Transcript BCS-BEC crossover for trapped Fermi gases
Exploring the pseudogap phase of a strongly interacting Fermi gas
A. Perali, P. Pieri, F. Palestini, and G. C. Strinati
Dipartimento di Fisica, Università di Camerino, Italy
+ collaboration with JILA experimental group: J. Gaebler, J. Stewart, T. Drake, and D. Jin http://bcsbec.df.unicam.it
Outline
• The pseudogap in high-Tc superconductors.
• Pairing fluctuations and the pseudogap: results obtained by t-matrix theory for attractive fermions through the BCS BEC crossover.
• Momentum resolved RF spectroscopy.
• Comparison between theory and JILA experiments: evidence for pseudogap and remnant Fermi surface in the normal phase of a strongly interacting Fermi gas.
High-Tc superconductors: phase diagram
La 2-x Sr x CuO 4 Pseudogap: competing order parameter or precursor of superconducting gap?
Pseudogap vs gap: density of states Precursor effect?
Gap and pseudogap in underdoped LaSrCuO
Pseudogap in underpoded superconducting cuprates: pairing above Tc and/or other mechanisms ?
M. Shi, … Campuzano..
Mesot EPL
88
, 27008 (2009) “Spectroscopic evidence for preformed Cooper pairs in the pseudogap phase of cuprates” ARPES spectra for underdoped La 1.895
Sr 0.105
CuO 4 at T=49K >
Tc=30 K
The dispersions in the gapped region of the zone obtained from the
Fermi-function-divided spectra
. The
full circles
are the two branches of the dispersion derived from (d) at
49K
,
open circles
correspond to the same cut (cut 1 in (e)) but at
12K
.
The curves indicated by triangles and diamonds are the dispersions at 49K along cuts closer to the anti-nodal points (cuts 2 and 3 in Fig. 1(j), respectively).
The BCS to BEC crossover problem at finite temperature: inclusion of pairing fluctuations above Tc
T-matrix self-energy : (
k
)
d
P
(2 ) 3 1 0 (
P
)
G
0 (
P
k
) where
G
(
k
)
G
0 (
k
) 1 (
k
) 1 0 (
P
) 1 1 v 0
m
4
a
d
p
(2 ) 3 1
d
p
(2 ) 3 1
l
l G
0 (
p
P
)
G
0 (
p
P
)
G
0 (
P
)
G
0 (
P
)
m p
2
k p
(
k
,
n
) ;
P
(
p
,
l
) (
P
, )
66
, 024510 (2002).
P. Pieri, L. Pisani, and G. Strinati, Phys. Rev. B
70
, 094508 (2004).
Why T-matrix diagrams?
Small parameter: • k F |a| << 1 for weak coupling • k F a << 1 for strong coupling • 1/T at high temperature
e
<< 1 In all these limits T-matrix recovers the corresponding asymptotic theory: • Galitskii theory (till order (k F |a|) 2 ) for the dilute Fermi gas in weak coupling • Dilute Bose gas in strong-coupling (zero order in k F a) • Virial expansion up to second virial coefficient
Phase diagram for the homogeneous and trapped Fermi gas as predicted by t-matrix
Tc from QMC at unitarity: Burovski et al. (2006), Bulgac et al. (2008), … C. Sa de Melo, M. Randeria and J. Engelbrecht, PRL
71
, 3202 (1993) (homogeneous) A. Perali, P. Pieri, L. Pisani, and G.C. Strinati, PRL
92
, 220404 (2004) (trap)
Single particle spectral function and density of states
Spectral function determined by analytic continuation to the real axis of the temperature Green’s function:
G
(
k
,
i
n
)
A
(
k
, )
G
(
k
, 1
i
0 ) Im
G R
(
k
, )
G R
(
k
, ) ( (
k
) ( 1/ )Im (
k
, ) Re (
k
, )) 2 Im (
k
, ) 2 (
k
) 2
k
2 2
m
i
n
i
0 The continuation to real axis can be perfomed
analitically
, without resorting to approximate methods (such as MaxEnt, Padé …)
A
(
k
, )
d
A
(
k
, )
f
1 ( )
d
N
( )
d
k
(2 ) 3
n k A
(
k
, )
Spectral function at T=Tc, unitary limit
Spectral function at T=Tc, (k F a) -1 =0.25
Temperature evolution at (k F a) -1 =0.25
Density of states
BCS-like equations for dispersions and weights
E k
2 (
k
2
k L
2 )/(2
m
) 2 2
v k
2 1 2 (1
k
/
E k
)
u k
2 1 2 (1
k
/
E k
)
“Remnant Fermi surface” in the pseudogap phase
“Luttinger” wave-vector
k L
How does the spectral function enters in RF spectroscopy?
