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

Exploring the pseudogap phase of a strongly interacting Fermi gas

http://bcsbec.df.unicam.it

Dipartimento di Fisica, Università di Camerino, Italy

+ collaboration with JILA experimental group: J. Gaebler, J. Stewart, T. Drake, and D. Jin

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

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

ARPES spectra for underdoped

La1.895Sr0.105CuO4 at T=49K > Tc=30 K

The dispersions in the gapped region of the zone obtained from the Fermi-function-divided spectra. The full circlesare 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-nodalpoints (cuts 2 and 3 in Fig. 1(j), respectively).

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”

T-matrix self-energy:

where

0(P) 1 1

v0

dp

(2 )3 1

G0(p P)G0( P)

l

m

4a dp

(2 )3 1

G0(p P)G0( P)

l

mp2

(k) dP

(2)3 1

0(P)G0(P k)

k (k,n ) ; P (P, )

p(p, l )

The BCS to BEC crossover problem at finite temperature:inclusion of pairing fluctuations above Tc

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).

G(k) G0(k) 1 (k) 1

Why T-matrix diagrams?

• kF|a| << 1 for weak coupling

• kFa << 1 for strong coupling

• 1/T at high temperature (better, fugacity )

z e << 1

• 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

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

Small parameter:

Phase diagram for the homogeneous and trapped Fermi gas as predicted by t-matrix

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)

Tc from QMC at unitarity:Burovski et al. (2006), Bulgac et al. (2008), …

Single particle spectral function and density of states

A(k, )d

1

A(k, ) f ( )d nk

N()

dk

(2 )3 A(k, )

Spectral function determined by analytic continuation to the real axisof the temperature Green’s function:

The continuation to real axis can be perfomed analitically, without resorting to approximate methods (such as MaxEnt, Padé …)

G(k,in ) G(k, i0)GR (k,)

A(k,)1

ImGR (k,)

( 1/ )Im(k,)

( (k) Re(k,))2 Im(k,)2

(k)2k 2

2m

in i0

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

E k 2(k 2 kL2) /(2m) 2 2

vk2

1

2(1 k /Ek ) uk

2 1

2(1k /Ek )

BCS-like description approximately valid close to Tc

“Remnant Fermi surface” in the pseudogap phase

kL

“Luttinger” wave-vector

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:

)]()2/([));()2/(,()2(

)( 22

22

3

33 rmkfrrmkkA

kdrdRF

where is the detuning of the RF probe with respect to the frequency of the atomic transition .

32

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

)]()2/([));()2/(,()2(

)( 22

22

3

33 rmkfrrmkkA

kdrdRF

Final state interaction strongly reduced in subsequent experiments with 6Liat MIT. In addition tomographic techinique introduced, eliminating trapaverage:

JILA experiment with 40K (final state interaction negligible) eliminated average over k (but not over r…)

)]([));(,();( 2232 rEfrrEkArdkEkRF sss

Momentum resolved RF spectrum proportional to:

E s k2 /(2m) where is the “single-particle energy”

Xbut average over k remains.

)]()2/([));()2/(,()2(

)( 22

22

3

33 rmkfrrmkkA

kdrdRF

X

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?

S. Nascimbene et al., Nature 463, 1057 (2010) and arXiv:1006.4052

A. Bulgac et al., PRL 96, 90 404 (2006)

Here T/TF < 0.03

For the normal unitaryFermi gas T/TF > 0.15

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).

Concluding remarks

Thank you!

Supplementary material

25

Spectral weight function below Tc

),( Im1

),( 11

kk RGA

5.0)( 1 FFak

1.0)( 1 FFak

5.0)( 1 FFak

Wave vector k chosen for each coupling at a value which minimizes the gap in the spectral function.'k

P. Pieri, L. Pisani, G.C. Strinati, PRL 92, 110401 (2004).

•In the superfluid phase: narrow “coherent peak” over a broad “pseudogap” feature.

• Pseudogap evolves into real gap when lowering temperature from T=Tc to T=0.

E.D. Kuhnle et al., arXiv:1012.2626

F. Palestini, A. Perali, P.P., G.C. Strinati, PRA 82, 021605(R) (2010).

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