look at the “low-temperature limit” of “top-down” holographic fundamental matter confined to d + 1 dimensions at: Quark mass Mq T > Tdeconf. d Baryon number density ρ T d > Tdeconf. ⇒ interesting scaling (see later...) MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization 2) Low-temperature limit Look at the “low-temperature limit” of “top-down” holographic fundamental matter confined to d + 1 dimensions at: Quark mass Mq T > Tdeconf. d Baryon number density ρ T d > Tdeconf. ⇒ interesting scaling (see later...) At Nf 6= 1 only at vanishing isospin density MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization 2) Low-temperature limit The Experiment MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization 2) Low-temperature limit The Experiment MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Outline 1 Massive Probe Branes MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Outline 1 Massive Probe Branes 2 Finding the Scaling Solution MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Outline 1 Massive Probe Branes 2 Finding the Scaling Solution 3 Thermodynamics Preliminaries Free Energy Response Functions Subleading Terms MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Outline 1 Massive Probe Branes 2 Finding the Scaling Solution 3 Thermodynamics Preliminaries Free Energy Response Functions Subleading Terms 4 Generalization General Issues 3+1 D3-D7 1+1 D3-D3 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization D3/SYM background Nc D3 branes in AdS5 × S 5 ⇔ N = 4 U (Nc ) SYM on the AdS boundary String coupling gs AdS radius L Schwarzschild radius r0 6= 0 Do supergravity, L2 /ls2 1 ⇔ ⇔ ⇔ ⇔ Yang-Mills coupling gY2 M = 4πgs L4 t’Hooft coupling λ = gY2 M Nc = 2l 4 s r0 thermal SYM with T = √ πL2 Consider strong coupling λ 1 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization D3/SYM background Nc D3 branes in AdS5 × S 5 ⇔ N = 4 U (Nc ) SYM on the AdS boundary String coupling gs AdS radius L Schwarzschild radius r0 6= 0 Do supergravity, L2 /ls2 1 ⇔ ⇔ ⇔ ⇔ Yang-Mills coupling gY2 M = 4πgs L4 t’Hooft coupling λ = gY2 M Nc = 2l 4 s r0 thermal SYM with T = √ πL2 Consider strong coupling λ 1 The setup: Put D3s at r = 0, then in dimensionless coordinates (factors of πT ) and u = rr0 , we have » – d u2 L2 4 2 2 2 2 ds = 2 −(1 − u ) d t + + d x + d y + d z + L2 dΩ25 u (1 − u4 ) 2 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Creating the Defect Add fundamental matter ↔ probe branes to the D3 background: MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Creating the Defect Add fundamental matter ↔ probe branes to the D3 background: background probe (1 + 1) probe (2 + 1) probe (3 + 1) : : : : D3 D3 D5 D7 0 t × × × × 1 x × × × × 2 y × × × MCW 3 z × × 4 r 5 6 × × × × × × × × 7 8 9 θ or or × Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Creating the Defect Add fundamental matter ↔ probe branes to the D3 background: background probe (1 + 1) probe (2 + 1) probe (3 + 1) : : : : D3 D3 D5 D7 0 t × × F × 1 x × × F × MCW 2 y × F × 3 z × × 4 r 5 6 × F × × F × F × 7 8 9 θ or or × Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Creating the Defect Add fundamental matter ↔ probe branes to the D3 background: background probe (1 + 1) probe (2 + 1) probe (3 + 1) : : : : D3 D3 D5 D7 0 t × × F × 1 x × × F × 2 y × F × 3 z × × 4 r 5 6 × F × × F × F × 7 8 9 θ or or × D5 Half supersymmetry broken ⇒ expect stability → N = 2 gauge theory of Nf hypermultiplets confined to the 2+1 defect; in the fundamental representation of the original SU (Nc ) SYM in the 3+1 bulk [A. Karch and L. Randall:’01] [J. Erdmenger, Z. Guralnik