Chaos in chiral condensates in gauge theories

Abstract

Assigning a chaos index for dynamics of generic quantum field theories is a challenging problem, because the notion of Lyapunov exponent, which is useful for singling out chaotic behaviors, works only in classical systems. We address the issue by using the AdS/CFT correspondence, as the large $N_c$ limit provides a classicalization (other than the standard $\hslash \to 0$) while keeping nontrivial quantum condensation. We demonstrate the chaos in the dynamics of quantum gauge theories: Time evolution of homogeneous quark condensates $⟨\bar{q}q⟩$ and $⟨\bar{q}{\gamma }_{5}q⟩$ in an $\mathcal{N}=2$ supersymmetric QCD with the $SU(N_c)$ gauge group at large $N_c$ and at large 't Hooft coupling $\lambda \equiv N_c{g}_{\mathrm{YM}}^2$ exhibits a positive Lyapunov exponent. The chaos dominates the phase space for energy density $E\gtrsim (6\times 10^2)\times m_q^4(N_c/{\lambda }^2)$ where $m_q$ is the quark mass. We evaluate the largest Lyapunov exponent as a function of $(N_c,\lambda ,E)$ and find that the $\mathcal{N}=2$ supersymmetric QCD is more chaotic for smaller $N_c$.

Comments: 9 pages, 5 figures. v2: Analysis of Lyapunov exponent of linear sigma model added, references and a footnote added. v3: Published version. An appendix added for explaining relation between classical chaos and out-of-time-ordered correlator. Title modified, abstract refined v4: Typos and grammatical errors corrected