Presentation at BEC center in November 2015.

1. Spectral properties of the Goldstino in
supersymmetric Bose-Fermi mixtures
Daisuke Satow (ECT*, !)
Collaborators: Jean-Paul Blaizot (Saclay CEA, ")
Yoshimasa Hidaka (RIKEN, #)
J. P. Blaizot, Y. Hidaka, D. S., arXiv:1510.06525 [cond-mat. quant-gas].

2. Outline
2
•Properties of goldstino in cold atom system
•Free Case
•Interacting Case
•How can we observe the goldstino
experimentally in cold atom system?
•Introduction (Supersymmetry, goldstino in
QCD…)
•Summary

3. 3
Symmetry related to interchanging a
boson and a fermion
Supersymmetry (SUSY)
fb fb
=

4. 4
(Quasi) SUSY in hot QCD
V. V. Lebedev and A. V. Smilga, Annals Phys. 202, 229 (1990)
There is SUSY approximately if we neglect the
interaction.
q g
=
Both of the quark and the gluon are regarded as
massless at high T

5. 5
SUSY can be realized also in Cold Atom System
Cold atom systems are simple and
clean many-body systems that
can feature aspects of SUSY.
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
Picture: Ferlaino group, Innsbruck.
cf: T. Ozawa, Nature Physics 11, 801 (2015),
Wess-Zumino model: Y. Yu, and K. Yang, PRL 105, 150605 (2010),
Dense QCD: K. Maeda, G. Baym and T. Hatsuda, PRL 103, 085301 (2009),
Relativistic QED: Kapit and Mueller, PRA 83, 033625 (2011).
(The detail will be explained later)

6. 6
Supercharge operator: annihilate one fermion
and create one boson (and its inverse process)
SUSY and supercharge
SUSY if [Q, H]=0
fb
Q: supercharge
fb

7. 7
SUSY breaking
E
nf
E
nb
E
nf
∫ddk
Q†
εk
E
nb
εk
µf
(T=0, weak coupling)
SUSY breaking due to the medium
δE=Δµ=µf -µb>0
Only this state is realized
as ground state.

8. 8
NG “fermion” which is related to SUSY breaking
Generally, symmetry breaking generates zero energy
excitation (Nambu-Goldstone (NG) mode).
symmetry breaking
order parameter
NG mode
We expect that SUSY breaking
also generates NG mode.

9. 9
“When the order parameter is finite, the propagator
in the left-hand side has a pole at k→0.”
Order parameterNG mode
I.
−ikµ d4
xeik·(x−y)
⟨T Jµ
(x)O(y)⟩ = ⟨{Q, O(y)}⟩,
−ikµ d4
xeik·(x−y)
⟨T Jµ
(x)O(y)⟩ = ⟨{Q, O}⟩
Broken symmetry
Nambu-Goldstone theorem:
(fermion ver.)
Jµ: supercurrent
Q=J0: supercharge
NG “fermion” which is related to SUSY breaking
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989)

10. 10
If we set O=Q, order parameter is energy-momentum
tensor (T µν) in the relativistic system.
Order parameterNG mode
I.
−ikµ d4
xeik·(x−y)
⟨T Jµ
(x)O(y)⟩ = ⟨{Q, O(y)}⟩,
−ikµ d4
xeik·(x−y)
⟨T Jµ
(x)O(y)⟩ = ⟨{Q, O}⟩
Broken symmetry
single fermion intermediate state. Hence, we can write, in momentum space
Γµν
JJ (k) ≡ d4
x eik(x−y)
⟨TJµ
(x)Jν
(y)⟩
= Γµ
JΨ(k)S(k)Γν
ΨJ (k).
By making use of the Ward-Takahashi identity (3.39), we find
−ikµΓµν
JJ (k) = −im⟨A⟩Γν
ΨJ (k).
This can now be calculated by using our earlier results for Γµ
JΨ(k).
First, in the limit of high temperature, we make use of (3.37) and obtai
the left hand side of the Ward-Takahashi identity as
−ikµΓµν
JJ (k) = m2
⟨A⟩2 π2
g2
γ0
−1
3 γi
=
π2
4
T4
γ0
−1
3 γi
.
For the right hand side, we must evaluate the thermal expectation valu
of the energy-momentum tensor. In a thermal equilibrium state, it reads
⟨Tµν
⟩ = diag(ρ, p, p, p),
where in the high-temperature limit the pressure is given by p = 1
3ρ, an
the energy density can be calculated as
(µ=0)
NG “fermion” which is related to SUSY breaking
In the presence of medium, SUSY is always broken.
V. V. Lebedev and A. V. Smilga, Nucl. Phys. B 318, 669 (1989)

