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Fermions at unitarity as a nonrelativistic CFT

Fermions at unitarity as a nonrelativistic CFT. Yusuke Nishida (INT, Univ. of Washington) in collaboration with D. T. Son (INT) Ref: Phys. Rev. D 76, 086004 (2007) [arXiv:0708.4056] 15 November, 2007 @ Harvard University. Contents of this talk Fermions at infinite scattering length

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Fermions at unitarity as a nonrelativistic CFT

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  1. Fermions at unitarityas a nonrelativistic CFT Yusuke Nishida (INT, Univ. of Washington) in collaboration with D. T. Son (INT) Ref: Phys. Rev. D 76, 086004 (2007) [arXiv:0708.4056] 15 November, 2007 @ Harvard University

  2. Contents of this talk • Fermions at infinite scattering length • scale free system realized using cold atoms • Operator-State correspondence • scaling dimensions in NR-CFT energy eigenvalues in a harmonic potential • Results using e(=d-2,4-d) expansions • scaling dimensions near d=2 and d=4 • extrapolations to d=3 • Summary and outlook

  3. Fermions at infinite scattering length Introduction

  4. Symmetry of nonrelativistic systems • Nonrelativistic systems are invariant under • Translations in time (1) and space (3) • Rotations (3) • Galilean transformations (3) • Two additional symmetries under • Scale transformation (dilatation) : • Conformal transformation : If the interaction is scale free Not only theoretically interesting Experimental realization of scale free system !

  5. Feshbach resonance C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90 (2003) add a V0(a) r0 Cold atom experiments high designability and tunability Attraction is arbitrarily tunable by magnetic field scattering length : a (rBohr) zero binding energy a>0 bound molecules = unitarity limit |a| a<0 No bound state add=0.6a >0 40K B (Gauss)

  6. Scale invariant systems a= • Fermions at unitarity • Strong coupling limit : |a| • Cold atoms @ Feshbach resonance • 0r0 << lde Broglie << |a| • Scale invariant Nonrelativistic CFT l Cf. neutrons : r0~1.4 fm << |aNN|~18.5 fm Mehen, Stewart, Wise, PLB(’00) • Fermions with two- and three-body resonances • Y.N., D.T. Son, and S. Tan, arXiv:0711.1562 • Particles obeying fractional statistics in d=2 (anyons) • R. Jackiw and S.Y. Pi, Phys. Rev. D42, 3500 (1990) • Resonantly interacting anyons Y.N., arXiv:0708.4056 External potential breaks scale invariance Isotropic harmonic potential NR-CFT in free space

  7. NR-CFT and operator-statecorrespondence Part I Scaling dimension of operator in NR-CFT Energy eigenvalue in a harmonic potential

  8. Nonrelativistic CFT C.R.Hagen, Phys.Rev.D (’72) U.Niederer, Helv.Phys.Acta.(’72) • Two additional symmetries under • scale transformation (dilatation) : • conformal transformation : Corresponding generators in quantum field theory D, C, and Hamiltonian form a closed algebra : SO(2,1) Continuity eq. If the interaction is scale invariant !

  9. Commutator [D, H] • E.g. Hamiltonian with two-body potential V(r) Generator of dilatation : scale invariance

  10. Primary operator Local operator has • scaling dimension • particle number Primary operator E.g., primary operator : nonprimary operator :

  11. Proof of correspondence Hamiltonian with a harmonic potential is Construct a state using a primary operator : is an eigenstate of particles in a harmonic potential with the energy eigenvalue !!!

  12. Trivial examples of • Noninteracting particles in d dimensions operator state N=1 : Lowest operator . . . 2nd lowest operator N=3 : Interacting case corrections byanomalous dimensions!

  13. Ladders of eigenstates . . . . . . . . . • Raising and lowering operators F.Werner and Y.Castin, Phys.Rev.A 74 (2006) E . . . breathing modes Each state created by the primary operator has a semi-infinite ladder with energy spacing Cf. Equivalent result derived from Schrödinger equation S. Tan, arXiv:cond-mat/0412764

  14. Operator-state correspondence Energy eigenvalues of N-particle state in a harmonic potential Scaling dimensions of N-body composite operator in NR-CFT Computable using diagrammatic techniques ! • Particles interacting via a 1/r2 potential • Fermions with two- and three-body resonances • Anyons / resonantly interacting anyons • expansions by statistics parameter near boson/fermion limits • Spin-1/2 fermions at infinite scattering length e (=d-2, 4-d) expansions near d=2 or d=4

  15. e expansion for fermions at unitarity Part II • Field theories for fermions at unitarity • perturbative near d=2 or d=4 • Scaling dimensions of operators • up to 6 fermions • expansions over e=d-2 or 4-d • Extrapolations to d=3

  16. Specialty of d=4 and 2 Z.Nussinov and S.Nussinov, cond-mat/0410597 2-body wave function Normalization at unitarity a diverges at r→0 for d4 Pair wave function is concentrated at its origin Fermions at unitarity in d4 form free bosons At d2, any attractive potential leads to bound states Zero binding energy “a” corresponds to zero interaction Fermions at unitarity in d2 becomes free fermions How to organize systematic expansions near d=2 or d=4 ?

