More on chaos at weak coupling

The quantum Lyapunov exponent $\lambda_{L}$ is an intriguing and complicated observable in quantum field theories. Of particular interest are theories where $\lambda_{L}$ saturates an upper bound [2], since this might indicate that they have semiclassical gravity duals. However, in this paper we will be interested in the opposite limit, and would like to study how the chaos exponent behaves as we turn on some small coupling $\mathcal{J}$. The classical version of this question has been studied extensively, and many interesting behaviors have been found for chaos at weak coupling. For example, the KAM theorem schematically states that an integrable system deformed by a small (integrability-breaking) deformation remains non-chaotic even for a finite but small enough deformation. We will ask an analogous question in a quantum setting: starting with $N$ decoupled theories and slowly increasing the strength of a random interaction between them, how does $\lambda_{L}$ behave?

Our setup is the following. The basic building block is a conformal field theory (CFT) $\mathcal{C}$, called the “core CFT”, which contains a primary $\Phi$. Next, we take $N$ copies of the core CFT, and deform it by a random interaction:

The couplings $J_{i_{1}...i_{q}}$ are random with Gaussian measure and variance $\langle J_{i_{1}...i_{q}}^{2}\rangle=(q-1)!\frac{\mathcal{J}^{2}}{N^{q-1}}$ (with no sum over repeated indices).¹¹1We emphasize that our disorder is spacetime-independent. The deformation should be understood in terms of conformal perturbation theory around $N$ copies of the CFT $\mathcal{C}$, and we will be studying these theories in the large-$N$ limit. We will call such theories “disordered CFTs”.

Disordered CFTs are a generalization of ideas appearing in disordered free theories, which can be obtained by setting $\mathcal{C}$ to be a free field theory, and setting $\Phi$ to be the corresponding free field. A famous example of disordered free theories is the SYK model [3, 4], and some additional examples can be found in [5, 6, 7, 8, 9, 10, 11, 12, 13]. In general, the random interactions allow for some exact computations in the IR for disordered free fields, assuming the theory flows to an interacting CFT [3, 14, 15]. This was generalized to arbitrary core CFTs in [1, 16]. In these theories, $\lambda_{L}$ can be read off from a certain out-of-time-ordered four-point function (or alternatively, the double-commutator) [4, 17, 18, 2]. This is a difficult computation in general, but it is aided by the large-$N$ limit and conformal invariance in the IR. In particular, we must also restrict ourselves to dimensions $d\leq 2$, since we will be interested in the theory at finite temperature, but in small dimensions a conformal transformation can map such theories to flat space.

An especially interesting subsector of disordered CFTs can be obtained by demanding that the interaction term in (1.1) is exactly marginal, such that $\mathcal{J}$ parametrizes a line of CFTs. This behavior is not generic, but can be obtained by using supersymmetry or by considering chiral theories, both of which we will discuss in this paper.²²2Similar behavior can be obtained in QM without these assumptions [11]. Having a line of CFTs allows us to follow interesting observables as we continuously move between different CFTs, and in particular will allow us to study $\lambda_{L}$ as a function of $\mathcal{J}$.

In [1], the chaos exponent was studied in the weak-coupling limit $\mathcal{J}\to 0$ in disordered CFTs where $\mathcal{J}$ is exactly marginal. In particular, in the examples that were discussed it was observed the chaos exponent in the limit $\mathcal{J}\to 0$ is also given by the leading exponential behavior of the double-commutator (DC) in a single core CFT. In other words, if the DC of a single core CFT behaves at large times as $\exp(\lambda_{L}^{0}t)$, then the chaos exponent of the interacting theory in the limit $\mathcal{J}\to 0$ is given by

It was conjectured that this result is general; we will call it the continuity conjecture.

The equality (1.2) is surprising for two main reasons. First, $\lambda_{L}^{0}$ cannot be interpreted as a chaos exponent in a single core CFT (since some form of a large-$N$ limit is required for this interpretation), and yet in the interacting theory it dictates the behavior of the chaos exponent at weak coupling. Second, it is possible for $\lambda_{L}^{0}$ to be negative. A negative chaos exponent seems counter-intuitive, and indeed the result cannot be trusted in the standard method of computing $\lambda_{L}$ due to some assumptions that are required. A more precise version of the continuity conjecture is then

Despite this fact, if $\lambda_{L}^{0}$ is negative then one can still determine that the chaos exponent of the theory is at most zero for a finite range of values of $\mathcal{J}$. As a result, it is enough to show that in a single core CFT $\lambda_{L}^{0}<0$ in order to find a discontinuous transition into chaos, see figure 1. This is reminiscent of classical KAM theory.

