The Phases of Chaos

Abstract

We develop a novel physical picture to understand certain universal properties of the GUE matrix model which are typically ascribed to quantum chaos, i.e. the ramp and the plateau. We argue that these features should instead be associated with a pattern of spontaneous (or weak explicit) symmetry breaking. In this language, the GUE matrix model corresponds to an effective theory that describes the symmetry-broken phase, and where the Hermitian matrix of the GUE should be understood as a massive $σ$ field. The physics of this symmetry-broken phase governs certain particular features of the ramp such as its length and shape. However, the simple existence of a ramp is more universal and phase independent; it is related to sum rules obeyed by a large class of matrix models that constrain the interpolation to the plateau regime. Finally, the plateau is controlled by the symmetry-restored phase, which we call confined chaos.

Figures (10)

• Figure 1: Log-log plot of the exact $L=30$(red) and infinite-$L$(yellow) squared disk amplitude $\tfrac{1}{L}\langle Z(it)\rangle_{\rm GUE}$ as a function of complex temperature $\beta=it$. At small $t\lambda$, the disk amplitude oscillates around zero with a $(t\lambda)^{-3/2}$ envelope. In this range, it is clear that the exact and infinite-$L$ answers agree to very high accuracy. Noticeable differences begin to accumulate around $t\lambda\sim L$. Once $t\lambda\sim 2L$, the exact amplitude decays exponentially to zero, indicative of a phase transition. We plot the square of the observable because $\left\langle Z(it)\right\rangle_{\rm GUE}$ is not sign definite, meaning that, unless squared, its log will generally be complex.
• Figure 2: Unitary Matrix Model potential \ref{['eq:potentialSYM']} for $L=2$. The potential is minimized when the eigenvalues are equal and maximized when they are diametrically opposed.
• Figure 3: Effective action over the coupling $g$ for the Yang-Mills model at various values of $a$
• Figure 4: Comparison between the numerically-evaluated exact (red) and large-$L$ (yellow) (including the first non-planar order) free energies in the Yang-Mills matrix model as a function of $a$. The spike in the large-$L$ free energy at $a=1$ is the tell-tale signal of a first-order phase transition. As expected, the first-order phase transition in the large-$L$ expression gets smoothed out at finite-$L$.
• Figure 5: We compare the numerically-evaluated one-point function in the Yang-Mills matrix model $\tfrac{1}{L}\langle\text{Tr}\,U^k\rangle_{\rm YM}$ with the equivalent observable in the GUE and the GWW ensembles in the deconfined phase ($a>1$). For $a$ slightly larger than 1 (at least for small values of $L$), YM and GWW expectation values do not match with the GUE. However, for $a\gg1$, we observe a nice agreement between these functions in the oscillatory phase. These plots are made using discrete data in Mathematica's ListPlot command with options Joined$\rightarrow$True, InterpolationOrder$\rightarrow$3.