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## **The Dirac Spectrum and the BEC-BCS Crossover in QCD at Nonzero Isospin Asymmetry**

#### **Bastian B. Brandt, Francesca Cuteri \*, Gergely Endr ˝odi and Sebastian Schmalzbauer**

Institute for Theoretical Physics, Goethe University, Max-von-Laue-Strasse 1, 60438 Frankfurt am Main, Germany; brandt@itp.uni-frankfurt.de (B.B.B.); endrodi@itp.uni-frankfurt.de (G.E.); schmalzbauer@itp.uni-frankfurt.de(S.S.)

 **\*** Correspondence:cuteri@itp.uni-frankfurt.de

Received: 14 December 2019; Accepted: 18 January 2020; Published: 4 February 2020

**Abstract:** For large isospin asymmetries, perturbation theory predicts the quantum chromodynamic (QCD) ground state to be a superfluid phase of *u* and ¯ *d* Cooper pairs. This phase, which is denoted as the Bardeen-Cooper-Schrieffer (BCS) phase, is expected to be smoothly connected to the standard phase with Bose-Einstein condensation (BEC) of charged pions at *μI* ≥ *<sup>m</sup>π*/2 by an analytic crossover. A first hint for the existence of the BCS phase, which is likely characterised by the presence of both deconfinement and charged pion condensation, comes from the lattice observation that the deconfinement crossover smoothly penetrates into the BEC phase. To further scrutinize the existence of the BCS phase, in this article we investigate the complex spectrum of the massive Dirac operator in 2+1-flavor QCD at nonzero temperature and isospin chemical potential. The spectral density near the origin is related to the BCS gap via a generalization of the Banks-Casher relation to the case of complex Dirac eigenvalues (derived for the zero-temperature, high-density limits of QCD at nonzero isospin chemical potential).

**Keywords:** lattice QCD; isospin; BCS phase
