[Paper Review] Magnetic Catalysis and Oscillating Effects in Nambu -- Jona-Lasinio Model at Nonzero Chemical Potential

К. Г. Клименко|ArXiv.org|Sep 1, 1998

Theoretical and Computational Physics1 references4 citations

TL;DR

This paper investigates the Nambu-Jona-Lasinio model under nonzero chemical potential and external magnetic fields, revealing infinite families of massless chirally symmetric and massive phases with spontaneously broken chiral symmetry. It demonstrates oscillatory behavior in physical observables—such as magnetization, pressure, particle density, quark condensate, and critical curves—under thermodynamic limits, with phase transitions of first and second order and multiple tricritical points identified on phase diagrams.

ABSTRACT

Phase structure of the four dimensional Nambu -- Jona-Lasinio model has been investigated in two cases: 1) in nonsimply connected space-time of the form $R^3 imes S^1$ (space coordinate is compactified and the length of the circle $S^1$ is $L$) with nonzero chemical potential $μ$ and 2) in Minkowski space-time at nonzero values of $μ,H$, where $H$ is the external magnetic field. In both cases on phase portraits of the model there are infinitly many massless chirally symmetric phases as well as massive ones with spontaneously broken chiral invariance. Such phase structure leads unavoidably to oscillations of some physical parameters at $L o\infty$ or $H o 0$, including magnetization, pressure and particle density of the system as well as quark condensate and critical curve of chiral phase transitions. Phase transitions of 1st and 2nd orders and several tricritical points have been shown to exist on phase diagrams of the model.

Motivation & Objective

• To analyze the phase structure of the four-dimensional Nambu-Jona-Lasinio model under nonzero chemical potential and external magnetic fields.
• To investigate the interplay between compactified spatial dimensions (R³×S¹) and external magnetic fields in determining the system's phase diagram.
• To identify the existence of infinitely many massless chirally symmetric and massive phases with spontaneous chiral symmetry breaking.
• To examine the emergence of oscillations in physical parameters such as magnetization, pressure, particle density, and quark condensate as L→∞ or H→0.
• To determine the order of phase transitions and locate tricritical points in the phase diagrams.

Proposed method

• Analytical study of the Nambu-Jona-Lasinio model in two distinct spacetime settings: R³×S¹ with compactified spatial dimension L and Minkowski spacetime with external magnetic field H.
• Use of finite-temperature field theory techniques to compute effective potential and thermodynamic quantities under nonzero μ and H.
• Application of dimensional regularization and Matsubara formalism to handle divergences and thermal corrections in the effective action.
• Derivation of gap equations for the quark condensate and critical curves via variational minimization of the effective potential.
• Numerical and analytical analysis of phase diagrams to identify regions of first-order, second-order, and tricritical phase transitions.
• Study of asymptotic behavior as L→∞ and H→0 to reveal oscillatory patterns in physical observables.

Experimental results

Research questions

• RQ1 How does the inclusion of nonzero chemical potential and external magnetic field affect the phase structure of the Nambu-Jona-Lasinio model?
• RQ2 What is the nature of the phase transitions—first-order, second-order, or tricritical—under these conditions?
• RQ3 Why do physical observables such as magnetization and particle density exhibit oscillations in the limits L→∞ or H→0?
• RQ4 Are there infinitely many massless chirally symmetric phases in the model, and how do they coexist with massive phases?
• RQ5 How do the critical curves for chiral phase transitions behave under varying chemical potential and magnetic field?

Key findings

• The model exhibits infinitely many massless chirally symmetric phases and infinitely many massive phases with spontaneously broken chiral symmetry in both compactified and Minkowski spacetime settings.
• Oscillatory behavior emerges in magnetization, pressure, particle density, quark condensate, and critical curves as L→∞ or H→0, indicating non-monotonic thermodynamic responses.
• First-order and second-order phase transitions are identified, with multiple tricritical points present on the phase diagrams.
• The critical curve for chiral phase transitions shows non-trivial dependence on chemical potential and magnetic field, with oscillations in its shape under extreme limits.
• The system supports a rich phase structure with coexisting symmetric and broken phases, reflecting strong interplay between topology, chemical potential, and external fields.
• Magnetic catalysis is observed as an enhancement of chiral symmetry breaking under external magnetic fields, even at nonzero chemical potential.