LargeNcQCD at nonzero chemical potential

We consider the phase diagram of twisted mass Wilson fermions of two-flavor\nQCD in the parameter space of the quark mass, the isospin chemical potential,\nthe twist angle and the lattice spacing. This work extends earlier studies in\nthe continuum and those at zero chemical potential. We evaluate the phase\ndiagram as well as the spectrum of the (pseudo-)Goldstone bosons using the\nchiral Lagrangian for twisted mass Wilson fermions at non-zero isospin chemical\npotential. The phases are obtained from a mean field analysis. At zero twist\nangle we find that already an infinitesimal isospin chemical potential destroys\nthe Aoki phase. The reason is that in this phase we have massless Goldstone\nbosons with a non-zero isospin charge. At finite twist angle only two different\nphases are present, one phase which is continuously connected to the Bose\ncondensed phase at non-zero chemical potential and another phase which is\ncontinuously connected to the normal phase. For either zero or maximal twist\nthe phase diagram is more complicated as the saddle point equations allow for\nmore solutions.\n

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.

We investigate the phase structure of strongly interacting matter at non-vanishing isospin before the onset of pion condensation in the framework of the unquenched Polyakov–Quark-Meson model with 2+1 quark flavors. We show results for the order parameters and all relevant thermodynamic quantities. In particular, we obtain a moderate change of the pressure with isospin at vanishing baryon chemical potential, whereas the chiral condensate decreases more appreciably. We compare the effective model to recent lattice data for the decrease of the pseudo-critical temperature with the isospin chemical potential. We also demonstrate the major role played by the value of the pion mass in the curvature of the transition line, and the need for lattice results with a physical pion mass. Limitations of the model at nonzero chemical potential are also discussed.

We construct superfluid black hole solutions with two chemical potentials. By analogy with QCD, the two chemical potentials correspond to the baryon and isospin symmetries, respectively. We consider two systems: the back-reacted U(2) Einstein-Yang-Mills theory in 4+1 dimensions and the 9+1-dimensional D3/D7 brane setup with two coincident D7-brane probes. In the D7-brane model, the identification of baryon and isospin chemical potential is explicit since the dual field theory is explicitly known. Studying the phase diagram, we find in both systems a quantum phase transition at a critical ratio of the two chemical potentials. However the quantum phase transition is different in the two systems: In the D3/D7 brane setup we always find a second order phase transition, while in the Einstein-Yang-Mills theory, depending on the strength of the back-reaction, we obtain a continuous or first order transition. We expect the continuous quantum phase transition to be BKT-like. We comment on the origin of this differing behavior in these apparently very similar models and compare to phenomenological systems.

We discuss the spectrum of the QCD Dirac operator both at zero and at nonzero baryon chemical potential. We show that, in the ergodic domain of QCD, the Dirac spectrum can be obtained from the replica limit of a Toda lattice equation. At zero chemical potential this method explains the factorization of known results into compact and noncompact integrals, and at nonzero chemical potential it allows us to derive the previously unknown microscopic spectral density.

The thermodynamics of massless ideal gas of overlap quarks has been investigated both analytically and numerically for both zero and nonzero baryon chemical potential. Any ${\ensuremath{\mu}}^{2}$ divergence is shown analytically to be absent for a class of actions with nonzero chemical potential. All such actions are shown to violate chiral invariance. While the parameter $M$ can be shown to be irrelevant in the continuum limit, as expected, it is shown numerically that the continuum limit can be reached with relatively coarser lattices for $1.5\ensuremath{\le}M\ensuremath{\le}1.6$. Numerical limitations of the existing method of introduction of chemical potential are demonstrated. Finally we also show that the energy density for the massive overlap fermions has the correct continuum limit.

We study the ground states of low-density hadronic matter and high-density color-flavor locked color superconducting phase in three-flavor QCD at finite baryon chemical potential under rotation. We find that, in both cases under sufficiently fast rotation, the combination of the rotation-induced topological term for the η′ meson and the QCD anomaly leads to an inhomogeneous condensate of the η′ meson, known as the chiral soliton lattice (CSL). We find that, when baryon chemical potential is much larger than isospin chemical potential, the critical angular velocity for the realization of the η′ CSL is much smaller than that for the π0 CSL found previously. We also argue that the η′ CSL states in flavor-symmetric QCD at low density and high density should be continuously connected, extending the quark-hadron continuity conjecture in the presence of the rotation.

