Nambu-Jona-Lasinio model description of weakly interacting Bose condensate and BEC-BCS crossover in dense QCD-like theories

QCD-like theories possess a positively definite fermion determinant at finite baryon chemical potential $\mu_{\text B}$ and the lattice simulation can be successfully performed. While the chiral perturbation theories are sufficient to describe the Bose condensate at low density, to describe the crossover from Bose-Einstein condensation (BEC) to BCS superfluidity at moderate density we should use some fermionic effective model of QCD, such as the Nambu-Jona-Lasinio model. In this paper, using two-color two-flavor QCD as an example, we examine how the Nambu-Jona-Lasinio model describes the weakly interacting Bose condensate at low density and the BEC-BCS crossover at moderate density. Near the quantum phase transition point $\mu_{\text B}=m_\pi$ ($m_\pi$ is the mass of pion/diquark multiplet), the Ginzburg-Landau free energy at the mean-field level can be reduced to the Gross-Pitaevskii free energy describing a weakly repulsive Bose condensate with a diquark-diquark scattering length identical to that predicted by the chiral perturbation theories. The Goldstone mode recovers the Bogoliubov excitation in weakly interacting Bose condensates. The results of in-medium chiral and diquark condensates predicted by chiral perturbation theories are analytically recovered. The BEC-BCS crossover and meson Mott transition at moderate baryon chemical potential as well as the beyond-mean-field corrections are studied. Part of our results can also be applied to real QCD at finite baryon or isospin chemical potential.Comment: 29 pages + 9 figures. Published version in PR

Finite range and upper branch effects on itinerant ferromagnetism in repulsive Fermi gases: Bethe-Goldstone ladder resummation approach

We investigate the ferromagnetic transition in repulsive Fermi gases at zero temperature with upper branch and effective range effects. Based on a general effective Lagrangian that reproduces precisely the two-body $s$-wave scattering phase shift, we obtain a nonperturbative expression of the energy density as a function of the polarization by using the Bethe-Goldstone ladder resummation. For hard sphere potential, the predicted critical gas parameter $k_{\rm F}a=0.816$ and the spin susceptibility agree well with the results from fixed-node diffusion Monte Carlo calculations. In general, positive and negative effective ranges have opposite effects on the critical gas parameter $k_{\rm F}a$: While a positive effective range reduces the critical gas parameter, a negative effective range increases it. For attractive potential or Feshbach resonance model, the many-body upper branch exhibits an energy maximum at $k_{\rm F}a=\alpha$ with $\alpha=1.34$ from the Bethe-Goldstone ladder resummation, which is qualitatively consistent with experimental results. The many-body T-matrix has a positive-energy pole for $k_{\rm F}a>\alpha$ and it becomes impossible to distinguish the bound state and the scattering state. These positive-energy bound states become occupied and therefore the upper branch reaches an energy maximum at $k_{\rm F}a=\alpha$. In the zero range limit, there exists a narrow window ($0.86<k_{\rm F}a<1.56$) for the ferromagnetic phase. At sufficiently large negative effective range, the ferromagnetic phase disappears. On the other hand, the appearance of positive-energy bound state resonantly enhances the two-body decay rate around $k_{\rm F}a=\alpha$ and may prevent the study of equilibrium phases and ferromagnetism of the upper branch Fermi gas.Comment: Published version, typos correcte

Interaction energy and itinerant ferromagnetism in a strongly interacting Fermi gas in the absence of molecule formation

We investigate the interaction energy and the possibility of itinerant ferromagnetism in a strongly interacting Fermi gas at zero temperature in the absence of molecule formation. The interaction energy is obtained by summing the perturbative contributions of Galitskii-Feynman type to all orders in the gas parameter. It can be expressed by a simple phase space integral of an in-medium scattering phase shift. In both three and two dimensions (3D and 2D), the interaction energy shows a maximum before reaching the resonance from the Bose-Einstein condensate side, which provides a possible explanation of the experimental measurements of the interaction energy. This phenomenon can be theoretically explained by the qualitative change of the nature of the binary interaction in the medium. The appearance of an energy maximum has significant effects on the itinerant ferromagnetism. In 3D, the ferromagnetic transition is reentrant and itinerant ferromagnetism exists in a narrow window around the energy maximum. In 2D, the present theoretical approach suggests that itinerant ferromagnetism does not exist, which reflects the fact that the energy maximum becomes much lower than the energy of the fully polarized state.Comment: Published versio

Chiral Phase Transition beyond Mean Field Approximation

Based on the analogy between the Nambu--Jona-Lasinio model of chiral symmetry breaking and the BCS theory of superconductivity, we investigate the effect of $\bar q q$ pair fluctuations on the chiral phase transition. We include uncondensed $\bar q q$ pairs at finite temperature and chemical potential in a self-consistent T-matrix formalism, the so-called $G_0 G$ scheme. The pair fluctuations reduce significantly the critical temperature and make quarks massive above the critical temperature.Comment: 5 pages, 4 figure

