Out-of-Time-Order Correlation Functions for Unitary Minimal Models

TL;DR

The paper analytically evaluates Out-of-Time-Order Correlators (OTOC) and Normal-Order Correlators (NOC) in 2D unitary minimal models across the full time range, using conformal symmetry and the Coulomb gas formalism. It shows that early-time dynamics are governed by WV OPEs with fractional-power corrections, while late-time dynamics are governed by WW/OPEs with monodromy tied to the modular S-matrix, revealing no large-N time-scale separation. The authors provide explicit calculations for specific operator choices, illustrate how the OTOC/NOC ratio f(t) encodes scrambling strength, and demonstrate how the late-time plateau connects to modular data, with the scrambling weakening as the central charge approaches the free-boson limit. The results extend to rational CFTs and illuminate the braid/monodromy structure underlying information spreading in strongly interacting 1+1D systems.

Abstract

We analytically study the Out-of-Time-Order Correlation functions (OTOC) for two spatially separated primary operators in two-dimensional unitary minimal models. Besides giving general arguments using the conformal symmetry, we also use the Coulomb gas formalism to explicitly calculate the OTOC across the full time regime. In contrast to large-$N$ chaotic systems, these models do not display a separation of time scales, due to the lack of a large parameter. We find that the physics at early times ($0<t-x\llβ$) and late times ($t-x\ggβ$) are controlled by different OPE channels, which are related to each other via the braiding matrix. The normalized OTOC obeys a power-law decay with a fractional power at early times and approaches a generically nonzero value in an exponential way at late times. The late time value is related to the modular $S$-matrix and is in agreement with earlier calculations. All of the results above are readily generalized to rational conformal field theories.

Figures (7)

• Figure 1: Trajectories of $\overline z_\text{OTOC}$ and $\overline z_\text{NOC}$ when the real time $t$ increases from $0$ to a large enough value. $\overline z_\text{OTOC}$ has a non-trivial winding around $\overline z=1$ while $\overline z_\text{NOC}$ doesn't. We choose $\beta=2\pi$ and $\epsilon=0.1$ to make this effect clear to see on the plot.
• Figure 2: Normal-order correlator for $W=V=\phi_{1,2}$ in different minimal models. We choose $x=1$, $\beta=2\pi$ and $\epsilon=0.0001$. The red line is constant $1$, plotted for convenience. $C_2(t)$ diverges at $t=x$, which is the light cone singularity. Because we drop the prefactor, $C_2(t)$ approaches $1$, i.e., the equilibrium value.
• Figure 3: OTOCs and NOC for $W=V=\phi_{1,2}$. (a) OTOC for $M(5,4)$. (b) OTOC for different different models labeled by their central charges. We choose $x=1$, $\beta=2\pi$ and $\epsilon=0.0001$. The red line is constant $1$, plotted for convenience.
• Figure 4: Ratio between OTOC and NOC for $W=V=\phi_{1,2}$ and different central charges. We choose $x=1$, $\beta=2\pi$ and $\epsilon=0.0001$.
• Figure 5: Fitting of the early-time and late-time behavior of the ratio $f(t)$. (a) Early-time behavior. The slope of the fitting curve is $1.25$, which is close to $2(h_{1,3}-h_{1,1})=1.2$. (b) Late-time behavior. The slope of the fitting curve is $0.60$, which is close to $h_{1,3}=0.6$. We choose $x=1$, $\beta=2\pi$ and $\epsilon=0.0001$.