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 $d \ge 2$ 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 decay 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 consider growth of local operators under Euclidean time evolution in lattice systems with local interactions. We derive rigorous bounds on the operator norm growth and then proceed to establish an analog of the Lieb-Robinson bound for the spatial growth. In contrast to the Minkowski case when ballistic spreading of operators is universal, in the Euclidean case spatial growth is system-dependent and indicates if the system is integrable or chaotic. In the integrable case, the Euclidean spatial growth is at most polynomial. In the chaotic case, it is the fastest possible: exponential in 1D, w…
We consider operator growth for generic large-N gauge theories at finite temperature. Our analysis is performed in terms of Fourier modes, which do not mix with other operators as time evolves, and whose correlation functions are determined by their two-point functions alone, at leading order in the large-N limit. The algebra of these modes allows for a simple analysis of the operators with whom the initial operator mixes over time, and guarantees the existence of boundary CFT operators closing the bulk Poincaré algebra, describing the experience of infalling observers. We discuss several exis…
We investigate and characterize the dynamics of operator growth in irrational two-dimensional conformal field theories. By employing the oscillator realization of the Virasoro algebra and CFT states, we systematically implement the Lanczos algorithm and evaluate the Krylov complexity of simple operators (primaries and the stress tensor) under a unitary evolution protocol. Evolution of primary operators proceeds as a flow into the 'bath of descendants' of the Verma module. These descendants are labeled by integer partitions and have a one-to-one map to Young diagrams. This relationship allows u…
We study random quantum circuits with symmetry, where the local 2-site unitaries are drawn from a quotient or subgroup of the full unitary group $U(d)$. Random quantum circuits are minimal models of local quantum chaotic dynamics and can be used to study operator growth and the emergence of diffusive hydrodynamics. We derive the transition probabilities for the stochastic process governing the growth of operators in four classes of symmetric random circuits. We then compute the butterfly velocities and diffusion constants for a spreading operator by solving a simple random walk in each class o…
In this work we probe the operator growth for systems with Lie symmetry using tools from quantum information. Namely, we investigate the Krylov complexity, entanglement negativity, von Neumann entropy and capacity of entanglement for systems with SU(1,1) and SU(2) symmetry. Our main tools are two-mode coherent states, whose properties allow us to study the operator growth and its entanglement structure for any system in a discrete series representation of the groups under consideration. Our results verify that the quantities of interest exhibit certain universal features in agreement with the …
We study the operator growth in open quantum systems with dephasing dissipation terms, extending the Krylov complexity formalism of Phys. Rev. X 9, 041017. Our results are based on the study of the dissipative $q$-body Sachdev-Ye-Kitaev (SYK$_q$) model, governed by the Markovian dynamics. We introduce a notion of ''operator size concentration'' which allows a diagrammatic and combinatorial proof of the asymptotic linear behavior of the two sets of Lanczos coefficients ($a_n$ and $b_n$) in the large $q$ limit. Our results corroborate with the semi-analytics in finite $q$ in the large $N$ limit,…
We develop statistical mechanics for stochastic growth processes as applied to Laplacian growth by using its remarkable connection with a random matrix theory. The Laplacian growth equation is obtained from the variation principle and describes adiabatic (quasi-static) thermodynamic processes in the two-dimensional Dyson gas. By using Einstein's theory of thermodynamic fluctuations we consider transitional probabilities between thermodynamic states, which are in a one-to-one correspondence with planar domains. Transitions between these domains are described by the stochastic Laplacian growth e…
Stochastic Krylov Dynamics: Revisiting Operator Growth in Open Quantum Systems
Indian Institute of Technology Gandhinagar（印度理工学院甘德纳加尔分校）；University of Cape Town（开普敦大学）；National Institute of Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）
We regularize the Laplacian growth problem with zero surface tension by introducing a short-distance cutoff $\hbar$, so that the change of the area of domains is quantized and equals an integer multiple of the area quanta $\hbar$. The domain can be then considered as an aggregate of tiny particles (area quanta) obeying the Pauli exclusion principle. The statistical theory of Laplacian growth is introduced by using Laughlin's description of the integer quantum Hall effect. The semiclassical evolution of the aggregate is similar to classical deterministic Laplacian growth. However, the quantizat…
