Nonzero Isospin Chemical Potential Research Articles

Effective Lagrangian at nonzero isospin chemical potential

We revisit the most general effective Lagrangian within chiral perturbation theory at nonzero isospin chemical potential ${\ensuremath{\mu}}_{I}$ up to $\mathcal{O}({p}^{4})$. In addition to the contributions already considered in the literature, we discuss the effects of new terms allowed by the symmetries derived within the external source method including spurion fields, as well as linear-field corrections relevant to $\mathcal{O}({p}^{4})$. We study the influence of those new contributions to the free energy density at zero temperature and observables derived from it, such as the pion and quark condensates and the isospin density. Corrections are shown to be compatible with lattice results, which favor nonzero values for the low-energy constants (LECs) multiplying the new $\mathcal{O}({p}^{2})$ and $\mathcal{O}({p}^{4})$ field operators in the Lagrangian. In particular, the $\mathcal{O}({p}^{4})$ LECs are renormalized to render the free energy density finite. Constraints on the LECs arise from preserving the physical condition ${n}_{I}({\ensuremath{\mu}}_{I}<{\ensuremath{\mu}}_{c})=0$, while ${\ensuremath{\mu}}_{c}={M}_{\ensuremath{\pi}}$ still holds to leading order and can be maintained to next-to-leading order through an additional constraint requiring the new LECs.

Competition of chiral soliton lattice and Abrikosov vortex lattice in QCD with isospin chemical potential

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.

Isospin asymmetry in holographic baryonic matter

We study baryonic matter with isospin asymmetry, including fully dynamically its interplay with pion condensation. To this end, we employ the holographic Witten-Sakai-Sugimoto model and the so-called homogeneous ansatz for the gauge fields in the bulk to describe baryonic matter. Within the confined geometry and restricting ourselves to the chiral limit, we map out the phase structure in the presence of baryon and isospin chemical potentials, showing that for sufficiently large chemical potentials condensed pions and isospin-asymmetric baryonic matter coexist. We also present first results of the same approach in the deconfined geometry and demonstrate that this case, albeit technically more involved, is better suited for comparisons with and predictions for real-world QCD. Our study lays the ground for future improved holographic studies aiming towards a realistic description of charge neutral, beta-equilibrated matter in compact stars, and also for more refined comparisons with lattice studies at nonzero isospin chemical potential.

Gluon propagators in 2+1 lattice QCD with nonzero isospin chemical potential

The static longitudinal and transverse gluon propagators in the Landau gauge are studied in 2+1 lattice QCD with nonzero isospin chemical potential mu _I. Parameterization of the momentum dependence of the propagators is provided for all values of the chemical potential under study. We find that the longitudinal propagator is infrared suppressed at nonzero mu _I with suppression increasing with increasing mu _I. It is found, respectively, that the electric screening mass is increasing with increasing mu _I. Additionally, we analyze the difference between two propagators as a function of the momentum and thus compare interactions in chromoelectric and chromomagnetic sectors.

Pion and kaon condensation at zero temperature in three-flavor \u03c7PT at nonzero isospin and strange chemical potentials at next-to-leading order

We consider three-flavor chiral perturbation theory (χPT) at zero temperature and nonzero isospin (μI) and strange (μS) chemical potentials. The effective potential is calculated to next-to-leading order (NLO) in the π±-condensed phase, the K±-condensed phase, and the {K}^0/{overline{K}}^0 -condensed phase. It is shown that the transitions from the vacuum phase to these phases are second order and take place when, left|{mu}_Iright|={m}_{pi },left|frac{1}{2}{mu}_I+{mu}_Sright|={m}_K , and left|-frac{1}{2}{mu}_I+{mu}_Sright|={m}_K , respectively at tree level and remains unchanged at NLO. The transition between the two condensed phases is first order. The effective potential in the pion-condensed phase is independent of μS and in the kaon-condensed phases, it only depends on the combinations pm frac{1}{2}{mu}_I+{mu}_S and not separately on μI and μS. We calculate the pressure, isospin density and the equation of state in the pion-condensed phase and compare our results with recent (2 + 1)-flavor lattice QCD data. We find that the three-flavor χPT results are in good agreement with lattice QCD for μI< 200 MeV, however for larger values χPT produces values for observables that are consistently above lattice results. For μI> 200 MeV, the two-flavor results are in better agreement with lattice data. Finally, we consider the observables in the limit of very heavy s-quark, where they reduce to their two-flavor counterparts with renormalized couplings. The disagreement between the predictions of two and three flavor χPT can largely be explained by the differences in the experimental values of the low-energy constants.