and I. Kirsch,: ’02] [O. DeWolfe, D.Z. Freedman and H. Ooguri:’02] MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Creating the Defect Add fundamental matter ↔ probe branes to the D3 background: background probe (1 + 1) probe (2 + 1) probe (3 + 1) : : : : D3 D3 D5 D7 0 t × × F × 1 x × × F × 2 y × F × 3 z × × 4 r 5 6 × F × × F × F × 7 8 9 θ or or × D5 Half supersymmetry broken ⇒ expect stability → N = 2 gauge theory of Nf hypermultiplets confined to the 2+1 defect; in the fundamental representation of the original SU (Nc ) SYM in the 3+1 bulk Turn on flux on the S 2 q units of flux ⇔ funny things happen MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Aside: MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Aside: What is the defect like in the 3+1 bulk? Like a sheet of semiconductor, plus: “Cloud of gluey excitations” around the defect R Turn on f ⇒ “ P [C 4 ] ∧ F ” sources C 4 ⇒ change Nc → Nc + q towards one side. Degrees of freedom turn on gradually over z = 0 . . . zmax (q) Can verify from quasiparticle spectrum [Rob Myers, MW: ’08, MW: ’09] and thermodynamics [MW: ’09] MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Density “Quarks” ↔ Endpoints of D3-D5 strings MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Density DBI Action Z q −det(P [G] + 2πls 2 F ) SDBI = −T5 Nf D5 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Density DBI Action Z q −det(P [G] + 2πls 2 F ) SDBI = −T5 Nf D5 Density δS Jˆ ↔ A, so hJ µ i = δA µ √ ⇒ Find: ρ = λNf Nc T 2 limu→0 ∂u Ãt (u) ⇒ turn on F = ∂u At (u) du ∧ dt MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Density DBI Action Z q −det(P [G] + 2πls 2 F ) SDBI = −T5 Nf D5 Density δS Jˆ ↔ A, so hJ µ i = δA µ √ ⇒ Find: ρ = λNf Nc T 2 limu→0 ∂u Ãt (u) ⇒ turn on F = ∂u At (u) du ∧ dt Also µ = δS ρ = limu→0 At (u), so ⇒ Boundary: A(u) ∼ µ̃ − ρ̃u + O(u2 ) Horizon: A(u) ∼ −q(u − 1) + O(u − 1)2 MCW – ρ̃ = q Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Quark Mass “Quarks” ↔ Endpoints of D3-D5 strings MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Quark Mass Separate D3 and D5 on the sphere Embedding scalar Ψ: dΩ25 = dθ2 + sin2 θ dΩ22 + cos2 θ dΩ22 ⇒ Mass: Mq = limu→0 (rH /u) sin θ 2πα0 MCW =: √ T m̃ λ 23/2 Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Quark Mass Separate D3 and D5 on the sphere Embedding scalar Ψ: dΩ25 = dθ2 + sin2 θ dΩ22 + cos2 θ dΩ22 ⇒ Mass: Mq = limu→0 (rH /u) sin θ 2πα0 =: √ T m̃ λ 23/2 Find dual condensate: C = < ψψ > = MCW δS δMq =: 1 T 2 Nf Nc c̃ 4π Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Quark Mass Separate D3 and D5 on the sphere Embedding scalar Ψ: dΩ25 = dθ2 + sin2 θ dΩ22 + cos2 θ dΩ22 ⇒ Mass: Mq = limu→0 (rH /u) sin θ 2πα0 =: √ T m̃ λ 23/2 Find dual condensate: C = < ψψ > = δS δMq =: 1 T 2 Nf Nc c̃ 4π Define: Ψ := sin θ - Asymptotically: Ψ ∼ m̃ u + c̃ u2 + . . . Ψ (1−Ψ2 )2 Horizon: Ψ ∼ Ψ0 + 21 ρ̃2 0+(1−Ψ02 )2 + . . . 0 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Quark Mass Massive embeddings MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Quark Mass Massive embeddings MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Quark Mass Massive embeddings MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Equations... Now, Action: Z p p S = 4πL2 T5 Nf dσ 4 − det P [G]4 1 − (2πls2 ∂u At (u))2 P [G]tt P [G]uu (1−Ψ2 ) MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Equations... Now, Action: Z p p S = 4πL2 T5 Nf dσ 4 − det P [G]4 1 − (2πls2 ∂u At (u))2 P [G]tt P [G]uu (1−Ψ2 ) Induced metric: „ „ « « u2 Ψ0 (u)2 L2 1 2 2 2 2 + du + u (1 − Ψ(u) )dΩ ds2 = 2 −(1 − u4 )dt̃2 + d~x̃22 + 2 u 1 − u4 1 − Ψ(u)2 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Equations... Now, Action: Z p p S = 4πL2 T5 Nf dσ 4 − det P [G]4 1 − (2πls2 ∂u At (u))2 P [G]tt P [G]uu (1−Ψ2 ) Induced metric: „ „ « « u2 Ψ0 (u)2 L2 1 2 2 2 2 + du + u (1 − Ψ(u) )dΩ ds2 = 2 −(1 − u4 )dt̃2 + d~x̃22 + 2 u 1 − u4 1 − Ψ(u)2 Equations of motion: √ ∂u At (u) = p 1 − Ψ2 (u) + u2 h(u)Ψ0 (u)2 p λT p 1 − Ψ(u)2 ρ̃2 u4 + (1 − Ψ(u)2 )2 ρ̃ and ` ´ 2(1 − Ψ2 )3 + u2 (1 − u4 ) ρ̃2 u4 + (1 − Ψ2 )2 Ψ02 q ` ´` ´ u4 (1−Ψ2 ) (1−Ψ2 ) 1−Ψ2 + (u2 − u6 )Ψ02 1 + ρ̃2 u4 + Ψ2 (Ψ2 − 2) s ! 