Fermions at unitarity as a nonrelativistic CFT

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) arXiv0708.4056 18
March, 2008 _at_ TRIUMF
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 d2 and d4
• extrapolations to d3
• 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 invariant
5
Scale invariant interactions

1. Interaction via 1/r2 potential

• Zero-range potential at infinite scattering
  length
• Spin-1/2 fermions at infinite scattering length
• Fermions with two- and three-body resonances
• Y.N., D.T. Son, and S. Tan, Phys. Rev. Lett.,
  100 (2008)

• Interaction due to fractional statistics in d2
• Anyons R. Jackiw and S.Y. Pi, Phys. Rev.
  D42, 3500 (1990)
• Resonantly interacting anyons Y.N., Phys.
  Rev. D (2008)

6
Feshbach resonance
C.A.Regal and D.S.Jin, Phys.Rev.Lett. 90 (2003)
Cold atom experiments high designability and
tunability
Attraction is arbitrarily tunable by magnetic
field
scattering length a (rBohr)
zero binding energy
agt0 bound molecules
unitarity limit a??
alt0 No bound state
add 0.6 a gt0
40K
B (Gauss)
7
Fermions in the unitarity limit
Strong interaction
?
-?
0
weak repulsion
weak attraction

• Fermions at unitarity
• Strong coupling limit a??
• Cold atoms _at_ Feshbach resonance
• 0?r0 ltlt lde Broglie ltlt a??
• Scale invariant Nonrelativistic CFT

l
Cf. neutrons r01.4 fm ltlt aNN18.5 fm
External potential breaks scale invariance
Isotropic harmonic potential NR-CFT in free
space
8
NR-CFT and operator-statecorrespondence
Part I
Scaling dimension of operator in NR-CFT
Energy eigenvalue in a harmonic potential
9
Trivial examples of

• Noninteracting particles in d dimensions

operator
state
N1 Lowest operator
. . .
2nd lowest operator
N3
Valid for any nonrelativistic scale invariant
systems !
10
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 !
11
Commutator D, H

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

Generator of dilatation
scale invariance
12
Primary operator
Local operator has

• scaling dimension
• particle number

Primary operator
E.g., primary operator nonprimary
operator
13
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
!!!
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 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 d2 or d4
15
e expansion for fermions at unitarity


Part II

• Field theories for fermions at unitarity
• perturbative near d2 or d4
• Scaling dimensions of operators
• up to 6 fermions
• expansions over e d-2 or 4-d
• Extrapolations to d3

16
Specialty of d4 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 d2 or
d4 ?
17
Field theories at unitarity 1

• Field theory becoming perturbative near d2

Renormalization of g
RG equation
Fixed point
The theory at fixed point is NR-CFT for fermions
at unitarity
Near d2, weakly-interacting fermions
perturbative expansion in terms of ed-2
Y.N. and D.T.Son, PRL(06) PRA(07) P.Nikolic
and S.Sachdev, PRA(07)
18
Field theories at unitarity 2

• Field theory becoming perturbative near d4

WF renormalization of ?
RG equation
Fixed point
The theory at fixed point is NR-CFT for fermions
at unitarity
Near d4, weakly-interacting fermions and
bosons perturbative expansion in terms of e4-d
Y.N. and D.T.Son, PRL(06) PRA(07) P.Nikolic
and S.Sachdev, PRA(07)
19
Scaling dimensionsnear d2 and d4
Strong coupling
d4
d2
d3
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

• Anomalous dimension near d2

• Anomalous dimension near d4

Ground state energy of N2 is exactly in any
2?d?4
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3-fermion operators near d2

• Lowest operator has L1 ground state

?
?
?
O(e)
O(e)
?
?
?
N3 L1
N3 L0

• Lowest operator with L0 1st excited state

22
3-fermion operators near d4

• Lowest operator has L0 ground state

?
O(e)
O(e)
?
?
?
?
?
N3 L0
N3 L1

• Lowest operator with L1 1st excited state

23
Operators and dimensions

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

e.g. N5
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 d3

• Naïve extrapolations of NLO results to d3

) S. Tan, cond-mat/0412764 ) D. Blume et
al., arXiv0708.2734
Extrapolated results are reasonably close to
values in d3
But not for N4,6 from d4 due to huge NLO
corrections
26
3 fermion energy in d dimensions
2d
4d
4d
2d
Fit two expansions using Padé approx.
Interpolations to d3
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 L0 and L1 states at d
3.3277
Ground state at d3 has L1
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 d2 or d4
• 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 d2 (weakly-interacting
  fermions)
• and d4 (weakly-interacting bosons fermions)
• Exact results for N2, 3 fermions in any
  dimensions d
• Padé fits of NLO expansions agree well with
  exact values
• Underestimate values in d3 as N is increased

How to improve e expanions ?
Accurate predictions in 3d

• Calculations of NNLO corrections
• Are expansions convergent ? (Yes, when N3 !)
• What is the best function to fit two expansions ?

31
Backup slides
32
BCS-BEC crossover
Eagles (1969), Leggett (1980) Nozières and
Schmitt-Rink (1985)
Strong interaction
?
Superfluidphase
-B
?
-?
0
BEC of molecules BCS of atoms
Unitary Fermi gas

• Strong coupling limit a kF??
• Atomic gas _at_ Feshbach resonance
• Simple scaling and universality

33
Measurement of 2 fermion energy
T. Stöferle et al., Phys.Rev.Lett. 96 (2006)
a??
Energy in a harmonic potential

• Schrödinger eq.
• CFT calculation

34
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, arXivcond-mat/0412764
35
5 fermion energy in d dimensions
2d
2d
4d
4d

• Level crossing between L0 and L1 states at d gt
  3
• Padé interpolations to d3

span in a small interval but underestimate
numerical values at d3
36
4 fermion and 6 fermion energy
2d
2d
4d
4d

• Ground state has L0 both near d2 and d4
• Padé interpolations to d3

4/0, 0/4 Padé are off from others due to huge
4d NLO
37
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-Schrödingerper
turbation
New analyticresultsconsistentwithnumericalres
ults
Cf. anyon field interacts via Chern-Simons
gauge field
38
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-Schrödingerper
turbation
New analyticresultsconsistentwithnumericalres
ults
Cf. anyon field interacts via Chern-Simons
gauge field
39
Extrapolation to d3 from d4-e

• Keep LO NLO results and extrapolate to e1

NLO corrections are small 5 35
Good agreement with recent Monte Carlo data
J.Carlson and S.Reddy, Phys.Rev.Lett.95, (2005)
cf. extrapolations from d2e
NLO are 100
40
Matching of two expansions in x

• Borel transformation Padé approximants

Expansion around 4d
x E / Efree
?0.42
2d boundary condition
2d
4d

• Interpolated results to 3d

d
41
Critical temperature

• Critical temperature from d4 and 2

NLO correctionis small 4

• Interpolated results to d3

MC simulations

• Bulgac et al. (05) Tc/eF 0.23(2)
• Lee and Schäfer (05) Tc/eF lt 0.14
• Burovski et al. (06) Tc/eF 0.152(7)
• Akkineni et al. (06) Tc/eF ? 0.25

42
NNLO correction for x

• NNLO correction for x

Arnold, Drut, Son, Phys.Rev.A (2006)
x
Nishida, Ph.D. thesis (2007)
Fit two expansions using Padé approximants

• Interpolations to 3d
• NNLO 4d NNLO 2d
• cf. NLO 4d NLO 2d

d