Spin Models and Gravity

Transcription

1 Spin Models and Gravity Umut Gürsoy (CERN) 6th Regional Meeting in String Theory, Milos June 25, 2011 Spin Models and Gravity p.1

2 Spin systems Spin Models and Gravity p.2

3 Spin systems Many condensed matter systems modeled by spin systems Spin Models and Gravity p.2

4 Spin systems Many condensed matter systems modeled by spin systems Consider Heisenberg type models H = J ij s i s j +, J > 0 with a discrete (e.g. the Ising model, Z 2 ) or continuous (e.g. the XY model, U(1)) spin symmetry. Spin Models and Gravity p.2

5 Spin systems Many condensed matter systems modeled by spin systems Consider Heisenberg type models H = J ij s i s j +, J > 0 with a discrete (e.g. the Ising model, Z 2 ) or continuous (e.g. the XY model, U(1)) spin symmetry. Paramagnet T > T c to ferromagnet transition T < T c as the system cools down. The U(1) XY model in 2D (Kosterlitz-Thouless model) or 3D and the O(3) quantum rotor in 3D, the Hubbard model. etc canonical models for super-fluidity/super-conductivity. Spin Models and Gravity p.2

6 Spin systems Many condensed matter systems modeled by spin systems Consider Heisenberg type models H = J ij s i s j +, J > 0 with a discrete (e.g. the Ising model, Z 2 ) or continuous (e.g. the XY model, U(1)) spin symmetry. Paramagnet T > T c to ferromagnet transition T < T c as the system cools down. The U(1) XY model in 2D (Kosterlitz-Thouless model) or 3D and the O(3) quantum rotor in 3D, the Hubbard model. etc canonical models for super-fluidity/super-conductivity. Non-trivial critical exponents at T c only computable by Monte-Carlo for D > 2. Spin Models and Gravity p.2

7 Generalities Spin Models and Gravity p.3

8 Generalities Landau approach: m(x) = a δ(x x a) s a The partition function Z = D me βf L(m) Spin Models and Gravity p.3

9 Generalities Landau approach: m(x) = a δ(x x a) s a The partition function Z = D me βf L(m) Around T c F L = d d 1 x ( α 0 (T) m(x) 2 + α 1 (T) m(x) 2 + α 2 (T) m(x) 4) Spin Models and Gravity p.3

10 Generalities Landau approach: m(x) = a δ(x x a) s a The partition function Z = D me βf L(m) Around T c F L = d d 1 x ( α 0 (T) m(x) 2 + α 1 (T) m(x) 2 + α 2 (T) m(x) 4) Zero modes of the system: 1. Goldstone mode (phase fluctuations) for T < T c. 2. Longitudinal flat direction arises as T T c Goldstone Longitidunal Spin Models and Gravity p.3

11 Spin Models and Gravity p.4

12 Mean field: M ˆv a saddle-point of FL M T T c 1 2 Spin Models and Gravity p.4

13 Mean field: M ˆv a saddle-point of FL M T T c 1 2 Gaussian fluctuations: m(x) = M + δ m(x). Spin Models and Gravity p.4

14 Mean field: M ˆv a saddle-point of FL M T T c 1 2 Gaussian fluctuations: m(x) = M + δ m(x). The spin-spin correlator - ordered phase: m i (L) m j (0) = M 2 v i v j + e L/ξ (T) L d 3+η v iv j + 1 L d 3+η (δ ij v i v j ) Diverging ξ (T) for T < T c Goldstone mode. Spin Models and Gravity p.4

15 Mean field: M ˆv a saddle-point of FL M T T c 1 2 Gaussian fluctuations: m(x) = M + δ m(x). The spin-spin correlator - ordered phase: m i (L) m j (0) = M 2 v i v j + e L/ξ (T) L d 3+η v iv j + 1 L d 3+η (δ ij v i v j ) Diverging ξ (T) for T < T c Goldstone mode. The spin-spin correlator - disordered phase: m i (L) m j (0) = e L/ξ(T) L d 3+η δ ij Divergence of ξ(t) and ξ (T) as T T c longitudinal zero mode. Spin Models and Gravity p.4

16 Mean field: M ˆv a saddle-point of FL M T T c 1 2 Gaussian fluctuations: m(x) = M + δ m(x). The spin-spin correlator - ordered phase: m i (L) m j (0) = M 2 v i v j + e L/ξ (T) L d 3+η v iv j + 1 L d 3+η (δ ij v i v j ) Diverging ξ (T) for T < T c Goldstone mode. The spin-spin correlator - disordered phase: m i (L) m j (0) = e L/ξ(T) L d 3+η δ ij Divergence of ξ(t) and ξ (T) as T T c longitudinal zero mode. Mean-field Approx: η = 0 and ξ(t) T T c 1 2 Spin Models and Gravity p.4

17 Mean field: M ˆv a saddle-point of FL M T T c 1 2 Gaussian fluctuations: m(x) = M + δ m(x). The spin-spin correlator - ordered phase: m i (L) m j (0) = M 2 v i v j + e L/ξ (T) L d 3+η v iv j + 1 L d 3+η (δ ij v i v j ) Diverging ξ (T) for T < T c Goldstone mode. The spin-spin correlator - disordered phase: m i (L) m j (0) = e L/ξ(T) L d 3+η δ ij Divergence of ξ(t) and ξ (T) as T T c longitudinal zero mode. Mean-field Approx: η = 0 and ξ(t) T T c 1 2 Sound-speed of Goldstone mode: m m e iψ then F L M 2 (δψ) 2. Spin Models and Gravity p.4