4 ρ̃2 u4 + (1 − Ψ2 )2 01−u = ∂u Ψ u2 (1 − Ψ2 )(1 − Ψ2 + (u2 − u6 )Ψ02 ) MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Consequences from Equations... Motivation IIa) At large ρ̃: m̃ ∝ ρ̃1/d ⇒ Mq T ∼ α(Ψ0 ) ` ´ ρ 1/d Td ⇒ Temperature-independent! MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Some Preliminaries Work at fixed Mq , rather than fixed T : T̄ = ρ̄ = c̄ = √ 1 T = λ 3/2 m̃ 2 Mq ρ λ ρ̃ √ = m̃2 Nf Nc λ 8Mq2 c̃ πλ C = , m̃2 2Nf Nc Mq2 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Some Preliminaries Work at fixed Mq , rather than fixed T : T̄ = ρ̄ = c̄ = √ 1 T = λ 3/2 m̃ 2 Mq ρ λ ρ̃ √ = m̃2 Nf Nc λ 8Mq2 c̃ πλ C = , m̃2 2Nf Nc Mq2 10. Ρ = 1, T = 1, 12, ... Ρ = 1, m = 1, 2, ... T = 15, 110, ... m = 5, 10, ... sin HΘL u 8. 6. 4. 2. BH 0. 0. 5. 10. 15. MCW 20. 25. H1uL30. cosHΘL Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Some Preliminaries How it looks like MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Some Preliminaries How it looks like MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Some Preliminaries Motivation IIb): Studied Phase Diagram... [MW: ’09] (ρ̃ = 0.5) ...and found... MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Some Preliminaries Motivation IIb): ... and found [MW: ’09]: ⇒ temperature-independent entropy MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Scaling solution Split in two Regimes MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Funnel Region Linear regime, Ψ(u) = Ψ0 + ψ(u) Equation of motion for Ψ(u) = Ψ0 + ψ(u): ` ´ 2Ψ0 (1 − Ψ20 ) = ρ̃2 u6 ∂u (1 − u4 )∂u ψ(u) + O(ψ 2 ) MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Funnel Region Linear regime, Ψ(u) = Ψ0 + ψ(u) Equation of motion for Ψ(u) = Ψ0 + ψ(u): ` ´ 2Ψ0 (1 − Ψ20 ) = ρ̃2 u6 ∂u (1 − u4 )∂u ψ(u) + O(ψ 2 ) → Solution ψ(u) = −Ψ0 (1 − Ψ20 ) 10ρ̃2 u4 „ 1 − u4 − 2u4 tan−1 MCW (1 + u2 )(1 + u)2 1−u + u4 ln 1+u 8 u4 « Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Funnel Region Linear regime, Ψ(u) = Ψ0 + ψ(u) Equation of motion for Ψ(u) = Ψ0 + ψ(u): ` ´ 2Ψ0 (1 − Ψ20 ) = ρ̃2 u6 ∂u (1 − u4 )∂u ψ(u) + O(ψ 2 ) → Solution ψ(u) = −Ψ0 (1 − Ψ20 ) 10ρ̃2 u4 ⇒ Ψ(u) = Ψ0 − region. „ 1 − u4 − 2u4 tan−1 Ψ0 (1−Ψ2 0) 10 ρ̃2 u4 (1 + u2 )(1 + u)2 1−u + u4 ln 1+u 8 u4 « is the boundary condition for the asymptotic MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Trumpet Region √ Define ξ := u ρ̃ ; u ∈]0, 1] → ξ ∈]0, MCW √ ρ̃] Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Trumpet Region √ Define ξ := u ρ̃ ; u ∈]0, 1] → ξ ∈]0, √ ρ̃] Observables ρ̄ = (∂ξ Ψ|ξ→0 )−2 2 ∂ξ Ψ|ξ→0 1 c̄ = ρ̄ ∂ξ2 Ψ|ξ→0 = 2 2(∂ξ Ψ|ξ→0 )2 p 1 T̄ = ρ̄/ρ̃ = √ ρ̃∂ξ Ψ|ξ→0 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Trumpet Region √ Define ξ := u ρ̃ ; u ∈]0, 1] → ξ ∈]0, √ ρ̃] Obtaining the solution Equation of motion: „ „ «« p p Ψ(ξ) 2 4 2 2 0 = ∂ξ 1 − Ψ(ξ) ξ + (1 − Ψ(ξ) ) ∂ξ ξ „ „ ««2 “ ” p p Ψ(ξ) ∂ξ Ψ(ξ) 1 − Ψ(ξ)2 ξ 4 + (1 − Ψ(ξ)2 )2 + ξ∂ξ ∂Ψ(ξ) Ψ(ξ) ξ solve numerically, but find ∂ξ2 Ψ|ξ→0 = −2Ψ0 ↔ c̄ = −ρ̄ Ψ0 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Trumpet Region √ Define ξ := u ρ̃ ; u ∈]0, 1] → ξ ∈]0, √ ρ̃] Obtaining the solution Equation of motion: „ „ «« p p Ψ(ξ) 2 4 2 2 0 = ∂ξ 1 − Ψ(ξ) ξ + (1 − Ψ(ξ) ) ∂ξ ξ „ „ ««2 “ ” p p Ψ(ξ) ∂ξ Ψ(ξ) 1 − Ψ(ξ)2 ξ 4 + (1 − Ψ(ξ)2 )2 + ξ∂ξ ∂Ψ(ξ) Ψ(ξ) ξ solve numerically, but find ∂ξ2 Ψ|ξ→0 = −2Ψ0 ↔ c̄ = −ρ̄ Ψ0 Linear regime, ρ̄ 1: 0 = 2Ψ − 2ξ∂ξ Ψ + ξ 2 (1 + ξ 4 )∂ξ2 Ψ ⇒ Ψ0 ∼ √ 1 √ ρ̄ K(1/ 2) ρ̄ 1, (1 − Ψ0 ) 1: 1 − Ψ ∝ ρ̄2 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Scaling solution How it looks like MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Scaling solution How it looks like 1. Ρ = 0.01, 0.02, ... Ρ = 0.1, 0.2, ... Ρ = 1, 2, ... Y 0.8 0.6 0.4 0.2 0. 0. 0.2 0.4 0.6 MCW 0.8 1. 1.2 1.4 Ξ Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Scaling solution How it looks like MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Scaling solution How it looks like MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Thermodynamics Obtain contribution only from the fundamental matter, can be discussed independently. MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Thermodynamics Obtain contribution only from the fundamental matter, can be discussed independently. General strategy: 1 Ru Compute Euclidean action Ie = u max Le + Ibdy ; min √ √ boundary terms are unique, here Ibdy. = − 31 γ + 12 Ψ2 γ [Karch, O’Bannon, Skenderis, Kostas: ’04] 2 Vary fields ⇒ Thermodynamic variables X i , T δIe = µi δX i 3 Do appropriate Legendre transformations MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Thermodynamics Obtain contribution only from the fundamental matter, can be discussed independently. General strategy: 1 Ru Compute Euclidean action Ie = u max Le + Ibdy ; min √ √ boundary terms are unique, here Ibdy. = − 31 γ + 12 Ψ2 γ [Karch, O’Bannon, Skenderis, Kostas: ’04] 2 Vary fields ⇒ Thermodynamic variables X i , T δIe = µi δX i 3 Do appropriate Legendre transformations Want canonical ensemble F (ρ, T ; M ): Z umax F = T I˜e , I˜e = Le + Ibdy + ρAt , umin Here use F̄Mq (ρ̄, T̄ ): F̄ := F λ Mq3 29/2 Nc Nf MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Thermodynamics Obtain contribution only from the fundamental matter, can be discussed independently. General strategy: 1 Ru Compute Euclidean action Ie = u max Le + Ibdy ; min √ √ boundary terms are unique, here Ibdy. = − 31 γ + 12 Ψ2 γ [Karch, O’Bannon, Skenderis, Kostas: ’04] 2 Vary fields ⇒ Thermodynamic variables X i , T δIe = µi δX i 3 Do appropriate Legendre transformations Want canonical ensemble F (ρ, T ; M ): Z umax F = T I˜e , I˜e = Le + Ibdy + ρAt , umin Here use F̄Mq (ρ̄, T̄ ): F̄ := F λ Mq3 29/2 Nc Nf Aside: At large temperatures T̄ 1, MCW T̄ ρ̄ = − 31 T̄ 3 Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Obtain the free energy Find F̄ = −ρ̄ 3/2 √ ρ̃ Z 0 dξp 4 ξ + (1 − Ψ2 )2 ξ4 MCW ! ˛ ξ 2 Ψ02 2 − 3Ψ2 ˛˛ + O(ρ̃−1/2 ) 1+ − 1 − Ψ2 6ξ 3 ˛bdy. r Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Obtain the free energy Find F̄ = −ρ̄ 3/2 √ ρ̃ Z 0 dξp 4 ξ + (1 − Ψ2 )2 ξ4 ! ˛ ξ 2 Ψ02 2 − 3Ψ2 ˛˛ + O(ρ̃−1/2 ) 1+ − 1 − Ψ2 6ξ 3 ˛bdy. r √ √ ρ̃] → ξ ∈]0, ∞[ − ξ ∈] ρ̃, ∞[ R∞ √ Integrate ξ ∈] ρ̃, ∞[: √ρ̃ d ξ (ξ −2 + O(ξ −6 )) Split ξ ∈]0, MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Obtain the free energy Find F̄ = −ρ̄ 3/2 √ ρ̃ Z 0 dξp 4 ξ + (1 − Ψ2 )2 ξ4 ! ˛ ξ 2 Ψ02 2 − 3Ψ2 ˛˛ + O(ρ̃−1/2 ) 1+ − 1 − Ψ2 6ξ 3 ˛bdy. r √ √ ρ̃] → ξ ∈]0, ∞[ − ξ ∈] ρ̃, ∞[ R∞ √ Integrate ξ ∈] ρ̃, ∞[: √ρ̃ d ξ (ξ −2 + O(ξ −6 )) Split ξ ∈]0, √ F̄ = ρ̄3/2 Φ(ρ̄) − T̄ ρ̄ + O(T̄ 4 / ρ̄) - Φ(ρ̄) := R∞ 0 dξ 2ξ4 „ q p 2 ξ 4 + (1 + Ψ2 )2 1 + MCW ξ2 Ψ02 1−Ψ2 − 2 + 3Ψ2 − 2ξΨΨ0 « Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Obtain the free energy Find F̄ = −ρ̄ 3/2 √ ρ̃ Z 0 dξp 4 ξ + (1 − Ψ2 )2 ξ4 ! ˛ ξ 2 Ψ02 2 − 3Ψ2 ˛˛ + O(ρ̃−1/2 ) 1+ − 1 − Ψ2 6ξ 3 ˛bdy. r √ √ ρ̃] → ξ ∈]0, ∞[ − ξ ∈] ρ̃, ∞[ R∞ √ Integrate ξ ∈] ρ̃, ∞[: √ρ̃ d ξ (ξ −2 + O(ξ −6 )) Split ξ ∈]0, √ F̄ = ρ̄3/2 Φ(ρ̄) − T̄ ρ̄ + O(T̄ 4 / ρ̄) „ « q p 2 Ψ02 2 0 2 ξ 4 + (1 + Ψ2 )2 1 + ξ1−Ψ 2 − 2 + 3Ψ − 2ξΨΨ √ ρ̄ 1: Φ(ρ̄) = 32 K(1/ 2) + 0.2697 + O(ρ̄−4 ) ρ̄ - Φ(ρ̄) := R∞ 0 dξ 2ξ4 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Obtain the free energy Find F̄ = −ρ̄ 3/2 √ ρ̃ Z 0 dξp 4 ξ + (1 − Ψ2 )2 ξ4 ! ˛ ξ 2 Ψ02 2 − 3Ψ2 ˛˛ + O(ρ̃−1/2 ) 1+ − 1 − Ψ2 6ξ 3 ˛bdy. r √ √ ρ̃] → ξ ∈]0, ∞[ − ξ ∈] ρ̃, ∞[ R∞ √ Integrate ξ ∈] ρ̃, ∞[: √ρ̃ d ξ (ξ −2 + O(ξ −6 )) Split ξ ∈]0, √ F̄ = ρ̄3/2 Φ(ρ̄) − T̄ ρ̄ + O(T̄ 4 / ρ̄) „ « q p 2 Ψ02 2 0 2 ξ 4 + (1 + Ψ2 )2 1 + ξ1−Ψ 2 − 2 + 3Ψ − 2ξΨΨ √ ρ̄ 1: Φ(ρ̄) = 32 K(1/ 2) + 0.2697 + O(ρ̄−4 ) ρ̄ - Φ(ρ̄) := R∞ 0 dξ 2ξ4 ρ̄ 1: Φ(ρ̄) = 1 √ ρ + O(ρ̄3/2 ) MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms F FHΡL 5. small-Ρ: FHΡL ~ Ρ 2 0.270 KH 1 2 L + large-Ρ: FHΡL ~ 3 Ρ 4. 3. 2. 1. 0. 0. 0.2 0.4 MCW 0.6 0.8 Ρ Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization 5 F 4 Preliminaries Free Energy Response Functions Subleading Terms FHΡL small-Ρ: FHΡL ~ Ρ 2 KH 1 2 L large-Ρ: FHΡL ~ 3 3 2 1.5 1.0 0.05 0.1 0.2 0.5 1 MCW 2 5 10 20 Ρ50 Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Small densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 MCW Mq 23/2 √ λ meson mass Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Small densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 Mq 23/2 √ λ meson mass S = ρ U = ρ µ = Mq 23/2 √ λ Mq 23/2 √ λ −T MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Small densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 Mq 23/2 √ λ meson mass S = ρ U = ρ µ = Mq 23/2 √ λ Mq 23/2 √ λ −T ⇒ Gas of deeply bound particles with ground state degeneracy ρ MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Small densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 Mq 23/2 √ λ meson mass S = ρ U = ρ µ = Mq 23/2 √ λ Mq 23/2 √ λ −T ⇒ Gas of deeply bound particles with ground state degeneracy ρ P̄ = −F̄ + ρ̄∂ρ̄ F̄ ˛ ` ´ ˛ K̄T = −V ∂∂VP̄ ˛ = ρ̄∂ρ̄ −F̄ + ρ̄∂ρ̄ F̄ = ρ̄2 ∂ρ̄2 F̄ , K̄T = K̄S T,N “ ”−1 ˛ 2 ∂P ∂P ˛ vs = ∂U S = ∂U = ρ∂Kρ U ∂ρ ∂ρ “ ” √ ρ3 λ All vanish ∝ O √λN N 23/2 M c q f C.f. ideal classical gas P ∝ ρT MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Small densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 Mq 23/2 √ λ meson mass S = ρ U = ρ µ = Mq 23/2 √ λ Mq 23/2 √ λ −T ⇒ Gas of deeply bound particles with ground state degeneracy ρ P̄ = −F̄ + ρ̄∂ρ̄ F̄ ˛ ` ´ ˛ K̄T = −V ∂∂VP̄ ˛ = ρ̄∂ρ̄ −F̄ + ρ̄∂ρ̄ F̄ = ρ̄2 ∂ρ̄2 F̄ , K̄T = K̄S T,N “ ”−1 ˛ 2 ∂P ∂P ˛ vs = ∂U S = ∂U = ρ∂Kρ U ∂ρ ∂ρ “ ” √ ρ3 λ All vanish ∝ O √λN N 23/2 M c q f C.f. ideal classical gas P ∝ ρT ⇒ Pressureless gas MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Large densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 MCW Mq 23/2 √ λ meson mass Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Large densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 Mq 23/2 √ λ meson mass S = ρ U = 2 K( √12 ) ρ 3 q √ ρ λNc Nf q µ = K( √12 ) √λNρ N − T c f √ √ ρ λNc Nf = 2 K( √1 ) 2 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Large densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 Mq 23/2 √ λ meson mass S = ρ U = 2 K( √12 ) ρ 3 q √ ρ λNc Nf q µ = K( √12 ) √λNρ N − T c f √ √ ρ λNc Nf = 2 K( √1 ) 2 2d Fermi liquid: µ ∝ √ ρ but cv =∝ T MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Large densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 Mq 23/2 √ λ meson mass S = ρ U = 2 K( √12 ) ρ 3 q √ ρ λNc Nf q µ = K( √12 ) √λNρ N − T c f √ √ ρ λNc Nf = 2 K( √1 ) 2 2d Fermi liquid: µ ∝ √ ρ but cv =∝ T 3/2 P = 1 K( √12 ) 3 √√ρ K = 1 K( √12 ) 2 √√ρ vs2 = 1 2 λNc Nf 3/2 λNc Nf MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Large densities (Induced length scale)−1 “ √ ρ λNc Nf ”1/2 Mq 23/2 √ λ meson mass S = ρ U = 2 K( √12 ) ρ 3 q √ ρ λNc Nf q µ = K( √12 ) √λNρ N − T c f √ √ ρ λNc Nf = 2 K( √1 ) 2 2d Fermi liquid: µ ∝ √ ρ but cv =∝ T 3/2 P = 1 K( √12 ) 3 √√ρ K = 1 K( √12 ) 2 √√ρ vs2 = 1 2 λNc Nf 3/2 λNc Nf Consistent with induced length scale l ∼ MCW “√ λNc Nf ρ ”1/2 ! Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Heat capacity Question cv = 0 + ?? MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Heat capacity Question cv = 0 + ?? Answer Do precision numerics of the full equations to obtain δF : F̄ = ρ̄3/2 Φ(ρ̄) − T̄ ρ̄ + δF cv = T ∂T2 δF Parametrize δ F̄ = −aT̄ 4 − bT̄ 5 or δ F̄ = −αT̄ β Not consistent with δ F̄ = e−Eg /T ρ̄ 1 consistent with Mq = 0 case [Karch, Son, Starinets: ’08] [MW: ’09] MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Heat capacity Answer, δ F̄ = −aT̄ 4 − bT̄ 5 0.100 0.050 0.020 0.010 0.005 5 0.002 0.001 T 4 coefficient T coefficient 1H10 ΡL 0.2 Ρ 0.1 0.5 1 MCW 5 10 100 Ρ Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization Preliminaries Free Energy Response Functions Subleading Terms Heat capacity Answer, δ F̄ = −aT̄ 4 − bT̄ 5 Β 0.1 0.1 Α 0.05 ΑT : Α Hlog scale, rightL Β Hlinear scale, leftL 1H10 ΡL 0.05 5 Β 0.01 0.005 4.5 4 0.001 0.05 0.1 0.5 1 5 MCW 10 50 100 Ρ Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Preliminaries background probe (1 + 1) probe (2 + 1) probe (3 + 1) : : : : D3 D3 D5 D7 0 t × F × F 1 x × F × F 2 y × × F MCW 3 z × F 4 r 5 6 F × F F × F × F 7 8 9 θ or or F Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Preliminaries background probe (1 + 1) probe (2 + 1) probe (3 + 1) : : : : D3 D3 D5 D7 0 t × F × F 1 x × F × F 2 y × × F 3 z × F 4 r 5 6 F × F F × F × F 7 8 9 θ or or F ˜ and (·) ¯ Dimensionless quantities: (·) d d Radial coordinate ξd := u/ρ̃1/d MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Preliminaries background probe (1 + 1) probe (2 + 1) probe (3 + 1) : : : : D3 D3 D5 D7 0 t × F × F 1 x × F × F 2 y × × F 3 z × F 4 r 5 6 F × F F × F × F 7 8 9 θ or or F ˜ and (·) ¯ Dimensionless quantities: (·) d d Radial coordinate ξd := u/ρ̃1/d 1+1: Ψ ∼ m̃1 u + c̃1 ln u and Ã1 t ∼ µ̃1 − ρ̃1 ln u MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Preliminaries background probe (1 + 1) probe (2 + 1) probe (3 + 1) : : : : D3 D3 D5 D7 0 t × F × F 1 x × F × F 2 y × × F 3 z × F 4 r 5 6 F × F F × F × F 7 8 9 θ or or F ˜ and (·) ¯ Dimensionless quantities: (·) d d Radial coordinate ξd := u/ρ̃1/d 1+1: Ψ ∼ m̃1 u + c̃1 ln u and Ã1 t ∼ µ̃1 − ρ̃1 ln u 3+1: Ψ ∼ m̃3 u + c̃3 u3 and Ã3 t ∼ µ̃3 − ρ̃3 u2 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Preliminaries background probe (1 + 1) probe (2 + 1) probe (3 + 1) : : : : D3 D3 D5 D7 0 t × F × F 1 x × F × F 2 y × × F 3 z × F 4 r 5 6 F × F F × F × F 7 8 9 θ or or F ˜ and (·) ¯ Dimensionless quantities: (·) d d Radial coordinate ξd := u/ρ̃1/d 1+1: Ψ ∼ m̃1 u + c̃1 ln u and Ã1 t ∼ µ̃1 − ρ̃1 ln u 3+1: Ψ ∼ m̃3 u + c̃3 u3 and Ã3 t ∼ µ̃3 − ρ̃3 u2 Again: c̄3 = ρ̄3 Ψ20 and c̄1 = ρ̄1 Ψ0 General free energy (d+1)/d F̄d = ρ̄d Φd (ρ̄d ) − ρ̄d T̄ + O(ρ̃(1−d)/d ) MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Embeddings Solve for Ψ0 1 1+1 HD3-D3L 2+1 HD3-D5L 3+1 HD3-D7L µ Ρ1 -1 µ Ρ2 -12 µ Ρ3 -13 Y0 0.5 0.1 0.1 0.5 1 5 MCW 10 50 100 Ρd Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Free energy Integrate for Φd (ρ̄d ) FHΡL 1+1 HD3-D3L 