This paper includes two main results. First we will prove the continuity conjecture (1.3) in quantum mechanics (QM). We then discuss two examples in $2d$ and show that they obey the continuity conjecture, providing further evidence for the $2d$ version of the conjecture. The first example is the chiral SYK model discussed in [12], where the chaos exponent was already found for all $\mathcal{J}$. We compare the result at small $\mathcal{J}$ to $\lambda_{L}^{0}$ and find agreement. Next we discuss the disordered $\mathcal{A}_{3}$ minimal model. Since this theory is dual to a free CFT, we can compute all $n$-point functions for its primaries, and we use this result to compute the chaos exponent at small $\mathcal{J}$. Comparing the result to $\lambda_{L}^{0}$ we again find agreement.

In the examples discussed in this paper, $\lambda_{L}^{0}$ always turns out to be non-negative, so that the physical picture is that of figure 1(a). We will discuss ideas for how to generate cases with a negative value and a discontinuous transition into chaos.

This paper is organized as follows. In section 2 we introduce the theories we consider in this paper and the methods for computing the chaos exponent. We then prove the continuity conjecture in QM in section 3. In section 4 we discuss our first $2d$ example, the chiral SYK model, and show that the continuity conjecture is obeyed. In section 5 we discuss our second example, the disordered $\mathcal{A}_{3}$ minimal model. We compute the chaos exponent at small $\mathcal{J}$ and again show that the continuity conjecture is obeyed. We discuss some future directions in section 6.

In this section we review the basic method discussed in [1] for obtaining the chaos exponent $\lambda_{L}$ at weak coupling for a general disordered CFT.

The continuity conjecture discussed in 2.3 relates the chaos exponent in disordered CFTs at small $\mathcal{J}$ to the late-time behavior of the DC in a single core CFT. It states that assuming that $\mathcal{G}_{R}$ defined in (2.14) (or equivalently, the double-commutator) behaves at large times as $\exp(\lambda_{L}^{0}t)$, then the chaos exponent as $\mathcal{J}\to 0$ approaches $\lambda_{L}^{0}$. We will now prove this statement under the assumption that perturbation theory in $\mathcal{J}$ is valid for the eigenvalues of the retarded kernel, so that the leading order is obtained by considering just the contribution of the four-point function to the retarded kernel, see section 2.2.

Consider the leading contribution to $k_{R}$, which comes from the four-point function diagram in figure 2. In $1d$ this contribution simplifies to (see Appendix B for a derivation):

Here, we performed the change of variables $z=e^{-t}$ on the “left rail” (points $1,3$) and $z=-e^{-t}$ on the “right rail” (points $2,4$), see e.g. [5] for details. The conformal cross-ratio is

Next we perform another change of coordinates to the coordinates $\kappa,\chi$, where $\kappa$ is defined as $z_{3}=\kappa/\chi$. The integral becomes is

The chaos exponent $\lambda_{L}$ is now given by the largest value of $\lambda$ for which the integral diverges in the limit $t_{3},t_{4}\to-\infty$, which in these coordinates corresponds to $\chi\rightarrow 0$. Note that the $\kappa$ integral is finite as long as $\lambda>-1$ (corresponding to $\lambda_{L}^{0}>-1$). At $\chi=0$ the $\kappa$ integral is smooth; it does not vanish or diverge. Therefore we can expand $\mathcal{G}_{R}(\chi)$ around $\chi=0$ inside the integral. Let us denote the leading order term by $\mathcal{G}_{R}(\chi)=c_{0}\chi^{-\lambda_{L}^{0}}+...$. Note that since at large times we have $\chi\sim e^{-t}$, we can identify $\lambda_{L}^{0}$ with the rate of growth of the DC of a single core CFT as in (2.20). Plugging in this expansion, we see that the integral converges only for $\lambda>\lambda_{L}^{0}$, which means that the chaos exponent at small $\mathcal{J}$ is $\lambda_{L}^{0}$. We have thus proved the continuity conjecture in $1d$.

