This week

Symmetry plays an important role in quantum systems: it can constrain the dynamics, give rise to selection rules, and provide computation methods in quantum computers. In recent years there are also new types of symmetries called generalized symmetries discovered in many quantum systems, including non-invertible symmetry and higher group symmetry. These lectures will be about symmetries in various quantum systems and their applications such as constraints on the low energy dynamics. Examples will be discussed in the lectures include quantum mechanics systems, gauge theories, lattice models, and the symmetry includes ordinary and higher form symmetry as well non-invertible symmetry.

Rare macroscopic fluctuations leading to large deviations in many-body systems have attracted significant attention in recent years. In this talk we present an exact solution to such a problem - the emptiness formation in one-dimensional quantum polytropic gases characterized by an arbitrary polytropic index \gamma, which defines the equation of state P ~ \rho^\gamma, where P is the pressure and \rho is the density. The problem consists of determining the probability of spontaneous formation of an empty interval in the ground state of the gas. In the limit of a macroscopically large interval, this probability is dominated by an instanton configuration. By solving the hydrodynamic equations in imaginary time, we derive the analytic form of the emptiness instanton. This solution is expressed as an integral representation analogous to those used for correlation functions in Conformal Field Theory. Prominent features of the spatiotemporal profile of the instanton are obtained directly from this representation and will be discussed in the context of limit shape phenomena. [1] A.G. Abanov and D.M. Gangardt, arXiv:2412.1168