11. 11
SUSY breaking
SUSY is fermionic symmetry,
so the NG mode is fermion (Goldstino).
Rare fermionic zero mode
fb
Q
fb
NG mode appears in <QQ>.
NG mode
−ikµ d4
xeik·(x−y)
⟨T Jµ
(x)O(y)⟩ = ⟨{Q, O(y)}⟩,
−ikµ d4
xeik·(x−y)
⟨T Jµ
(x)O(y)⟩ = ⟨{Q, O}⟩

12. Quasi-goldstino in hot QCD
12
In weak coupling regime, we established the
existence of the (quasi) goldstino in QCD.
Y. Hidaka, D. S., and T. Kunihiro, Nucl. Phys. A 876, 93 (2012),
D. S., PRD 87, 096011 (2013).
Dispersion relation Reω＝p/3
Damping rate Imω=ζq+ζg=O(g2T)
Residue
components are different in both gauges.
We can solve the self-consistent equation in the same way as in QED. The solution is
as follows: We expand the vertex function in terms of ˜p/g2T as in Eq. (2.26). The zeroth
order solution is
A =
1
1 + Cf λ
1, B(k) =
1
k0
2Cf λ
1 + Cf λ
γ0
, C(k) = 0. (2.49)
Here A, B(k), and C(k) are defined by Eq. (2.27), and λ is defined by Eq. (2.7). The
self-consistent equation which determines the vertex function at the first order δΓµ(p, k)
is written as
δΓµ
(p, k) = Cf
d4k′
(2π)4
˜X(k′
)
kνγµ + γνk′µ
k · k′
/k′
PT
νρ(k′
) −2
˜p · k′
δm2
Aγρ
+ δΓρ
(p, k′
) .
(2.50)
Then δΠ(p) defined by Eq. (2.33) becomes
δΠ(p) =
16π2A2λ3Cf
g2
(γ0
(p0
+ iζ) + vp · γ). (2.51)
Owing to Eq. (2.43), the retarded quark self-energy is found to be
ΣR
(p) = Cf
d4k
(2π)4
˜X(k)
γµ/kPT
µν(k)Γν(p, k)
1 + 2˜p · k/δm2
= −
1
Z
(γ0
(p0
+ iζ) + vp · γ),
(2.52)
where the expression of the residue Z will be given shortly. We note that this expression
is the same as that in QED except for the numerical factor. Thus, the expression for the
pole position of the ultrasoft mode in QCD is the same as in QED, while the residue of
that mode is not: The residue is
Z =
g2
16π2λ2Cf
(1 + Cf λ)2
=
g2N
8π2(N2 − 1)
5
6
N +
1
2N
+
2
3
Nf
2
. (2.53)
We can also show that the analytic solution of the self-consistent equation satisfies
the WT identity in the leading order as in QED, by checking that the counterparts of
Eqs. (2.40) and (2.41) are satisfied.

13. Motivation of this research
13
What is the properties
of the goldstino
in cold atom system?
Cold atom system is
easier to realize, and cleaner.

14. 14
SUSY in Cold Atom System
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
Trap two kinds of fermion (f, F) and their
bound state (boson: b) on optical lattice.
f F
b

15. 15
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
=
(tf =tb=th)
f b
How to tune t: M. Snoek, S. Vandoren, and H. T. C. Stoof, PRA 74, 033607 (2006)
H = tf
X
hiji
f†
i fj tb
X
hiji
b†
i bj
SUSY in Cold Atom System

16. 16
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
ry.
atoms
lattice
ns and
Y [3].
d state
nergy
e b-f
mixture can be adjusted to negligibly small. In the tight-
binding approximation, one has
Hα = −
⟨ij⟩
tαa
α†
i aα
j − µα
i
a
α†
i aα
i ,
(1)
V =
Ubb
2 i
nb
i nb
i − 1 + Ubf
i
nb
i n
f
i ,
011604-1 ©2010 The American Physical Society
=
F decouples.
(Ubb =Ubf =U)
When tf =tb, Ubb =Ubf, Q =bf † commutes with the
Hamiltonian.
SUSY in Cold Atom System

17. 17
If we set O=Q†, NG mode appears in <QQ†>.
Order parameter is density (<{Q, Q†}>=ρ) in this case.
NG “fermion” which is related to SUSY breaking
Order parameterNG mode
I.
−ikµ d4
xeik·(x−y)
⟨T Jµ
(x)O(y)⟩ = ⟨{Q, O(y)}⟩,
−ikµ d4
xeik·(x−y)
⟨T Jµ
(x)O(y)⟩ = ⟨{Q, O}⟩
Broken symmetry
Q =bf †
Q† =b†f
SUSY is always broken when ρ is finite.