18 Mean field: M ˆv a saddle-point of FL M T T c 1 2 Gaussian fluctuations: m(x) = M + δ m(x). The spin-spin correlator - ordered phase: m i (L) m j (0) = M 2 v i v j + e L/ξ (T) L d 3+η v iv j + 1 L d 3+η (δ ij v i v j ) Diverging ξ (T) for T < T c Goldstone mode. The spin-spin correlator - disordered phase: m i (L) m j (0) = e L/ξ(T) L d 3+η δ ij Divergence of ξ(t) and ξ (T) as T T c longitudinal zero mode. Mean-field Approx: η = 0 and ξ(t) T T c 1 2 Sound-speed of Goldstone mode: m m e iψ then F L M 2 (δψ) 2. In the MFA c ψ M 2 T T c Spin Models and Gravity p.4

19 Spin Models and Gravity p.5

20 Can one model these basic features in GR? Can one go beyond the MFA? Spin Models and Gravity p.5

21 Can one model these basic features in GR? Can one go beyond the MFA? The answer to both questions is in the affirmative. Spin Models and Gravity p.5

22 Can one model these basic features in GR? Can one go beyond the MFA? The answer to both questions is in the affirmative. Map spin-models Gauge theories Gauge theories GR! Spin Models and Gravity p.5

23 Can one model these basic features in GR? Can one go beyond the MFA? The answer to both questions is in the affirmative. Map spin-models Gauge theories Gauge theories GR! A new approach to holographic super-fluids/super-conductors Spin Models and Gravity p.5

24 Outline Mapping spin-models to gauge theories Realization of confinement-deconfinement transition in GR Continuous Hawking-Page transitions in GR Near transition region: linear-dilaton background Calculation of observables: Second-sound Spin-spin correlators from string theory Discussion Spin Models and Gravity p.6

25 Lattice gauge theory and Spin-models Polyakov 78; Susskind 79 Spin Models and Gravity p.7

26 Lattice gauge theory and Spin-models Polyakov 78; Susskind 79 Any LGT with arbitrary gauge group G in d-dimensions with arbitrary adjoint matter Integrate out gauge invariant states generate effective theory for the Polyakov loop Z LGT (P;T) Z SpM ( s;t 1 ) Ferromagnetic spin model H = J ij s i s j + in d 1 dimensions with spin symmetry C = Center(G) Spin Models and Gravity p.7

27 Spin Models and Gravity p.8

28 Inversion of temperature: Deconfined (high T) phase in LGT Ordered (low T) phase of Spin-model Confined (low T) phase in LGT Disordered (high T) phase of Spin-model Spin Models and Gravity p.8

29 Inversion of temperature: Deconfined (high T) phase in LGT Ordered (low T) phase of Spin-model Confined (low T) phase in LGT Disordered (high T) phase of Spin-model P (L)P(0) conf e ml, P(x) = 0 P (L)P(0) deconf 1 + e ml, P(x) = 0 Spin Models and Gravity p.8

30 LGT - SpM equivalence Polyakov 78; Susskind 79; Svetitsky and Yaffe 82 Spin Models and Gravity p.9

31 LGT - SpM equivalence Polyakov 78; Susskind 79; Svetitsky and Yaffe 82 Consider a LGT with non-trivial center symmetry Spin Models and Gravity p.9

32 LGT - SpM equivalence Polyakov 78; Susskind 79; Svetitsky and Yaffe 82 Consider a LGT with non-trivial center symmetry Lagrangian of LGT: electric U r,0 and magnetic U r,i link variables Spin Models and Gravity p.9

33 LGT - SpM equivalence Polyakov 78; Susskind 79; Svetitsky and Yaffe 82 Consider a LGT with non-trivial center symmetry Lagrangian of LGT: electric U r,0 and magnetic U r,i link variables Typical phase diagram: T P 0 P 0 (*) Polyakov loop P N t 1 n=0 U r+nˆt,0 parameter is the order g Spin Models and Gravity p.9

34 LGT - SpM equivalence Polyakov 78; Susskind 79; Svetitsky and Yaffe 82 Consider a LGT with non-trivial center symmetry Lagrangian of LGT: electric U r,0 and magnetic U r,i link variables Typical phase diagram: T P 0 P 0 (*) Polyakov loop P N t 1 n=0 U r+nˆt,0 is the order parameter (*) Svetitsky and Yaffe 82: At all T the magnetic fluctuations are gapped. g Spin Models and Gravity p.9

35 LGT - SpM equivalence Polyakov 78; Susskind 79; Svetitsky and Yaffe 82 Consider a LGT with non-trivial center symmetry Lagrangian of LGT: electric U r,0 and magnetic U r,i link variables Typical phase diagram: T P 0 P 0 (*) Polyakov loop P N t 1 n=0 U r+nˆt,0 is the order parameter (*) Svetitsky and Yaffe 82: At all T the magnetic fluctuations are gapped. g No long-range magnetic fluctuations integrate out U r,j Spin Models and Gravity p.9

36 LGT - SpM equivalence Polyakov 78; Susskind 79; Svetitsky and Yaffe 82 Consider a LGT with non-trivial center symmetry Lagrangian of LGT: electric U r,0 and magnetic U r,i link variables Typical phase diagram: T P 0 P 0 (*) Polyakov loop P N t 1 n=0 U r+nˆt,0 is the order parameter (*) Svetitsky and Yaffe 82: At all T the magnetic fluctuations are gapped. g No long-range magnetic fluctuations integrate out U r,j The resulting theory L[P] describes long-range fluctuations at criticality Polyakov 78; Susskind 79: Can be mapped onto a spin-model with P s (explicitly shown in the limit g 1) Spin Models and Gravity p.9

