Thermalization and chaos in ${\mathrm{QED}}_3$

We study the real time dynamics of $N_f$ flavors of fermions coupled to a $U(1)$ gauge field in $2+1$ dimensions to leading order in a $1/N_f$ expansion. For large enough $N_f$, this is an interacting conformal field theory and describes the low energy properties of the Dirac spin liquid. We focus on thermalization and the onset of many-body quantum chaos which can be diagnosed from the growth of initially anticommuting fermion field operators. We compute such anticommutators in this gauge theory to leading order in $1/N_f$. We find that the anticommutator grows exponentially in time and compute the quantum Lyapunov exponent. We briefly comment on chaos, locality, and gauge invariance.

Figure 1

1 represents the time ordered correlation function on the two fold contour and 1 represents the OTOC. The red represent $A^{\mu }$ and blue represents $A^{\nu }$. These are two of the terms in the squared commutator for the photon field. There are two more times, one which is time ordered and one which is out of time order.

Figure 2

The fermion polarization bubble is the $O(1)$ contribution to the gauge field propagator and dominates over the bare term at low energies.

Figure 3

The imaginary and real parts of the full polarization component ${\mathrm{\Pi }}^{xx}(k^0,\mathit{k})$ are plotted against the high temperature limits in Figs. 3 and 3, respectively.

Figure 4

The leading contribution to the fermion self-energy is $O(\frac{1}{N_f})$ so the correction to the mass can be neglected.

Figure 5

A plot of the fermion decay rate ${\mathrm{\Gamma }}_{\mathit{k}}$ in units of temperature which vanishes as $\mathit{k}\to 0$.

Figure 6

A diagrammatic picture of the square of the fermion anticommutator. The lower rails and retarded propagators and upper rails are advanced propagators and the rails are Wightman functions.

Figure 7

We get four types of terms from expanding the squared current-current commutator to first order. The first nontrivial diagrams can be divided into two groups and are shown in (a) and (b). The black dots are current insertions. Each group contains a diagram corresponding to each direction of particle flow. Only the diagrams with uncrossed rungs will contribute to chaos.

Figure 8

Terms contributing to the current current commutator at higher order.

Figure 9

The decay rate is related to the imaginary part of the fermion self-energy via the optical theorem.