Pion Superfluidity Research Articles

Kaon superfluidity in the early Universe

Previously, it was found that pion superfluidity could be realized in the quantum chromodynamics (QCD) epoch of the early Universe, when lepton flavor asymmetry |le+lμ| is large enough to generate a charge chemical potential |μQ| larger than vacuum pion mass. By following the same logic, kaon superfluidity might also be possible when |le+lμ| is so large that |μQ| becomes larger than vacuum kaon mass. Such a possibility is checked by adopting Ginzburg-Landau approximation within the three-flavor Polyakov–Nambu–Jona-Lasinio model. Consider the case with full chemical balance, though kaon superfluidity could be stable compared to the chiral phases with only σ condensations, it would get killed by the more favored homogeneous pion superfluidity. If we introduce mismatch between s and d quarks, kaon superfluidity would require so large s quark density that such a state is impossible in the early Universe. Published by the American Physical Society 2024

First-order QCD transition in a primordial magnetic field

Recalling the expectation of an extremely strong primordial magnetic field $H$, we recheck transitions among the phases of chiral symmetry restoration ($\chi SR$), chiral symmetry breaking ($\chi SB$), and pion superfluidity ($\pi SF$)in the QCD epoch of the early Universe. For homogeneous phases in a finite $H$, a sensible scheme is adopted to determine the phase boundaries of $\pi SF$, which is also superconductivity phase itself. In the first part, the QCD phase diagrams are studied in detail within the chiral effective Polyakov-Nambu--Jona-Lasinio model and the transitions involving $\pi SF$ are found to be of first order at relatively small $H$. As expected from the Meissner effect, the regime of $\pi SF$ shrinks with increasing $H$ and completely vanishes beyond a threshold value. In the second part, the bubble dynamics is illuminated for the stronger first-order transition, $\chi SR\rightarrow\pi SF$, in the more convenient Polyakov-quark-meson model. The coupled equations of motion of pion condensate and magnetic field are solved consistently to give the bubble structure. Then, based on bubble collisions, we explore gravitational wave (GW) emission by developing a simple toy model in advance; and the characteristic frequency of the relic GW is estimated to be of the order $0.1-1\,{\rm K}$ or $10^9-10^{10}\,{\rm Hz}$ in our galaxy.

Pion superfluid phase transition at finite isospin chemical potential

The pion superfluidity phase transition at $T-\mu_I$ and $\mu_q-\mu_I$ planes are studied in the framework of Dyson-Schwinger equations. The rainbow truncation and Gaussian effective gluon propagator are employed to calculate pion condensate, $\langle\pi\rangle$, as a function of $\mu_I$ at finite $T$ and $\mu_q$. At $T=(0, 80, 113, 120)$~MeV, the $\langle\pi\rangle$ keeps zero when $\mu_I$ is less than a critical value $\mu_{Ic}(T)$. The $\langle\pi\rangle$ at $T=113$~MeV agrees with the lattice QCD result. At the $T-\mu_I$ plane, the pion superfluid phase appears at low temperature and high isospin chemical potential region. At finite $\mu_q$, $\langle\pi\rangle$ varying with $\mu_I$ almost coincide for $\mu_q<400$~MeV. At $\mu_q-\mu_I$ plane, the pion superfluid phase appears at $\mu_I\gtrsim m_\pi$ when $\mu_q<400$~MeV.

Reentrant pion superfluidity and cosmic trajectories within a PNJL model

In this work, we self-consistently explore the possibility of charged pion superfluidity and cosmic trajectories in early Universe under the framework of Polyakov-Nambu--Jona-Lasinio model. By taking the badly constrained lepton flavor asymmetries $l_{\rm e}$ and $l_\mu$ as free parameters, the upper boundaries of pion superfluidity phase are consistently found to be around the pseudocritical temperature at zero chemical potentials. So the results greatly support the choice of $T=0.16~{\rm GeV}$ as the upper boundary of pion superfluidity in the previous lattice QCD study. Take $l_{\rm e}+l_\mu=-0.2$ as an example, we demonstrate the features of pion condensation and the associated cosmic trajectories with the evolution of early Universe. While the trajectory of electric chemical potential reacts strongly at both the lower and upper boundaries of reentrant pion superfluidity, the trajectories of other chemical potentials only respond strongly at the upper boundary.

QCD equation of state and the structure of up-down quark stars in NJL model

Chiral and pion superfluidity phase transitions are studied within NJL model. The model parameters are fitted by pion mass and decay constant together with recent isospin density from lattice QCD. Dense and cold QCD matter suffered the first order chiral phase transition during the quark chemical potential increase. Based on the studies of phase structures, the equation of states (EoS) of dense QCD matter is calculated with the conditions of β equilibrium and electric charge neutrality. Take the EoS as input, the mass-radius relation of up-down quark stars is predicted, which is consistent with recent observations of heaviest mass and radius of 1.4M⊙.