18. 18
Sum rules
Canonical (anti-)commutation relations:
Q ≡ dd
x q
where q(x) annihilates one boson and creates one fermion at
q(x) ≡ f†
(x
By using the canonical commutation and anticommutation re
[b(x), b†
(y)] = δ(d)
(x − y), {f
the following operator
Q ≡ dd
x q(x),
where q(x) annihilates one boson and creates one fermion at point x
q(x) ≡ f†
(x)b(x).
By using the canonical commutation and anticommutation relations
[b(x), b†
(y)] = δ(d)
(x − y), {f(x), f†
(y)} = δ(d)
(x − y),
Model-independent sum rules
symmetry breaking: the location of the pole in Eq. (2.2
vanishes. Because of this analogy with the Nambu-Gold
“Goldstino”. It is a collective excitation, as reflected for i
to the total density ρ. But in contrast to most collecti
nature, the Goldstino carries fermionic quantum numbe
parameter for SUSY breaking (recall that the action of t
number of particle in that state, see Eq. (2.10)). Note
breaking induced by the difference of chemical potentials
C. Spect
The spectral function σ(ω, p) = 2ImGR
(ω, p):
σ(ω, p) =
∞
−∞
dt eiωt
d
obeys sum rules which can be used to obtain model ind
finite momentum. We shall consider here more specifical
∞
−∞
dω
2π
σ(ω, p) = dd
x e
where in the last step, we have used the (anti)-commutat
the residue of the pole at ω = ∆µ in Eq. (2.21). In othe
we have already observed.
leading after some calculation
where ˜ω ≡ ω − ∆µ0
, ˜ωf ≡ ˜ω −
temperature, µf ≃ ϵF and µb
from the region of ω values wh
1.
2.
Z 1
1
d˜!
2⇡
˜! (˜!, p) = thp2
(⇢f ⇢b)
Z 1
1
d˜!
2⇡
(˜!, p) = ⇢f + ⇢b

19. Free case (U=0)
19
Calculate the spectrum of the Goldstino.
(2 dimensions No BEC.)
(continuum, not lattice model)
Since Q =bf †, the propagator
in free limit is <Q†(x)Q(0)>=
f
b

20. Result (U=T=0)
20
Continuum+Pole.
-10
-5
0
5
10
0 0.5 1 1.5 2 2.5 3
~
/th
p
ρf=0.5, ρb=1.0

21. 21
Results (T=0)
21
Landau damping
Cut
Continuum.
εk=thk2, |k|<kF at T=0
(ω, p) (ω+thk2, p+k)
should be on-shell: ω=thp2+2thp k cosθ
Width is Δω=2thpkF
-1~10~kF

22. Results (T=0)
22
Landau damping
These pole and continuum satisfies the NG theorem.
Cut
Pole, No width.
εk, k=0 at T=0
(-ω, -p)
(-ω, -p)
should be on-shell: ω=-thp2
Other value of ω is not allowed!
Dispersion Relation ω＝Δµ -th p2
Strength ρb

23. Results (T=0)
23
Both the continuum and the pole
contribute to the sum rules (same order).
-10
-5
0
5
10
0 0.5 1 1.5 2 2.5 3
~
/th
p
Z 1
1
d˜!
2⇡
˜! (˜!, p) = thp2
(⇢f ⇢b)
Z 1
1
d˜!
2⇡
(˜!, p) = ⇢f + ⇢b

24. Finite T effect (U=0)
24
-10
-5
0 0.5 1 1.5 2 2.5 3
p
FIG. 4: Color online. The range of the support for the fermion hole continuum is indicated by the red shaded area. The black
curve indicates the location of the boson particle pole position (for p < kF ) and the boson hole pole for p > kF . The parameters
are, T = U = 0, ρb = 1, ρf = 0.5, corresponding to kF =
√
2π ≃ 2.5. The dashed horizontal line indicates the value ˜ω = −∆µ0
,
corresponding to ω = 0. It touches the minimum of the continuum at p = kF . When p > kF , the boson hole pole lies outside
the fermion continuum, and its energy ω < 0 corresponds to minus the excitation energy of the corresponding Goldstino.
0
1
2
-4 -2 0 2 4 6
σth
ω
~
/th
0
40
80
120
-1.4 -1.2 -1 -0.8 -0.6
σth
ω
~
/th
FIG. 5: Color online. The free (U/th = 0) spectral function at |p| = 1.0. The solid (black), dashed (red), and dotted (blue)
lines correspond to temperatures T/th = 0, 0.2, and 1.0, respectively. Left panel: excitations induced by q†
p. The large width
is due to the fermion continuum (ρf = 0.5, ρb = 1). Right panel: excitations induced by qp, with vanishing fermion density
(ρf = 0, ρb = 1).
(ρf =0)
T/th =0, 0.2, 1
Even at (unrealistically) high T (T/th=1),
the width of the peak is not large.
( )
At finite T, k can be finite,
so ω can have a width
(-ω, -p)
(-ω+εk, -p+k)
(p=1)

25. Analysis with Resummed Perturbation
25
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
The bare dispersion relations of the fermion and the boson are
the same (tf =tb=th), so the loop integral diverges at , p→0.
(pinch singularity)
˜!
(fermion term)= →∞
Z
d2
k
(2⇡)2
nF (✏k)
˜! th(2k · p + p2)
f
Switch on the interaction.

