Fate of Chiral Symmetries in the Quark-Gluon Plasma from an Instanton-Based Random Matrix Model of QCD

We propose a new way of understanding how chiral symmetry is realized in the high temperature phase of QCD. Based on the finding that a simple free instanton gas precisely describes the details of the lowest part of the spectrum of the lattice overlap Dirac operator, we propose an instanton-based random matrix model of QCD with dynamical quarks. Simulations of this model reveal that even for small quark mass the Dirac spectral density has a singularity at the origin, caused by a dilute gas of free instantons. Even though the interaction, mediated by light dynamical quarks, creates small instanton–anti-instanton molecules, those do not influence the singular part of the spectrum, and this singular part is shown to dominate Banks-Casher type sums in the chiral limit. By generalizing the Banks-Casher formula for the singular spectrum, we show that in the chiral limit the chiral condensate vanishes if there are at least two massless flavors. Our model also indicates a possible way of resolving a long-standing debate, as it suggests that for two massless quark flavors the $U(1{)}_{\mathrm{A}}$ symmetry is likely to remain broken up to arbitrarily high finite temperatures.

Figure 1

The distribution of the log of the lowest Dirac eigenvalue in two different volumes on the lattice and in the random matrix model. The smaller volume was used to fit $A$, and for the larger volume the prediction of the model is shown, without any further fitting.

Figure 2

The topological susceptibility and the instanton density normalized by $m^2{\chi }_0$, and the chiral condensate normalized by $m{\chi }_0$ as a function of the quark mass. The fit is of the form $c/m^2+d$, where $c$ and $d$ are constants.

Figure 3

The logarithmic spectral density of the matrix model with two flavors of quarks of mass $ma=0.05$ compared to the spectral density of the quenched matrix model with the same topological susceptibility. The inset is an enlargement of the largest eigenvalues where the two spectra significantly differ.

Figure 4

The distribution of the distance of the closest opposite charged object on the dynamical and quenched matrix model configurations used for Fig. 3.

Figure 5

The spectral density (density of $\log \lambda$) for different system sizes in the quenched matrix model (parameters given in the text).