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 $\hbar \to 0$) while keeping nontrivial quantum condensation. We demonstrate the chaos in the dynamics of quantum gauge theories: Time evolution of homogeneous quark condensates $\langle \bar{q}q\rangle$ and $\langle \bar{q} γ_5 q\rangle$ in an ${\cal N}=2$ supersymmetric QCD with the $SU(N_c)$ gauge group at large $N_c$ and at large 't Hooft coupling $λ\equiv N_c g_{\rm YM}^2$ 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/λ^2) $ where $m_q$ is the quark mass. We evaluate the largest Lyapunov exponent as a function of $(N_c,λ,E)$ and find that the ${\cal N}=2$ supersymmetric QCD is more chaotic for smaller $N_c$.