26. 26
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
(1) Resum density correction to the dispersion relations.
the pinch singularity is
regularized.
Uρb
Uρf
+
fermion:
boson:
2Uρb
(fermion term)= →U -1
Analysis with Resummed Perturbation
Z
d2
k
(2⇡)2
nF (✏f
k)
˜! [th(2k · p + p2) + U⇢]

27. 27
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
All ring diagrams contributes at the same order.
We need to sum up infinite ring diagrams.
U-1 ×U ×U-1=U-1U-1
Analysis with Resummed Perturbation
(2) Random Phase Approximation

28. 28
Results (T=0)
Continuum+Pole (as U=0 case), but the continuum is
shifted so that the pole is out of the continuum at small p.
(U/th=0.1)
-8
-6
-4
-2
0
2
0 0.5 1 1.5 2 2.5
~
/th
p

29. 29
Results (T=0)
Shift of continuum:
Cut
εk=thk2+Uρb, k<kF at T=0
(ω, p) (ω+thk2+Uρb, p+k)
should be on-shell: ω=thp2+2thpkcosθ+Uρ
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0 0.01 0.02 0.03 0.04 0.05
~
/th
p
shift: Uρ

30. Three momentum regions
30
~
energy/momentum
dependent term (A)
density term (B)
3 momentum region:
small p, intermediate p, large p
A<<B A~B A>>B
+(boson term)
Z
d2
k
(2⇡)2
nF (✏f
k)
˜! [th(2k · p + p2) + U⇢]

31. Results (T=0)
31
small p
Expand in terms of energy, momentum
~
Z
d2
k
(2⇡)2
nF (✏f
k)
A + B
'
Z
d2
k
(2⇡)2
nF (✏f
k)
B
✓
1
A
B
◆

32. Results (T=0)
32
Dispersion Relation ω＝Δµ -αp2
Strength
(maximum value allowed by sum rule)
At small temperature, and in the vicinity of ˜ω = 0, the A term in Eq. (3.13) is of order thp /(U ρ) and
of order t2
hp2
k4
f /(U3
ρ3
) ∼ t2
hp2
/(U3
ρ), while the first term is of order 1/U. At weak coupling U ≪ th
therefore much smaller than the B term.
By using the expansion (3.16) in Eq. (3.11), we obtain the retarded propagator in the form
GR
(ω, p) ≃ −
Z
˜ω + αp2
.
Here α and Z are given respectively by
α ≡
1
ρ
A −
B
Uρ
=
th(ρb − ρf )
ρ
+
t2
h
πUρ2
∞
0
d|k||k|3
(nf (ϵf
k) + nb(ϵb
k)),
≈
th
ρ
(ρb − ρf ) +
4πthρ2
f
Uρ
.
nd
Z ≡ ρ 1 +
B
U2ρ3
p2
= ρ − p2 t2
h
πU2ρ2
∞
0
d|k||k|3
(nf (ϵf
k) + nb(ϵb
k))
≈ ρ − p2
4πt2
hρ2
f
U2ρ2
.
-0.05
0
0.05
0.1
0.15
0 0.01 0.02 0.03 0.04 0.05
~
/t
p
A = th(ρb − ρf ),
B = −
t2
h
π
∞
0
d|k||k|3
(nf (ϵf
k) +
Next, we expand Eq. (3.13) for small ˜ω, which yields
G0
=
1
U
1 +
˜ω
Uρ
+
˜ω
Uρ
2
+
p2
Uρ2
A −
B
Uρ
+ O(˜ω3
, ˜ω2
p2
, p4
).
At small temperature, and in the vicinity of ˜ω = 0, the A term in Eq.
is of order t2
hp2
k4
f /(U3
ρ3
) ∼ t2
hp2
/(U3
ρ), while the first term is of orde
is therefore much smaller than the B term.
By using the expansion (3.16) in Eq. (3.11), we obtain the retarded
GR
(ω, p) ≃ −
Z
˜ω + αp2
.
Here α and Z are given respectively by
α ≡
1
ρ
A −
B
Uρ
=
th(ρb − ρf )
ρ
+
t2
h
πUρ2
∞
0
d|k||k|3
(n
≈
th
ρ
(ρb − ρf ) +
4πthρ2
f
Uρ
.
and
Z ≡ ρ 1 +
B
U2ρ3
p2
= ρ − p2 t2
h
πU2ρ2
∞
0
d|k||k|3
(nf (ϵ
4πt2
ρ2
α=

33. 33
Results (T=0)
Contributions to the sum rules
In contrast to U=0 case, the pole
has almost all the spectral weight!
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.01 0.02 0.03 0.04
p
continuum
pole
total
expansion
1.
Z 1
1
d˜!
2⇡
(˜!, p) = ⇢ p2
4⇡t2
h⇢2
f
U2⇢2
+ p2
4⇡t2
h⇢2
f
U2⇢2

