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 BCSBEC 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
La2-xSrxCuO4
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
ARPES spectra
for underdoped
La1.895Sr0.105CuO4
at T=49K > Tc=30 K
EPL 88, 27008
(2009)
“Spectroscopic evidence
for preformed Cooper pairs
in the pseudogap phase
of cuprates”
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)  

dP
1
(2  ) 
3
  (P ) G
0
0

(P  k )
0
G ( k )  G (k )

1
 (k )
1

where
0 ( P )
1


1
v0
m
4a




dp
1
(2  ) 
3

p  (p,  l )
l

1

3
(2  ) 

dp
k  (k,  n ) ; P  (P,  )
0
0
G ( p P ) G ( P )

l

G ( p  P ) G ( P )  2 
p 

0
0
m
A. Perali, P. Pieri, G.C. Strinati, and C. Castellani, Phys. Rev.B 66, 024510 (2002).
P. Pieri, L. Pisani, and G. Strinati, Phys. Rev. B 70, 094508 (2004).
Why T-matrix diagrams?
Small parameter:
• kF|a| << 1 for weak coupling
• kFa << 1 for strong coupling
• 1/T at high temperature (better, fugacity
ze

<< 1)
In all these limits T-matrix recovers the corresponding
asymptotic theory:

• Galitskii theory for the dilute Fermi
gas in weak coupling
(till order (kF|a|)2)
• Dilute Bose gas in strong-coupling (zero order in kFa)
• 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 i  n    i0 
of the temperature Green’s function:

G ( k,i  n )  G ( k ,   i0 )  G ( k,  )
A ( k,  )  
2
 (k) 
k
2m
1

R
Im G ( k ,  ) 
( 1 /  ) Im  ( k,  )
R
2
2
(   ( k )  Re  ( k ,  ))  Im  ( k, )
2

The continuation to real axis can be perfomed analitically, without resorting
to approximate methods (such as MaxEnt, Padé …)







A ( k,  ) d   1
A ( k,  ) f ( ) d   n k
N ( ) 

dk
(2  )
3
A ( k,  )
Spectral function at T=Tc, unitary limit
Spectral function at T=Tc, (kFa)-1=0.25
Temperature evolution at (kFa)-1=0.25
Density of states
BCS-like equations for dispersions and weights
Ek  


2
2
( k  k ) /( 2 m )   
2
2
L
2
2
vk 
1
2
(1   k / E k )
2
uk 
BCS-like description approximately
valid close to Tc

1
2
(1   k / E k )
“Remnant Fermi surface” in the pseudogap phase
“Luttinger” wave-vector
kL

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:
3
RF (  ) 
 d r
3
d k
( 2 )
A ( k , k /( 2 m )      2 ( r ); r ) f [ k /( 2 m )      2 ( r )]
3
2
2
where   is the detuning of the RF probe with respect to the frequency
of the atomic transition 2  3 .
Final state interaction was large in first experiments with 6Li (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 6Li
at MIT. In addition tomographic techinique introduced, eliminating trap
average:
3
RF (  ) 
X
 d r
d k
3
( 2 )
A ( k , k /( 2 m )      2 ( r ); r ) f [ k /( 2 m )      2 ( r )]
3
2
2
but average over k remains.
JILA experiment with 40K (final state interaction negligible) eliminated
average over k (but not over r…)
3
RF (  ) 
 d r
d k
X
3
( 2 )
A ( k , k /( 2 m )      2 ( r ); r ) f [ k /( 2 m )      2 ( r )]
3
2
2
Momentum resolved RF spectrum proportional to:
RF ( k ; E s )  k
2
 d r A ( k , E s   2 ( r ); r ) f [ E s   2 ( r )]
3
where E s  k 2 /( 2 m )    is the “single-particle energy”
Comparison with momentum resolved RF spectra from JILA exp.
A. Perali, et al., Phys. Rev. Lett. 106, 060402 (2011)
Use sum rule (sum over  ,k,r equals N) to normalize exp data and
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/TF > 0.15
Here T/TF < 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 close to Tc, in the normal phase
of strongly-interacting ultracold fermions.
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
(k F aF )
A( k ,  )  
1

1
  0 .5
Im G 11 ( k ,  )
R
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.
(k F aF )
1
 0 .1
1
 0 .5
• Pseudogap evolves into real gap when lowering
temperature from T=Tc to T=0.
(kF aF )
P. Pieri, L. Pisani, G.C. Strinati, PRL 92, 110401 (2004).
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