Abstract
We study the effects of addition of Chern-Simons (CS) term in the minimal Yang Mills (YM) matrix model composed of two $2 \times 2$ matrices with $SU(2)$ gauge and $SO(2)$ global symmetry. We obtain the Hamiltonian of this system in appropriate coordinates and demonstrate that its dynamics is sensitive to the values of both the CS coupling, $\kappa$, and the conserved conjugate momentum, $p_\phi$, associated to the $SO(2)$ symmetry. We examine the behavior of the emerging chaotic dynamics by computing the Lyapunov exponents and plotting the Poincar\'{e} sections as these two parameters are varied and, in particular, find that the largest Lyapunov exponents evaluated within a range of values of $\kappa$ are above that is computed at $\kappa=0$, for $\kappa p_\phi < 0$. We also give estimates of the critical exponents for the Lyapunov exponent as the system transits from the chatoic to non-chaotic phase with $p_\phi$ approaching to a critical value.
Highlights
Setting the energy E 1⁄4 1, ħ 1⁄4 0.1, and letting pφ assume the values 0,1,2, which is convenient for ease in comparison with the pure YM matrix model results in [5], we obtain the largest Lyapunov exponents (LLE), λL, as the CS coupling takes on a range of values, in which the typical behavior of the LLE’s is captured
We have studied the chaotic structure of the minimal Yang-Mills Chern-Simons matrix model
We have studied the chaotic dynamics of the model, and in particular, probed the changes in the Lyapunov exponent as the values of both the CS coupling, κ, and the conserved conjugate momentum, pφ, are varied
Summary
There has been growing interest in exploring the structure of chaotic dynamics emerging from the matrix quantum mechanics [1,2,3,4,5,6,7,8,9,10,11,12], such as the Banks-FischlerShenker-Susskind (BFSS) and the Berestein-MaldacenaNastase (BMN) models [13,14,15,16,17,18,19] which appear in the discrete light-cone quantization of M theory in the flat and the pp-wave background, respectively. It is important to note that the BFSS, BMN matrix models, but even their subsectors at small values of N appear as nontrivial many-body systems, and we lack a complete solution to these or even for the smallest YangMills (YM) matrix model to date The latter may be described as being composed of two 2 × 2 Hermitian matrices with SUð2Þ gauge and SOð2Þ global symmetries. Yang-Mills Chern-Simons (YMCS) model in 2 þ 1 dimensions and reducing it to 0 þ 1.1 In a manner similar to the one followed in [5], while paying attention to the differences in the procedure due to the CS term, which is first order in time derivative, we obtain the Hamiltonian of the system The latter has the same degrees of freedom as the pure YM model, while the effective potential is governed by pφ, and the CS coupling κ, which enters into the effective potential via κpφ and another term ∝ κ2r2.
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