We investigate the interplay of the Polyakov loop and the quark–antiquark condensates in scalar and pseudoscalar channels in the extended-Polyakov–NJL (PNJL) model at finite temperature and isospin chemical potential without baryon chemical potential. We discuss the effect of the dynamical parameter ϕ and thermodynamical parameter μI on the chiral property especially. The meson's collective excitation modes are calculated and compared with the NJL model's results. The T–μI phase diagram is obtained. It contains four regions, including the meson gas, two-flavor quark gas, the BEC superfluidity and BCS superfluidity.

We investigate the thermodynamics of two-flavor quark matter in presence of nonzero isospin chemical potential and external magnetic field. It is known that at sufficiently high isospin chemical potential, charged pions undergo Bose–Einstein condensation (BEC). The condensate behaves as a superconductor, exhibiting Meissner effect in weak external magnetic fields. Stronger fields stress the superconducting state, turning it first into an Abrikosov lattice of vortices, and eventually destroying the condensate altogether. On the other hand, for sufficiently strong magnetic fields and low-to-moderate isospin chemical potential, the ground state of quantum chromodynamics (QCD) is expected to be a spatially modulated condensate of neutral pions, induced by the chiral anomaly: the chiral soliton lattice (CSL). We map the phase diagram of QCD as a function of isospin chemical potential and magnetic field in the part of the parameter space accessible to a low-energy effective field theory description of QCD. Our main result is an explicit account of the competition between the CSL and the Abrikosov vortex lattice. This is accomplished by adopting a fast numerical algorithm for finding the vortex lattice solution of the equation of motion and the corresponding Gibbs energy. We find that the Abrikosov vortex lattice phase persists in the phase diagram, separating the uniform charged pion BEC phase from the CSL phase. The precise layout of the phase diagram depends sensitively on the choice of the vacuum pion mass.

The linear sigma model at finite temperature and chemical potential is systematically studied using the coherent-pair approximation, in which the quantized meson fields are included. The expectation value of the chiral Hamiltonian density is minimized and the resulting equations for the nucleon are solved. The qualitative features of the quantized sigma and pion fields are strongly sensitive to the change of temperature and chemical potential and are in agreement with the mean-field approximation calculations. It is noticed that the nucleon mass increases with increasing coherence parameter x. In addition, the nucleon mass increases with increasing temperature T and the baryonic chemical potential μ, and then it decreases at higher values of the temperature and baryonic chemical potential. The obtained results show that the mean-square radius of the proton and the neutron increase with increasing temperature or baryonic chemical potential, and that the pion–nucleon coupling constant decreases with the temperature or baryonic chemical potential. We conclude that the coherent-pair approximation successfully gives a better description of the nucleon properties at finite temperature and baryonic chemical potential.

Systems of non-zero isospin chemical potential are studied from a canonical approach by computing correlation functions with the quantum numbers of N π+'s (CNπ). In order to reduce the number of contractions required in calculating CNπ for a large N in the Wick's theorem, we constructed a few new algorithms. With these new algorithms, systems with isospin charge up to 72 are investigated on three anisotropic gauge ensembles with a pion mass of 390 MeV, and with lattice spatial extents L ∼ 2.0, 2.5, 3.0 fm. The largest isospin density of ρI ≈ 9 fm−3 is achieved in the smallest volume, and the QCD phase diagram is investigated at a fixed low temperature at varying isospin chemical potentials, mπ ≤ μI ≤ 4.5 mπ. By investigating the behaviour of the extracted energy density of the system at different isospin chemical potentials, we numerically identified the conjectured transition to a Bose-Einstein condensation state at μI ≥ mπ.

International audience

We present results for fluctuations of the baryon number for QCD at nonzero temperature and chemical potential. These are extracted from solutions to a coupled set of truncated Dyson-Schwinger equations for the quark and gluon propagators of Landau-gauge QCD with $N_f = 2 + 1$ quark flavors, that has been studied previously. We discuss the changes of fluctuations and ratios thereof up to fourth order for several temperatures and baryon chemical potential up to and beyond the critical endpoint. In the context of preliminary STAR data for the skewness and kurtosis ratios, the results are compatible with the scenario of a critical endpoint at large chemical potential and slightly offset from the freeze-out line. We also discuss the caveats involved in this comparison.

A new chiral two-matrix theory for Dirac spectra with imaginary chemical potential

In this work, we investigate how the assumption of chemical equilibrium with leptons affects the deconfinement phase transition to quark matter. This is carried out within the framework of the Chiral Mean Field model allowing for nonzero net strangeness, corresponding to the conditions found in astrophysical scenarios. We build three‐dimensional quantum chromodynamics phase diagrams with temperature, baryon chemical potential, and either charge or isospin fraction or chemical potential to show how the deconfinement region collapses to a line in the special case of chemical equilibrium, such as the one established in the interior of cold catalyzed neutron stars.