Topological classification of non-Hermitian systems with reflection symmetry
We classify topological phases of non-Hermitian systems in the Altland-Zirnbauer classes with an additional reflection symmetry in all dimensions. By mapping the non-Hermitian system into an enlarged Hermitian Hamiltonian with an enforced chiral symmetry, our topological classification is thus equivalent to classifying Hermitian systems with both chiral and reflection symmetries, which effectively change the classifying space and shift the periodical table of topological phases. According to our classification tables, we provide concrete examples for all topologically nontrivial non-Hermitian classes in one dimension and also give explicitly the topological invariant for each nontrivial example. Our results show that there exist two kinds of topological invariants composed of either winding numbers or numbers. By studying the corresponding lattice models under the open boundary condition, we unveil the existence of bulk-edge correspondence for the one-dimensional topological non-Hermitian system characterized by winding numbers, however no bulk-edge correspondence is observed for the topological system characterized by numbers.
The band theory of topological insulators and superconductors has been greatly developed in the past decades Hassan (); Qi (); Wen1 (); Thouless (); Haldane (); Kane1 (); Kane2 (); Fu1 (); Fu2 (); ZhangSC (). Topologically non-trivial insulators or superconductors are characterized by either an integer () or a binary () topological index and have stable edge states Bernevig (); Moore (); Kitaev1 (). The well-known examples are the integer quantum Hall effect and spin Hall effect, for which different topological phases are characterized by Chern number Thouless (); Haldane () and topological number Kane1 (); Kane2 (); ZhangSC (), respectively. Traditionally, non-interacting fermionic systems can be divided into ten Altland-Zirnbauer (AZ) symmetry classes, in terms of the presence or absence of Time-reversal (T), charge-conjugation or particle-hole (C), and chiral or sublattice symmetry (S). In the framework of the ’ten-fold way’, it is well established that five of ten classes are topologically nontrivial in every spatial dimension Fu1 (); Fu2 (); AZ (); Kitaev2 (); Ludwig1 (); Ludwig2 (); Ludwig3 (); Ludwig4 (). When adding additional symmetries to the system, the classifying space of Hamiltonian is changed, and we can get different topological classification Fu1 (); Fu2 (); AZ (); Kitaev2 (); Ludwig1 (); Ludwig2 (); Ludwig3 (); Ludwig4 (); Furusaki (); Stone (); Chiu (); Wen2 (). Many theoretical works have been carried out on the classification of various symmetry-protected-topological phases Furusaki (); Stone (); Chiu (); Wen2 (), ranging from the reflection symmetry protected topological phases to the crystalline topological phases, and most of these works are focused on the Hermitian systems.
Recent experimental progress on the study of optical systems and electrical systems with gain and loss has unveiled that these systems can be effectively described by non-Hermitian Hamiltonians Ruter2010 (); Peng2014 (); Feng2014 (); Konotop2016 (); Xiao2017 (); Weimann2017 (); Menke (); Klett (); Bender1 (); Bender2 (), and motivates great interest in exploring topological phases in non-Hermitian systems Hu2011 (); Esaki2011 (); Rudner (); ZhuBG (); Yuce (); Xiong (); Xiao (); Weimann (); ElG (); TElee1 (); Shen (); TElee2 (); Leykam (); Yin (); Lieu (); Hatano (); Longhi (); Parto (); WangZhong1 (); WangZhong2 (); Gong (); Bergholtz (); Song (); Torres2 (); Torres1 (). Various topological non-Hermitian models have been studied Yin (); JiangHui (); TElee2 (); Leykam (); Lieu (); Shen (); WangZhong1 (); WangZhong2 (); Xiong (); Bergholtz (), and it has been demonstrated that topological non-Hermitian systems may exhibit quite different behaviors from their Hermitian counterparts, associated with some distinctive properties of the non-Hermitian Hamiltonian, e.g., the existence of exceptional points, biorthonormal eigenvectors, unusual bulk-edge correspondence and emergence of skin effect Heiss (); Dembowski (); Rotter (); Hu2017 (); Hassan2017 (); Bergholtz (); WangZhong1 (); WangZhong2 (); JiangHui (); Xiong ().
