Transcript Fermions at unitarity as a nonrelativistic CFT Yusuke Nishida (INT, Univ. of Washington) in collaboration with D.

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]
18 March, 2008 @ TRIUMF
Contents of this talk
1. 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 e ( = d-2, 4-d) expansions
 scaling dimensions near d=2 and d=4
 extrapolations to d=3
4. Summary and outlook
3/30
Introduction
Fermions at infinite scattering length
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
4/30
5/30
Scale invariant interactions
1. Interaction via 1/r2 potential
2. 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)
3. Interaction due to fractional statistics in d=2
• Anyons
R. Jackiw and S.Y. Pi, Phys. Rev. D42, 3500 (1990)
• Resonantly interacting anyons
Y.N., Phys. Rev. D (2008)
Experimental realization in cold atoms !
Feshbach resonance
Cold atom experiments
C.A.Regal and D.S.Jin,
Phys.Rev.Lett. 90 (2003)
6/30
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
a
add = 0.6 a >0
40K
B (Gauss)
V0(a)
r0
7/30
Fermions in the unitarity limit
0.6 a
+
Strong interaction
weak repulsion
0
a
weak attraction
Fermions at unitarity
• Strong coupling limit : |a|
• Cold atoms @ Feshbach resonance
• 0r0 << lde Broglie << |a|
• Scale invariant
-
a=
l
Nonrelativistic CFT
Cf. neutrons : r0~1.4 fm << |aNN|~18.5 fm
External potential breaks scale invariance
Isotropic harmonic potential
NR-CFT in free space
8/30
Part I
NR-CFT and operator-state
correspondence
Scaling dimension of operator in NR-CFT
Energy eigenvalue in a harmonic potential
9/30
Trivial examples of
• Noninteracting particles in d dimensions
operator
state
N=1 : Lowest operator
2nd lowest operator
N=3 :
Valid for any nonrelativistic scale invariant systems !
Nonrelativistic CFT
C.R.Hagen, Phys.Rev.D (’72)
U.Niederer, Helv.Phys.Acta.(’72)
10/30
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]
• E.g. Hamiltonian with two-body potential V(r)
Generator of dilatation :
scale invariance
11/30
12/30
Primary operator
Local operator
has
• scaling dimension
• particle number
Primary operator
E.g., primary operator :
nonprimary operator :
Proof of correspondence
13/30
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
!!!
Operator-state correspondence
14/30
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 d=2 or d=4
Part II
e expansion
for fermions at unitarity
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 e = d-2 or 4-d
3. Extrapolations to d=3
15/30
Specialty of d=4 and 2
Z.Nussinov and S.Nussinov, 16/30
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 ?
17/30
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)
18/30
Field theories at unitarity 2
• Field theory becoming perturbative near d=4
WF renormalization of 
RG equation :
p
p
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)
19/30
Scaling dimensions
near 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)
20/30
2-fermion operators
• Anomalous dimension near d=2
• Anomalous dimension near d=4
p
Ground state energy of N=2 is exactly
p
in any 2d4
21/30
3-fermion operators near d=2
• Lowest operator has L=1
ground state




• Lowest operator with L=0

N=3
L=1
O(e)

O(e)
1st excited state
N=3
L=0
22/30
3-fermion operators near d=4

N=3
L=0
O(e)


• Lowest operator with L=1
ground state

O(e)

• Lowest operator has L=0

1st excited state
N=3
L=1
23/30
Operators and dimensions





e.g. N=5
 
• NLO results of e =d-2 and e =4-d expansions
  
Operators and dimensions
24/30
• NLO results of e =d-2 and e =4-d expansions
O(e)
O(e2)
O(e)
Comparison to results in d=3
25/30
• 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
26/30
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 !
27/30
Exact 3 fermion energy
Padé fits have behaviors consistent with
exact 3 fermion energy in d dimension
Exact
is
computed from
=
+
28/30
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 and outlook 1
29/30
• 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
30/30
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 ?
31/30
Backup slides
BCS-BEC crossover
Eagles (1969), Leggett (1980)
32/30
Nozières and Schmitt-Rink (1985)
Strong interaction
Superfluid
phase
?
+ BEC of molecules
0
BCS of atoms -
Unitary Fermi gas
• Strong coupling limit : |akF|
• Atomic gas @ Feshbach resonance
• Simple scaling and universality
-B
Measurement of 2 fermion energy
33/30
T. Stöferle et al., Phys.Rev.Lett. 96 (2006)
|a|
Energy in a harmonic potential
• Schrödinger eq.
• CFT calculation
34/30
Ladders of eigenstates
...
E
...
F.Werner and Y.Castin, Phys.Rev.A 74 (2006)
...
• Raising and lowering operators
...
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
35/30
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
36/30
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
37/30
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
Coincide
with results
by RayleighSchrödinger
perturbation
Cf. anyon field
interacts via Chern-Simons gauge field
New analytic
results
consistent
with
numerical
results
38/30
Anyon spectrum to NLO
• Ground state4 energy
of N anyons in a harmonic potential
anyon spectrum
Sporre et al.,inPhys.Rev.B
PerturbativeM.
expansion
terms of(1992)
statistics parameter a
a0 : boson limit
a1 : fermion limit
Coincide
with results
by RayleighSchrödinger
perturbation
Cf. anyon field
interacts via Chern-Simons gauge field
New analytic
results
consistent
with
numerical
results
Extrapolation to d=3 from d=4-e
39/30
• Keep LO & NLO results and extrapolate to e=1
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 d=2+e
NLO are 100 %
Matching of two expansions in x
40/30
• Borel transformation + Padé approximants
Expansion around 4d
x = E/Efree
♦=0.42
2d boundary condition
2d
• Interpolated results to 3d
4d
d
41/30
Critical temperature
• Critical temperature from d=4 and 2
NLO correction
is small ~4 %
• Interpolated results to d=3
Tc / eF
4d
MC simulations
• Bulgac et al. (’05): Tc/eF = 0.23(2)
• Lee and Schäfer (’05): Tc/eF < 0.14
• Burovski et al. (’06): Tc/eF = 0.152(7)
• Akkineni et al. (’06): Tc/eF  0.25
2d
d
NNLO correction for x
42/30
• NNLO correction for x
Arnold, Drut, Son, Phys.Rev.A (2006)
Nishida, Ph.D. thesis (2007)
x
Fit two expansions
using Padé approximants
Interpolations to 3d
• NNLO 4d + NNLO 2d
cf. NLO 4d + NLO 2d
d