A Universal Operator Growth Hypothesis

We present a hypothesis for the universal properties of operators evolving under Hamiltonian dynamics in many-body systems. The hypothesis states that successive Lanczos coefficients in the continued fraction expansion of the Green’s functions grow linearly with rate $\alpha$ in generic systems, with an extra logarithmic correction in 1D. The rate $\alpha$—an experimental observable—governs the exponential growth of operator complexity in a sense we make precise. This exponential growth prevails beyond semiclassical or large-$N$ limits. Moreover, $\alpha$ upper bounds a large class of operator complexity measures, including the out-of-time-order correlator. As a result, we obtain a sharp bound on Lyapunov exponents ${\lambda }_L\le 2\alpha$, which complements and improves the known universal low-temperature bound ${\lambda }_L\le 2\pi T$. We illustrate our results in paradigmatic examples such as nonintegrable spin chains, the Sachdev-Ye-Kitaev model, and classical models. Finally, we use the hypothesis in conjunction with the recursion method to develop a technique for computing diffusion constants.

Popular Summary

Imagine a cup of tea just after a splash of milk has been added. The milk will swirl around until it spreads uniformly through the cup. This is irreversible; no matter how long you wait, the milk will never separate from the tea. This phenomenon, sometimes called the arrow of time, means that systems will proceed inevitably toward thermal equilibrium, where a small amount of information specifies the entire system. This process underpins classical thermodynamics, but the quantum case is much less clear. Though most quantum systems also march toward equilibrium, their time evolution is unitary, which means that no information can be forgotten. How, then, can quantum systems achieve thermal equilibrium? This puzzle is the focus of our work.

Progress in recent years, such as the celebrated eigenstate thermalization hypothesis, has led to a qualitative picture of how a quantum system “forgets” information—the process of operator growth. The idea is that any local operator (the rough quantum equivalent of a drop of milk) will spread out over time, becoming more and more complex. While this process is indeed unitary and reversible, the buildup of complexity means the probability of a reversal quickly becomes infinitesimally small. Our work makes this idea fully quantitative by fleshing out the mathematical idea of complexity and showing how reversible dynamics give rise to irreversible quantities using a mathematical tool called Lanczos coefficients.

We expect our results to help us understand further the notion of many-body quantum chaos and to compute properties of strongly correlated quantum matter.

Figure 1

Artist’s impression of the space of operators and its relation to the 1D chain defined by the Lanczos algorithm starting from a simple operator $\mathcal{O}$. The region of complex operators corresponds to that of large $n$ on the 1D chain. Under our hypothesis, the hopping amplitudes $b_n$ on the chain grow linearly asymptotically in generic thermalizing systems (with a log correction in one dimension; see Sec. 4c). This implies an exponential spreading ${(n)}_t\sim e^{2\alpha t}$ of the wave function ${\phi }_n$ on the 1D chain, which reflects the exponential growth of operator complexity under Heisenberg evolution, in a sense that we make precise in Sec. 5. The form of the wave function ${\phi }_n$ is only a sketch; see Fig. 5 for a realistic picture.

Figure 2

Lanczos coefficients in a variety of models demonstrating common asymptotic behaviors. “Ising” is $H={\sum }_i{X}_i{X}_{i+1}+Z_i$ with $\mathcal{O}={\sum }_j{e}^{i{q}_j}{Z}_j$ ($q=1/128$ here and below) and has $b_n\sim O(1)$. “X in XX” is $H={\sum }_i{X}_i{X}_{i+1}+Y_i{Y}_{i+1}$ with $\mathcal{O}={\sum }_j{X}_j$, which is a string rather than a bilinear in the Majorana fermion representation, so this is effectively an interacting integrable model that has $b_n\sim \sqrt{n}$. XXX is $H={\sum }_i{X}_i{X}_{i+1}+Y_i{Y}_{i+1}+Z_i{Z}_{i+1}$ with $\mathcal{O}={\sum }_j{e}^{i{q}_j}(X_j{Y}_{j+1}-Y_j{X}_{j+1})$ that appears to obey $b_n\sim \sqrt{n}$. Finally, SYK is Eq. (18), where $q=4$, $J=1$, and $\mathcal{O}=\sqrt{2}{\gamma }_1$ with $b_n\sim n$. The Lanczos coefficients are rescaled vertically for display purposes. Numerical details are given in Appendixes pp2 and pp3.