37 LGT - SpM equivalence at criticality Svetitsky and Yaffe 82 Spin Models and Gravity p.10

38 LGT - SpM equivalence at criticality Svetitsky and Yaffe 82 If criticality survives the continuum limit of the LGT then critical phenomena of the gauge theory and the Spin model are in same universality class. Spin Models and Gravity p.10

39 LGT - SpM equivalence at criticality Svetitsky and Yaffe 82 If criticality survives the continuum limit of the LGT then critical phenomena of the gauge theory and the Spin model are in same universality class. Some examples: 1. Pure SU(2) in d = 4 second order transition with Z 2 (Ising) critical exponents, 2. SU(N) in d dimensions with N > 2d d 2, Spin model with Z N fixed point flows to a U(1) XY model L Φ 2 + Φ N + Φ N + Φ 2 so the mass-term is relevant for N > 2d/(d 2): Z N U(1). d = 4 non-trivial critical exponents; d > 4 mean-field exponents. Spin Models and Gravity p.10

40 LGT - SpM equivalence at criticality Svetitsky and Yaffe 82 If criticality survives the continuum limit of the LGT then critical phenomena of the gauge theory and the Spin model are in same universality class. Some examples: 1. Pure SU(2) in d = 4 second order transition with Z 2 (Ising) critical exponents, 2. SU(N) in d dimensions with N > 2d d 2, Spin model with Z N fixed point flows to a U(1) XY model L Φ 2 + Φ N + Φ N + Φ 2 so the mass-term is relevant for N > 2d/(d 2): Z N U(1). d = 4 non-trivial critical exponents; d > 4 mean-field exponents. Focus on SU(N) with N, Spin model with Z N U(1) fixed point. Spin Models and Gravity p.10

41 Spontaneous breaking of U(1) in GR Witten 98 Spin Models and Gravity p.11

42 Spontaneous breaking of U(1) in GR Witten 98 Thermal Gas Black-hole x0 M x0 M ds 2 TG = b2 0 (r) dr 2 + dt 2 + dx 2 d 1 r ds 2 BH = b2 (r) dr 2 r r h f(r) + f(r)dt2 + dx 2 d 1 Spin Models and Gravity p.11

43 Spontaneous breaking of U(1) in GR Witten 98 Thermal Gas Black-hole x0 M x0 M ds 2 TG = b2 0 (r) dr 2 + dt 2 + dx 2 d 1 r ds 2 BH = b2 (r) dr 2 r r h f(r) + f(r)dt2 + dx 2 d 1 In addition pure gauge B µν -field: Ψ = M B = const. Spin Models and Gravity p.11

44 Spontaneous breaking of U(1) in GR Witten 98 Thermal Gas Black-hole x0 M x0 M ds 2 TG = b2 0 (r) dr 2 + dt 2 + dx 2 d 1 r ds 2 BH = b2 (r) dr 2 r r h f(r) + f(r)dt2 + dx 2 d 1 In addition pure gauge B µν -field: Ψ = M B = const. Topological shift symmetry Ψ Ψ + const: Spin Models and Gravity p.11

45 Spontaneous breaking of U(1) in GR Witten 98 Thermal Gas Black-hole x0 M x0 M ds 2 TG = b2 0 (r) dr 2 + dt 2 + dx 2 d 1 r ds 2 BH = b2 (r) dr 2 r r h f(r) + f(r)dt2 + dx 2 d 1 In addition pure gauge B µν -field: Ψ = M B = const. Topological shift symmetry Ψ Ψ + const: Only wrapped strings with S F (G + ib + ΦR (2) ) charged under the U(1) part of it Spin Models and Gravity p.11

46 Spontaneous breaking of U(1) in GR Witten 98 Thermal Gas Black-hole x0 M x0 M ds 2 TG = b2 0 (r) dr 2 + dt 2 + dx 2 d 1 r ds 2 BH = b2 (r) dr 2 r r h f(r) + f(r)dt2 + dx 2 d 1 In addition pure gauge B µν -field: Ψ = M B = const. Topological shift symmetry Ψ Ψ + const: Only wrapped strings with S F (G + ib + ΦR (2) ) charged under the U(1) part of it If e S F 0 then U(1) spontaneously broken! Spin Models and Gravity p.11

47 Spontaneous breaking of U(1) in GR Witten 98 Thermal Gas Black-hole x0 M x0 M ds 2 TG = b2 0 (r) dr 2 + dt 2 + dx 2 d 1 r ds 2 BH = b2 (r) dr 2 r r h f(r) + f(r)dt2 + dx 2 d 1 In addition pure gauge B µν -field: Ψ = M B = const. Topological shift symmetry Ψ Ψ + const: Only wrapped strings with S F (G + ib + ΦR (2) ) charged under the U(1) part of it If e S F 0 then U(1) spontaneously broken! Identify s P e S F : Hawking-Page spontaneous magnetization! Spin Models and Gravity p.11

48 Spontaneous breaking of U(1) in GR Witten 98 Thermal Gas Black-hole x0 M x0 M ds 2 TG = b2 0 (r) dr 2 + dt 2 + dx 2 d 1 r ds 2 BH = b2 (r) dr 2 r r h f(r) + f(r)dt2 + dx 2 d 1 In addition pure gauge B µν -field: Ψ = M B = const. Topological shift symmetry Ψ Ψ + const: Only wrapped strings with S F (G + ib + ΦR (2) ) charged under the U(1) part of it If e S F 0 then U(1) spontaneously broken! Identify s P e S F : Hawking-Page spontaneous magnetization! Fluctuations δψ Goldstone mode in the dual spin-model Spin Models and Gravity p.11

