Onset of many-body chaos in the $O(N)$ model

TL;DR

This work computes the onset of many-body chaos in the $(2+1)$-dimensional $O(N)$ model at finite temperature above its quantum critical point using a controlled $1/N$ expansion. By analyzing ladder diagrams for the squared commutator, the authors extract a chaos exponent $\lambda_L \approx 3.2\,\frac{T}{N}$ and a butterfly velocity $v_B$ approaching the speed of light, with conformal scaling set by the QCP. They provide a detailed Bethe-Salpeter framework, quantify the dephasing time $\tau_\varphi \sim N/T$, and discuss the spatial structure $\mathcal{C}(t,\boldsymbol{x}) \sim \exp(\lambda_L t - |\boldsymbol{x}|^2/(4 D_L t))$, including finite-momentum effects. The paper also comments on the behavior in proximate symmetry-broken and unbroken phases and situates the results within broader chaos-diffusion relations and potential experimental tests.

Abstract

The growth of commutators of initially commuting local operators diagnoses the onset of chaos in quantum many-body systems. We compute such commutators of local field operators with $N$ components in the $(2+1)$-dimensional $O(N)$ nonlinear sigma model to leading order in $1/N$. The system is taken to be in thermal equilibrium at a temperature $T$ above the zero temperature quantum critical point separating the symmetry broken and unbroken phases. The commutator grows exponentially in time with a rate denoted $λ_L$. At large $N$ the growth of chaos as measured by $λ_L$ is slow because the model is weakly interacting, and we find $λ_L \approx 3.2 T/N$. The scaling with temperature is dictated by conformal invariance of the underlying quantum critical point. We also show that operators grow ballistically in space with a "butterfly velocity" given by $v_B/c \approx 1$ where $c$ is the Lorentz-invariant speed of particle excitations in the system. We briefly comment on the behavior of $λ_L$ and $v_B$ in the neighboring symmetry broken and unbroken phases.

Figures (13)

• Figure 1: Sketch of the operator ordering in the out-of-time-order correlator $F(t)$, Eq. \ref{['otodef']}. The red dots correspond to $W$ operators and the blue dots correspond to $V$ operators. The two real time folds are separated by an imaginary time of $\beta/2$.
• Figure 2: A summary of the results for the growth exponent, $\lambda_L$, and butterfly velocity, $v_B$, in the different regions of the $g-T$ phase diagram (see Eq. \ref{['on']}). This paper is primarily concerned with the 'quantum critical' regime (blue shaded region) where temperature is the only relevant energy scale. The results for the symmetry-broken (unbroken) regions, corresponding to $g<g_c$ ($g>g_c$), are discussed briefly in Section \ref{['proximate']}.
• Figure 3: Resummed bubble diagrams, $\Pi(i\omega_n,{\boldsymbol{k}})$, renormalize the bare $\lambda$ propagator ${\cal G}_\lambda^{(0)}(i\omega_n,{\boldsymbol{k}})$ (blue dashed line) to yield ${\cal G}_\lambda(i\omega_n,{\boldsymbol{k}})$ (black dashed line).
• Figure 4: The self-energy corrections to the $\varphi_a$ propagator. The second diagram is independent of the external momentum/frequency.
• Figure 5: The top diagram represents a general uncrossed ladder diagram in which two kinds of rungs are allowed. The first rung (dashed black line---type-I rung) corresponds to the insertion of ${\cal G}_{W,\lambda}$ between the two sides of the ladder. The second rung (wavy black line---type-II rung) corresponds to the insertion of the box diagram shown at right into the ladder. The contribution of the box is denoted ${\cal G}_{\text{eff}}$ and is defined in Eq. \ref{['Geff']}. The bottom diagram represents the ladder sum over the two kinds of rungs, for which we write down a Bethe-Saltpeter equation in Eq. \ref{['resummed']} (see Figure \ref{['ladsum']}).