Operator growth, or operator spreading, describes the process where a "simple" operator acquires increasing complexity under the Heisenberg time evolution of a chaotic dynamics, therefore has been a key concept in the study of quantum chaos in both single-particle and many-body systems. An explicit way to quantify the complexity of an operator is the Shannon entropy of its operator coefficients over a chosen set of operator basis, dubbed "coefficient entropy". However, it remains unclear if the basis-dependency of the coefficient entropy may result in a false diagnosis of operator growth, or t…
In many-body chaotic systems, the size of an operator generically grows in Heisenberg evolution, which can be measured by certain out-of-time-ordered four-point functions. However, these only provide a coarse probe of the full underlying operator growth structure. In this article we develop a methodology to derive the full growth structure of fermionic systems, that also naturally introduces the effect of finite temperature. We then apply our methodology to the SYK model, which features all-to-all $q$-body interactions. We derive the full operator growth structure in the large $q$ limit at all…
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 c…
Operator scrambling denotes the evolution of a simple operator into a complicated one (in the Heisenberg picture), which characterizes quantum chaos in many-body systems. More specifically, a simple operator evolves into a linear superposition of many operators, most of which are many-body operators supported on a region of size much larger than $1$. In general, an operator does not have a definite size but is characterized by a probability distribution of size. The operator size is related to out-of-time-order correlation functions, but these are generically difficult to obtain from experimen…
Towards a Refinement of Krylov Complexity: Scrambling, Classical Operator Growth and Replicas
National Taiwan University（国立台湾大学）；National Center for Theoretical Sciences（国家理论科学中心）；National Sun Yat-sen University（国立中山大学）；Gwangju Institute of Science and Technology（光州科学技术院）
Recently, a hypothesis on the complexity growth of unitarily evolving operators was presented. This hypothesis states that in generic, non-integrable 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 the paper at hand, we numerically test this hypothesis for a variety of exemplary systems, including 1d and 2d Ising models as well as 1d Heisenberg models. While we find the hypothesis…
The structure of the Heisenberg evolution of operators plays a key role in explaining diverse processes in quantum many-body systems. In this paper, we discuss a new universal feature of operator evolution: an operator can develop a void during its evolution, where its nontrivial parts become separated by a region of identity operators. Such processes are present in both integrable and chaotic systems, and are required by unitarity. We show that void formation has important implications for unitarity of entanglement growth and generation of mutual information and multipartite entanglement. We …
The spreading of quantum information in closed systems, often termed scrambling, is a hallmark of many-body quantum dynamics. In open systems, scrambling competes with noise, errors and decoherence. Here, we provide a universal framework that describes the scrambling of quantum information in open systems: we predict that the effect of open-system dynamics is fundamentally controlled by operator size distributions and independent of the microscopic error mechanism. This framework allows us to demonstrate that open quantum systems exhibit universal classes of information dynamics that fundament…
We examine a lattice model of tumor growth where survival of tumor cells depends on the supplied nutrients. When such a supply is random, the extinction of tumors belongs to the directed percolation universality class. However, when the supply is correlated with distribution of tumor cells, which as we suggest might mimick the angiogenic growth, the extinction shows different, and most likely novel critical behaviour. Such a correlation affects also the morphology of the growing tumors and drastically raise tumor survival probability.
We characterize the growth and spreading of operators and entanglement in two paradigmatic non-thermalizing phases - the many-body localized phase and the random singlet phase - using out-of-time-ordered correlators, the entanglement contour, and operator entanglement. We contrast these phases with strongly thermalizing holographic conformal field theories and fully localized Anderson insulators. We obtain a phenomenological description of the operator and state dynamics of these phases and demonstrate the utility of the entanglement contour and operator entanglement measures as useful probes …