The chiral SYK model was introduced in [12] (see also [19]). The theory is a disordered free theory in $2d$, where the core CFT is a free chiral Majorana fermion. The theory was shown to have a line of fixed points characterized by $\mathcal{J}$, so that the chaos exponent can be found as a function of $\mathcal{J}$. Since the theory is chiral, instead of the standard chaos exponent it is more interesting to consider the velocity-dependent chaos exponent, given by considering a large time and large distance limit with the ratio $v=x/t$ kept constant. The corresponding chaos exponent is denoted $\lambda_{v}$. It was found that the chaos exponent always starts at zero as $\mathcal{J}\to 0$, and rises as we increase $\mathcal{J}$. In particular, choosing $v$ such that $\lambda_{v}$ is maximal, one finds that at infinite $\mathcal{J}$ the chaos exponent $\lambda_{v}$ saturates the bound on chaos [2]. We will be interested in the weak-coupling limit, where $\mathcal{J}$ is close to zero. We will show that the continuity conjecture is obeyed for the velocity-dependent chaos exponent, and discuss corrections in $\mathcal{J}$.

The velocity-dependent chaos exponent at weak coupling can be easily extracted from Appendix B of [12] (see equation (B.6)), and in the limit $\mathcal{J}\to 0$ one finds

where $u_{\pm}=1\pm\frac{\mathcal{J}}{2\pi}$ and

Note that since $u_{-}<v<u_{+}$, $\eta$ is finite in the limit $\mathcal{J}\to 0$.

Since the theory is chiral, it is not surprising that the chaos exponent vanishes outside of a cone around the speed of light. Taking the strict $\mathcal{J}\to 0$ limit, we find that $\lambda_{v}(\mathcal{J}\to 0)$ vanishes trivially unless $v=1$ (since $u_{-}=u_{+}=1$), but at $v=1$ it turns out to vanish as well. Thus we find $\lambda_{v}(\mathcal{J}\to 0)=0$.

We would like to compare this to $\lambda_{L}^{0}$, which describes the large-time behavior of the DC in a single copy of the free core CFT. The DC in the core CFT is given by $G_{R}(13)G_{R}(24)$, where $G_{R}(ij)$ is the retarded propagator between spacetime points $i,j$ and is given by

Next we take the large-time limit where $t=t_{1}=t_{2}$ and $x=x_{1}=x_{2}$ are large while $v=x/t$ is kept constant. We find (up to unimportant constants)

Plugging in $\mathcal{J}=0$ we find as a result that $\lambda_{v}^{0}=0$ always. Thus we have found

and so the continuity conjecture is obeyed.

We can also try to understand subleading corrections to the chaos exponent as discussed in section 2.2.1. The eigenvalues of the chiral SYK model were computed in [12], and we can expand the result in $\mathcal{J}$:

Expanding to higher orders one finds $\alpha=2$ and $\beta_{n}=n=n\alpha/2$ in the notation of section 2.2.1. As a result, the expansion is not in terms of $\mathcal{J}$, but in terms of $\mathcal{J}/\lambda$, and so a perturbative computation of $\lambda_{L}$ will fail at higher orders in $\mathcal{J}$.

We now discuss the case where the core CFT is the $\mathcal{N}=2$ supersymmetric $\mathcal{A}_{3}$ minimal model. This model can be constructed using a single chiral superfield $X$ with superpotential

The model has central charge $c=3/2$ and its spectrum includes a chiral primary of dimension $1/4$ which we will call $\Phi$. This theory can be identified with the theory of a free boson $H$ an a free fermion $\chi$, which combine into a single free $\mathcal{N}=1$ chiral multiplet [20]. The core CFT thus has a free field representation.

The disordered $\mathcal{A}_{3}$ minimal model with $q=4$ can then be constructed as

In particular, this deformation is marginal. In fact, as discussed in [1], it can be shown using standard arguments [21, 22, 23] that any realization of this theory is a CFT, so that the interaction is exactly marginal (even without averaging).

In order to compute $\lambda_{L}$ we need to know $n$-point functions of $\Phi$. Since the $\mathcal{A}_{3}$ minimal model has a free field realization in terms of $\mathcal{N}=1$ superfields, we should be able to identify the components of $\Phi$ with products of operators from the free boson and free fermion CFT. In practice, the various components will be mapped to products of vertex operators and fermionic twist fields, whose $n$-point functions are known. Plugging the results into the retarded kernel, we will be able to read off $\lambda_{L}$. Following the discussion above, we will focus on the four-point function which gives the leading contribution to $\lambda_{L}$ at small $\mathcal{J}$.

Our notations appear in appendix A. In particular, we use lightcone coordinates $x^{\pm}$ so that a two-point function takes the form $(x^{+}x^{-})^{\Delta}\equiv|x|^{2\Delta}$.