2+1 HD3-D5L 3+1 HD3-D7L Ρ1 -1 Ρ2 -12 14 Ρ3 -43 14 Ρ3 -43 + Ρ3 -13 10 5 1 0.5 0.05 0.1 0.5 MCW 1 5 10 50 Ρd Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 3+1 D3-D7 Generic Features ρ̄3 1: Deeply bound “gas” 1/3 ρ̄3 1: Lengthscale ∝ ρ̄3 ρ̄3 1: c2v = 1 3 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 3+1 D3-D7 Generic Features ρ̄3 1: Deeply bound “gas” 1/3 ρ̄3 1: Lengthscale ∝ ρ̄3 ρ̄3 1: c2v = 1 3 Instability at ρ̄3 1 Φ3 ∼ ⇒ F̄3 = 1 −4/3 ρ̄ 4 3 1 + ρ̄3 4 −1/3 + ρ̄3 . − T + ... ⇒Negative Pressure! MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 1+1 D3-D3 Renormalization issues Ψ ∼ m̃1 u + c̃1 ln u and Ã1 t ∼ µ̃1 − ρ̃1 ln u √ ⇒ Boundary term Ibdy. = 12 γΨ()2 (1 + 1/ ln()) [Karch, O’Bannon, Skenderis, Kostas: ’04] MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 1+1 D3-D3 Renormalization issues Ψ ∼ m̃1 u + c̃1 ln u and Ã1 t ∼ µ̃1 − ρ̃1 ln u √ ⇒ Boundary term Ibdy. = 12 γΨ()2 (1 + 1/ ln()) [Karch, O’Bannon, Skenderis, Kostas: ’04] Still divergent ∼ ρ̃2 2 ln → Only covariant term Aµ Aν γ µν 2 ln √ γ [Hung, Sinha: ’09] Evaluates to a Hawking-Ross term MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 1+1 D3-D3 Renormalization issues Ψ ∼ m̃1 u + c̃1 ln u and Ã1 t ∼ µ̃1 − ρ̃1 ln u √ ⇒ Boundary term Ibdy. = 12 γΨ()2 (1 + 1/ ln()) [Karch, O’Bannon, Skenderis, Kostas: ’04] Still divergent ∼ ρ̃2 2 ln → Only covariant term Aµ Aν γ µν 2 ln √ γ [Hung, Sinha: ’09] Evaluates to a Hawking-Ross term Thermodynamic variables Ambiguity how to consider δ Ã1 t ∼ δ µ̃ + δ ρ̃ ln u Turns out that T δIe = µδρ − Mq δC MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 1+1 D3-D3 Renormalization issues Ψ ∼ m̃1 u + c̃1 ln u and Ã1 t ∼ µ̃1 − ρ̃1 ln u √ ⇒ Boundary term Ibdy. = 12 γΨ()2 (1 + 1/ ln()) [Karch, O’Bannon, Skenderis, Kostas: ’04] Still divergent ∼ ρ̃2 2 ln → Only covariant term Aµ Aν γ µν 2 ln √ γ [Hung, Sinha: ’09] Evaluates to a Hawking-Ross term Thermodynamic variables Ambiguity how to consider δ Ã1 t ∼ δ µ̃ + δ ρ̃ ln u Turns out that T δIe = µδρ − Mq δC ⇒ Need to do Legendre transformation in C Mq MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension “ ”1/2 M 23/2 Properties controlled by Mρ2 : √λNρ N Q q√λ q MCW c f Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension “ ”1/2 M 23/2 Properties controlled by Mρ2 : √λNρ N Q q√λ q c f Suggest ∃ quantum liquid MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension “ ”1/2 M 23/2 Properties controlled by Mρ2 : √λNρ N Q q√λ q c f Suggest ∃ quantum liquid S = Nbrayon ⇒ large ground state degeneracy; chemical potential has a term −T , consistent with Bose-Einstein distribution of degeneracy given by the baryon number. cv = 0 + O(T 3 . . . T 4 ) – higher order than a classical Fermi or Bose gas – but no energy gap MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension “ ”1/2 M 23/2 Properties controlled by Mρ2 : √λNρ N Q q√λ q c f Suggest ∃ quantum liquid S = Nbrayon ⇒ large ground state degeneracy; chemical potential has a term −T , consistent with Bose-Einstein distribution of degeneracy given by the baryon number. cv = 0 + O(T 3 . . . T 4 ) – higher order than a classical Fermi or Bose gas – but no energy gap ρ̄ 1: U ∼ Nbaryon × Mmeson ; P, K, cv = 0 ⇒ interacting gas of deeply bound quarks. MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension “ ”1/2 M 23/2 Properties controlled by Mρ2 : √λNρ N Q q√λ q c f Suggest ∃ quantum liquid S = Nbrayon ⇒ large ground state degeneracy; chemical potential has a term −T , consistent with Bose-Einstein distribution of degeneracy given by the baryon number. cv = 0 + O(T 3 . . . T 4 ) – higher order than a classical Fermi or Bose gas – but no energy gap ρ̄ 1: U ∼ Nbaryon × Mmeson ; P, K, cv = 0 ⇒ interacting gas of deeply bound quarks. „√ «1/2 λNc Nf −1 −1 ρ̄ 1: U ∼ Nbaryon × lind (ρ); µ ∼ lind (ρ) — lind (ρ) = ρ - large P, K; vs → 1 √ . 