  17. Field theories at unitarity 1 • Field theory becoming perturbative near d=2 Renormalization of g RG equation : Fixed point : The theory at fixed point is NR-CFT for fermions at unitarity Near d=2, weakly-interacting fermions perturbative expansion in terms of e=d-2 Y.N. and D.T.Son, PRL(’06) & PRA(’07); P.Nikolić and S.Sachdev, PRA(’07)

  18. Field theories at unitarity 2 p p • Field theory becoming perturbative near d=4 WF renormalization of RG equation : Fixed point : The theory at fixed point is NR-CFT for fermions at unitarity Near d=4, weakly-interacting fermions and bosons perturbative expansion in terms of e=4-d Y.N. and D.T.Son, PRL(’06) & PRA(’07); P.Nikolić and S.Sachdev, PRA(’07)

  19. Scaling dimensionsnear d=2 and d=4 g g Strong coupling d=2 d=3 d=4 Cf. Applications to thermodynamics of fermions at unitarity Y.N. and D.T.Son, PRL 97 (’06) & PRA 75 (’07); Y.N., PRA 75 (’07)

  20. 2-fermion operators p p • Anomalous dimension near d=2 • Anomalous dimension near d=4 Ground state energy of N=2 is exactly in any 2d4

  21. 3-fermion operators near d=2 • Lowest operator has L=1 ground state    O(e) O(e)    N=3 L=1 N=3 L=0 • Lowest operator with L=0 1st excited state

  22. 3-fermion operators near d=4 • Lowest operator has L=0 ground state  O(e) O(e)      N=3 L=0 N=3 L=1 • Lowest operator with L=1 1st excited state

  23. Operators and dimensions           • NLO results of e=d-2 and e=4-d expansions e.g. N=5

  24. Operators and dimensions • NLO results of e=d-2 and e=4-d expansions O(e) O(e2) O(e)

  25. Comparison to results in d=3 • Naïve extrapolations of NLO results to d=3 *) S. Tan, cond-mat/0412764 †) D. Blume et al., arXiv:0708.2734 Extrapolated results are reasonably close to values in d=3 But not for N=4,6 from d=4 due to huge NLO corrections

  26. 3 fermion energy in d dimensions 2d 4d 4d 2d Fit two expansions using Padé approx. Interpolations to d=3 span in a small interval very close to the exact values !

  27. Exact 3 fermion energy Padé fits have behaviors consistent withexact 3 fermion energy in d dimension Exact is computed from = +

  28. Energy level crossing     Level crossing between L=0 and L=1 states at d=3.3277 Ground state at d=3 has L=1 Excited state         Ground state

  29. Summary and outlook 1 • Operator-state correspondence in nonrelativistic CFT Energy eigenvalues of N-particle state in a harmonic potential Scaling dimensions of N-body composite operator in NR-CFT Exact relation for any nonrelativistic systems if the interaction is scale invariant and the potential is harmonic and isotropic • e(=d-2,4-d) expansions near d=2 or d=4 • for spin-1/2 fermions at infinite scattering length • Statistics parameter expansions for anyons

  30. Summary and outlook 2 e (=d-2, 4-d) expansions for fermions at unitarity • Clear picture near d=2 (weakly-interacting fermions) • and d=4 (weakly-interacting bosons & fermions) • Exact results for N=2,3 fermions in any dimensions d • Padé fits of NLO expansions agree well with exact values • Underestimate values in d=3 as N is increased How to improve e expanions? Accurate predictions in 3d • Calculations of NN…LO corrections • Are expansions convergent ? (Yes, when N=3 !) • What is the best function to fit two expansions ? • Exact result for N=4 fermions

  31. Backup slides

  32. 5 fermion energy in d dimensions 2d 2d 4d 4d • Level crossing between L=0 and L=1 states at d>3 • Padé interpolations to d=3 span in a small interval but underestimate numerical values at d=3

  33. 4 fermion and 6 fermion energy 2d 2d 4d 4d • Ground state has L=0 both near d=2 and d=4 • Padé interpolations to d=3 [4/0], [0/4] Padé are off from others due to huge 4d NLO

  34. Anyon spectrum to NLO • Ground state energy of N anyons in a harmonic potential • Perturbative expansion in terms of statistics parameter a • a0 : boson limit a1 : fermion limit Coincidewith resultsby Rayleigh-Schrödingerperturbation New analyticresultsconsistentwithnumericalresults Cf. anyon field interacts via Chern-Simons gauge field

  35. Anyon spectrum to NLO 4 anyon spectrum M. Sporre et al., Phys.Rev.B (1992) • Ground state energy of N anyons in a harmonic potential • Perturbative expansion in terms of statistics parameter a • a0 : boson limit a1 : fermion limit Coincidewith resultsby Rayleigh-Schrödingerperturbation New analyticresultsconsistentwithnumericalresults Cf. anyon field interacts via Chern-Simons gauge field

  36. Trivial examples of • Noninteracting spin-1/2 fermions in d dimensions operator state N=1 : Lowest operator . . . 2nd lowest operator     N=2 : N=3 : • Noninteracting N bosons in d dimensions ground state 1st excited state Interacting case corrections byanomalous dimensions !

  37. Approach of e expansion g g • Fermions at unitarity as a function of d Strong coupling d=2 d=3 d=4 d2 : g~(d-2) weakly-interacting fermions d4 : g2~(4-d) weakly-interacting fermions & bosons Systematic expansions of scaling dimension D in terms of “d-2” or “4-d” Cf. Applications to thermodynamics of fermions at unitarity Y.N. and D.T.Son, PRL 97 (’06) & PRA 75 (’07); Y.N., PRA 75 (’07)

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