For systematic understanding of non-Hermitian topological phases, it is highly desirable to carry out topological classification for non-Hermitian systems. Regardless of spatial symmetry, non-Hermitian systems can be classified into 43 categories Bernard (), which is more complicate than the Hermitian systems. Very recently, Gong et. al. have classified non-Hermitian phases for 10 of the 43 categories by considering T, C and S symmetries Gong (), which can be viewed as an extension of topological classification of the non-Hermitian counterparts of AZ classes. Inspired by the periodic classification table for Hermitian topological phases protected by the reflection symmetry Furusaki (); Chiu (), in this work we study the topological classification of non-Hermitian systems with reflection symmetry. Our classification is based on K theory and Clifford algebras Kitaev2 () and includes two steps. The first step is Hermitianization of the non-Hermitian Hamiltonian, and the second step is to represent the classifying space as Clifford algebra extension. After the two steps, we get a richer classification with periodic classification tables for the complex and real classes shown in table II and III, respectively. Then we construct the topological invariant for each of the one-dimensional (1D) non-trivial classes and discuss the bulk-edge correspondence for various models.
The paper is organized as follows. In section II, we first give a general description of classification of non-Hermitian system in the AZ classes with additional reflection symmetry and then give the classification tables. In section III, focusing on the one-dimensional non-Hermitian systems, we consider all topologically nontrivial examples according to our classification tables and construct topological invariants for all example Hamiltonians. In Section IV, we study the bulk-edge correspondence by considering several typical example systems. A summary is given in the last section.
Ii Classification of non-Hermitian system with reflection symmetry
The topological classification of non-Hermitian systems with time reverse symmetry , pseudo particle-hole symmetry and chiral symmetry was recently studied by Gong et.al. Gong () For a given AZ class which fulfills or symmetry, its Hamiltonian fulfills
where and is an anti-unitary operator with . We can always represent the anti-unitary operator as with being an unitary matrix and being a complex conjugate operator. The operator can represent and with and , respectively. For a Hamiltonian with the constraint , corresponding to having no zero eigen energy, one can always make a polar decomposition
where is a reversible matrix, is a positive-definite Hermitian matrix and is an unitary matrix. It was proved that can continuously transform to under , or symmetry without the change of topological properties Gong (). So classifying equals to classifying the . To classify the non-Hermitian Hamiltonian, it is convenient to introduce an enlarged Hamiltonian
It is obvious that is a Hermitian matrix, which fulfills a chiral symmetry
where and . Now the problem transforms to classify a Hermitian Hamiltonian with an addition symmetry . Under such a scheme, the classification has been given in Ref. Gong (). For convenience, we also list the results here (table I).
|Cartan class||Classifying Space||1||2||3||4||5||6||7|
When the system has a reflection symmetry, we have
where the reflection operator fulfills . By using the polar decomposition , we can prove that can continuously transform to under the reflection symmetry (see the appendix A), and thus classifying H(k) equals to classifying U(k). Similarly, we can define an enlarged Hermitian Hamiltonian defined by Eq.(3). Besides the chiral symmetry given by Eq.(4), also fulfills the following symmetries:
Here can represent or , i.e., and . Classifying U(k) equals to classifying Hermitian Hamiltonian with two addition symmetry and . Since is Hermitian, we transform the problem into a classification problem of Hermitian Hamiltonian.
where denotes the momentum in the th direction and the gamma matrices obey anticommunication relations
where J=i is a complex structure. It follows that and satisfy
Classifying U(k) equals to classifying Hermitian Hamiltonian with two addition symmetry and . Since is Hermitian, we transform the problem into a classification problem of Hermitian Hamiltonian.
|Cartan class||Classifying Space||1||2||3||4||5||6||7|
Topological classification of AZ classes with additional symmetries has been studied in Furusaki (); Chiu (). Considering a Hermitian Hamiltonian, if we add multiple additional symmetries to the classification (), and the additional symmetries anticommute with each other , we have the following conclusions by Clifford algebras and their extensions Furusaki (): For Class A, if the number of additional symmetries is , the classifying space shifts to ; For Class AIII, if additional symmetries and chiral symmetry have the following relations: for and for , the classifying space shifts to . With the help of the above conclusions, we can get the non-Hermitian classification with reflection symmetry for the complex classes as shown in Table II:
Class A: Two symmetries and are added to the system (), and thus the classifying space shifts to . Consequently, the non-Hermitian system of Class A with reflection symmetry is characterized by for even (odd) d.