The concepts of operator size and computational complexity play important roles in the study of quantum chaos and holographic duality because they help characterize the structure of time-evolving Heisenberg operators. It is particularly important to understand how these microscopically defined measures of complexity are related to notions of complexity defined in terms of a dual holographic geometry, such as complexity-volume (CV) duality. Here we study partially entangled thermal states in the Sachdev-Ye-Kitaev (SYK) model and their dual description in terms of operators inserted in the inter…
In an isolated system, the time evolution of a given observable in the Heisenberg picture can be efficiently represented in Krylov space. In this representation, an initial operator becomes increasingly complex as time goes by, a feature that can be quantified by the Krylov complexity. We introduce a fundamental and universal limit to the growth of the Krylov complexity by formulating a Robertson uncertainty relation, involving the Krylov complexity operator and the Liouvillian, as generator of time evolution. We further show the conditions for this bound to be saturated and illustrate its val…
For one-dimensional growth processes we consider the distribution of the height above a given point of the substrate and study its scale invariance in the limit of large times. We argue that for self-similar growth from a single seed the universal distribution is the Tracy-Widom distribution from the theory of random matrices and that for growth from a flat substrate it is some other, only numerically determined distribution. In particular, for the polynuclear growth model in the droplet geometry the height maps onto the longest increasing subsequence of a random permutation, from which the he…
Systems far from equilibrium approach stability slowly due to "anti-mixing" characterized by regions of the phase-space that remain disconnected after prolonged action of the flow. We introduce the Optimal Growth Mode (OGM) to capture this slow initial relaxation. The OGM is calculated from Markov matrices approximating the action of the Fokker-Planck operator onto the phase space. It is obtained as the mode having the largest growth in energy before decay. Important nuances between the OGM and the more traditional slowest decaying mode are detailed in the case of the Lorenz 63 model. The impl…
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 $α$ in generic systems, with an extra logarithmic correction in 1d. The rate $α$ --- an experimental observable --- governs the exponential growth of operator complexity in a sense we make precise. This exponential growth even prevails beyond semiclassical or large-$N$ limits. Moreover, $α$ upper bounds a large class of operator c…
Inspired by the universal operator growth hypothesis, we extend the formalism of Krylov construction in dissipative open quantum systems connected to a Markovian bath. Our construction is based upon the modification of the Liouvillian superoperator by the appropriate Lindbladian, thereby following the vectorized Lanczos algorithm and the Arnoldi iteration. This is well justified due to the incorporation of non-Hermitian effects due to the environment. We study the growth of Lanczos coefficients in the transverse field Ising model (integrable and chaotic limits) for boundary amplitude damping a…
The time evolution of local operators in quantum compass models is characterized by simplicity as it can be represented as expanding and contracting strings of operators. Here we present an analytical solution to the problem of growth of a local energy operator in a quantum compass model on a Bethe lattice. We find a linear increase in time of the average operator length and a diffusive spreading of the operator length distribution. By a moment method we evaluate the local energy autocorrelation function that shows a Lorentzian shape at low frequencies. Furthermore, by a stochastic method we v…
This dissertation develops, for several families of statistical mechanical and random growth models, techniques for analyzing infinite-volume asymptotics. In the statistical mechanical setting, we focus on the low-temperature phases of spin glasses and directed polymers, wherein the ensembles exhibit localization which is physically phenomenological. We quantify this behavior in several ways and establish connections to properties of the limiting free energy. We also consider two popular zero-temperature polymer models, namely first- and last-passage percolation. For these random growth models…
Dynamics of Loschmidt echoes from operator growth in noisy quantum many-body systems
All Souls College, Oxford（牛津大学圣灵魂学院）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·皮尔斯理论物理中心）；TCM Group, Cavendish Laboratory, Ray Dolby Centre, University of Cambridge（剑桥大学卡文迪许实验室雷·多尔比中心TCM小组）
A theoretical framework is developed to describe experiments on the structure of epitaxial thin films, particularly niobium on sapphire. We extend the hypothesis of dynamical scaling to apply to the structure of thin films from its conventional application to simple surfaces. We then present a phenomenological continuum theory that provides a good description of the observed scattering and the measured exponents. Finally the results of experiment and theory are compared.
