Generalized crossover in strongly interacting fermions
with finite effective range
Paper: HT, J. Phys. Soc. Jpn. 88, 093001 (2019).; Phys. Rev. A 97, 043613 (2018).
Hiroyuki Tajima1
Department of Mathematics and Physics, Kochi University, Kochi, 780-8520 Japan―The purpose of this poster―We generalize BCS-BEC crossover physics with zero-range interactions to the case with arbitrary effective range andscattering length. Using a Nozieres-Schmitt-Rink approach, we proposed a universal phase diagram of two-component
fermions with respect to low-energy constants. A novel crossover induced by the change of effective ranges is found at theunitarility limit. Our results cover an ultracold Fermi gas as well as dilute neutron matter.

Introduction

Results“Similarity” between ultracold Fermi gas and pure neutron matter
BCS-BEC crossover of 2-com. Fermi gas

𝑘F 𝑎𝑠 −1 < 0Strong
couplinginteraction

𝑎𝑠 : scattering length𝑟e : effective range
Cold atom

Neutrons𝑎𝑠−∞~∞-18.5 fm𝑟e~0 (broad res.)

2.8 fmdensity~1015 cm−3~0.16 fm−3

𝑘F 𝑎𝑠 −1−∞~∞−∞~0Assuming that 𝑘F 𝑎𝑠 −1 is afunction of the density at fixed 𝑎𝑠

Weakcoupling

6Li (exp.) PRX, 7, 041004 (2017)

We can accessneutron star
physics on Earthvia cold atom
experiments
APR: PRC, 58, 1804 (1998)
Simulation for neutron matter“Difference” between ultracold Fermi gas and pure neutron matter

BD: the region where effective
range expansion breaks down

Feshbach resonance in 6Li atoms● Chemical potential

Cutoff parameter1 + 1 − 2𝑟e /𝑎𝑝c =𝑟e

Two-body binding energy

11𝐸𝑏 =
𝑚𝑎2 1 − 1/(𝑎𝑝c ) 2
Bound for effective range(≃ Wigner’s causality bound)

The effective range correction plays an important role in pairing properties
Phase shift with different 𝑟e● Superfluid critical temperature

S-wave phase shift
1 1𝑘cot𝛿𝑠 = − + 𝑟e 𝑘 2
𝑎𝑠 2

S-wave scattering is
dominant even in a low
density nuclear matter

http://astro.riken.jp/wordpress/?page_id=1425● Generalized phase diagram𝑟𝑒 ≤ 𝑎/2

PNM: pure neutron matter

Pairing gap● Effective range induced crossover
Zero-range crossover withchanging the scattering length
Y. Ohashi, HT, P. van Wyk, Prog. Part.
Nucl. Phys. 111, 103739 (2020).

QMC at T = 0
A. Gezerlis and J. Carlson, PRC, 77, 032801(R) (2008).

C.L. Blackley, et al., PRA, 89, 042701 (2014).

We investigate pairing properties with finite effective range
and show the superfluid critical temperature at arbitrary
scattering length and effective range.

Formulation†𝐻 = ෍ 𝜀𝒌 − 𝜇 𝑐𝒌,𝜎𝑐𝒌,𝜎 + ෍ 𝜀𝒒 Τ2 + 𝜈 − 2𝜇 𝑏𝒒† 𝑏𝒒 + ෍ 𝑔𝒑 𝑏𝒒† 𝑐𝒑+𝒒Τ2,↑ 𝑐−𝒑+𝒒Τ2,↓ + H. c.
𝒌,𝝈𝒒𝒑,𝒒𝑔r𝑔r
Two-component Fermi atoms

closed-channel molecule

Feshbach coupling𝑘2● kinetic energy: 𝜀𝒌 = 2𝑚
● pseudospin: 𝜎 =↑, ↓● diatomic annihilation operator: 𝑏𝒒● chemical potential: 𝜇● threshold energy of a closed-channel molecule: 𝜈● atomic annihilation operator: 𝑐𝒌,𝜎4𝜋𝜈
● scattering length: 𝑎 = −+ 𝑝c2𝑚𝑔−1MF*: mean-field results with renormalized dimer level● atom-dimer coupling:
𝑔𝒑 =𝑔

8𝜋
211−
● effective range: 𝑟e = − 2 2 +𝑚 𝑔𝑝c𝑝c 𝑎𝒒𝑓(𝑏): Fermi (Bose) distribution function● Thouless criterion (condition for 𝑇c ):(gapless bosonic excitation)

𝑘F 𝑎 −1 → −∞ (BCS)𝑘F 𝑎 −1 → ∞ (BEC)

𝑇c /𝑇F𝜋~exp − 𝑘F 𝑟e
4→ 0.204𝑇F

𝜋~exp −2𝑘F 𝑎→ 0.218𝑇F

𝜇c /𝜀F→ 𝜀F→0→ 𝜀F→ −𝐸b /2𝑘F 𝑟e → ∞ (BCS)

Summary𝜕
𝑁 = 2 ෍ 𝑓 𝜀𝒑 − 𝜇 + 2 ෍ 𝑏(𝜀𝒒 + 𝜈 − 2𝜇) − 𝑇 ෍ln 1 − 𝐷 𝒒, 𝑖𝜁ℓ Π(𝒒, 𝑖𝜁ℓ )𝜕𝜇𝐷 𝒒 = 0, 𝑖𝜁𝑙 = 0 = ∞Scattering length induced crossover𝑘F 𝑟e → −∞ (BEC)

Coupling parameters

1 + 𝑝Τ𝑝c 2● Particle number equation:𝒑

Effective range induced crossover𝒒,𝑖𝜁ℓ

𝐷: dimer propagator𝜁ℓ : fermion (boson) Matsubara frequency● Particle-particle bubble for Cooper channel
Π 𝒒, 𝑖𝜈ℓ= ෍ 𝑔𝒑2
𝒑1 − 𝑓 𝜀𝒑+𝒒Τ𝟐 − 𝜇 − 𝑓(𝜀−𝒑+𝒒Τ𝟐 − 𝜇)𝑖𝜈ℓ − 𝜀𝒑+𝒒Τ𝟐 − 𝜀−𝒑+𝒒Τ𝟐 + 2𝜇

● We have investigated a generalized crossover of superfluid criticaltemperature in a two-component fermions with arbitrary scattering length andeffective range within the Nozieres-Schmitt-Rink approach.
● We have proposed a generalized phase diagram of BCS-BEC crossover andpresented the second crossover induced by the effective range.
Future work: low-dimensions, population or mass imbalances, etc…
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