34. 34
Results (T=0)
The contributions from the pole and the continuum have
the same order, but in a different way from U=0 case.
-0.02
-0.01
0
0.01
0.02
0 0.01 0.02 0.03 0.04
p
continuum
pole
total
expansion
2.
Z 1
1
d˜!
2⇡
˜! (˜!, p) = p2
"
th(⇢f ⇢b)
4⇡t2
h⇢2
f
U2⇢2
#
+ p2
4⇡t2
h⇢2
f
U2⇢2
Pole: ~p2 ~p0
Continuum: ~p2~p0
Z 1
1
d˜!
2⇡
˜! (˜!, p) = p2
"
Continuum: ~p0~p2
cf: U=0 case

35. Results (T=0)
35
intermediate p
The pole is absorbed by the continuum,
No sharp peak.
0
10
20
30
40
50
60
-0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
th
~
/th
p=0.05

36. Results (T=0)
36
density effect is negligible→free theory
~
Z
d2
k
(2⇡)2
nF (✏f
k)
A + B
'
Z
d2
k
(2⇡)2
nF (✏f
k)
A
large p

37. Results (T=0)
37
Pole+Continuum
Dispersion Relation ω＝Δµ -th p2
Strength ρb (<ρ)
(as in U=0 case)
-8
-7.5
-7
-6.5
-6
-5.5
-5
2 2.2 2.4 2.6 2.8 3
~
/th
p

38. 38
Results (T=0)
medium effect: large smallintermediate
Crossover from small p to large p
-8
-6
-4
-2
0
2
0 0.5 1 1.5 2 2.5
~
/th
p
ω＝Δµ-αp2
Z＝ρ
ω＝Δµ-th p2
Z＝ρb

39. 39
Finite T effect
T/th =0, 0.2, 1
13
0
10
20
30
0 0.1 0.2 0.3
σth
ω
~
/th
0
20
40
60
-0.2 0 0.2 0.4
σth
ω
~
/th
4
p=0.002
13
0
10
20
30
0 0.1 0.2 0.3
σth
ω
~
/th
0
20
40
60
-0.2 0 0.2 0.4
σth
ω
~
/th
4
p=0.05
0
10
20
0 0.1 0.2 0.3
σth
ω
~
/th
0
20
40
-0.2
σth
0
200
-0.4 -0.2 0 0.2 0.4
σth
ω
~
/th
0
2
4
-1
σth
FIG. 8: Color online. The spectral function at various temperature. The results at |p| =
upper-left (0.002), upper-right (0.05), and lower panels (0.25). The large (small) σ part is
panel so that the shape of the peak (continuum) becomes clearer. The solid (black) line
(red) line to T/th = 0.2, and the dotted (blue) line to T/th = 1, respectively. The plot on
the lower left panel, to show the structure of the peak near ˜ω = 0.
These two expressions (3.18) and (3.19) determine the spectral properties of the G
last approximate equalities in the equations above are valid at small temperatu
neglected. The dispersion relation is quadratic [4, 5], but the coefficient of p2
,
from Eq. (2.32) under the assumption that the Goldstino pole exhausts the sum
does not exist in Eq. (2.32). In fact, at weak coupling, the Goldstino ceases to e
momentum becomes finite. Equation (3.19) reveals indeed that the residue decr
from its maximum value Z = ρ achieved at |p| = 0. The Goldstino continues to
in Fig. 7, left panel, its quadratic dispersion being accurately reproduced with the
|p| 0.02.
The spectral strength missing in the residue is moved to the continuum, in agree
This is illustrated by the plot of the spectral function (the upper-left panel of Fig. 8)
of the pole and the continuum to the sum rule are displayed in Fig. 9, upper pa
|p| smaller than 0.02, the spectral weight of the pole is well described by the resu
Eq. (3.19). At such small |p|, the spectral weight of the continuum is almost negl
|p| increases. Above |p| = 0.04, where the pole hits the continuum, all the spectral
A similar analysis can be performed for the second sum-rule, Eq. (2.31). For co
p=0.25
Modestly smeared.
-1.5
-1
-0.5
0
0.5
1
0 0.05 0.1 0.15 0.2 0.25 0.3
~
/th
p

40. 40
The coefficient α increases when T increases.
14
15
16
17
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
α/t
T/t
U/t=0.1, ρf=0.5, ρb=1.0
If typical value of k (~T) increases, this term becomes larger.
To get the expression of the dispersion relation of the goldstino at small
ω − ∆µ and |p|, we expand G0
in terms of ω − ∆µ and p: G0
≃ −1/U +
A(ω − ∆µ + iϵ) + Bp2
, where the term that is proportional to |p| vanishes
due to the rotational symmetry. The coefficients are given by
A = −
1
U2ρ
, (36)
B =
t(ρf − ρb)
(Uρ)2
−
t2
(Uρ)3π
∞
0
d|k||k|3
(nf (ϵf
k) + nb(ϵb
k)). (37)
Then, Eq. (28) becomes
GR
(p) ≃ −
Z
ω − ∆µ + αp2 + iϵ
. (38)
We see that the pole and the residue of GR
(p) has a form predicted by the
projection operator method, Eq. (??). α and Z are given by
α ≡
B
A
(39)
=
t(ρb − ρf )
ρ
+
t2
πUρ2
∞
0
d|k||k|3
(nf (ϵf
k) + nb(ϵb
k)),
Z ≡ −
1
U2A
= ρ. (40)
We note that, reflecting the fact that the small momentum expansion we
α＝
T-dep.T-indep.
Finite T effect