Recent progresses on QCD phases in a strong magnetic field: views from Nambu–Jona-Lasinio model

In this review, we summarize recent progress on the possible phases of quantum chromodynamics (QCD) in the presence of a strong magnetic field, mainly from the views of the chiral effective Nambu--Jona-Lasinio model. Four kinds of phase transitions are explored in detail: chiral symmetry breaking and restoration, neutral pseudoscalar superfluidity, charged pion superfluidity and charged rho superconductivity. In particular, we revisit the unsolved problems of inverse magnetic catalysis effect and competition between the chiral density wave and solitonic modulation phases. It is shown that useful results can be obtained by adopting self-consistent schemes.

QCD phase diagram at finite isospin and baryon chemical potentials with the self-consistent mean field approximation * *Supported by National Natural Science Foundation of China (12075117, 11535005, 11775118, 11690030, 11905104) and National Major state Basic Research and Development of China (2016YEF0129300)

The self-consistent mean field approximation of the two-flavor NJL model, with a free parameter to reflect the competition between the "direct" channel and the "exchange" channel, is employed to study the QCD phase structure at finite isospin chemical potential , finite baryon chemical potential and finite temperature T, and especially to study the location of the QCD critical point. Our results show that in order to match the corresponding lattice results of isospin density and energy density, the contributions of the "exchange" channel need to be considered in the framework of the NJL model, and a weighting factor should be taken. It is also found that for fixed isospin chemical potentials, the lower temperature of the phase transition is obtained with increasing in the plane, and the largest difference of the phase transition temperature with different 's appears at . At the temperature of the QCD critical end point (CEP) decreases with increasing , while the critical baryon chemical potential increases. At high isospin chemical potential ( MeV), the temperature of the QCD tricritical point (TCP) increases with increasing , and in the low temperature regions the system will transition from the pion superfluidity phase to the normal phase as increases. At low density, the critical temperature of the QCD phase transition with different 's rapidly increases with at the beginning, and then increases smoothly around MeV. In the high baryon density region, the increase of the isospin chemical potential will raise the critical baryon chemical potential of the phase transition.

Dual meson condensates in the Polyakov-loop enhanced linear sigma model

The extent to which the dual meson condensates obtained in the Polyakov-loop enhanced linear sigma model can indicate deconfining transition is investigated by imposing the twisted boundary conditions. The influence of the improved unquenching effect, the dependence on the glue potential parametrization, and the role of the fermion vacuum contribution are focused on at the mean field level. At zero density, the rapid rise of the dual sigma condensate with $T$ is confirmed, which is more sensitive to the chiral transition than the increase of the Polyakov loop. For finite isospin chemical potential ${\ensuremath{\mu}}_{I}&gt;{m}_{\ensuremath{\pi}}/2$, the dual sigma condensate shows abnormal thermal behavior which decreases with $T$ below the melting temperature ${T}_{c}^{{I}_{3}}$ of pion superfluidity. In contrast, the dual pion condensate always increases with $T$, with the maximum slope located at ${T}_{c}^{{I}_{3}}$ rather than ${T}_{c}^{P}$, as determined by the Polyakov loop. The dual vector meson condensate for ${\ensuremath{\mu}}_{I}&gt;{m}_{\ensuremath{\pi}}/2$ is also more sensitive to the chiral restoration when considering the fermion vacuum contribution. The study suggests that the dual condensates calculated in this model are not appropriate indicators of deconfinement due to some limitations and uncertainty.

Charged rho superconductor in the presence of magnetic field and rotation

In this work, we mainly explore the possibility of charged rho (rho ^pm ) superconductor in the presence of parallel magnetic field and rotation within three-flavor Nambu–Jona-Lasino model. By following similar schemes as in the previous studies of charged pion (pi ^pm ) superfluid, the rho ^pm superconductor is found to be favored for both choices of Schwinger phase in Minkowski and curved spaces. Due to the stability of the internal spin structure, charged rho begins to condensate at a smaller threshold of angular velocity than charged pion for the given large magnetic fields. Even the axial vector meson condensation is checked – the conclusion is that rho ^pm superconductor is the robust ground state at strong magnetic field and fast rotation, which actually sustains to very large angular velocity.