2 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension “ ”1/2 M 23/2 Properties controlled by Mρ2 : √λNρ N Q q√λ q c f Suggest ∃ quantum liquid Also looked at 3+1 (D3-D7) and 1+1 (D3-D3) dimensions MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension “ ”1/2 M 23/2 Properties controlled by Mρ2 : √λNρ N Q q√λ q c f Suggest ∃ quantum liquid Also looked at 3+1 (D3-D7) and 1+1 (D3-D3) dimensions Similar, but: Thermodynamic instability P < 0 in 3+1 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension “ ”1/2 M 23/2 Properties controlled by Mρ2 : √λNρ N Q q√λ q c f Suggest ∃ quantum liquid Also looked at 3+1 (D3-D7) and 1+1 (D3-D3) dimensions Similar, but: Thermodynamic instability P < 0 in 3+1 Issues because of divergences in 1+1 MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Conclusions Studied ρ T 2 at Mq 6= 0 for fundamental matter → Temperature-independent scaling solution for probe brane embeddings Aside: Brane tension matches string tension “ ”1/2 M 23/2 Properties controlled by Mρ2 : √λNρ N Q q√λ q c f Suggest ∃ quantum liquid Also looked at 3+1 (D3-D7) and 1+1 (D3-D3) dimensions Look at correlators Transport properties, quasiparticle spectrum Issues with operator mixing MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 Y. Kitazawa, “Effective lagrangian for open superstring from five point function,” Nucl. Phys. B 289, 599 (1987). C. P. Bachas, P. Bain and M. B. Green, “Curvature terms in D-brane actions and their M-theory origin,” JHEP 9905, 011 (1999) [arXiv:hep-th/9903210]. N. R. Constable, R. C. Myers and O. Tafjord, “Non-Abelian brane intersections,” JHEP 0106, 023 (2001) [arXiv:hep-th/0102080]. P. Kovtun, D.T. Son and A.O. Starinets, “Holography and hydrodynamics: Diffusion on stretched horizons,” JHEP 0310, 064 (2003) [arXiv:hep-th/0309213]. MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 lisa A. Karch and L. Randall, “Open and closed string interpretation of SUSY CFT’s on branes with boundaries,” JHEP 0106 (2001) 063 [arXiv:hep-th/0105132] hirosi O. DeWolfe, D.Z. Freedman and H. Ooguri, “Holography and defect conformal field theories,” Phys. Rev. D 66 (2002) 025009 [arXiv:hep-th/0111135] D. Mateos, R.C. Myers and R.M. Thomson, “Thermodynamics of the brane,” arXiv:hep-th/0701132. R. C. Myers, A. O. Starinets and R. M. Thomson, “Holographic spectral functions and diffusion constants for fundamental matter,” JHEP 0711, 091 (2007) [arXiv:0706.0162 [hep-th]]. J. Erdmenger, Z. Guralnik and I. Kirsch, “Four-dimensional superconformal theories with interacting boundaries or Phys. Rev. D 66, 025020 (2002) [arXiv:hep-th/0203020]. D. Mateos, R. C. Myers and R. M. Thomson, “Holographic phase transitions with fundamental matter,” Phys. Rev. Lett. 97 (2006) 091601 [arXiv:hep-th/0605046]. MCW Massive Quantum Liquids from holographic Angel’s Massive Probe Branes Finding the Scaling Solution Thermodynamics Generalization General Issues 3+1 D3-D7 1+1 D3-D3 sl2z E. Witten, “SL(2,Z) action on three-dimensional conformal field theories with Abelian symmetry,” arXiv:hep-th/0307041. C.P. Herzog, P. Kovtun, S. Sachdev and D.T. Son, “Quantum critical transport, duality, and M-theory,” arXiv:hep-th/0701036. S.A. Hartnoll and P. Kovtun, “Hall conductivity from dyonic black holes,” arXiv:0704.1160 [hep-th]. MCW Massive Quantum Liquids from holographic Angel’s
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