Topological Classification Of Non-hermitian Systems

In this dissertation,I will introduce our work on non-Hermitian symmetry and topological classification during my Ph.D.These works include the construction of the54-fold symmetry classes in non-Hermitian systems,topological classification of nonHermitian systems with reflection symmetry,topological classification of defects in non-Hermitian systems,and topological classification of Floquet non-Hermitian system.The first chapter is the foundation of other chapters.First,we introduce the process of deepening the understanding of phase transitions.In the beginning,we use discontinuous quantity to define phase and phase transition.But it will lead to a conclusion.The conclusion contradicts that phase only depends on the state of the matter.Thus,we use the symmetries of the state to define the phase and phase transition.The definition and its deduction is the Ginzburg-Landau theory.In the Ginzburg-Landau theory,phases only depend on the state of the matter,and there are some universal features in phases and phase transition.But the Ginzburg-Landau theory is not the only definition of phase that satisfies these two conditions.Then we introduce the KT phase transition.And we use topological invariants to define the phase and phase transition.It is being called the topological phase transition.The topological phase transition also satisfies these two conditions.And we will discuss more examples to understand it concretely.Then,we introduce the symmetry class,some basic methods and concepts in topological classification of phases of matter,and non-Hermitian systems.To understand the topological phase systematically and religiously,we introduce the topological classification of the non-interaction Hermitian Fermi system in 10-fold Altland-Zirnbauer(AZ)symmetry class.The mathematical foundation of the topological classification is homotopy group,relative homotopy group,Clifford algebra,and K-theory.We will introduce these mathematical concepts and a topological classification method which we call it four-step method.The method is useful to get the topological classification of the Hermitian system under some symmetries.Then we introduce the significance of studying the non-Hermitian system.Because it has some applications in classical optical,mechanics,and electric circuits systems,open quantum systems,and so on.And we will introduce some basic concepts in the non-Hermitian system,including exceptional points,non-Hermitianskin effect,38-fold Bernard-Le Clair(BL)classes,and 54-fold generalized Bernard-Le Clair(GBL)classes.GBL class can be regarded as the non-Hermitian generalization of the Hermitian AZ class.It is the total number of possible groups constituted by non-spatial symmetries.Thus,there is no self-inconsistence if we use the GBL class to describe the non-Hermitian system.When → 4)( is Hamiltonian)can be regarded as an equivalence relation for non-spatial symmetries,some classes are equivalent to each other in the 54 GBL class.After subtracting these classes,the nonHermitian system can effectively be described by 38-fold BL classes.If the Hamiltonian is time-independent and we only care about point gap topology,the equivalent relation is valid.If the Hamiltonian is time-dependent or we care about line-like gap topology,the equivalent relation is invalid.Thus,sometimes there are inconsistencies if we use the 38-fold BL symmetry class to describe the non-Hermitian symmetry class.And one of most important motivations for the author to construct the 54-fold symmetry class is the inconsistencies.Although 54-fold GBL class is more than 38-fold BL class.The GBL class won’t lead to inconsistent results(for the time-dependent,time-independent,point-like gap,line-like gap).And we only use group theory when constructing of 54-fold GBL class(without any assumptions).Its construction principle is similar to the construction principle of AZ classes or space groups.In the second chapter,we introduce the topological classification of non-Hermitian systems with reflection symmetry.In the third chapter,we introduce the topological classification of defects in non-Hermitian systems.In the fourth chapter,we introduce the symmetry and topological classification of Floquet non-Hermitian systems.To get these topological classifications,we transform these problems into the topological classification of Hermitian systems under certain symmetries by mappings or continuous deformations.Then,these problems can be solved by the four steps method.To understand these topological classification conclusions better,we give some examples in each chapter of the second,third,and fourth chapter.Finally,we conclude this dissertation and discuss possible research directions in the future.