49 Identification of the symmetries Spin Models and Gravity p.12

50 Identification of the symmetries T T c Gravity Gauge theory Spin model BH, U(1) / B Deconf. U(1) / C S.fluid U(1) / S T 1 c TG, U(1) B Conf. U(1) C Normal U(1) S T Spin Models and Gravity p.12

51 Identification of the symmetries T T c Gravity Gauge theory Spin model BH, U(1) / B Deconf. U(1) / C S.fluid U(1) / S T 1 c TG, U(1) B Conf. U(1) C Normal U(1) S T Another condition for superfluidity: Second speed c ψ 0 as T T c iff a continuous phase transition Spin Models and Gravity p.12

52 Identification of the symmetries T T c Gravity Gauge theory Spin model BH, U(1) / B Deconf. U(1) / C S.fluid U(1) / S T 1 c TG, U(1) B Conf. U(1) C Normal U(1) S T Another condition for superfluidity: Second speed c ψ 0 as T T c iff a continuous phase transition F L M 2 ( δψ) 2 + (δ m ) 2 + Spin Models and Gravity p.12

53 Identification of the symmetries T T c Gravity Gauge theory Spin model BH, U(1) / B Deconf. U(1) / C S.fluid U(1) / S T 1 c TG, U(1) B Conf. U(1) C Normal U(1) S T Another condition for superfluidity: Second speed c ψ 0 as T T c iff a continuous phase transition F L M 2 ( δψ) 2 + (δ m ) 2 + Continuous Hawking-Page Normal-to-superfluid transition GRAVITY/SPIN-MODEL CORRESPONDENCE Spin Models and Gravity p.12

54 Continuous HP in dilaton-einstein U.G. 10 Spin Models and Gravity p.13

55 Continuous HP in dilaton-einstein U.G. 10 Specify S N 2 d d+1 x g ( ) R ξ( Φ) 2 + V (Φ) e d 1 Φ (db) 2 Spin Models and Gravity p.13

56 Continuous HP in dilaton-einstein U.G. 10 Specify S N 2 d d+1 x ( ) g R ξ( Φ) 2 + V (Φ) e d 1 Φ (db) 2 Look for solutions of the type: ) ds 2 TG = b2 0 (r) ( dr 2 + dt 2 + dx 2 d 1 ( ) ds 2 BH = b2 (r) dr 2 f(r) + f(r)dt2 + dx 2 d 1 Requirements for a second order Hawking-Page transition: Spin Models and Gravity p.13

57 Continuous HP in dilaton-einstein U.G. 10 Specify S N 2 d d+1 x ( ) g R ξ( Φ) 2 + V (Φ) e d 1 Φ (db) 2 Look for solutions of the type: ) ds 2 TG = b2 0 (r) ( dr 2 + dt 2 + dx 2 d 1 ( ) ds 2 BH = b2 (r) dr 2 f(r) + f(r)dt2 + dx 2 d 1 Requirements for a second order Hawking-Page transition: i.) There is a finite T c at which: ii.) F(T c ) = 0. TG(BH) dominates for T < T c (T > T c ). iii.) S(T c ) = 0 iv.) Make sure that this happens between the thermodynamically favored BH and TG branches. Spin Models and Gravity p.13

58 Solution to the constraints Spin Models and Gravity p.14

59 Solution to the constraints All can be solved if as T T c horizon marginally traps the singularity! Spin Models and Gravity p.14

60 Solution to the constraints All can be solved if as T T c horizon marginally traps the singularity! This happens iff V (Φ) V e 2 q ξ d 1 Φ (1 + V sub (Φ)) Spin Models and Gravity p.14

61 Solution to the constraints All can be solved if as T T c horizon marginally traps the singularity! This happens iff V (Φ) V e 2 q ξ d 1 Φ (1 + V sub (Φ)) Nature of the transition is determined by V sub. Spin Models and Gravity p.14

62 Solution to the constraints All can be solved if as T T c horizon marginally traps the singularity! This happens iff V (Φ) V e 2 q ξ d 1 Φ (1 + V sub (Φ)) Nature of the transition is determined by V sub. Define t = T T c T c. nth order transition F t n : when V sub (Φ) = e κφ, with κ = ζ(d 1) n 1 for n 2 Spin Models and Gravity p.14

63 Solution to the constraints All can be solved if as T T c horizon marginally traps the singularity! This happens iff V (Φ) V e 2 q ξ d 1 Φ (1 + V sub (Φ)) Nature of the transition is determined by V sub. Define t = T T c T c. nth order transition F t n : when V sub (Φ) = e κφ, with κ = BKT scaling F e ct 1 α : when V sub (Φ) = Φ α. ζ(d 1) n 1 for n 2 Spin Models and Gravity p.14

64 Linear-dilaton near T c Spin Models and Gravity p.15

65 Linear-dilaton near T c Study vicinity of T c in a Einstein-dilaton system 3D XY model: Spin Models and Gravity p.15

66 Linear-dilaton near T c Study vicinity of T c in a Einstein-dilaton system 3D XY model: Universal result: The geometry becomes linear-dilaton background at criticality: ds 2 dt 2 + dx 2 d 1 + dr2 ; Φ(r) 3r 2l Spin Models and Gravity p.15

67 Linear-dilaton near T c Study vicinity of T c in a Einstein-dilaton system 3D XY model: Universal result: The geometry becomes linear-dilaton background at criticality: ds 2 dt 2 + dx 2 d 1 + dr2 ; Φ(r) 3r 2l Exact solution to string theory to all orders in l s! Spin Models and Gravity p.15

