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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] 22 January, 2008 @ UW particle theory group. Contents of this talk Fermions at infinite scattering length

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fermions at unitarity as a nonrelativistic cft

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]

22 January, 2008 @ UW particle theory group

slide2

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
symmetry of nonrelativistic systems
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 !

feshbach resonance
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)

scale invariant systems
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

    • 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

measurement of 2 fermion energy
Measurement of 2 fermion energy

T. Stöferle et al., Phys.Rev.Lett. 96 (2006)

  • |a|

Energy in a harmonic potential

  • Schrödinger eq.
  • CFT calculation
nr cft and operator state correspondence
NR-CFT and operator-statecorrespondence

Part I

Scaling dimension of operator in NR-CFT

Energy eigenvalue in a harmonic potential

nonrelativistic cft
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 !

commutator d h
Commutator [D, H]
  • E.g. Hamiltonian with two-body potential V(r)

Generator of dilatation :

scale invariance

primary operator
Primary operator

Local operator has

  • scaling dimension
  • particle number

Primary operator

E.g., primary operator :

nonprimary operator :

proof of correspondence
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 !!!

trivial examples of
Trivial examples of
  • Noninteracting particles in d dimensions

operator

state

N=1 : Lowest operator

. . .

2nd lowest operator

N=3 :

Interacting case corrections byanomalous dimensions!

ladders of eigenstates
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

operator state correspondence
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

e expansion for fermions at unitarity
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
specialty of d 4 and 2
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 ?

field theories at unitarity 1
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)

field theories at unitarity 2
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)

scaling dimensions near d 2 and d 4
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)

2 fermion operators
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

3 fermion operators near d 2
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
3 fermion operators near d 4
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
operators and dimensions
Operators and dimensions

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

e.g. N=5

operators and dimensions1
Operators and dimensions
  • NLO results of e=d-2 and e=4-d expansions

O(e)

O(e2)

O(e)

comparison to results in d 3
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

3 fermion energy in d dimensions
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 !

exact 3 fermion energy
Exact 3 fermion energy

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

Exact is

computed from

= +

energy level crossing
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

summary
Summary
  • 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
summary and outlook 2
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
5 fermion energy in d dimensions
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

4 fermion and 6 fermion energy
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

anyon spectrum to nlo
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

anyon spectrum to nlo1
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