Cold atoms and AdS/CFT

Transcription

1 Cold atoms and AdS/CFT D. T. Son Institute for Nuclear Theory, University of Washington Cold atoms and AdS/CFT p.1/20

2 What is common for strong coupled cold atoms and QGP? Cold atoms and AdS/CFT p.2/20

3 ``Two plasmasó workshop at RIKEN/BNL, Dec What is common for strongly coupled atoms and QGP? Edward Shuryak Department of Physics and Astronomy University at Stony Brook Cold atoms and AdS/CFT p.3/20

4 History/motivation BCS/BEC crossover Unitarity regime Schrödinger symmetry Plan Geometric realization of Schrödinger symmetry Conclusion Refs.: D.T. Son, (PRD 2008) also Balasubramanian and McGreevy (PRL 2008) Part of the results obtained in collaboration with Yusuke Nishida Cold atoms and AdS/CFT p.4/20

5 BCS mechanism Consider a gas of spin-1/2 fermions ψ a, a =, No interactions: Fermi sphere: states with k < k F filled Cooper phenomenon: any attractive interaction leads to condensation ψ ψ 0 k y k k x k Cold atoms and AdS/CFT p.5/20

6 Critical temperature: BCS/BEC crossover T c ɛ F e 1/g, g 1 Leggett (1980): If one increases the interaction strength g, how large can one make T c /ɛ F? Cold atoms and AdS/CFT p.6/20

7 Critical temperature: BCS/BEC crossover T c ɛ F e 1/g, g 1 Leggett (1980): If one increases the interaction strength g, how large can one make T c /ɛ F? BCS BEC Strong attraction: leads to bound states, which form a dilute Bose gas T c = temperature of Bose-Einstein condensation 0.22ɛ F. Cold atoms and AdS/CFT p.6/20

8 Unitarity regime Consider in more detail the process of going from BCS to BEC Assume a square-well potential, with fixed but small range r 0. V 0 < 1/mr 2 0: no bound state V 0 = 1/mr 2 0: one bound state appears, at first with zero energy V 0 > 1/mr 2 0: at least one bound state V r Cold atoms and AdS/CFT p.7/20

9 Unitarity regime Consider in more detail the process of going from BCS to BEC Assume a square-well potential, with fixed but small range r 0. V 0 < 1/mr 2 0: no bound state V 0 = 1/mr 2 0: one bound state appears, at first with zero energy V 0 > 1/mr 2 0: at least one bound state V r Unitarity regime: take r 0 0, keeping one bound state at zero energy. In this limit: no intrinsic scale associated with the potential In the language of scattering theory: infinite scattering length a Cold atoms and AdS/CFT p.7/20

10 Consider the following model S = Field theory interpretation Z Dimensional analysis: dtd d x iψ t ψ 1 «2m ψ 2 c 0 ψ ψ ψ ψ [t] = 2, [x] = 1, [ψ] = d 2, [c 0] = 2 d Contact interaction is irrelevant at d > 2 Cold atoms and AdS/CFT p.8/20

11 Consider the following model Field theory interpretation S = Z dtd d x iψ t ψ 1 «2m ψ 2 c 0 ψ ψ ψ ψ Dimensional analysis: [t] = 2, [x] = 1, [ψ] = d 2, [c 0] = 2 d Contact interaction is irrelevant at d > 2 RG equation in d = 2 + ɛ: Two fixed points: c 0 = 0: trivial, noninteracting c 0 s = ɛc 0 c2 0 2π c 0 = 2πɛ: interacting fixed point: unitarity regime When ɛ = 1 interacting fixed point is strongly coupled Cold atoms and AdS/CFT p.8/20

12 Systems near unitarity Neutrons: a = 20 fm, a 1 fm Helium-4 atoms: two-atom bound state with energy 1mK 10 K but are bosons, many-body systems not universal Trapped atom gases, with scattering length a controlled by magnetic field Cold atoms and AdS/CFT p.9/20

13 George Bertsch asked the question: Ground state energy What is the ground state energy of a gas of spin-1/2 particles with infinite scattering length, zero range interaction? Cold atoms and AdS/CFT p.10/20

14 George Bertsch asked the question: Ground state energy What is the ground state energy of a gas of spin-1/2 particles with infinite scattering length, zero range interaction? Dimensional analysis: no intrinsic length/energy scale, only density n. So: ɛ E V = #n5/3 m The same parametric dependence as the energy of a free gas ɛ(n) = ξɛ free (n) Cold atoms and AdS/CFT p.10/20

15 George Bertsch asked the question: Ground state energy What is the ground state energy of a gas of spin-1/2 particles with infinite scattering length, zero range interaction? Dimensional analysis: no intrinsic length/energy scale, only density n. So: ɛ E V = #n5/3 m The same parametric dependence as the energy of a free gas ɛ(n) = ξɛ free (n) George Bertsch announced that he would give a money price for the best estimate of ξ Cold atoms and AdS/CFT p.10/20