68 Linear-dilaton near T c Study vicinity of T c in a Einstein-dilaton system 3D XY model: Universal result: The geometry becomes linear-dilaton background at criticality: ds 2 dt 2 + dx 2 d 1 + dr2 ; Φ(r) 3r 2l Exact solution to string theory to all orders in l s! Similar to Little string theory in a double scaling limit Giveon, Kutasov 99 Spin Models and Gravity p.15

69 Large N and α Spin Models and Gravity p.16

70 Large N and α Boundary value of the dilaton Φ 0 Take Φ 0, N such that e Φ 0 N = e Φ 0 = const. In the large N limit it is dominated by the sphere diagrams. Spin Models and Gravity p.16

71 Large N and α Boundary value of the dilaton Φ 0 Take Φ 0, N such that e Φ 0 N = e Φ 0 = const. In the large N limit it is dominated by the sphere diagrams. Expectation: strong correlations α corrections suppressed The correlation length ξ T T c ν near T c In fact α R s e 2Φ h vanishes precisely when T T c. Spin Models and Gravity p.16

72 Large N and α Boundary value of the dilaton Φ 0 Take Φ 0, N such that e Φ 0 N = e Φ 0 = const. In the large N limit it is dominated by the sphere diagrams. Expectation: strong correlations α corrections suppressed The correlation length ξ T T c ν near T c In fact α R s e 2Φ h vanishes precisely when T T c. However, another invariant g s µν µ Φ ν Φ const l 2 s T T c One has to take into account α corrections. as Spin Models and Gravity p.16

73 Large N and α Boundary value of the dilaton Φ 0 Take Φ 0, N such that e Φ 0 N = e Φ 0 = const. In the large N limit it is dominated by the sphere diagrams. Expectation: strong correlations α corrections suppressed The correlation length ξ T T c ν near T c In fact α R s e 2Φ h vanishes precisely when T T c. However, another invariant g s µν µ Φ ν Φ const l 2 s T T c One has to take into account α corrections. as Can be done because this regime is governed by a linear-dilaton CFT on the world-sheet! Spin Models and Gravity p.16

74 Embedding in string theory? Spin Models and Gravity p.17

75 Embedding in string theory? Consider d 1 = 3, n = 2. Spin Models and Gravity p.17

76 Embedding in string theory? Consider d 1 = 3, n = 2. Simplest example: V = V e 4 3 Φ ( 1 + 2e 2Φ 0 e 2Φ) A consistent truncation of IIB with single scalar! Pilch-Warner 00 N = 4 sym softly broken by mass-term for a hyper-multiplet. Near AdS minimum: V (0) = m 2 l 2 = 4 = (4 ) consistent with mass-deformation. Spin Models and Gravity p.17

77 Embedding in string theory? Consider d 1 = 3, n = 2. Simplest example: V = V e 4 3 Φ ( 1 + 2e 2Φ 0 e 2Φ) A consistent truncation of IIB with single scalar! Pilch-Warner 00 N = 4 sym softly broken by mass-term for a hyper-multiplet. Near AdS minimum: V (0) = m 2 l 2 = 4 = (4 ) consistent with mass-deformation. An analytic kink solution from asymptotically AdS at r = 0, Φ = Φ 0 : ds 2 TG e Φ(r) = e 4 3 Φ 0 cosh2 3( 3r 2l ) sinh 2 ( 3r 2l ) ( dt 2 + dx 2 d 1 + dr2), = e Φ 0 cosh( 3r 2l ). Spin Models and Gravity p.17

78 Unfortunately this does not do the job: Spin Models and Gravity p.18

79 Unfortunately this does not do the job:! Background is NOT linear-dilaton! Spin Models and Gravity p.18

80 Unfortunately this does not do the job:! Background is NOT linear-dilaton!! 2nd order HP at T c happens in a sub-dominant branch F T 1 2 Spin Models and Gravity p.18

81 Unfortunately this does not do the job:! Background is NOT linear-dilaton!! 2nd order HP at T c happens in a sub-dominant branch F T 1 2 Very generic in Einstein-scalar system Confinementdeconfinement transition is generically first order in gauge theories. Spin Models and Gravity p.18

82 Second speed of sound Spin Models and Gravity p.19

83 Second speed of sound Landau theory: fluctuations of the order parameter M e iψ F L M 2 ( δψ) 2 + Spin Models and Gravity p.19

84 Second speed of sound Landau theory: fluctuations of the order parameter M e iψ F L M 2 ( δψ) 2 + Second sound vanishes as c 2 ψ M 2 (T c T) 2β Spin Models and Gravity p.19

85 Second speed of sound Landau theory: fluctuations of the order parameter M e iψ F L M 2 ( δψ) 2 + Second sound vanishes as c 2 ψ M 2 (T c T) 2β Gravity/Spin-Model correspondence: F L A gr on-shell, at large N Expect mean-field scaling c 2 ψ (T c T). Spin Models and Gravity p.19

86 Second speed of sound Landau theory: fluctuations of the order parameter M e iψ F L M 2 ( δψ) 2 + Second sound vanishes as c 2 ψ M 2 (T c T) 2β Gravity/Spin-Model correspondence: F L A gr on-shell, at large N Expect mean-field scaling c 2 ψ (T c T). Equate the Landau free energy and the regulated on-shell action: F L (T) = A(T) = A BH (T) A TG (T) Spin Models and Gravity p.19

