We quantify the emergent complexity of quantum states near quantum critical points on regular 1D lattices, via complex network measures based on quantum mutual information as the adjacency matrix, in direct analogy to quantifying the complexity of EEG/fMRI measurements of the brain. Using matrix product state methods, we show that network density, clustering, disparity, and Pearson's correlation obtain the critical point for both quantum Ising and Bose-Hubbard models to a high degree of accuracy in finite-size scaling for three classes of quantum phase transitions, Z2, mean field superfluid/Mott insulator, and a BKT crossover.Classical statistical physics has developed a powerful set of tools for analyzing complex systems, chief among them complex networks, in which connectivity and topology predominate over other system features [1]. Complex networks model systems as diverse as the brain and the internet; however, up till now they have been obtained in quantum systems by explicitly enforcing complex network structure in their quantum connections [2][3][4][5][6][7], e.g. entanglement percolation on a complex network [4]. In contrast, complexity measures on the brain observe emergent complexity arising out of, e.g., a regular array of EEG electrodes placed on the scalp, via an adjacency matrix formed from the classical mutual information calculated between them [8]. We apply the quantum generalization of this measure, an adjacency matrix of the quantum mutual information calculated on quantum states [9], to well known quantum many-body models on regular 1D lattices, and uncover emergent quantum complexity which clearly identifies quantum critical points (QCPs) [10,11]. Quantum mutual information bounds two-point correlations from above [12], measurable in a precise and tunable fashion in e.g. atom interferometry in 1D Bose gases [13], among many other quantum simulator architectures. Using matrix product state (MPS) computational methods [14,15], we demonstrate rapid finite size-scaling for both transverse Ising and Bose-Hubbard models, including Z 2 , mean field, and BKT quantum phase transitions.As we move toward more and more complex quantum systems in materials design and quantum simulators, involving a hierarchy of scales, diverse interacting components, and a structured environment, we expect to observe long-lived dynamical features, fat-tailed distributions, and other key identifiers of complexity [16][17][18]. Such systems include quantum simulator technologies based on ultracold atoms and molecules [19], trapped ions [20], and Rydberg gases [21], as well as superconducting Josephson-junction based nanoelectromechanical systems in which different quantum subsystems form compound quantum machines with both electrical and mechanical components [22]. A key area in which we have taken a first step beyond phase diagrams and ground state properties is non-equilibrium quantum dynamics, where critical exponents and renormalization group theory are only weakly applicable at best, e.g. in the KibbleZurek mechanism,...

We use network analysis to describe and characterize an archetypal quantum system -an Ising spin chain in a transverse magnetic field. We analyze weighted networks for this quantum system, with link weights given by various measures of spin-spin correlations such as the von Neumann and Rényi mutual information, concurrence, and negativity. We analytically calculate the spinspin correlations in the system at an arbitrary temperature by mapping the Ising spin chain to fermions, as well as numerically calculate the correlations in the ground state using matrix product state methods, and then analyze the resulting networks using a variety of network measures. We demonstrate that the network measures show some traits of complex networks already in this spin chain, arguably the simplest quantum many-body system. The network measures give insight into the phase diagram not easily captured by more typical quantities, such as the order parameter or correlation length. For example, the network structure varies with transverse field and temperature, and the structure in the quantum critical fan is different from the ordered and disordered phases. I. INTRODUCTIONNetwork analysis is a powerful technique to characterize the structure of connections between agents in a network [1,2]. Studies have shown that classical systems as diverse as the brain and the Internet have a complex network structure [3][4][5][6][7][8]. Quantum systems also show a wide variety of complexity emerging due to inter-particle interactions. Like classical systems, quantum systems have an interconnected web of correlations, and network analysis provides a powerful set of tools to study them. However, while complex networks are ubiquitous in classical systems with a sufficiently rich set of interacting components, it is an open question what the minimal interacting quantum many-body system is in which complex network structures can appear.In this paper, we address this question by studying the network of correlations that arises in the simplest of interacting quantum models, the one-dimensional transverse field Ising model (TIM). We introduce and calculate networks whose links are weighted by various measures of correlations and entanglement, and quantify their complexity. The emergence of network complexity illuminates the richness of the quantum system.Earlier works have studied complex networks in the context of quantum systems, but by enforcing complex network structure in the Hamiltonian, e.g, in interactions [9][10][11][12][13][14]. However, there is no need for this explicit enforcement, as one finds network structure already in quantum states even for simple models such as the nearest-neighbor TIM.

We uncover signatures of quantum chaos in the many-body dynamics of a Bose-Einstein condensate-based quantum ratchet in a toroidal trap. We propose measures including entanglement, condensate depletion, and spreading over a fixed basis in many-body Hilbert space, which quantitatively identify the region in which quantum chaotic many-body dynamics occurs, where random matrix theory is limited or inaccessible. With these tools, we show that many-body quantum chaos is neither highly entangled nor delocalized in the Hilbert space, contrary to conventionally expected signatures of quantum chaos.

We investigate the dimension of the phase-space attractor of a quantum chaotic many-body ratchet in the meanfield limit. Specifically, we explore a driven Bose-Einstein condensate in three distinct dynamical regimes-Rabi oscillations, chaos, and self-trapping regimes-and for each of them we calculate the correlation dimension. For the ground state of the ratchet formed by a system of field-free noninteracting particles, we find four distinct pockets of chaotic dynamics throughout these regimes. We show that a measurement of local density in each of the dynamical regimes has an attractor characterized by a higher fractal dimension, D R = 2.59 ± 0.01, D C = 3.93 ± 0.04, and D S = 3.05 ± 0.05, compared to the global measure of current, D R = 2.07 ± 0.02, D C = 2.96 ± 0.05, and D S = 2.30 ± 0.02. The deviation between local and global measurements of the attractor's dimension corresponds to an increase towards higher condensate depletion, which remains constant for long time scales in both Rabi and chaotic regimes. The depletion is found to scale polynomially with particle number N, namely, as N β with β R = 0.51 ± 0.004 and β C = 0.18 ± 0.004 for the two regimes. Thus, we find a strong deviation from the mean-field results, especially in the chaotic regime of the quantum ratchet. The ratchet also reveals quantum revivals in the Rabi and self-trapping regimes but not in the chaotic regime, with revival times scaling linearly in particle number for Rabi dynamics. Based on the obtained results, we outline pathways for the identification and characterization of emergent phenomena in driven many-body systems. This includes the identification of many-body localization from the many-body measures of the system, the influence of entanglement on the rate of the convergence to the mean-field limit, and the establishment of a polynomial scaling of the Ehrenfest time at which the mean-field description fails to describe the dynamics of the system.

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