We analyze a simple model of quantum dynamics, which is a discrete-time deterministic version of the Frederickson-Andersen model. We argue that this model is integrable, with a quasiparticle description related to the classical hard-rod gas. Despite the integrability of the model, commutators of physical operators grow as in generic chaotic models, with a diffusively broadening front, and local operators obey the eigenstate thermalization hypothesis (ETH). However, large subsystems violate ETH; as a function of subsystem size, eigenstate entanglement first increases linearly and then saturates…
In the realm of Boltzmann-Gibbs statistical mechanics there are three well known isomorphic connections with random geometry, namely (i) the Kasteleyn-Fortuin theorem which connects the $λ\to 1$ limit of the $λ$-state Potts ferromagnet with bond percolation, (ii) the isomorphism which connects the $λ\to 0$ limit of the $λ$-state Potts ferromagnet with random resistor networks, and (iii) the de Gennes isomorphism which connects the $n \to 0$ limit of the $n$-vector ferromagnet with self-avoiding random walk in linear polymers. We provide here strong numerical evidence that a similar isomorphism…
Which properties of metabolic networks can be derived solely from stoichiometric information about the network's constituent reactions? Predictive results have been obtained by Flux Balance Analysis (FBA), by postulating that cells set metabolic fluxes within the allowed stoichiometry so as to maximize their growth. Here, we generalize this framework to single cell level using maximum entropy models from statistical physics. We define and compute, for the core metabolism of Escherichia coli, a joint distribution over all fluxes that yields the experimentally observed growth rate. This solution…
We present a simple non-equilibrium model of mass condensation with Lennard-Jones interactions between particles and the substrate. We show that when some number of particles is deposited onto the surface and the system is left to equilibrate, particles condense into an island if the density of particles becomes higher than some critical density. We illustrate this with numerically obtained phase diagrams for three-dimensional systems. We also solve a two-dimensional counterpart of this model analytically and show that not only the phase diagram but also the shape of the cross-sections of thre…
Three models from statistical physics can be analyzed by employing space-time determinantal processes: (1) crystal facets, in particular the statistical properties of the facet edge, and equivalently tilings of the plane, (2) one-dimensional growth processes in the Kardar-Parisi-Zhang universality class and directed last passage percolation, (3) random matrices, multi-matrix models, and Dyson's Brownian motion. We explain the method and survey results of physical interest.
We study the growth of the operator entanglement entropy (EE) of the time evolution operator in chaotic, many-body localized and Floquet systems. In the random field Heisenberg model we find a universal power law growth of the operator EE at weak disorder, a logarithmic growth at strong disorder, and extensive saturation values in both cases. In a Floquet spin model, the saturation value after an initial linear growth is identical to the value of a random unitary operator (the Page value). We understand these properties by mapping the operator EE to a global quench problem evolved with a simil…
Approaches from statistical physics are applied to investigate the structure of network models whose growth rules mimic aspects of the evolution of the world-wide web. We first determine the degree distribution of a growing network in which nodes are introduced one at a time and attach to an earlier node of degree k with rate A_ksim k^gamma. Very different behaviors arise for gamma<1, gamma=1, and gamma>1. We also analyze the degree distribution of a heterogeneous network, the joint age-degree distribution, the correlation between degrees of neighboring nodes, as well as global network propert…
In this paper, we give a general discussion on the calculation of the statistical distribution from a given operator relation of creation, annihilation, and number operators. Our result shows that as long as the relation between the number operator and the creation and annihilation operators can be expressed as $a^{\dagger}b=Λ\left(N\right) $ or $N=Λ^{-1} \left( a^{\dagger}b\right)$, where $N$, $a^{\dagger}$, and $b$ denote the number, creation, and annihilation operators, i.e., $N$ is a function of quadratic product of the creation and annihilation operators, the corresponding statistical dis…
The generalization of the Zubarev nonequilibrium statistical operator method for the case of Renyi statistics is proposed when the relevant statistical operator (or distribution function) is obtained based on the principle of maximum for the Renyi entropy. The nonequilibrium statistical operator and corresponding generalized transport equations for the reduced-description parameters are obtained. A consistent description of kinetic and hydrodynamic processes in the system of interacting particles is considered as an example.
The lifetime of statistical system is introduced. It is supposed that the nonequilibrium statistical operator implicitly contains the lifetime. The operations of taking of invariant part, averaging on initial conditions used in works of D.N. Zubarev, temporary coarse-graining (Kirkwood), choose of the direction of time are replaced by averaging on lifetime distribution. The expression for lifetime change at transitions from quasi-equilibrium system to nonequilibrium one is derived. A sort of the nonequilibrium statistical operator for systems of the finite sizes is suggested.
We discuss the methods to calculate the roughness exponent alpha and the dynamic exponent z from the scaling properties of the local roughness, which is frequently used in the analysis of experimental data. Through numerical simulations, we studied the Family, the restricted solid-on-solid (RSOS), the Das Sarma-Tamborenea (DT) and the Wolf-Villain (WV) models in one- and two dimensional substrates, in order to compare different methods to obtain those exponents. The scaling at small length scales do not give reliable estimates of alpha, suggesting that the usual methods to estimate that expone…
The Zubarev nonequilibrium statistical operator (NSO) method in Renyi statistics is discussed. The solution of $q$-parametrized Liouville equation within the NSO method is obtained. A statistical approach for a consistent description of reaction-diffusion processes in "gas-adsorbate-metal" system is proposed using the NSO method in Renyi statistics.