87 Second speed of sound Landau theory: fluctuations of the order parameter M e iψ F L M 2 ( δψ) 2 + Second sound vanishes as c 2 ψ M 2 (T c T) 2β Gravity/Spin-Model correspondence: F L A gr on-shell, at large N Expect mean-field scaling c 2 ψ (T c T). Equate the Landau free energy and the regulated on-shell action: F L (T) = A(T) = A BH (T) A TG (T) Associate δψ with fluctuations of the B-field: ψ = M B Spin Models and Gravity p.19

88 Second speed of sound Landau theory: fluctuations of the order parameter M e iψ F L M 2 ( δψ) 2 + Second sound vanishes as c 2 ψ M 2 (T c T) 2β Gravity/Spin-Model correspondence: F L A gr on-shell, at large N Expect mean-field scaling c 2 ψ (T c T). Equate the Landau free energy and the regulated on-shell action: F L (T) = A(T) = A BH (T) A TG (T) Associate δψ with fluctuations of the B-field: ψ = M B One finds c 2 ψ e V r h (T T c ). Second sound indeed vanishes at T c with the mean-field exponent! Spin Models and Gravity p.19

89 Critical exponents from probe strings Spin Models and Gravity p.20

90 Critical exponents from probe strings Identification: m(x) P[x] W F Spin Models and Gravity p.20

91 Critical exponents from probe strings Identification: m(x) P[x] W F In the superfluid (BH) phase m = M v then m ReP, m ImP. Spin Models and Gravity p.20

92 Critical exponents from probe strings Identification: m(x) P[x] W F In the superfluid (BH) phase m = M v then m ReP, m ImP. For the two-point function: m i (x) m j (0) = m (x) m (0) v i v j + m (x) m (0) (δ ij v i v j ). Spin Models and Gravity p.20

93 Critical exponents from probe strings Identification: m(x) P[x] W F In the superfluid (BH) phase m = M v then m ReP, m ImP. For the two-point function: m i (x) m j (0) = m (x) m (0) v i v j + m (x) m (0) (δ ij v i v j ). m m RePReP m m ImPImP Spin Models and Gravity p.20

94 Quantum computation Spin Models and Gravity p.21

95 Quantum computation Division of paths: r (0, r m ) UV, r (r m, r h ) IR Spin Models and Gravity p.21

96 Quantum computation Division of paths: r (0, r m ) UV, r (r m, r h ) IR For r h and r m large enough, the IR region governed by the linear-dilaton CFT:. T(z) = 1 α : X µ X µ : +v µ 2 X µ with v µ = V 2 δ µ,r Spin Models and Gravity p.21

97 Quantum computation Division of paths: r (0, r m ) UV, r (r m, r h ) IR For r h and r m large enough, the IR region governed by the linear-dilaton CFT:. T(z) = 1 α : X µ X µ : +v µ 2 X µ V 2 δ µ,r with v µ = The mass ( spectrum: ) m 2 2 α N + Ñ 2 + p 2 + (2πkT)2 + ( ) w 2 2πTα and level matching kw + N Ñ = 0. Spin Models and Gravity p.21

98 Quantum computation Division of paths: r (0, r m ) UV, r (r m, r h ) IR For r h and r m large enough, the IR region governed by the linear-dilaton CFT:. T(z) = 1 α : X µ X µ : +v µ 2 X µ V 2 δ µ,r with v µ = The mass ( spectrum: ) m 2 2 α N + Ñ 2 + p 2 + (2πkT)2 + ( ) w 2 2πTα and level matching kw + N Ñ = 0. Semi-classical approximation: Keep only the lowest mode in the path-integral. Spin Models and Gravity p.21

99 Quantum computation Division of paths: r (0, r m ) UV, r (r m, r h ) IR For r h and r m large enough, the IR region governed by the linear-dilaton CFT:. T(z) = 1 α : X µ X µ : +v µ 2 X µ V 2 δ µ,r with v µ = The mass ( spectrum: ) m 2 2 α N + Ñ 2 + p 2 + (2πkT)2 + ( ) w 2 2πTα and level matching kw + N Ñ = 0. Semi-classical approximation: Keep only the lowest mode in the path-integral. One-point function vanishes as M(T) T T c 1 2 in the semiclassical approx! Spin Models and Gravity p.21

100 Quantum computation Division of paths: r (0, r m ) UV, r (r m, r h ) IR For r h and r m large enough, the IR region governed by the linear-dilaton CFT:. T(z) = 1 α : X µ X µ : +v µ 2 X µ V 2 δ µ,r with v µ = The mass ( spectrum: ) m 2 2 α N + Ñ 2 + p 2 + (2πkT)2 + ( ) w 2 2πTα and level matching kw + N Ñ = 0. Semi-classical approximation: Keep only the lowest mode in the path-integral. One-point function vanishes as M(T) T T c 1 2 in the semiclassical approx! Spin Models and Gravity p.21

101 & Two-point function Three types of paths: X1 X1 X1 L L L r h r r f r h r r r (a) (b) (c) Spin Models and Gravity p.22

102 Two-point function Three types of paths: X1 X1 X1 L L L r r r f r r r r (a) (b) (c) (a) m(l) m(0) a = M 2. Finite in BH, 0 for TG. Spin Models and Gravity p.22

103 Two-point function Three types of paths: X1 X1 X1 L L L r r r f r r r r (a) (b) (c) (a) m(l) m(0) a = M 2. Finite in BH, 0 for TG. (b) S F1 m T L + 4 m(l) m(0) b e m T L+ for L 1. Spin Models and Gravity p.22

104 Two-point function, cont ed (c) bulk exchange diagrams: m (L) m (0) c ReP[L]ReP[0] e m + L L d 3 m (L) m (0) c ImP[L]ImP[0] e m L L d 3 Spin Models and Gravity p.23

