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

Contents of this talk1. Fermions at infinite scattering length

scale free system realized using cold atoms

2. Operator-State correspondence scaling dimensions in NR-CFT

energy eigenvalues in a harmonic potential

3. Results using ( = d-2, 4-d) expansions scaling dimensions near d=2 and d=4

extrapolations to d=3

4. Summary and outlook

3/30

Fermions at infinite scattering length

Introduction

4/30

Two additional symmetries under

• Scale transformation (dilatation) :

• Conformal transformation :

Symmetry of nonrelativistic systems

Nonrelativistic systems are invariant under• Translations in time (1) and space (3)

• Rotations (3)• Galilean transformations (3)

If the interaction is scale free

Not only theoretically interesting Experimental realization of scale free system !

5/30

40K

Feshbach resonance

Attraction is arbitrarily tunable by magnetic field

C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90 (2003)

B (Gauss)

Cold atom experiments high designability and tunability

V0(a)

r0

a

a<0 No bound state

a>0bound

molecules

add

add = 0.6 a >0

scattering length : a (rBohr) zero binding energy= unitarity limit

|a|

6/30

Fermions at unitarity• Strong coupling limit : |a|• Cold atoms @ Feshbach resonance• 0r0 << de Broglie << |a|

• Scale invariant Nonrelativistic CFT

Scale invariant systems

External potential breaks scale invariance

Isotropic harmonic potential NR-CFT in free space

a=

l

• Fermions with two- and three-body resonancesY.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

Cf. neutrons : r0~1.4 fm << |aNN|~18.5 fm Mehen, Stewart, Wise, PLB(’00)

7/30

NR-CFT and operator-statecorrespondence

Part I

Energy eigenvalue in a harmonic potential

Scaling dimension of operator in NR-CFT

8/30Nonrelativistic CFTTwo additional symmetries under• scale transformation (dilatation) :• conformal transformation :

C.R.Hagen, Phys.Rev.D (’72) U.Niederer, Helv.Phys.Acta.(’72)

Corresponding generators in quantum field theory

Continuity eq.

If the interaction is scale invariant !

D, C, and Hamiltonian form a closed algebra : SO(2,1)

9/30Commutator [D, H]

Generator of dilatation :

scale invariance

• E.g. Hamiltonian with two-body potential V(r)

10/30Primary operator

Local operator has

Primary operator

E.g., primary operator :

nonprimary operator :

• scaling dimension

• particle number

11/30Proof 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 !!!

:

12/30Trivial examples of

• Noninteracting particles in d dimensions

2nd lowest operator

N=3 :

. . .

Interacting case corrections by anomalous dimensions !

N=1 : Lowest operator

operator state

13/30

. . .

Ladders of eigenstates• Raising and lowering operators

F.Werner and Y.Castin, Phys.Rev.A 74 (2006) . . .

. . .

. . .

E

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

breathing modes

14/30

Energy eigenvalues of N-particle state in a harmonic potential

Operator-state correspondence

• 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

Scaling dimensions of N-body composite operator in NR-CFT

Computable using diagrammatic techniques !

( = d-2, 4-d) expansions near d=2 or d=4

15/30

expansion for fermions at unitarity

Part II

1. Field theories for fermions at unitarity perturbative near d=2 or d=4

2. Scaling dimensions of operators up to 6 fermions expansions over = d-2 or 4-d

3. Extrapolations to d=3

16/30Specialty of d=4 and 2

2-body wave function

Z.Nussinov and S.Nussinov, cond-mat/0410597

Pair wave function is concentrated at its origin

Fermions at unitarity in d4 form free bosons

Normalization at unitarity a

diverges at r→0 for d4

At d2, any attractive potential leads to bound states

Zero binding energy “a” corresponds to zero interaction

Fermions at unitarity in d2 becomes free fermions

How to organize systematic expansions near d=2 or d=4 ?

