Time Evolution of Quantum Mechanical Many-Body Systems

Tomasz Szołdra, Piotr Sierant, Korbinian Kottmann +2 more

The transition between ergodic and many-body localized (MBL) phases is expected to occur via an avalanche mechanism, in which ergodic bubbles that arise due to local fluctuations in system properties thermalize their surroundings leading to delocalization of the system, unless the disorder is sufficiently strong to stop this process. We propose an algorithm based on neural networks that allows us to detect the ergodic bubbles using experimentally measurable two-site correlation functions. Investigating the time evolution of the system, we observe a logarithmic in time growth of the ergodic bubbles in the MBL regime. The distribution of the size of ergodic bubbles converges during time evolution to an exponentially decaying distribution in the MBL regime, and a power-law distribution with a thermal peak in the critical regime, supporting thus the scenario of delocalization through the avalanche mechanism. Our algorithm permits us to pinpoint the quantitative differences in the time evolution of systems with random and quasiperiodic potentials, as well as to identify rare (Griffiths) events. Our results open different pathways in studies of the mechanisms of thermalization of disordered many-body systems and beyond.

Hadi Sobhani, HASSAN HASSANABADI

In this article, a quantum many-body system is considered. Then two time-dependent interactions have been added to the system. Changing of them is assumed in general form. After that, by using algebraic method, time evolution of this many-body system has been investigated. In order to study the time evolution, Lewis-Riesenfeld dynamical invariant and time evolution operator method have been used. Appropriate dynamical invariants are constructed and their Eigenvalues are derived as well as appropriate time evolution operators are constructed. These calculations have been done in general form so there are no limiting assumptions on changing of time-dependent functions.

Benjamin Schlein

In these notes, we review some recent mathematical results concerning the derivation of effective evolution equations from many body quantum mechanics. In particular, we discuss the emergence of the Hartree equation in the so-called mean field regime (for example, for systems of gravitating bosons), and we show that the Gross-Pitaevskii equation approximates the dynamics of initially trapped Bose-Einstein condensates. We explain how effective evolution equations can be derived, on the one hand, by analyzing the so called BBGKY hierarchy, describing the time-evolution of reduced density matrices, and, on the other hand, by studying the dynamics of coherent initial states in a Fock-space representation of the many body system.

Arno Böhm, Piotr Kielanowski, G. B. Mainland