Non-Hermitian (NH) systems may have quite different physics properties from that of Hermitian counterparts. For example, in the NH systems with nonreciprocal hopping, there exists (single-body version of) skin effect-The eigenstates are exponentially
localized at the boundaries. An interesting problem is about the generalization of NH skin effect to a many-body NH system. In this paper, we studied many-body physics in the quantum systems with nonreciprocal hoppings and obtained analytical results. In these many-body NH systems, the single-body NH skin effect upgrades to quantum (Maxwell s) pressure-demon effect, which leads to band-width renormalization and a uniform pressure gradient to the system. In particular, according to the quantum pressure-demon effect, in many-body Bosonic/Fermionic Hotano-Nelson model, there exist new physical phenomena compared with their Hermitian counterparts: Liouvillian Bose-Einstein condensation and Liouvillian Fermi-surface in real space, respectively. This discovery will open a door to learn many-body physics for NH quantum systems.
The breakdown of the bulk-boundary correspondence in non-Hermitian (NH) topological systems is an open, controversial issue. In this paper, to resolve this issue, we ask the following question: Can a (global) topological invariant completely describe
the topological properties of a NH system as its Hermitian counterpart? Our answer is no. One cannot use a global topological invariant (including non-Bloch topological invariant) to accurately characterize the topological properties of the NH systems. Instead, there exist a new type of topological invariants that are absence in its Hermitian counterpart -- the state dependent topological invariants. With the help of the state-dependent topological invariants, we develop a new topological theory for NH topological system beyond the general knowledge for usual Hermitian systems and obtain an exact formulation of the bulk-boundary correspondence, including state-dependent phase diagram, state-dependent phase transition and anomalous transport properties (spontaneous topological current). Therefore, these results will help people to understand the exotic topological properties of various non-Hermitian systems.
