A. Perali, P. Pieri, F. Palestini, and G. C. Strinati

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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
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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.

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High-Tc superconductors phase diagram
La2-xSrxCuO4
Pseudogap competing order parameter or precursor
of superconducting gap?
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Pseudogap vs gap density of states
Precursor effect?
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Gap and pseudogap in underdoped LaSrCuO
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ARPES spectra for underdoped La1.895Sr0.105CuO4
at T49K gt Tc30 K
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
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).
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The BCS to BEC crossover problem at finite
temperature inclusion of pairing fluctuations
above Tc
T-matrix self-energy
where
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).
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Why T-matrix diagrams?
Small parameter

• kFa ltlt 1 for weak coupling
• kFa ltlt 1 for strong coupling
• 1/T at high temperature (better, fugacity
  )

In all these limits T-matrix recovers the
corresponding asymptotic theory

• Galitskii theory for the dilute Fermi gas in
  weak coupling (till order (kFa)2)
• Dilute Bose gas in strong-coupling (zero order
  in kFa)
• Virial expansion up to second virial
  coefficient

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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)
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Single particle spectral function and density of
states
Spectral function determined by analytic
continuation to the real axis of the temperature
Greens function The continuation to real
axis can be perfomed analitically, without
resorting to approximate methods (such as
MaxEnt, Padé )
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Spectral function at TTc, unitary limit
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Spectral function at TTc, (kFa)-10.25
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Temperature evolution at (kFa)-10.25
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Density of states
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BCS-like equations for dispersions and weights
BCS-like description approximately valid close to
Tc
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Remnant Fermi surface in the pseudogap phase
Luttinger wave-vector
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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
where is the detuning of the RF probe with
respect to the frequency of the atomic
transition .
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)
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Momentum-resolved RF spectroscopy
Final state interaction strongly reduced in
subsequent experiments with 6Li at MIT. In
addition tomographic techinique introduced,
eliminating trap average
X
but average over k remains.
JILA experiment with 40K (final state
interaction negligible) eliminated average over
k (but not over r)
X
Momentum resolved RF spectrum proportional to
where is the
single-particle energy
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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.
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Quasi-particle dispersions and widths
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Is the unitary Fermi gas in the normal phase a
Fermi liquid?
For the normal unitary Fermi gas T/TF gt 0.15
Here T/TF lt 0.03
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Concluding remarks

• A pairing gap at TTc (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).

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Thank you!
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Supplementary material
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Spectral weight function below Tc
Wave vector k chosen for each coupling at a
value 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 TTc to T0.

P. Pieri, L. Pisani, G.C. Strinati, PRL 92,
110401 (2004).
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The contact
F. Palestini, A. Perali, P.P., G.C. Strinati, PRA
82, 021605(R) (2010).
E.D. Kuhnle et al., arXiv1012.2626