Pion quasiparticles and QCD phase transitions at finite temperature and isospin density from holography

Spectra of pions, which are known as the pseudo-Goldstone bosons of spontaneous chiral symmetry breaking, as well as their relationship with chiral phase transition and pion superfluidity phase transition, have been investigated in the framework of soft-wall AdS/QCD. In chiral limit, it is proved both numerically and analytically that pions are massless Goldstone bosons even at finite temperature, which was usually considered as an assumption in soft-wall models. Above $T_c$, at which chiral condensate $\langle \bar{q}q\rangle$ vanishes, the spectra of pions and scalar mesons merge together, showing the evidence of the restored chiral symmetry in hadronic spectrum level. Extending to finite quark mass, pion masses increase with quark mass. Further, it is more interesting to observe that the pole masses of pions decrease with temperature below $T_c$, which agrees with the analysis in Phys.Rev.Lett.88(2002)202302. Meanwhile, symmetry restoration above $T_c$ could be seen in the spectra of scalar and pseudo-scalar mesons. With finite temperature and isospin chemical potential $\mu_I$, it is shown that the masses of charged pions would split. The mass of positive charged pion $\pi^+$ decreases almost linearly to zero when $\mu_I$ grows to $\mu_{I}^c$, where pion condensation starts to form. This reveals the Goldstone nature of $\pi^+$ after pion superfluidity transition, which are closely related to the experimental observation.

Magnetic field effect on the pion superfluid

Magnetic field effect on pion superfluid phase transition is investigated in frame of a Pauli-Villars regularized NJL model. Instead of directly dealing with charged pion condensate, we apply the Goldstone's theorem (massless Goldstone boson $\pi^+$) to determine the onset of pion superfluid phase, and obtain the phase diagram in magnetic field, temperature, isospin and baryon chemical potential space. At weak magnetic field, it is analytically proved that the critical isospin chemical potential of pion superfluid phase transition is equal to the mass of $\pi^+$ meson in magnetic field. The pion superfluid phase is retarded to higher isospin chemical potential, and can survive at higher temperature and higher baryon chemical potential under external magnetic field.

New matter phases at high temperature and density

It is widely accepted that a new phase structure will emerge in nuclear matter at high temperature and density. In this article the QCD (quantum chromodynamic) phase transitions are briefly reviewed from both theoretical and experimental aspects. Basing on the general principle of QCD and modern phase transition theory, we analyze potential phases in the strong interacting nuclear matter. The phase structure could be understood intuitively through the breaking and restoration of fundamental symmetries of QCD. In the space spanned by the temperature, baryon and isospin chemical potential there are different phases corresponding to three important symmetries, i.e. the color, chiral and isospin. At low temperature and density, the color and isospin symmetries are preserved, while the chiral one is spontaneously broken. At high temperature or density, the color or isospin or both of them would be broken as the chiral restoration. In order to explore the detailed structure of the phase diagram various theoretical approaches of studying the transitions are introduced, such as lattice QCD, mean field calculation basing on effective models, functional renormalization group and Dyson-Schwinger equation. In experiments there are two kinds of physics systems to realize conditions of QCD phase transitions: the relativistic heavy ion collisions and compact stars. Nuclear collision is the only way to approach hot QCD in laboratories on the earth. We focus on some robust probes to the quark gluon plasma formed in the early stage of the collisions. There are generally two types of the traditional signals which are focusing on the profile of the fire ball and the internal interaction of it respectively. For the first case we introduce collective flows as observables for the shape of fire ball. While for the second we review the jet quenching and the heavy flavor suppression. In the hot fire ball the classical transport theory should be modified by including quantum effects induced by the chiral restoration. This novel transport phenomenon of QCD at high temperature is named by the chiral anomalous transport. On the phase transitions at high density, we review the recent progress on color superconductivity and pion superfluidity. Color superconductivity is formed at high baryon chemical potential when quarks pair with each other into diquarks and condense. While pion superfluidity would emerge at high isospin chemical potential with pion condensate. Model studies suggest these states are able to exist in the core of compact star. By modifying the structure of them, such as the mass-radius relation, these states are expected to be identified by astronomical observation at higher precision. Currently dense systems are difficult to produce at main facilities of the world, such as RHIC (relativistic heavy ion collider) and LHC (large hadron collider) because of the designed high energy scale of the accelerators. Therefore it is a critical need for the facilities which could cover the low and medium energy region where the potential critical point and various phase structures would locate. Considering the running and constructing heavy ion facilities, the CSR (cooler-storage-ring) in Lanzhou and the high intensity HIAF (heavy-ion accelerator facility) in Huizhou, the properties of the new QCD phases and their realizations in nuclear collisions will be the new frontier of nuclear physics in China.

Meson spectral functions at finite temperature and isospin density with the functional renormalization group

The pion superfluid and the corresponding Goldstone and soft modes are investigated in two-flavor quark-meson model with functional renormalization group. By solving the flow equations for the effective potential and the meson two-point functions at finite temperature and isospin density, the critical temperature for the superfluid increases sizeably in comparison with solving the flow equation for the potential only. The spectral function for the soft mode shows clearly a transition from meson gas to quark gas with increasing temperature and a crossover from BEC to BCS pairing of quarks with increasing isospin density.