Ginzburg-Landau free energy of crystalline color superconductors: A matrix formalism from solid-state physics

The Ginzburg-Landau (GL) free energy of crystalline color superconductors is important for understanding the nature of the phase transition to the normal quark matter and predicting the preferred crystal structure. So far the GL free energy at zero temperature has only been evaluated up to the sixth order in the condensate. To give quantitative reliable predictions we need to evaluate the higher-order terms. In this work, we present a new derivation of the GL free energy by using the discrete Bloch representation of the fermion field. This derivation introduces a simple matrix formalism without any momentum constraint, which may enable us to calculate the GL free energy to arbitrary order by using a computer.Comment: Published version, references adde

Dynamic density structure factor of a unitary Fermi gas at finite temperature

We present a theoretical investigation of the dynamic density structure factor of a strongly interacting Fermi gas near a Feshbach resonance at finite temperature. The study is based on a gauge invariant linear response theory. The theory is consistent with a diagrammatic approach for the equilibrium state taking into account the pair fluctuation effects and respects some important restrictions like the $f$-sum rule. Our numerical results show that the dynamic density structure factor at large incoming momentum and at half recoil frequency has a qualitatively similar behavior as the order parameter, which can signify the appearance of the condensate. This qualitatively agrees with the recent Bragg spectroscopy experiment results. We also present the results at small incoming momentum.Comment: 7 pages, 4 figure

BCS-BEC crossover in three-dimensional Fermi gases with spherical spin-orbit coupling

We present a systematic theoretical study of the BCS-BEC crossover problem in three-dimensional atomic Fermi gases at zero temperature with a spherical spin-orbit coupling which can be generated by a synthetic non-Abelian gauge field coupled to neutral fermions. Our investigations are based on the path integral formalism which is a powerful theoretical scheme for the study of the properties of the bound state, the superfluid ground state, and the collective excitations in the BCS-BEC crossover. At large spin-orbit coupling, the system enters the BEC state of a novel type of bound state (referred to as rashbon) which possesses a non-trivial effective mass. Analytical results and interesting universal behaviors for various physical quantities at large spin-orbit coupling are obtained. Our theoretical predictions can be tested in future experiments of cold Fermi gases with three-dimensional spherical spin-orbit coupling.Comment: V3: published version in PR

BCS-BEC crossover in relativistic Fermi systems

We review the BCS-BEC crossover in relativistic Fermi systems, including the QCD matter at finite density. In the first part we study the BCS-BEC crossover in a relativistic four-fermion interaction model and show how the relativistic effect affects the BCS-BEC crossover. In the second part, we investigate both two-color QCD at finite baryon density and pion superfluid at finite isospin density, by using an effective Nambu--Jona-Lasinio model. We will show how the model describes the weakly interacting diquark and pion condensates at low density and the BEC-BCS crossover at high density.Comment: 44 Pages, 20 figures, accepted by International Journal of Modern Physics A. arXiv admin note: substantial text overlap with arXiv:1007.1920, arXiv:1202.3490, arXiv:hep-ph/070304

BCS-BEC quantum phase transition and collective excitations in two-dimensional Fermi gases with p- and d-wave pairings

It is generally believed that the BCS-BEC evolution in fermionic systems with s-wave pairing is a smooth crossover. However, for nonzero orbital-angular-momentum pairing such as p- or d-wave pairing, the system undergoes a quantum phase transition at the point where the chemical potential $\mu$ vanishes. In this paper, we study the BCS-BEC quantum phase transition and the collective excitations associated with the order-parameter fluctuations in two-dimensional fermionic systems with p- and d-wave pairings. We show that the quantum phase transition in such systems can be generically traced back to the infrared behavior of the fermionic excitation at $\mu=0$: $E_{\bf k}\sim k^l$, where $l=1,2$ is the quantum number of the orbital angular momentum. The nonanalyticity of the thermodynamic quantities is due to the infrared divergence caused by the fermionic excitation at $\mu=0$. As a result, the evolution of the Anderson-Bogoliubov mode is not smooth: Its velocity is nonanalytical across the quantum phase transition.Comment: V2: published versio

Phase Diagram of Neutron-Proton Condensate in Asymmetric Nuclear Matter

We investigate the phase structure of homogeneous and inhomogeneous neutron-proton condensate in isospin asymmetric nuclear matter. At extremely low nuclear density the condensed matter is in homogeneous phase at any temperature, while in general case it is in Larkin-Ovchinnikov-Fulde -Ferrell phase at low temperature. In comparison with the homogeneous superfluid, the inhomogeneous superfluid can survive at higher nuclear density and higher isospin asymmetry.Comment: 4 pages, 2 figures, arguments and Fig.2 changed, references adde