Fermions at unitarity as a nonrelativistic CFT Yusuke Nishida (INT, Univ. of Washington) in collaboration with D.
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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
• 0r0 << 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 d4
Pair wave function is concentrated at its origin
Fermions at unitarity in d4 form free bosons
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 ?
17/30
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 2d4
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
a0 : boson limit
a1 : 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
a0 : boson limit
a1 : 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