Quarksonic matter at high isospin density**Supported by Thousand Young Talents Program of China, Shanghai Natural Science Foundation, (14ZR1403000) and NSFC (11535012), China Postdoctoral Science Foundation (KLH1512072)

Analogous to the quarkyonic matter at high baryon density in which the quark Fermi seas and the baryonic excitations coexist, it is argued that a “quarksonic matter” phase appears at high isospin density where the quark (antiquark) Fermi seas and the mesonic excitations coexist. We explore this phase in detail in both large Nc and asymptotically free limits. In the large Nc limit, we sketch a phase diagram for the quarksonic matter. In the asymptotically free limit, we study the pion superfluidity and thermodynamics of the quarksonic matter by using both perturbative calculations and an effective model.

Critical behavior and dimension crossover of pion superfluidity

We investigate the critical behavior of pion superfluidity in the framework of the functional renormalization group (FRG). By solving the flow equations in the SU(2) linear sigma model at finite temperature and isospin density, and making comparison with the fixed point analysis of a general $O(N)$ system with continuous dimension, we find that the pion superfluidity is a second order phase transition subject to an $O(2)$ universality class with a dimension crossover from ${d}_{c}=4$ to ${d}_{c}=3$. This phenomenon provides a concrete example of dimension reduction in thermal field theory. The large-$N$ expansion gives a temperature independent critical exponent $\ensuremath{\beta}$ and agrees with the FRG result only at zero temperature.

Effects of chiral imbalance and magnetic field on pion superfluidity and color superconductivity

The effects of chiral imbalance and external magnetic field on pion superfluidity and color superconductivity are investigated in extended Nambu--Jona-Lasinio models. We take Schwinger approach to treat the interaction between charged pion condensate and magnetic field at finite isospin density and include simultaneously the chiral imbalance and magnetic field at finite baryon density. For the superfluidity, the chiral imbalance and magnetic field lead to catalysis and inverse catalysis effects, respectively. For the superconductivity, the chiral imbalance enhances the critical baryon density, and the magnetic field results in a de Haas--van Alphan oscillation on the phase transition line.

Structure of compact stars in a pion superfluid phase

The gross structure of compact stars composed of pion superfluid quark matter is investigated in the frame of Nambu-Jona-Lasinio model. Under the Pauli-Villars regularization scheme, the uncertainty of the thermodynamic functions for inhomogeneous states is cured, and the LOFF state appeared in the hard cutoff scheme is removed from the phase diagram of pion superfluid. Different from the unpaired quark matter and color superconductor, the strongly coupled pion superfluid in the state close to the BCS limit is a possible candidate of compact stars with mass $M\simeq 3M_{\odot }$ and radius $R\simeq 14$ km.

Meson Scattering Identifying the BCS-BEC Crossover at Quark Level

Instead of the fermion-fermion scattering which identifies the BCS-BEC crossover in cold atom systems, boson-boson scattering is measurable and characterizes the BCS-BEC crossover in QCD superfluid. We study π-π scattering in a pion superfluid described by the Nambu–Jona-Lasinio model. We found that the amplitude of the scattering length drops down monotonously with decreasing isospin density and finally vanishes at the boundary of the phase transition. This indicates clearly a BCS-BEC crossover in the pion superfluid.

Meson scattering in a pion superfluid

Instead of the fermion-fermion scattering which identifies the BCS-BEC crossover in cold atom systems, boson-boson scattering is measurable and characterizes the BCS-BEC crossover at quark level. We study $\pi$-$\pi$ scattering in a pion superfluid described by the Nambu--Jona-Lasinio model. We found that the scattering amplitude drops down monotonically with decreasing isospin density and finally vanishes at the boundary of the phase transition. This indicates a BCS-BEC crossover in the pion superfluid.

Spontaneous parity and charge-conjugation violations at real isospin and imaginary baryon chemical potentials

The phase structure of two-flavor QCD is investigated at real isospin and imaginary quark chemical potentials by using the Polyakov-loop extended Nambu--Jona-Lasinio model. In the region, parity symmetry is spontaneously broken by the pion superfluidity phase transition, whereas charge-conjugation symmetry is spontaneously violated by the Roberge-Weiss transition. The chiral (deconfinement) crossover at zero isospin and quark chemical potentials is a remnant of the parity (charge-conjugation) violation. The interplay between the parity and charge-conjugation violations are analyzed, and it is investigated how the interplay is related to the correlation between the chiral and deconfinement crossovers at zero isospin and quark chemical potentials.