Lianyi He and Pengfei Zhuang Physics Department, Tsinghua U. BCS-BEC Crossover in Relativistic System and Its Possible Realization in QCD Lianyi He and Pengfei Zhuang Physics Department, Tsinghua U. References: Phys.Rev.D75:096003,2007 Phys.Rev.D75:096004,2007 Phys.Rev.D76:056003,2007 E-print Arxiv: 0709.2246 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Impact of Superconductivity Phenomena Order parameter Pair Breaking effect Metallic superconductivity Zeeman Splitting Superfluidity in Ultracold Fermi atoms Population imbalance Chiral symmetry breaking in QCD Baryon chemical potential Color superconductivity in dense quark matter Isospin chemical potential Strange quark mass flavor superfluidity in isospin matter 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Theoretical History of Crossover: Introduction to BCS-BEC Crossover Theoretical History of Crossover: Leggett (1980) noted that BCS T=0 wavefunction could be generalized to arbitrary attraction: a smooth BCS-BEC crossover ! Let us consider the history of theoretical studies of BCS-BEC crossover. Of course, the first theory is a mean-field theory. Here is the phase diagram at zero temperature. As the attractive interaction U is changed from a small negative value to a large negative value, we go from the BCS regime to the BEC regime. For the BCS regime, the BCS wave-function is valid. While for the BEC regime, we have the BEC variational wave-function. The important discovery by Eagles and Leggett is that these two wave-functions actually are connected. For example, in the BEC limit, the vk in the BCS wave-function becomes very small, we can rewrite it into this form. Therefore, the BCS wave-function can be transformed into the BEC wave-function. This observation suggests that from BCS to BEC it is not an abrupt phase transition, rather, we have a smooth crossover. The mean-field theory has been generalized by Holland and Levin to the cold atomic gases. strong coupling weak coupling BEC BCS 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Eagles (1969): Pairing without superconductivity So, what is the BCS-BEC crossover? Consider a two-component Fermi gas with attractive interactions. When the interaction is weak, we have the BCS fermionic superfluidity of Cooper-pairs, and we know that these Cooper-pairs are long-range objects. As the interaction increases, the size of Cooper-pairs becomes smaller and smaller, and finally can be regarded as a point-like structure, or molecules. We then can have a condensate of molecules. These two limits, BCS and BEC limits, are actually well-understood, however, the regime between these limits, is not so clear, and is to explored. To access the crossover, of course, the interaction strength should be enhanced enough large. In HTSC, the root of PG is not quite clear. The pre-formed boson mechanism is only one explanation. 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Experimental evidence for pseudogap in ultracold fermions Science 305, 1128 (2004) Science 307, 1296 (2005) 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

m T BEC? Relativistic BCS-BEC Crossover Motivation: 1) QCD phase diagram at moderate temperature and density, strong coupling nature! 2) BCS-BEC crossover in CSC? Pseudogap of CSC 3) NJL model: BCS theory of chiral symmetry breaking. Chiral pesudogap above Tc? T m Quark Spectrum and pseudogap, M. Kitazawa, et.al. 2005 SQGP? Structure change of Cooper pairs, H.Abuki, et.al.2002 pseudogap? SB CSC BEC? 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Progress in this field of research H.Abuki and Y.Nishida: NSR approach, new crossover: BCS-BEC-RBEC L.He and P.Zhuang: BCS-Leggett theory at zero T and pesudogap theory at finite T BCS-BEC vs. Chiral Restoration J.Deng and Q.Wang: Boson-fermion model in mean field and beyond mean field M.Kitazawa, D.Rischke and I.Shovkovy: Possible Diquark BEC in NJL T.Brauner: Collective Excitation 1/N expansion including quantum fluctuation (with H.Abuki)

Zero Temperature: BCS-Leggett mean Field Theory Model Lagrangian Mean Field Equations 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Fermionic Excitation Spectrum anti-fermion Non-relativistic limit Criterion for BCS and BEC: Minimum of the dispersion BCS: at , existence of Fermi surface BEC: at , molecule condensation The role of fermion mass (new aspect in QCD) BCS-BEC Chiral symmetry breaking& restoration 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

The Non-Relativistic Limit For small value of the parameter , the NR version of BCS-Leggett result is recovered The condition or is needed 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Relativistic Effect I: BCS-NBEC-RBEC Crossover Physical picture: RBEC: , anti-particles available! 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Relativistic Effect II: Density Induced Crossover 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Collective Mode Evolution Effective action Dispersion of the collective mode Goldstone mode A-B mode BCS NBEC Bogoliubov theory RBEC Ultra-relativistic BEC 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

K.Levin and Qijin Chen, et.al, Nonzero Temperature: Pseudogap theory K.Levin and Qijin Chen, et.al, Phys.Rept. 412, 1(2005) 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Tc Calculation NBEC regime RBEC regime Bound states above Tc 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

BCS-BEC Crossover in QCD? Two color QCD QCD at finite isospin density In the BEC region, quark energy gap is different from the order parameter! Can be confirmed in Lattice! Phys.Rev.D75:096004,2007 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Chiral restoration vs. BCS-BEC crossover 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Condensation and Chiral Phase Transition Pseudogap in QCD? Condensation and Chiral Phase Transition NJL Model Nc dependence Quark mass is still large above chiral phase transition! 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Diquark condensation and color superconductivity NJL model, 2SC phase Quark pesudogap remarkable! A high Tc superconductor! 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Non-Fermi liquid behavior at low T! (High T, see M.Kitazawa, 2005)

1. BCS-BEC crossover theory has been generalized Summary 1. BCS-BEC crossover theory has been generalized to relativistic system, both at zero and at finite temperature. 2. Proper NR limit is recovered and new relativistic effects are found. 3. Possible BCS-BEC crossover in diquark and isospin matter is discussed. 4. The size of quark pseudogap in chiral and CSC phase transition is calculated. Outlook Determine the pseudogap phase in QCD phase diagram Signals of diquark fluctuation and pesudogap in HIC Effect of the pseudogap on compact star cooling More precise theory, including quantum fluctuation 2017/4/24 ATHIC2008, TSUKUBA, JAPAN

Thank You! 2017/4/24 ATHIC2008, TSUKUBA, JAPAN