41. Effect on Observable Quantity
41
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
Photoassociation (PA) process
b
cesses which contributes to the PA variation of F-fermion number. The
Q†
)/
√
2, i(Q − Q†
)/
√
2, Nf + Nb, and Nf − N
Q + Q†
√
, i
Q − Q†
two lasers
f F
Binding energy: Eb
cesses which contributes to the PA variation of F-fermion number. The
Q†
)/
√
2, i(Q − Q†
)/
√
2, Nf + Nb, and Nf − Nb instead
Q + Q†
√
2 , i
Q − Q†
√
xample, the PA spectrum which displays the molecular formation rate
PACS number(s): 67.85.Pq, 11.30.Pb, 37.10.Jk
ersymmetry
atoms have
mic system,
i atom, its
onrelativity,
ergy SUSY
latter, such
example, a
decays to a
pton) [4].
he mixture,
ced. For a
ndensation,
since they
n operators
ulated by a
mixture [7]. To probe the SUSY behaviors, we load a free
Fermi atom F gas, which does not interact with both b and f
directly. In a photoassociation (PA) process [8], the transitions
between two atoms and one molecule, that is, Ff ↔ b, are
induced by two laser beams with frequencies ω1 and ω2. For
the SUSY b-f mixture, this resembles a high-energy physics
process: a quark or a lepton (f ) absorbs a fermionic gaugino
(“absorbs” an F and emits a photon) and decays to a squark
or a slepton (b) or vice verse. (One can also consider f to be
a Fermi molecule formed by the bound state of a Bose atom b
and a Fermi atom F, i.e., processes Fb ↔ f . We study these
processes separately.)
For a negative detuning, δ0 = ω2 − ω1 − Eb, we show that
the molecule dissociation process b → Ff is forbidden. In
the formation process Ff → b, two types of new fermionic
excitations, an individual (bosonic) particle–(fermionic) hole

42. 42
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
F couples with goldstino through PA process.
Production rate of F should contain
information of spectrum of goldstino.
Effect on Observable Quantity
transit to the molecule b by emitting a photon with frequency
ω2. Meanwhile, the molecule b may also be excited to |1⟩
by the ω2 beam and then is unbound with some probability
by emitting a photon with frequency ω1. For large detuning
0, the state |1⟩ can be eliminated adiabatically, so that the PA
is modeled by the tight-binding Hamiltonian
Hex =
i
(gib
†
i fiFieiδ0t
+ H.c.), (2)
where the detuning δ0 = ωc − Eb, with the effective driven
frequency ωc = ω2 − ω1, and gj = g0 exp(−ik0 · rj ), with
g0 ∝ dd
r exp(−ik0 · r)w∗
b(r)wf (r)wF (r) being the coupling
intensity independent of the site.
In the k space, the Hamiltonian Hex is rewritten as
Hex = g0
√
ρ
k
Q
†
k−k0
Fkeiδ0t
+ H.c. , (3)
where Ekp
Ubf and ρ
lated pole
and indivi
Next w
dimension
ρf = 0.5.
δt = tb −
Ec(k) ≃
poles of th
where µ
Ec(|k|, θ)
θ = arctan
as |k| incr
011604-2
goldstino
α=b,f
Hα = −tα
⟨ij⟩
aα†
i aα
j − µα
i
aα†
i aα
i ,
V =
Ubb
2 i
b†
i b†
i bibi + Ubf
i
b†
i bif†
i fi,
Hex = HF + g0
i
b†
i fiFieiδ0t
+ H.c. ,
(3.1)
where aα
i = bi, fi, and Fi are the annihilation operator of
b, f, and F at site i. We introduced detuning δ0, which is
determined by the laser frequency, and assumed that the
laser momentum is negligible [13]. H0 is the same form
as the Hamiltonian of the Hubbard model. For detail of
the setup, see Ref. [13].
The annihilation operator of the goldstino is defined as
qi ≡ bif†
i , and the supercharge is defined as Q ≡ i qi.
of the gol
Eq. (2.43)
this expre
We calc
operator i
done at T
by adoptin
a clear vie
for determ
the retard
kinetic equ
the case t
temperatu
is µb (µf )