Class AIII() and AIII(): Here represents that with . Adding two symmetries and to the system of the , while always commutes with , anti-commutes with for Class AIII() or commutes with for Class AIII() due to anticommutes with both and . It follows that for the Class AIII(), and the classifying space shifts to , suggesting that the system is characterized by for even (odd) d. On the other hand, we have for the Class AIII(), and thus the classifying space shifts to , suggesting that the system is characterized by for even (odd) d.
|Cartan class||Classifying Space||1||2||3||4||5||6||7|
Next we consider the classification of the real classes. For the Hermitian Hamiltonian with multiple additional symmetries , where and , following Ref.Furusaki (), we know conclusions for the classifying space for the Class AI and AII ( only) or C and D ( only): Given that or , we denote the relations as or , and the number of and as and . For class AI and AII, the classifying space shifts to . For class C and D, the classifying space shifts to . Taking advantage of the above conclusions, we can get the classifying space for the corresponding non-Hermitian systems with reflection symmetry. In the following discussion, for convenience, we shall used , and to represent , and . Because () commute with both and in each class, we don’t need to discuss their commutation relations separately in each class. For class AI, AII, C and D, represent that ().
Class AI(): Two symmetries and are added to the system of . Since anticommutes with , and , anticommutes with . It follows that , and . The classifying space is still .
Class AI(): Since anticommutes with and , commutes with . Thus we have , and . The classifying space shifts to .
Class AII(): Since anticommutes with , and , anticommutes with . Thus we have , and . The classifying space is still .
Class AII(): Since anticommutes with and , commutes with . Thus we have , and . The classifying space shifts to .
Class C(): Since anticommutes with and , commutes with . It follows that , and . The classifying space shifts to .
Class C(): Since anticommutes with , anticommutes with . Thus we have , and . The classifying space is still .
Class D(): Since anticommutes with and , commutes with . It follows , and . The classifying space shifts to .
Class D(): Since anticommutes with , anticommutes with . Thus we have , and . The classifying space is still .
Once the classifying space is known, for example given by , we can get that the system is characterized by . The classification results are shown in table III. Similarly, we can analyze classes BDI, DIII, CI and CII with reflection symmetry and get the classifying space for these non-Hermitian systems (see the appendix B). The classification results are also listed in table III.
Now we discuss some examples of non-Hermitian topological phases protected by reflection symmetry. Particularly, we confine our study on 1D systems and give a complete list of all topologically nontrivial types in one dimension. We also construct explicitly topology invariants for our example Hamiltonians. For convenience, we use to represent in this section.
1. Class AIII (). For this class, the Hamiltonian should satisfy that and , and we have . Consider a 2-band Hamiltonian having the following form:
It is easy to check that the chiral and reflection operators are given by and , respectively. Taking
we get the non-Hermitian Hamiltonian
From the table II, we know that the topological phase of the -type system can be characterized by a winding number. Define the topological invariant
It is easy to check that for , , and , and for , , and , which indicates the system in different topological phases.
2. Class AI(). According to the definition, the system has time-reversal and reflection symmetry. Consider a two-band Hamiltonian given by
with . It is easy to check that the Hamiltonian fulfills and , where , and . We also have
which suggests that the eigenvalues are either real or complex with conjugate pairs. It follows that is real, and we can define the topological invariant as
For the example Hamiltonian:
Consider the case with . If , we have . If , then . The topological invariant or characterizes topologically different phases.
3. Class D(). Consider the Hamiltonian given by
The Hamiltonian should fulfill the pseudo particle-hole and reflection symmetry, i.e., and . Taking and , the symmetries enforces . By observing
we see the eigen energies appearing in pairs with the form of and , where and are real. Since is real, similarly we can define topological invariant as . An example Hamiltonian is given by
It is straightforward to get
Consider the case with . If , we have . If , then .
4. Class BDI(). For all classes labeled by which shall be discussed in the following text, the Hamiltonian satisfies that , , , and . Consider the Hamiltonian
It is easy check that the chiral and reflection are fulfilled by taking and . Taking and , we see that the T and C symmetry are fulfilled if . The additional constraint means that the two bands is a Hermitian Hamiltonian, which suggests the two band BDI() class must be a Hermitian system. A well known example is the SSH model described by
for which the topological phases can be characterized by the winding number .
