Chiral Condensation and Chiral Phase Diagram under Combined Rotation and Chemical Potential in Holographic QCD

Abstract: We investigate the combined effects of rotation and finite quark chemical potential on inhomogeneous chiral condensation and the chiral phase diagram within the soft-wall holographic QCD model. Using the five-dimensional AdS-RN metric, we study the spatial profile of the chiral condensate and the resulting $T - Ω$ phase diagram under Neumann and Dirichlet boundary conditions. Increasing $Ω$ induces strong spatial inhomogeneity: the condensate is suppressed more strongly near the edge than at the center, and vanishes at the boundary when $Ω$ exceeds a critical value. The chemical potential $μ$ acts as a global suppression factor, reducing the condensate magnitude without altering the spatial pattern. The $T - Ω$ phase diagrams are investigated for different chemical potentials .For the case $μ$ = 0, they are also studied at different distances from the rotation axis. It is found that both $Ω$ and $μ$ lower the chiral phase transition temperature, and their suppression effects are additive. In a rotating system, the critical temperature becomes position-dependent, decreasing with increasing distance from the rotation axis. These findings reveal a rich, spatially dependent phase structure in rotating QCD matter, relevant for non-central heavy-ion collisions.

• The paper demonstrates that rotation induces spatial inhomogeneity in the chiral condensate, particularly suppressing it near the plasma edge.
• The paper shows that chemical potential produces a uniform suppression of the chiral condensate across all spatial coordinates.
• The paper finds that the combined effects of rotation and chemical potential lower the pseudocritical temperature, offering key insights for heavy-ion collision experiments.

Chiral Condensation and Chiral Phase Structure Under Rotation and Chemical Potential in Holographic QCD

Introduction

This study examines the interplay between rotation and finite quark chemical potential in the context of inhomogeneous chiral condensation and the associated phase diagram within the soft-wall AdS/QCD framework. The motivation is rooted in the phenomenology of non-central heavy-ion collisions, where rotating quark-gluon plasma (QGP) is generated, and finite baryon densities are realized. Traditional lattice QCD suffers from the sign problem at finite chemical potential, motivating recourse to holographic descriptions for accessing the non-perturbative regime with both rotation and density.

Theoretical Framework

The analysis is based on the soft-wall holographic QCD model, which successfully reproduces a variety of chiral symmetry breaking phenomena and can accommodate first-order, second-order, and crossover transitions by tuning potential parameters and the dilaton profile. The background spacetime is a five-dimensional AdS-Reissner-Nordström (AdS-RN) black hole geometry, encoding temperature and finite quark chemical potential via black hole charge and horizon placement. Rotation is incorporated as an angular component of a $U(1)$ gauge field within the bulk, dual to the boundary theory's polarization and angular momentum densities.

The physical system is spatially confined in the radial direction to ensure causality, with the edge of the plasma identified at $r=R$. Both Neumann and Dirichlet boundary conditions for the scalar condensate are explored, corresponding to free versus rigid boundaries that impact the condensate’s spatial behavior near the edge.

The scalar field’s near-boundary behavior defines the chiral condensate $\sigma$ and, through solving coupled PDEs for the scalar and gauge fields, the spatial and thermodynamic dependence of $\sigma$ is mapped out.

Results and Numerical Findings

Spatial Inhomogeneity Induced by Rotation

Rotation manifests as a spatially modulated suppression of the chiral condensate, most pronounced near the edge of the system. At angular velocities $\Omega$ exceeding a critical threshold $\Omega_c$, the condensate vanishes at the system’s boundary while remaining finite toward the center. This spatial inhomogeneity stands in stark contrast with the uniform profile found in non-rotating, finite temperature/density cases. Dirichlet boundary conditions accentuate the abruptness of this vanishing near the edge relative to Neumann conditions.

Chemical Potential as a Uniform Suppression Factor

Under fixed rotation and temperature, the chemical potential $\mu$ leads to a global, uniform decrease in the condensate’s magnitude across all spatial coordinates; however, it does not alter the spatial profile imprinted by rotation. The normalized shape $\sigma(r)/\sigma(0)$ remains essentially unchanged for varying $\mu$, indicating that baryon density acts as a homogeneous weakening factor for chiral symmetry breaking.

Additivity and Position Dependence in the Phase Diagram

Both rotation and chemical potential lower the pseudocritical chiral transition temperature $T_c$. Notably, their effects are additive, so that the $r=R$0 phase boundary shifts downward as either parameter increases. Importantly, in a rotating system $r=R$1 becomes a local function of radius $r=R$2, decreasing monotonically with distance from the rotation axis. Consequently, different radial regions of the plasma can simultaneously reside in broken and restored chiral symmetry phases within the same global $r=R$3 environment.

Quantitatively, boundary conditions impact the magnitude of $r=R$4 at the center, with Dirichlet boundaries consistently yielding lower critical temperatures compared to Neumann.

Phase Diagrams

The $r=R$5 and $r=R$6 phase diagrams constructed from the scalar condensate at the center ($r=R$7) reveal:

• For fixed $r=R$8, increasing $r=R$9 monotonically suppresses $\sigma$0.
• For fixed $\sigma$1, increasing $\sigma$2 lowers $\sigma$3, and the phase boundary moves to lower temperatures.
• At finite $\sigma$4 and $\sigma$5, the suppression in $\sigma$6 is stronger due to their additive influence.

Furthermore, the spatial dependence of the $\sigma$7 phase boundary confirms that at larger radii, symmetry restoration occurs at lower temperatures, implying inhomogeneous phase coexistence under rotation.

Implications and Future Directions

The findings have direct phenomenological relevance for the QCD matter created in non-central heavy-ion collisions, where both rotational flow and finite baryon density are realized. The spatially-dependent suppression of chiral condensation suggests that experimental observables sensitive to the chiral transition—such as fluctuations and correlations of low-mass hadrons—may possess radial dependence correlated with angular momentum and local density distributions. The work underscores the need for refined modeling of realistic boundary conditions and finite-size effects in hydrodynamic and holographic simulations of the QGP.

Theoretically, the clear, additive analogy between rotation and chemical potential in regulating chiral symmetry provides a pathway for further investigations of critical phenomena and possible emergent phases (e.g., vortex-ordered states or localized chiral domains) under combined extreme conditions. Incorporating backreaction of matter fields, spatially varying angular velocities, and more sophisticated modeling of the vortex sector remain compelling open directions.

Conclusion

This study systematically elucidates the combined impact of rotation and finite chemical potential on chiral condensation and phase structure in holographic QCD. The principal outcomes include: (1) the emergence of spatial inhomogeneity in the condensate profile under rotation, (2) the additive suppression of the chiral transition temperature by rotation and baryon density, and (3) the boundary-condition sensitivity of critical behavior, especially near system edges. These results provide a richer understanding of the chiral phase structure of QCD matter under realistic, extreme conditions and offer testable predictions for future experimental and theoretical investigations.