105 Two-point function, cont ed (c) bulk exchange diagrams: m (L) m (0) c ReP[L]ReP[0] e m + L L d 3 m (L) m (0) c ImP[L]ImP[0] e m L L d 3 m + minimum of the CT + modes: G µν,φ, m minimum of the CT modes: B µν, Spin Models and Gravity p.23

106 Two-point function, cont ed (c) bulk exchange diagrams: m (L) m (0) c ReP[L]ReP[0] e m + L L d 3 m (L) m (0) c ImP[L]ImP[0] e m L L d 3 m + minimum of the CT + modes: G µν,φ, m minimum of the CT modes: B µν, Spectrum analysis U.G., Kiritsis, Nitti 07: CT + bounded from below for any T. Spin Models and Gravity p.23

107 Two-point function, cont ed (c) bulk exchange diagrams: m (L) m (0) c ReP[L]ReP[0] e m + L L d 3 m (L) m (0) c ImP[L]ImP[0] e m L L d 3 m + minimum of the CT + modes: G µν,φ, m minimum of the CT modes: B µν, Spectrum analysis U.G., Kiritsis, Nitti 07: CT + bounded from below for any T. CT include a zero-mode: m = 0 as ψ = M B is modulus: Goldstone mode! Correct qualitative behavior: m (L) m (0) e m + L +e m T L L d 3 m (L) m (0) 1 L d 3 Spin Models and Gravity p.23

108 Two-point function, cont ed (c) bulk exchange diagrams: m (L) m (0) c ReP[L]ReP[0] e m + L L d 3 m (L) m (0) c ImP[L]ImP[0] e m L L d 3 m + minimum of the CT + modes: G µν,φ, m minimum of the CT modes: B µν, Spectrum analysis U.G., Kiritsis, Nitti 07: CT + bounded from below for any T. CT include a zero-mode: m = 0 as ψ = M B is modulus: Goldstone mode! Correct qualitative behavior: m (L) m (0) e m + L +e m T L L d 3 m (L) m (0) 1 L d 3 Precisely the expected behavior from the XY model, with ξ 1 min(m T, m + ) for L 1. Spin Models and Gravity p.23

109 Correlation length ξ Spin Models and Gravity p.24

110 Correlation length ξ (b) Connected paths: X1 I f L Ψ f χ r r 0 Ψ i I i IR CFT Spin Models and Gravity p.24

111 Correlation length ξ (b) Connected paths: X1 I f L Ψ f χ r r 0 Ψ i I i IR CFT Propagator in the IR: IR (χ) dp r d d 2 p e ip x(χ)l Spin Models and Gravity p.24

112 Correlation length ξ (b) Connected paths: X1 I f L Ψ f χ r r 0 Ψ i I i IR CFT Propagator in the IR: IR (χ) dp r d d 2 p e ip x(χ)l Dominant mode is the winding tachyon : ( ξ = i/p x = 4 α + ( ) ) πTα Spin Models and Gravity p.24

113 Correlation length ξ (b) Connected paths: X1 I f L Ψ f χ r r 0 Ψ i I i IR CFT Propagator in the IR: IR (χ) dp r d d 2 p e ip x(χ)l Dominant mode is the winding tachyon : ( ξ = i/p x = 4 α + ( ) ) πTα Indeed diverges if identify with Hagedorn a la Atick-Witten T c = 1 4πl s and ξ l s 2 2 ( ) T Tc T c 1 2 Spin Models and Gravity p.24

114 Correlation length ξ (b) Connected paths: X1 I f L Ψ f χ r r 0 Ψ i I i IR CFT Propagator in the IR: IR (χ) dp r d d 2 p e ip x(χ)l Dominant mode is the winding tachyon : ( ξ = i/p x = 4 α + ( ) ) πTα Indeed diverges if identify with Hagedorn a la Atick-Witten T c = 1 4πl s and ξ l s 2 2 Mean-field scaling again! ( ) T Tc T c 1 2 Spin Models and Gravity p.24

115 Summary Spin Models and Gravity p.25

116 Summary A general connection between gravity and spin-models. Normal-to-superfluid transition continuous HP in GR. Spin Models and Gravity p.25

117 Summary A general connection between gravity and spin-models. Normal-to-superfluid transition continuous HP in GR. Role of large N clarified: number of spin-states at a site in case of SU(N). Spin Models and Gravity p.25

118 Summary A general connection between gravity and spin-models. Normal-to-superfluid transition continuous HP in GR. Role of large N clarified: number of spin-states at a site in case of SU(N). A specific case: SU(N) at large N XY-type models. Two-derivative approximation fails near T c. Physics around T c governed by linear-dilaton CFT Spin Models and Gravity p.25

119 Summary A general connection between gravity and spin-models. Normal-to-superfluid transition continuous HP in GR. Role of large N clarified: number of spin-states at a site in case of SU(N). A specific case: SU(N) at large N XY-type models. Two-derivative approximation fails near T c. Physics around T c governed by linear-dilaton CFT Probe strings spin fluctuations Scaling in second sound and other critical exponents β and ν from GR as expected. Spin Models and Gravity p.25

120 Outlook Spin Models and Gravity p.26

121 Outlook Top-down approach to AdS/CMT: D-brane constructions, embedding in critical string theory Spin Models and Gravity p.26

122 Outlook Top-down approach to AdS/CMT: D-brane constructions, embedding in critical string theory Corrections to critical exponents: 1/N and α corrections beyond the semi-classical approximation? Spin Models and Gravity p.26