43. 43
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
are not qualitatively affected by T and nor is
ar formation rate. For simplicity, we take a zero-
approximation in our calculation. It follows from
he rate R = Rb→Ff − RFf →b contains two parts:
=
k
2πg2
0ρA k − k0, εF
k − δ0 θ δ0 − εF
k ,
(6)
=
k
2πg2
0ρA k − k0, εF
k − δ0 θ −εF
k ,
ctively, are the dissociation rate for b → Ff and
n rate for Ff → b.
e and individual fermionic modes. In order to
the weak interactions, we perturbatively calculate
3
. 1: Two processes which contributes to the PA variation of F-fermion number. The information on the goldstino spectrum
mbedded.
we use (Q + Q†
)/
√
2, i(Q − Q†
)/
√
2, Nf + Nb, and Nf − Nb instead of Q, Q†
, Nf , and Nb, we get
Q + Q†
√
2
, i
Q − Q†
√
2
= −i[Q, Q†
] = −i −2Q†
Q + Nf + Nb , (2.22)
Q + Q†
√
2
, i
Q − Q†
√
2
=
Q + Q†
√
2
,
Q + Q†
√
2
= i
Q − Q†
√
2
, i
Q − Q†
√
2
= 0, (2.23)
Q + Q†
√
2
,
Q + Q†
√
2
= i
Q − Q†
√
2
, i
Q − Q†
√
2
= {Q, Q†
} = Nf + Nb, (2.24)
Q + Q†
√
2
,
Nf + Nb
√
2
= i
Q − Q†
√
2
,
Nf − Nb
√
2
= 0, (2.25)
Q + Q†
N − N
(εF
K-δ0, k-k0)
(δ0, k0)
RΠ
f
attice with cold
ts denote Fermi
Lower: The PA
ding energy Eb.
ory. The Green
mation. Upper:
the free Green
he solid (green)
en functions of
d F) are the
; and µα and
mber operators
litudes tα and
unction wα(r)
nhomogeneity
cules has been
miltonian Hbf
µb = µf [1,3].
ameters obey,
here ρf is the
meters of the
ously shining
nto the lattice
k
where the retarded Green function is given by
DR(k, ω) =
∞
−∞
dxA(k − k0, x)
nf (x) − nf εF
k
x − εF
k − ω − i0+
, (5)
in terms of one loop calculations [Fig. 1(b)]. The single-
particle dispersions are εα
k = −2tα
d
s=1 cos ks − µα, where
the lattice spacings are set to be the unit. nf (x) is the
Fermi distribution at temperature T and the spectral function
A(k, ω) = −Im R(k, ω)/π is defined by the retarded Green
function R(k, ω) = −i
∞
0 dt⟨g|{Qk(t), Q
†
k(0)}|g⟩eiωt
.
At sufficiently low temperature, the pole and branch cut
in Eq. (5) are not qualitatively affected by T and nor is
the molecular formation rate. For simplicity, we take a zero-
temperature approximation in our calculation. It follows from
Eq. (5) that the rate R = Rb→Ff − RFf →b contains two parts:
Rb→Ff =
k
2πg2
0ρA k − k0, εF
k − δ0 θ δ0 − εF
k ,
(6)
RFf →b =
k
2πg2
0ρA k − k0, εF
k − δ0 θ −εF
k ,
which, respectively, are the dissociation rate for b → Ff and
the formation rate for Ff → b.
Collective and individual fermionic modes. In order to
obtain R for the weak interactions, we perturbatively calculate
R(k, ω) = ρ−1
[ −1
0 (k, ω) + Ubf ]−1
, which formally results
from the equation of motion of Qk. It then follows from the
random phase approximation (RPA) illustrated by the “bubble”
goldstino spectral function Fermi distribution
RΠ
bf
attice with cold
ots denote Fermi
Lower: The PA
ding energy Eb.
eory. The Green
imation. Upper:
the free Green
he solid (green)
een functions of
d F) are the
i; and µα and
mber operators
litudes tα and
unction wα(r)
nhomogeneity
cules has been
miltonian Hbf
µb = µf [1,3].
rameters obey,
here ρf is the
evolution from the ground state |G⟩ = |g⟩|F⟩ of H0. It follows
from the linear response theory that
R = 2g2
0ρ
k
ImDR(k, −δ0), (4)
where the retarded Green function is given by
DR(k, ω) =
∞
−∞
dxA(k − k0, x)
nf (x) − nf εF
k
x − εF
k − ω − i0+
, (5)
in terms of one loop calculations [Fig. 1(b)]. The single-
particle dispersions are εα
k = −2tα
d
s=1 cos ks − µα, where
the lattice spacings are set to be the unit. nf (x) is the
Fermi distribution at temperature T and the spectral function
A(k, ω) = −Im R(k, ω)/π is defined by the retarded Green
function R(k, ω) = −i
∞
0 dt⟨g|{Qk(t), Q
†
k(0)}|g⟩eiωt
.
At sufficiently low temperature, the pole and branch cut
in Eq. (5) are not qualitatively affected by T and nor is
the molecular formation rate. For simplicity, we take a zero-
temperature approximation in our calculation. It follows from
Eq. (5) that the rate R = Rb→Ff − RFf →b contains two parts:
Rb→Ff =
k
2πg2
0ρA k − k0, εF
k − δ0 θ δ0 − εF
k ,
(6)
RFf →b =
k
2πg2
0ρA k − k0, εF
k − δ0 θ −εF
k ,
which, respectively, are the dissociation rate for b → Ff and
the formation rate for Ff → b.
Collective and individual fermionic modes. In order to
Effect on Observable Quantity
(εF
k=tFk2-µF)
tF: hopping parameter of Flaser