5. Class BDI(). For the Hamiltonian given by
with , it is easy to check all symmetries are fulfilled with , , and . We also have and . The topological phase can be characterized by . For the example Hamiltonian
it follows that when and , and when and .
6. Class CI(). For the Hamiltonian given by
we have , and . It is straightforward to check that the reflection operator is given by , fulfilling that , and . Then we get , which leads to , suggesting that is a real matrix. According to the classification table, the topological phase should be characterized by a number, which can be defined as . For the example Hamiltonian with given by
Considering the case with , we have for and for .
7. Class DIII(). Consider the Hamiltonian given by
It is not hard to find the T, C, and S symmetries are fulfilled by taking , and . Furthermore, we find that the reflection operator is given by , satisfying , and . It follows , which leads to , i.e., is real matrix. According to the classification table, the topological phase is characterized by a topological invariant, which is defined by . For the example Hamiltonian
Considering the case with , we have for and for .
8. Class BDI(). For the Hamiltonian given by
with , it is straightforward to check the T, C and S symmetries are fulfilled if we take , and . Introducing the reflection operator as , we see that and . The reflection symmetry requires , which leads to . It suggests that is a reflection operator for and is the time reversal operator. Then both and belong to the class AI(), and each of them is characterized by a topological invariant. Correspondingly, is characterized by a topological invariant. Consider the example Hamiltonian
Consider the case with . If , we have . If , then (). So the topological phase classified by a topological invariant.
9. Class CII(). Considering the Hamiltonian given by Eq.(19) and taking , and , we find that T, C and S symmetries are fulfilled if there exists an additional constraint:
The above constraint condition suggests that the Hamiltonian and fulfill the time reverse symmetry separately. If we do not consider the reflection symmetry, belong to the class AII with a topological invariant Gong (), and consequently is characterized by a invariant. Adding reflection symmetry with the reflection operator given by , we can check and . The reflection symmetry leads to , suggesting that only one of is independent. So the topological phase of is characterized by a topological invariant. For the example Hamiltonian
we can define the topological invariant as
It is straightforward to get for and for (we requires that ).
10. Class CII(). Similar to the Class CII(), for the Hamiltonian given by Eq.(19), we have , and , and the Hamiltonian satisfies T, C and S symmetries if the constrain condition Eq.(20) is fulfilled. Introducing the reflection operator , we have and . The reflection symmetry leads to , and thus only one of is independent. Similarly, the topological phase is also characterized by a topological invariant. Consider the example Hamiltonian
Iv Bulk-edge correspondence
In the previous section, we construct model Hamiltonians in momentum space for all 1D non-trivial classes and give the definitions of corresponding topological numbers. In this section we discuss the bulk-edge correspondence by studying several examples of non-trivial classes. To study the bulk-edge correspondence, we need consider the model in the coordinate space with open boundary condition (OBC).
iv.1 Class AIII()
Consider a lattice version of the model (11) described by the Hamiltonian:
where with and , which is identical to Eq.(11). The eigenvalue of the Hamiltonian is given by
and the gap closes at . The topological invariant is
We show the topological invariant versus in Fig.1a, which indicates a topological transition from the region of (with ) to of (with ). In Fig1b, we display the spectrum of the system under the OBC versus . In the topologically non-trivial phase, we find that the system has double degenerate zero mode edge states as shown in Fig1c, whereas no zero mode edge state exists in the topologically trivial regime. It is clear that there is a bulk-edge correspondence in this class. In Fig1d, we also display the mean inverse partition ration (MIPR), which is the average of inverse partition ratios () of all eigen states. For a given state, its is defined as
where is the wave function’s amplitude at the site i. If the state is an extended state, its shall tend to zero as the lattice size increases to the infinity limit. On the other hand, the IPR for a localized state remains to be finite even in the infinite size limit. We have checked that the increase of MIPR in the region is coming from the contribution of IPRs of zero mode edge states and no skin effect is found for other bulk states.
iv.2 Class CII() and Class CII()
Consider the lattice model described by the Hamiltonian:
In the momentum space, the Hamiltonian can be represented as