QCD at finite isospin density: Chiral perturbation theory confronts lattice data

We consider the thermodynamics of three-flavor QCD in the pion-condensed phase at nonzero isospin chemical potential (μI) and vanishing temperature using chiral perturbation theory in the isospin limit. The transition from the vacuum phase to a superfluid phase with a Bose-Einstein condensate of charged pions is shown to be second order and takes place at μI=mπ. We calculate the pressure, isospin density, and energy density to next-to-leading order in the low-energy expansion. Our results are compared with recent high-precision lattice simulations as well as previously obtained results in two-flavor chiral perturbation theory. The agreement between the lattice results and the predictions from three-flavor chiral perturbation theory is very good for μI<200 MeV. For larger values of μI, the agreement between lattice data and the two-flavor predictions is surprisingly good and better than with the three-flavor predictions. Finally, in the limit ms≫mu=md, we show that the three-flavor observables reduce to the two-flavor observables with renormalized parameters. The disagreement between the results for two-flavor and three-flavor χPT can largely be explained by the differences in the measured low-energy constants.

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

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 μ I ≥ m 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).

Meson properties in asymmetric matter

In this work we study dynamic and thermodynamic (at T = 0) properties of mesons in asymmetric matter in the framework of Chiral Perturbation Theory. We consider a system at vanishing temperature with nonzero isospin chemical potential and strangeness chemical potential; meson masses and mixing in the normal phase, the pion condensation phase and the kaon condensation phase are described. We find differences with previous works, but the results presented here are supported by both theory group analysis and by direct calculations. Some pion decay channels in the normal and the pion condensation phases are studied, finding a nonmonotonic behavior of the decay width as a function of µI. Furthermore, pressure, density and equation of state of the system at T = 0 are studied, finding remarkable agreement with analogue studies performed by lattice calculations.

Meson physics in asymmetric matter

This paper describes dynamic and thermodynamic (at T = 0) properties of mesons in asymmetric matter in the framework of Chiral Perturbation Theory. We consider the effect of nonzero isospin and strangeness chemical potentials on a mesonic system and report on the corresponding phase diagram. We also study meson masses and mixing in the resulting normal phase, pion condensation phase and kaon condensation phase. We find differences with previous papers regarding meson masses and mixing in the condensed phases; the results presented here are supported by theory group analysis and direct calculations. Pressure, density and equation of state of the system at T = 0 and nonzero μI are calculated, finding remarkable agreement with analogue studies performed by lattice calculations.

Classification of sign-problem-free relativistic fermions on the basis of the Majorana positivity

We classify the sign-problem-free relativistic fermion actions on the basis of the Majorana representation. In the Majorana representation, the sign-problem-free condition is given by the semi-positivity of a Pfaffian. We show that the known sign-problem-free actions of the Dirac fermions, which are usually understood from the semi-positivity of a determinant, e.g., the action of quantum chromodynamics with nonzero chiral chemical potential or nonzero isospin chemical potential, can also be understood from the semi-positivity of a Pfaffian. We also derive new classes of the sign-problem-free relativistic fermion actions with Majorana-type source terms.

Phase diagram of dynamical twisted-mass Wilson fermions at finite isospin chemical potential

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

Aspects of Meson Condensation

In this work pion and kaon condensation in the framework of chiral perturbation theory is studied. I consider a system at vanishing temperature with nonzero isospin chemical potential and strangeness chemical potential; meson masses and mixing in the normal phase, the pion condensation phase and the kaon condensation phase are described. There are differences with previous works, but the results presented here are supported by both theory group analysis and by direct calculations. Some pion decay channels in the normal and the pion condensation phases are studied, finding a nonmonotonic behavior of the $\Gamma$ decay width as a function of $\mu_I$.

Lattice QCD at non-zero isospin 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.

Lattice QCD at nonzero isospin chemical potential

Quantum chromodynamics (QCD) at non-zero isospin chemical potential is studied in a canonical approach by analyzing systems of fixed isospin number density. To construct these systems, we develop a range of new algorithms for performing the factorially large numbers of Wick contractions required in multi-hadron systems. We then use these methods to study systems with the quantum numbers of up to 72 $\pi^+$'s on three ensembles of gauge configurations with spatial extents $L\sim$ 2.0, 2.5 and 3.0 fm, and light quark masses corresponding to a pion mass of {390 MeV}. The ground state energies of these systems are extracted and the volume dependence of these energies is utilized to determine the two- and three- body interactions amongst $\pi^+$'s. The systems studied correspond to isospin densities of up to $\rho_I\sim 9\ {\rm fm}^{-3}$ and probe isospin chemical potentials, $\mu_I$, in the range $m_\pi\ \lsim \mu_I\ \lsim 4.5\ m_\pi$, allowing us to investigate aspects of the QCD phase diagram at low temperature and for varying isospin chemical potential. By studying the energy density of the system, we provide numerical evidence for the conjectured transition of the system to a Bose-Einstein condensed phase at $\mu_I\ \gsim m_\pi$.

