Abstract It was recently conjectured that in generic quantum many-body systems, the spectral density of local operators has the slowest high-frequency decay as permitted by locality. We show that the infinite-temperature version of this ‘universal operator growth hypothesis’ holds for the quantum Ising spin model in d ⩾ 2 dimensions, and for the chaotic Ising chain (with longitudinal and transverse fields) in one dimension. Moreover, the disordered chaotic Ising chain that exhibits many-body localization can have the same high-frequency spectral density asymptotics as thermalizing models. Our argument is statistical in nature, and is based on the observation that the moments of the spectral density can be written as a sign-problem-free sum over paths of Pauli string operators.
Abstract It was recently conjectured that in generic quantum many-body systems, the spectral density of local operators has the slowest high-frequency decay as permitted by locality. We show that the infinite-temperature version of this ‘universal operator growth hypothesis’ holds for the quantum Ising spin model in d ⩾ 2 dimensions, and for the chaotic Ising chain (with longitudinal and transverse fields) in one dimension. Moreover, the disordered chaotic Ising chain that exhibits many-body localization can have the same high-frequency spectral density asymptotics as thermalizing models. Our argument is statistical in nature, and is based on the observation that the moments of the spectral density can be written as a sign-problem-free sum over paths of Pauli string operators.
We put forward a Monte Carlo algorithm that samples the Euclidean time operator growth dynamics at infinite temperature. Crucially, our approach is free from the numerical sign problem for a broad family of quantum many-body spin systems, allowing for numerically exact and unbiased calculations. We apply this methodological headway to study the high-frequency dynamics of the mixed-field quantum Ising model in one and two dimensions. The resulting quantum dynamics display rapid thermalization, supporting the recently proposed operator growth hypothesis. Physically, our findings correspond to an exponential falloff of generic response functions of local correlators at large frequencies. Remarkably, our calculations are sufficiently sensitive to detect subtle logarithmic corrections of the hypothesis in one dimension. In addition, in two dimensions, we uncover a nontrivial dynamical crossover between two large-frequency decay rates. Last, we reveal spatiotemporal scaling laws associated with operator growth, which are found to be strongly affected by boundary contributions.
We investigate the operator growth dynamics of the transverse field Ising spin chain in one dimension as varying the strength of the longitudinal field. An operator in the Heisenberg picture spreads in the extended Hilbert space. Recently, it has been proposed that the spreading dynamics has a universal feature signaling chaoticity of underlying quantum dynamics. We demonstrate numerically that the operator growth dynamics in the presence of the longitudinal field follows the universal scaling law for one-dimensional chaotic systems. We also find that the operator growth dynamics satisfies a crossover scaling law when the longitudinal field is weak. The crossover scaling confirms that the uniform longitudinal field makes the system chaotic at any nonzero value. We also discuss the implication of the crossover scaling on the thermalization dynamics and the effect of a nonuniform local longitudinal field.
Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, nonintegrable many-body systems, the so-called Lanczos coefficients associated with an autocorrelation function grow asymptotically linear, with a logarithmic correction in one-dimensional systems. In contrast, the growth is expected to be slower in integrable or free models. In this paper, we numerically test this hypothesis for a variety of exemplary systems, including one-dimensional and two-dimensional Ising models as well as one-dimensional Heisenberg models. While we find the hypothesis to be practically fulfilled for all considered Ising models, the onset of the hypothesized universal behavior could not be observed in the attainable numerical data for the Heisenberg model. The proposed linear bound on operator growth associated with the hypothesis eventually stems from geometric arguments involving the locality of the Hamiltonian as well as the lattice configuration. We derive and investigate a related geometric bound, and we find that while the bound itself is not sharply achieved for any considered model, the hypothesis is nonetheless fulfilled in most cases.
For the disordered Ising chain with transverse and longitudinal fields, where couplings and fields are drawn from strictly positive distributions, Cao~\cite{Cao} has shown that the moments μ2k=∥[H,σ0z](k)∥22 grow almost factorially, μ2k1/(2k)∼k/lnk, and thus asymptotically at the maximal allowed rate. We generalize this result by resolving the operator norm in support length and show that the weight at length ℓk∼k/lnk already exhibits almost factorial growth, ∥[H,σ0z]ℓk(k)∥2≳(k/lnk)k. This implies maximal spatial delocalization of local operators and, in particular, rules out dynamical locality---the strongest form of many-body localization---at any disorder strength. We further establish rigorously a finite-size crossover scale L∼(W/J)2, where W is the disorder and J the coupling strength. For L≲(W/J)2 numerical studies only access a pre-asymptotic regime. Finally, we identify a structural path-entropy obstruction to perturbative LIOM constructions, based on the almost factorial branching of operator content and independent of resonance effects; the same mechanism strongly suggests ballistic real-time operator spreading, so sub-ballistic or localized dynamics would require a presently unidentified cancellation principle acting on almost factorially many disorder-dependent paths with random amplitudes.
We consider the spreading of a local operator A in one-dimensional many-body systems with Hamiltonian H by calculating the k -fold commutator [ H , [ H , [ . . . , [ H , A ] ] ] ] . We derive bounds for the operator norm of this commutator in free and interacting systems with and without disorder thus directly connecting the with questions of localization. We analytically show, in particular, that an almost-factorial growth of the operator norm—as recently proven for the random Ising model and strongly suggested here to also hold for the Heisenberg model with random fields—is inconsistent with an exponential localization of A . Assuming there exists a quasilocal unitary U which maps H onto an effective Hamiltonian H ̃ = U H U † = ∑ n E n τ n z + ∑ i , j J i j τ i z τ j z + ⋯ , we show that A ̃ = U A U † is a quasilocal operator which, in contrast to the Anderson case, indeed in the general many-body case, leading to an almost-factorial growth of the commutator norm. Therefore, either the unitary U in many-body systems with maximal norm growth does not exist, and such systems are always ergodic, or unusual nonergodic phases described by H ̃ exist which violate the operator growth hypothesis and in which local operators spread over the entire lattice, implying that transport will eventually set in. To investigate this issue further, we concentrate on the chain with random magnetic fields. We analytically and symbolically verify our general results for the noninteracting Anderson and Aubry-André models. For the case, the symbolic calculations are consistent with a maximal norm growth. Furthermore, we find no indication of a weakened exponential localization of A , expected for strong disorder and low commutator orders if the unitary U does exist. Finally, we study the differences between the interacting and noninteracting cases when trying to perturbatively construct U by consecutive Schrieffer-Wolff transformations. While it is straightforward to show that this construction converges in the Anderson case, we find no indications for a convergence in the interacting case, suggesting that U does not exist and that many-body localization is absent.
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT).Most prominent features of such RMT behavior with respect to a random spectrum, both encompassed in spectral pair correlation function, are statistical suppression of small level spacings (correlation hole) and enhanced stiffness of the spectrum at large spectral ranges.For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry [Proc.R. Soc.London, A400, 229 (1985)] within the so-called diagonal approximation of semiclassical periodic-orbit sums, while the derivation of the full RMT spectral form factor K(t) (Fourier transform of spectral pair correlation function), from semiclassics has been completed by Müller et al. [Phys.Rev. Lett.93, 014103 (2004)].In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains.Such systems display two universal types of behaviour which are termed as the 'many-body localized phase' and 'ergodic phase'.In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder.Here we provide a clear theoretical explanation for these observations.We compute K(t) in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.In particular, we relate K(t) to partition functions of a class of twisted classical Ising models on a ring of size t, hence the leading order RMT behavior K(t)2t is a consequence of translation and reflection symmetry of the Ising partition function.