Spectral and Phase Structure of a QCD-Inspired Unitary Matrix Model with Fisher-Hartwig Singularities

TL;DR

The paper analyzes the large $N$ limit of a SU($N$) unitary matrix model with a complex potential containing Fisher-Hartwig singularities, motivated by QCD-inspired complexified potentials. Using the standard unitary-matrix formalism with a Lagrange multiplier to enforce det U = 1, it derives saddle-point equations for the eigenvalue density on a complex contour and classifies phases into four ungapped sectors and a single gapped sector; closed-form spectral densities exist in the ungapped phases, while the gapped phase requires a resolvent-based, semi-analytic/numerical treatment. Phase transitions between ungapped phases are third order and proceed via the gapped phase, with the beta function and higher derivatives of the free energy exhibiting kink behavior; the results include a mapping to a low-temperature QCD-like model and insights into how Fisher-Hartwig singularities shape phase structure in complex unitary ensembles. These findings offer a potential window into complex actions in QCD-like theories and highlight the methodological role of resolvent techniques in handling complex potentials. The work also paves the way for extensions to finite-N effects and additional complex deformations of the potential.

Abstract

We investigate the large $N$ limit of a complex action unitary matrix model with Fisher-Hartwig singularities, motivated by QCD-inspired models with complexified potentials. We show that the model exhibits multiple ungapped phases and a single gapped phase. The phases are characterized by Fisher-Hartwig singularities in the complex plane. We show that the phase transitions are third order, with transitions between ungapped phases forbidden. We also briefly discuss the implications for the QCD phase diagram at the end.

Figures (8)

• Figure 1: Left: support of the spectral function in the ungapped phase A (small $\alpha$ large $\gamma$ phase) for a few values of $\alpha$. The dot at $z=-1/\gamma$ is the pole which is inside $\mathcal{C}$, while the other pole (not shown) is outside. Right: phase boundary between the ungapped phase A and the gapped phase.
• Figure 2: Support of the spectral function (left panels) and the phase boundary between the gapped and ungapped phases (right panels). The panels from top to bottom correspond to ungapped phases B, C and D in the same order.
• Figure 3: Schematic of the support $\mathcal{C}$ of the spectral function in the gapped phase (dashed, both panels). The dots on the negative real axis are the poles of the potential ($z = -1/\alpha, -1/\gamma$). Solid closed curve in the left panel is the contour $\mathcal{C}^*$ used to solve integrals in the gapped phase. It can be deformed continuously to that shown in the right panel. See Fig. \ref{['fig:gappedContour']} for the actual contours in the gapped phase.
• Figure 4: Support of the spectral function $\rho(z)$ in the gapped phase.
• Figure 5: Phase diagram of the model for $\sigma=0.5$ and $\sigma=2$. The four ungapped phases are represented by various colors and the gapped phase by white. As $\sigma$ is increased the gapped phase grows outwards from the center at $\alpha=1, \gamma=1$.