Fermions at unitarity as a nonrelativistic CFT
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Transcript of 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]
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
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Fermions at infinite scattering length
Introduction
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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 !
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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|
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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)
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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
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. . .
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
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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
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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
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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)
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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
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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
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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
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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)