44. 44
T. Shi, Y. Yu, and C. P. Sun, PRA 81, 011604(R) (2010)
•Momentum of laser is small:
• diluteness: Uρ ≪ 1
• |p| ≪ Uρ/(kF ta2
)
In the evaluation of the effe
• F is dilute: µ′
F ≃ 0
• k0 ≃ 0
Effect on Observable Quantity
3
wo processes which contributes to the PA variation of F-fermion number. The information on the goldstino spectrum
ed.
se (Q + Q†
)/
√
2, i(Q − Q†
)/
√
2, Nf + Nb, and Nf − Nb instead of Q, Q†
, Nf , and Nb, we get
Q + Q†
√
2
, i
Q − Q†
√
2
= −i[Q, Q†
] = −i −2Q†
Q + Nf + Nb , (2.22)
Q + Q†
√ , i
Q − Q†
√ =
Q + Q†
√ ,
Q + Q†
√ = i
Q − Q†
√ , i
Q − Q†
√ = 0, (2.23)
(εF
k-δ0, k)
(δ0, 0)
•F is dilute: µF ' 0
3
FIG. 1: Two processes which contributes to the PA variation of F-fermion number. The information on the goldstino spectrum
is embedded.
If we use (Q + Q†
)/
√
2, i(Q − Q†
)/
√
2, Nf + Nb, and Nf − Nb instead of Q, Q†
, Nf , and Nb, we get
Q + Q†
√
2
, i
Q − Q†
√
2
= −i[Q, Q†
] = −i −2Q†
Q + Nf + Nb , (2.22)
Q + Q†
√
2
, i
Q − Q†
√
2
=
Q + Q†
√
2
,
Q + Q†
√
2
= i
Q − Q†
√
2
, i
Q − Q†
√
2
= 0, (2.23)
Q + Q†
√
2
,
Q + Q†
√
2
= i
Q − Q†
√
2
, i
Q − Q†
√
2
= {Q, Q†
} = Nf + Nb, (2.24)
Q + Q†
√
2
,
Nf + Nb
√
2
= i
Q − Q†
√
2
,
Nf − Nb
√
2
= 0, (2.25)
Q + Q†
√
2
,
Nf − Nb
√
2
= −(Q − Q†
), (2.26)
i
Q − Q†
√ ,
Nf − Nb
√ = −i(Q + Q†
), (2.27)
tF k2
0 = µ ↵k2
k2
=
µ + 0
tF + ↵
on-shell condition:
•δ0=-Δµ k=0
We can extract the information of the
goldstino at small momentum.

45. Effect on Observable Quantity (Result at T=0)
45
mation rate of b, and it can be observed via the PA variatio
SY, It is given by
R ≃
g2
0N
2tF + α
δ0.
D. Summary of conditions for parameters
um of the goldstino, the conditions are the following:
Π0(ω, p) ≃
(2π)2 ω − ∆µ − t(2k · p − p2)
.
Πf
0 (ω, p) ≃
d2
k
(2π)2
nF (ϵf
k)
ω − (∆µ + t(2k · p + p2) + Uρ)
,
Πb
0(ω, p) ≃
d2
k
(2π)2
nB(ϵb
k)
ω − (∆µ + t(2k · p − p2) + Uρ)
.
α ≡
t2
4πρ2
f
Uρ2
−
t(ρf − ρb)
ρ
α ≡
1
ρ
t2
4πρ2
f
Uρ
− t(ρf − ρb)
(T=0 case)
By tuning δ0, we can extract α from R!
( )N: total number

46. • We analyzed the spectral properties of the goldstino in the
absence/presence of interaction with RPA and sum rules.
• We observed the crossover from small p to large p region (from
interaction dominant to free case).
• We discussed the possibility for experimental detection of the
goldstino.
Summary
46
-8
-6
-4
-2
0
2
0 0.5 1 1.5 2 2.5
~
/th
p
ω＝Δµ-αp2
Z＝ρ
ω＝Δµ-th p2
Z＝ρb

47. Future perspectives
47
•BEC (spectrum of goldstino, symmetry
breaking pattern…)
•Beyond the one-loop approximation (damping
rate of the goldstino).