Heating Problems In Non-Hermi Systems

Thermalization refers to a system based on nonequilibrium which can reach a steady state after a long time evolution, and the observable of expectations can be described by traditional statistical system.Since Rigol et al. used the eigenstate thermalization hy-pothesis(ETH) to explain thethermalization and its mechanism,the relaxation dynamics of observables in isolated many-body quantum systems far from equilibrium, as well as their description after relaxation, have generated much interest in recent years. It has been shown that quenches in the nonintegrable model lead to thermalization and in the integrable model can’t.In the other words, the observables in quenches between noninte-grable systems relax to stationary values that can be predicted by traditional ensembles of statistical mechanics, such as Microcanonical ensemble. Instead, generalized Gibb-s ensembles need to be used to describe observables after relaxation in the integrable systems,also called the " generalized thermalization" .On the other hand,Hamiltonians that are non-Hermitian have traditionally been used to describe dissipative processes, such as the phenomenon of radioactive decay. Howev-er, these non-Hermitian Hamiltonians are only approximate, phenomenological descrip-tions of physical processes.Bender and Boettcher indicated that the spectrum of a non-Hermitian Hamiltonian with parity-time symmetry can be all real.In the non-Hermitian system, the mathematical axiom of Hermiticity is replaced by the physically transpar-ent condition of space - time reflection ( PT) symmetry.The the spectrum is real if the Hamiltonian has an unbroken PT symmetry. There is a new inner product which called CPT inner product in the non-Hermitian Hamiltonian having an unbroken PT symme-try.Recent experiments have proved that hermiticity may not be the fundamental axioms in the quantum mechanics.More and more non-Hermitian PT-symmetric Hamiltonian systems have been researched,such as the Anderson models,topological models,spin chain and the tight binding chain.In this letter,we show that the thermalization in the non-Hermitian system is unlike the hermitian system.First,we consider a one-dimensional tight-binding model with gain and loss.We finded that the PT symmetry of systems is spontaneously broken fastly as the nonintegrable date increases.We only show the thermalization as one approaches the integrable ponit.So we use a special transformation to make the hermitian Hamiltoni-an to the non-Hermitian Hamiltonian and the both have the same spectrum.After the transformation,the eigenvalues can be all real when the system away from integrability.We finded the quenches in this non-Hermitian nonintegrable model may be can’t lead to thermalization.A parameter A has been used to descibe the departure of the hermiticity in the model.When the λ is small,the thermalization in the non-Hermitian system is like as the hermitian system. On the contrary,The quenches in the nonintegrable model can’t lead to thermalization when the λ is bigger.