Semiclassical Approach to Many-Particle Interference: Quantum Signatures of Classical Chaos and Criticality

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The notion of many-particle interference has recently gained increasing attention in many-body physics in fields such as many-body localization, photonic quantum networks and cold atoms in optical lattices. In a closer sense many-particle interference refers to dynamical interference effects in Fock space describing correlations beyond mean field and is naturally implemented in the Feynman propagator. In a semiclassical regime it can be represented by a sum over "classical" paths in Fock space carrying interfering amplitudes. Complementary to semiclassical concepts form single-particle dynamics, valid in the short-wavelength limit, such a regime is reached in an N-particle system for large particle number N, usually referred to as thermodynamic limit. In this project we envisage to further develop such a many-body semiclassical approach, where quantum propagation and the many-body density of states can be regarded as being based on interference between different classical (mean-field) paths. This provides a link between many-body quantum behavior and properties of the corresponding classical system and, in particular, opens up the interesting question how instabilities in the classical dynamics (close to a critical point or if chaotic) affect quantum many-body features. The overreaching goal of our project is to describe signatures of classical singularities in the discrete quantum spectra of many-body systems with emphasis on excited state quantum phase transitions and fast scrambling of quantum correlations close to criticality or due to chaoticity. More specifically, (i) we will provide analytical estimates for the scaling of energy gaps and the critical interaction strength for different types of classical singularities for various many-body models and will benchmark our results against numerical simulations. (ii) We will explore the quantum role of the Lyapunov exponent, associated with the so-called quantum butterfly effect in many-body systems, currently understood as leading to fast scrambling and speeding up the generation of many-body quantum correlations due to classical instability. Such correlations are measured through so-called out-of-time-order correlators (OTOCs). Here our objectives are to find a rigorous many-body semiclassical derivation of the exponential growth rates of OTOCs, to explain the observed saturation of OTOCs in terms of many-body quantum interference, and to obtain insight into Maldacena’s universal bound on the exponential growth of OTOCs from a quantum dynamics perspective.