17/30Field theories at unitarity 1• Field theory becoming perturbative near d=2

RG equation :

The theory at fixed point is NR-CFT for fermions at unitarity

Fixed point :

Near d=2, weakly-interacting fermions perturbative expansion in terms of =d-2

Y.N. and D.T.Son, PRL(’06) & PRA(’07); P.Nikolić and S.Sachdev, PRA(’07)

Renormalization of g

18/30Field theories at unitarity 2• Field theory becoming perturbative near d=4

RG equation :

The theory at fixed point is NR-CFT for fermions at unitarity

Fixed point :

p p

Near d=4, weakly-interacting fermions and bosons perturbative expansion in terms of =4-d

Y.N. and D.T.Son, PRL(’06) & PRA(’07); P.Nikolić and S.Sachdev, PRA(’07)

WF renormalization of

19/30

Scaling dimensionsnear d=2 and d=4

Strong coupling

d=4d=2

g

d=3

g

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)

20/30

p p

2-fermion operators• Anomalous dimension near d=2

• Anomalous dimension near d=4

Ground state energy of N=2 is exactly in any 2d4

21/303-fermion operators near d=2

• Lowest operator has L=1 ground state

• Lowest operator with L=0 1st excited state

O()

O()N=3L=1

N=3L=0

22/303-fermion operators near d=4

• Lowest operator has L=0 ground state

• Lowest operator with L=1 1st excited state

O()

O()

N=3L=0

N=3L=1

23/30Operators and dimensions• NLO results of = d-2 and = 4-d expansions

e.g. N=5

24/30Operators and dimensions

O()

O()

O(2)

• NLO results of = d-2 and = 4-d expansions

25/30Comparison to results in d=3

*) S. Tan, cond-mat/0412764 †) D. Blume et al., arXiv:0708.2734

• Naïve extrapolations of NLO results to d=3

Extrapolated results are reasonably close to values in d=3

But not for N=4,6 from d=4 due to huge NLO corrections

26/303 fermion energy in d dimensions

Fit two expansions using Padé approx.

span in a small interval very close to the exact values !

Interpolations to d=3

2d

2d

4d

4d

27/30Exact 3 fermion energy

Padé fits have behaviors consistent withexact 3 fermion energy in d dimension

= +Exact is

computed from

28/30Energy level crossing

Level crossing betweenL=0 and L=1 states

at d = 3.3277

Ground state at d=3 has L=1

Ground state

Excited state

29/30

Energy eigenvalues of N-particle state in a harmonic potential

Summary and outlook 1

• ( = 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

Scaling dimensions of N-body composite operator in NR-CFT

• Operator-state correspondence in nonrelativistic CFT

Exact relation for any nonrelativistic systemsif the interaction is scale invariant

and the potential is harmonic and isotropic

30/30Summary and outlook 2

• 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

( = d-2, 4-d) expansions for fermions at unitarity

How to improve expanions ?

• 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

Accurate predictions in 3d

31/30

Backup slides

32/305 fermion energy in d dimensions

span in a small intervalbut underestimate numerical values at d=3

• Level crossing between L=0 and L=1 states at d > 3• Padé interpolations to d=3

2d

4d

2d

4d

33/304 fermion and 6 fermion energy

[4/0], [0/4] Padé are off from others due to huge 4d NLO

• Ground state has L=0 both near d=2 and d=4• Padé interpolations to d=3

2d

4d

2d

4d

34/30Anyon spectrum to NLO

• Ground state energy of N anyons in a harmonic potential Perturbative expansion in terms of statistics parameter 0 : boson limit 1 : fermion limit

Coincidewith resultsby Rayleigh-Schrödingerperturbation

New analyticresults

consistentwith

numericalresults

Cf. anyon field interacts via Chern-Simons gauge field

35/30Anyon spectrum to NLO

• Ground state energy of N anyons in a harmonic potential Perturbative expansion in terms of statistics parameter 0 : boson limit 1 : fermion limit

Coincidewith resultsby Rayleigh-Schrödingerperturbation

New analyticresults

consistentwith

numericalresults

Cf. anyon field interacts via Chern-Simons gauge field

4 anyon spectrumM. Sporre et al., Phys.Rev.B (1992)

36/30Trivial examples of• Noninteracting spin-1/2 fermions in d dimensions

• Noninteracting N bosons in d dimensions

2nd lowest operator

N=2 :

N=3 :

. . .

Interacting case corrections by anomalous dimensions !

ground state 1st excited state

N=1 : Lowest operator

operator state

37/30

Strong coupling

d2 : g~(d-2) weakly-interacting fermions

d4 : g2~(4-d) weakly-interacting fermions & bosons

Approach of expansion

d=4d=2

g

Systematic expansions of scaling dimension in terms of “d-2” or “4-d”

• Fermions at unitarity as a function of d

d=3

g

Cf. Applications to thermodynamics of fermions at unitarityY.N. and D.T.Son, PRL 97 (’06) & PRA 75 (’07); Y.N., PRA 75 (’07)