16 George Bertsch asked the question: Ground state energy What is the ground state energy of a gas of spin-1/2 particles with infinite scattering length, zero range interaction? Dimensional analysis: no intrinsic length/energy scale, only density n. So: ɛ E V = #n5/3 m The same parametric dependence as the energy of a free gas ɛ(n) = ξɛ free (n) George Bertsch announced that he would give a money price for the best estimate of ξ Since then, ξ is called the Bertsch parameter. Cold atoms and AdS/CFT p.10/20

17 Analogy with AdS/CFT ɛ(n) = ξɛ free (n) Similar to pressure in N = 4 SYM theory P(T) λ = 3 4 P(T) λ 0 Is there an useful AdS/CFT-type duality for unitarity Fermi gas? As in N = 4 SYM, perhaps we should start with the vacuum. Cold atoms and AdS/CFT p.11/20

18 AdS/CFT correspondence Is the correspondence between N = 4 super-yang-mills theory and type IIB string theory on AdS 5 S 5. a standard tool to investigate ESQGP. One of the early evidences: matching of symmetries SO(4, 2) SO(6). SO(4, 2): conformal symetry of 4-dim theories (CFT 4 ), isometry of AdS 5 SO(6): SU(4) is the R-symmetry of N = 4 SYM, isometry of S 5. Conformal algebra: P µ, M µν, K µ, D [D, P µ ] = ip µ, [D, K µ ] = ik µ [P µ, K ν ] = 2i(g µν D + M µν ) Cold atoms and AdS/CFT p.12/20

19 Schrödinger algebra Nonrelativistic field theory is invariant under: Phase rotation M: ψ ψe iα Time and space translations, H and P i Rotations M ij Galilean boosts K i Dilatation D: x λx, t λ 2 t Conformal transformation: x x 1 + λt, t t 1 + λt [D, P i ] = ip i, [D, K i ] = ik i, [P i, K j ] = δ ij M [D, H] = 2iH, [D, C] = 2iC,, [H, C] = id Note: introduction of density µψ ψ breaks the symmetry. Cold atoms and AdS/CFT p.13/20

20 Embedding the Schrödinger algebra Sch(d): is the symmetry of the Schrödinger equation i ψ t + 2 2m ψ = 0 CFT d+2 : is the symmetry of the Klein-Gordon equation 2 µφ = 0, µ = 0,1,, d + 1 In light-cone coordinates x ± = x 0 ± x d+1 the Klein-Gordon equation becomes 2 x + x φ + i i φ = 0, i = 1,,d Restricting φ to φ = e imx ψ(x +,x): Klein-Gordon eq. Schrödinger eq.: 2im x + ψ + 2 ψ = 0, 2 = dx i=1 2 i This means Sch(d) CFT d+2 Cold atoms and AdS/CFT p.14/20

21 Embedding (2) Sch(d) is the subgroup of CFT d+2 containing group elements which does not change the ansatz φ = e imx ψ(x +, x i ) Algebra: sch(d) is the subalgebra of the conformal algebra, containing elements that commute with P + [P +, O] = 0 One can identify the Schrödinger generators: M = P +, H = P, K i = M i, D nonrel = D rel + M +, C = 1 2 K+ Cold atoms and AdS/CFT p.15/20

22 Nonrelativistic dilatation D rel M + x + λx + λ 2 x + x λx x x i λx i λx i Cold atoms and AdS/CFT p.16/20

23 Geometric realization of Schrödinger algebra Start from AdS d+3 space: ds 2 = 2dx+ dx + dx i dx i + dz 2 z 2 Invariant under the whole conformal group, in particular with respect to relativistic scaling x µ λx µ, z λz and boost along the x d+1 direction: x + λx +, x λ 1 x Break the symmetry down to Sch(d): ds 2 = 2dx+ dx + dx i dx i + dz 2 2(dx+ ) 2 z 2 z 4 The additional term is invariant only under a combination of relativistic dilation and boost: x + λ 2 x +, x x, x i λx i Cold atoms and AdS/CFT p.17/20

24 Model ds 2 = 2dx+ dx + dx i dx i + dz 2 2(dx+ ) 2 z 2 z 4 Is there a model where this is a solution to the Einstein equation? The additional term gives rise to a change in R ++ z 4 : we need matter that provides T ++ z 4. Can be provided by A µ with A + 1/z 2 : has to be a massive gauge field. S = Z d d+3 x g 1 2 R Λ 1 «4 F µν 2 m2 2 A2 µ Cold atoms and AdS/CFT p.18/20

25 String theory construction Several groups constructed a string-theory background with Schrödinger symmetry Start from AdS 5 or AdS 5 black hole Perform Null Melvin Twist Get a metric dual to some dimensional At the effective field theory level: metric + massive gauge field + scalar Implications: η/s = 1/(4π). Equation of state: P = T 4 /µ 2 : still unrealistic Lacking superfluidity at small T/µ Cold atoms and AdS/CFT p.19/20

26 Conclusion Unitarity fermions is a nonrelativistic version of strongly coupled QGP Schrödinger symmetry: nonrelativistic conformal symmetry There is a metric with Schrödinger symmetry Starting point for dual phenomenology of unitarity fermions? Cold atoms and AdS/CFT p.20/20