In the absence of final state interaction, linear response theory yields for the
RF
experimental signal:
RF
( )
d
3
r
d
3
k
( 2 ) 3
A
(
k
,
k
2 /( 2
m
) 2 (
r
);
r
)
f
[
k
2 /( 2
m
) 2 (
r
)] Final state interaction was large in first experiments with 6 Li (Innsbruck,MIT), complicating the theoretical analysis (which showed, however, a beatiful connection with the theory of paraconductivity in superconductors!) [P. Pieri, A. Perali and G. Strinati, Nat. Phys
. 5
, 736 (2009)]
Momentum-resolved RF spectroscopy
Final state interaction strongly reduced in subsequent experiments with 6 Li at MIT. In addition tomographic techinique introduced, eliminating trap average:
RF
( )
d
3 X
r
d
3
k
( 2 ) 3
A
(
k
,
k
2 /( 2
m
) 2 (
r
);
r
)
f
[
k
2 /( 2
m
) 2 (
r
)] but average over k remains.
JILA experiment with 40 K (final state interaction negligible) eliminated average over k (but not over r…)
RF
( )
d
3
r
d
( 2 3 X
k
) 3
A
(
k
,
k
2 /( 2
m
) 2 (
r
);
r
)
f
[
k
2 /( 2
m
) 2 (
r
)]
Momentum resolved
RF spectrum proportional to:
RF
(
k
;
E s
)
k
2
d
3
r A
(
k
,
E s
2 (
r
);
r
)
f
[
E s
2 (
r
)]
E s
k
2 /(2
m
) “
single particle energy”
Comparison with momentum resolved RF spectra from JILA exp.
A. Perali, et al., Phys. Rev. Lett.
106
, 060402 (2011) theoretical spectra in an unbiased way. Eliminates freedom to adjust the relative heights of experimental and theoretical spectra.
“Quasi-particle” dispersions and widths
Is the unitary Fermi gas in the normal phase a Fermi liquid?
For the normal unitary Fermi gas T/T F > 0.15
Here T/T F < 0.03
S. Nascimbene et al., Nature
463
, 1057 (2010) and arXiv:1006.4052
A. Bulgac et al., PRL
96
, 90 404 (2006)
Concluding remarks
A
pairing gap
at T=Tc (
pseudogap
), from close to unitarity to the BEC regime, is present in the single-particle spectral function A(k,w).
Momentum resolved RF spectroscopy
: comparison between experiments and t-matrix calculations for EDCs, peaks and widths demonstrate the presence of a pseudogap of strongly-interacting ultracold fermions.
close to Tc, in the normal phase The pseudogap coexists with a “ remnant Fermi surface ” which approximately satisfies the Luttinger theorem in an extended coupling range.
The presence of a pseudogap in the unitary Fermi gas is consistent with recent thermodynamic measurements at ENS (that were interpreted in terms of a “Fermi liquid” picture).
Thank you!
Supplementary material
Spectral weight function below Tc
A
(
k
, ) 1 Im
G R
11 (
k
, ) Wave vector
k
chosen for each coupling at a value
k
' which minimizes the gap in the spectral function.
•In the superfluid phase: narrow “coherent peak” over a broad “pseudogap” feature.
• Pseudogap evolves into real gap when lowering temperature from T=T c to T=0.
P. Pieri, L. Pisani, G.C. Strinati, PRL
92,
110401 (2004).
(
k F a F
) 1 0 .
5 (
k F a F
) 1 0 .
1 (
k F a F
) 1 0 .
5 25
The contact
F. Palestini, A. Perali, P.P., G.C. Strinati, PRA
82,
021605(R) (2010).
E.D. Kuhnle et al., arXiv:1012.2626