Critical line of two-flavor QCD at finite isospin or baryon densities from imaginary chemical potentials

We determine the (pseudo)critical lines of QCD with two degenerate staggered fermions at nonzero temperature and quark or isospin density, in the region of imaginary chemical potentials; analytic continuation is then used to prolongate to the region of real chemical potentials. We obtain an accurate determination of the curvatures at zero chemical potential, quantifying the deviation between the case of finite quark and of finite isospin chemical potential. Deviations from a quadratic dependence of the pseudocritical lines on the chemical potential are clearly seen in both cases: we try different extrapolations and, for the case of nonzero isospin chemical potential, confront them with the results of direct Monte Carlo simulations. Finally we find that, as for the finite quark density case, an imaginary isospin chemical potential can strengthen the transition till turning it into strong first order.

SpontaneousCPviolation in the strong interaction atθ=π

Spontaneous $CP$ violation in the strong interaction is analyzed at $\ensuremath{\theta}=\ensuremath{\pi}$ within the framework of the two-flavor Nambu-Jona-Lasinio model. It is found that the occurrence of spontaneous $CP$ violation at $\ensuremath{\theta}=\ensuremath{\pi}$ depends on the strength of the 't Hooft determinant interaction, which describes the effect of instanton interactions. The dependence of the phase structure, and, in particular, of the $CP$-violating phase, on the quark masses, temperature, baryon and isospin chemical potential is examined in detail. When available a comparison to earlier results from chiral perturbation theory is made. From our results we conclude that spontaneous $CP$ violation in the strong interaction is an inherently low-energy phenomenon. In all cases we find agreement with the Vafa-Witten theorem, also at nonzero density and temperature. Meson masses and mixing in the $CP$-violating phase display some unusual features as a function of instanton interaction strength. A modification of the condition for charged pion condensation at nonzero isospin chemical potential and a novel phase of charged ${a}_{0}$ mesons are discussed.

Existence of the critical point in finite density lattice QCD

We propose a method to probe the nature of phase transitions in lattice QCD at finite temperature and density, which is based on the investigation of an effective potential as a function of the average plaquette. We analyze data obtained in a simulation of two-flavor QCD using p4-improved staggered quarks with bare quark mass m/T=0.4 and find that a first order phase transition line appears in the high density regime for {mu}{sub q}/T > or approx. 2.5. We also discuss the difference between the phase structures of QCD with nonzero quark chemical potential and nonzero isospin chemical potential.

Confinement-deconfinement phase transition at nonzero chemical potential

We present arguments suggesting that large size overlapping instantons are the driving mechanism of the confinement-deconfinement phase transition in QCD at nonzero chemical potential $\ensuremath{\mu}$. The arguments are based on the picture that instantons at very large chemical potential in the weak coupling regime are localized configurations with finite size $\ensuremath{\rho}\ensuremath{\sim}{\ensuremath{\mu}}^{\ensuremath{-}1}$. At the same time, the same instantons at smaller chemical potential in the strong coupling regime are well represented by the so-called instanton-quarks with fractional topological charge $1/{N}_{c}$. We estimate the critical chemical potential ${\ensuremath{\mu}}_{c}(T)$ where this phase transition takes place as a function of temperature in the domain where our approach is justified. In this picture, the long standing problem of the ``accidental'' coincidence of the chiral and deconfinement phase transitions at nonzero temperature (observed in lattice simulations) is naturally resolved. We also derive results at nonzero isospin chemical potential ${\ensuremath{\mu}}_{I}$ where direct lattice calculations are possible, and our predictions can be explicitly tested.

Diquark and pion condensation in random matrix models for two-color QCD

We introduce a random matrix model with the symmetries of QCD with two colors at nonzero isospin and baryon chemical potentials and temperature. We analyze its phase diagram and find phases with condensation of pion and diquark states in addition to the phases with spontaneously broken chiral symmetries. In the limit of small chemical potentials and quark masses, we reproduce the mean field results obtained from chiral Lagrangians. As in the case of QCD with three colors, the presence of two chemical potentials breaks the flavor symmetry and leads to phases that are characterized by different behaviors of the chiral condensates for each flavor. In particular, the phase diagram we obtain is similar to QCD with three colors and three flavors of quarks of equal masses at zero baryon chemical potential and nonzero isospin and strange chemical potentials. A tricritical point of the superfluid transitions found in lattice calculations and from an analysis in terms of chiral Lagrangians does not appear in the random matrix model. Remarkably, at fixed isospin chemical potential, for the regions outside of the superfluid phases, the phase diagram in the temperature - baryon chemical potential plane for two colors and three colors are qualitatively the same.

The Lattice NJL Model at Non-zero Baryon and Isospin Densities

We present initial results of a numerical investigation of the chiral symmetry restoring transition in the (3+1)-dimensional Nambu – Jona-Lasinio model with both non-zero baryon chemical potential ( μ B ≠ 0 ) and isospin chemical potential ( μ I ≠ 0 ). With non-zero isospin chemical potential, the model suffers from a sign problem. We proceed in two ways: ( i ) We perform “partially quenched” simulations in which μ I is made non-zero only during the measurement of chiral observables; ( ii ) We perform full simulations with imaginary isospin chemical potential with the aim to analytically continue results to real μ I .