123 Outlook Top-down approach to AdS/CMT: D-brane constructions, embedding in critical string theory Corrections to critical exponents: 1/N and α corrections beyond the semi-classical approximation? Generalization to other spin models e.g. discrete center: e.g. 3D Ising model from the GR dual of large-n Sp(N)? Spin Models and Gravity p.26

124 Outlook Top-down approach to AdS/CMT: D-brane constructions, embedding in critical string theory Corrections to critical exponents: 1/N and α corrections beyond the semi-classical approximation? Generalization to other spin models e.g. discrete center: e.g. 3D Ising model from the GR dual of large-n Sp(N)? Embedding in string theory - many examples with linear-dilaton geometries, NS5 branes, etc. Spin Models and Gravity p.26

125 Outlook Top-down approach to AdS/CMT: D-brane constructions, embedding in critical string theory Corrections to critical exponents: 1/N and α corrections beyond the semi-classical approximation? Generalization to other spin models e.g. discrete center: e.g. 3D Ising model from the GR dual of large-n Sp(N)? Embedding in string theory - many examples with linear-dilaton geometries, NS5 branes, etc. How about 2 spatial dimensions? vortex proliferation in Kosterlitz-Thouless. Spin Models and Gravity p.26

126 Outlook Top-down approach to AdS/CMT: D-brane constructions, embedding in critical string theory Corrections to critical exponents: 1/N and α corrections beyond the semi-classical approximation? Generalization to other spin models e.g. discrete center: e.g. 3D Ising model from the GR dual of large-n Sp(N)? Embedding in string theory - many examples with linear-dilaton geometries, NS5 branes, etc. How about 2 spatial dimensions? vortex proliferation in Kosterlitz-Thouless. Continuous HP transitions in string theory. Spin Models and Gravity p.26

127 THANK YOU! Spin Models and Gravity p.27

128 Solving for the conditions Spin Models and Gravity p.28

129 Solving for the conditions Condition iii): Entropy difference S = 1 4G D e (d 1)A(r h). can vanish only for BH TG, i.e. when M BH 0. Spin Models and Gravity p.28

130 Solving for the conditions Condition iii): Entropy difference S = 1 4G D e (d 1)A(r h). can vanish only for BH TG, i.e. when M BH 0. T c corresponds to the point r h horizon marginally traps the singularity! Spin Models and Gravity p.28

131 Solving for the conditions Condition iii): Entropy difference S = 1 4G D e (d 1)A(r h). can vanish only for BH TG, i.e. when M BH 0. T c corresponds to the point r h horizon marginally traps the singularity! Then condition ii) is automatic. Spin Models and Gravity p.28

132 Solving the conditions, cont ed Spin Models and Gravity p.29

133 Solving the conditions, cont ed For condition i) look at Einstein s equations: A A 2 + ξ 2 d 1 Φ = 0, f + (d 1)A f = 0, (d 1)A 2 f + A f + A f V d 1 e2a = 0. Spin Models and Gravity p.29

134 Solving the conditions, cont ed For condition i) look at Einstein s equations: A A 2 + ξ 2 d 1 Φ = 0, f + (d 1)A f = 0, (d 1)A 2 f + A f + A f V d 1 e2a = 0. One solves for the blackness function" f(r) = 1 R r 0 e (d 1)A R rh 0 e (d 1)A. Spin Models and Gravity p.29

135 Solving the conditions, cont ed For condition i) look at Einstein s equations: A A 2 + ξ 2 d 1 Φ = 0, f + (d 1)A f = 0, (d 1)A 2 f + A f + A f V d 1 e2a = 0. One solves for the blackness function" R r 0 e (d 1)A f(r) = 1 R rh 0 e (d 1)A. The Hawking temperature is: T 1 = 4πe (d 1)A(r h) r h 0 e (d 1)A(r) dr. Spin Models and Gravity p.29

136 Solving the conditions, cont ed For condition i) look at Einstein s equations: A A 2 + ξ 2 d 1 Φ = 0, f + (d 1)A f = 0, (d 1)A 2 f + A f + A f V d 1 e2a = 0. One solves for the blackness function" R r 0 e (d 1)A f(r) = 1 R rh 0 e (d 1)A. The Hawking temperature is: T 1 = 4πe (d 1)A(r h) r h 0 e (d 1)A(r) dr. T T c > 0 in the limit A(r h ) can only happen for A(r) A r + Spin Models and Gravity p.29

137 Solving the conditions, cont ed For condition i) look at Einstein s equations: A A 2 + ξ 2 d 1 Φ = 0, f + (d 1)A f = 0, (d 1)A 2 f + A f + A f V d 1 e2a = 0. One solves for the blackness function" R r 0 e (d 1)A f(r) = 1 R rh 0 e (d 1)A. The Hawking temperature is: T 1 = 4πe (d 1)A(r h) r h 0 e (d 1)A(r) dr. T T c > 0 in the limit A(r h ) can only happen for A(r) A r + Plug in Einstein: Φ(r) +A d 1 ξ r + and finally: Spin Models and Gravity p.29

138 Solving the conditions, cont ed For condition i) look at Einstein s equations: A A 2 + ξ 2 d 1 Φ = 0, f + (d 1)A f = 0, (d 1)A 2 f + A f + A f V d 1 e2a = 0. One solves for the blackness function" R r 0 e (d 1)A f(r) = 1 R rh 0 e (d 1)A. The Hawking temperature is: T 1 = 4πe (d 1)A(r h) r h 0 e (d 1)A(r) dr. T T c > 0 in the limit A(r h ) can only happen for A(r) A r + Plug in Einstein: Φ(r) +A d 1 ξ r + and finally: q V (Φ) V e 2 ξ d 1 Φ (1 + V sub (Φ)), Φ Spin Models and Gravity p.29