Figure 3

Illustration of the spectral function and the analytical structure of $C(t),t\in \mathbb{C}$. When the Lanczos coefficients have linear growth rate $\alpha$, $\mathrm{\Phi }(\omega )$ has exponential tails $\sim e^{-|\omega |/{\omega }_0}$ with ${\omega }_0=2\alpha /\pi$; $C(t)$ is analytical in a strip of half-width $1/{\omega }_0$, and the singularities closest to the origin are at $t=\pm i/{\omega }_0$. See Appendix pp1-s2 for further discussion.

Figure 4

(a) Lanczos coefficients in a variety of strongly interacting spin-half chains: $H_1={\sum }_i{X}_i{X}_{i+1}+0.709Z_i+0.9045X_i$, $H_2=H_1+{\sum }_{i}0.2Y_i$, and $H_3=H_1+{\sum }_{i}0.2Z_i{Z}_{i+1}$. The initial operator $\mathcal{O}$ is an energy density wave with momentum $q=0.1$. (b) Crossover to apparently linear growth as interactions are added to a free model. Here, $H={\sum }_i{X}_i{X}_{i+1}-1.05Z_i+h_X{X}_i$, and $\mathcal{O}\propto {\sum }_{i}1.05X_i{X}_{i+1}+Z_i$. The $b_n$’s are bounded when $h_X=0$ but appear to have asymptotically linear growth for any $h_X\ne 0$. Logarithmic corrections are not clearly visible in the numerical data. Numerical details are given in Appendix pp3.

Figure 5

The exact solution wave function (26) in the semi-infinite chain at various times. The wave function is defined only at $n=0,1,2\dots$ but is extrapolated to intermediate values for display.

Figure 6

(a) The growth rate $\alpha$ versus the classical Lyapunov exponent ${\lambda }_L/2$ in the classical Feingold-Peres model of coupled tops, Eq. (46). $\alpha \ge {\lambda }_L/2$, in general, with equality around the $c=0$ where the model is the most chaotic. The growth rate appears to be discontinuous at the noninteracting points $c=\pm 1$, similarly to Fig. 4. (b) The first 40 Lanczos coefficients of the quantum $s=2,\dots ,32$ and classical $(s=\infty )$ FP model, with $c=0$.

Figure 7

Numerical computation of the diffusion coefficient for the energy density operator $\mathcal{O}={\mathcal{E}}_q$ in $H=\sum _i{X}_i{X}_{i+1}-1.05Z_i+0.5X_i$. (a) The Lanczos coefficients for $q=0.15$ are fit to Eq. (50) with $\alpha =0.35$ and $\eta =1.74$. We actually find it better not to approximate $G^{(N)}(z)$ by ${\tilde{G}}^{(N)}(z)$ but instead by ${\tilde{G}}^{(N+\delta )}(z)$ for some integer offset $\delta$ so that $\eta \approx 1$ (in the example shown, $\delta =12$). Large $\eta$ or negative values lead to numerical pathologies. (b) The approximate Green’s function (52) at $q=0.15$. The arrow shows the “leading” pole that governs diffusion. (c) The locations of the leading poles for a range of $q$. One can clearly see the diffusive dispersion relation $z=iD{q}^2/2+O(q^4)$. Fitting yields a diffusion coefficient $D=3.3(5)$.

Figure 8

Exact Lyapunov exponent ${\lambda }_L(T)$ (b21) and growth rate $\alpha (T)$ with the Wightman inner product (b23) of the SYK model in the large-$q$ limit as a function of the temperature (in units of coupling constant $\mathcal{J}$). The conjectured bound ${\lambda }_L(T)\le 2\alpha (T{)}_W$ is exactly saturated at all temperatures, while the universal bound ${\lambda }_L(T)\le 2\pi T$ saturates only in the zero-temperature limit.

Figure 9

Change in the growth rate near integrability for the SYK model with $q=2$ and $q=4$ [Eq. (b9)]. The ratio of the $q=4$ to $q=2$ terms is given by $J$, and the model becomes free at $J=0$.

Figure 10

The size distribution of the Pauli strings in the Krylov vectors ${\mathcal{O}}_n$ for the Hamiltonian $H_1$ with parameters and initial operator as in Fig. 4. Though the distribution drops quickly after its peak, $P_n(s)$ is supported on $[0,\lfloor n/2\rfloor +2]$.