Huiping Wang(王會(huì)平) Li Ren(任莉) Liguo Qin(秦立國(guó)) and Yueyin Qiu(邱岳寅)

1School of Mathematics,Physics and Statistics,Shanghai University of Engineering Science,Shanghai 201620,China

2School of Science,Chongqing University of Posts and Telecommunications,Chongqing 400065,China

Keywords: edge states,transfer matrix method,long-range hopping

1. Introduction

There is ongoing interest in understanding the effects of electronic correlations on topological and nontopological quantum states due to the promise of uncovering new phenomena in condensed matter physics.Among such topological states of matter,topological superconductors[1,2]offer promising perspectives for realizing robust quantum devices[3–5]due to the presence of topologically protected edge states. In addition,much attention has been paid to nontopological states in ferroelectricABO3crystals[6]and some semiconductors (i.e.,Si, Ge)[7–9]owing to their potential applications. Quantum states are mainly governed by Hamiltonians with short-range terms. However, the addition of LR hopping amplitudes is important, because they are ubiquitous in real systems and their role can be crucial.[10–12]In recent years, LR classical and quantum systems have become a central focus for physical investigation. The existing theoretical works have reported that LR systems can present many interesting and unusual phenomena,essentially owing to the violation of locality.[13–19]In particular, some LR models can present new phases, manifesting striking properties which are absent in the short-range limit.[20–32]

Motivated by these ideas, we study the behavior of edge states in the presence of the LR hopping in LR interacting systems. For most of the conventional crystals with lowindex surfaces, the short-range terms will be dominant and the LR ones are relatively weak, which can be ignored. Previous works[33–37]have investigated the existence/absence of edge states based on some models with many(a few)electron modes per unit cell and with the short-range terms. However,for some strongly correlated materials and some possible naturally stable crystals with high-index surfaces and/or the strong spin-orbit coupling, the LR coupling cannot be ignored. It is natural to ask how LR terms affect edge states. With this motivation,the question we want to answer in this manuscript is what is the condition of the presence of edge states in the systems with LR hopping. As we know,an infiniten-dimensional crystal is made up of an infinite number of parallel (n ?1)-dimensional crystals which are periodically arrayed by coupling. For the sake of discussion,here we take a 2D crystal as an example.The(n?1)-dimensional crystal represents a crystal line (CL) whennis equal to 2. Here we mainly focus on such model crystals with three different type structures (type I:···–P–P–P–P–···, type II:···–P–Q–P–Q–···, and type III:···–P=P–P=P–···where P and Q denote CLs, and the signs–and=mark the distance between the nearest neighbor(NN)CLs).

In this work, we perform an analytical study of the behavior of edge states in LR interacting systems when LR hopping terms are considered. Our analysis includes LR coupling among CLs and all possible neighbor hoppings within each CL. For such model crystals with three-type structures, we determine analytically the condition of the existence of edge states in the presence of LR terms. Our findings are supported by numerical calculations for specific parameter choices. The conclusion that edge states can survive in such model systems with LR hopping terms derives from the transfer matrix in the bulk different from that in the boundary layers.

Before presenting the following demonstration,we give a definition of“edge state”firstly. Edge states can be described such that they can propagate along the direction of the cut boundary and their amplitudes decay exponentially in distance normal to the cut boundary. With the aid of the transfer matrix language,an edge state has the following decay relation:

whereγis a decay rate andais the distance between the 1NN CLs. Whenδ=1(γ=0),it corresponds to the extended state and must be associated with the bulk state.

2. Edge states in type I

In this section, we focus on such a semi-infinite crystal with type I P–P–P–P–···. For each CL, it is a lower dimensional crystal than the original one. The Fourier transformation can be used for each CL because the wave vectork||along the boundary is a good quantum number. When we consider LR coupling and utilize the diagonal representation of each CL Hamiltonian, our effective model Hamiltonian can be written as

whereEm×m=E ?εm×mandEis the eigen energy of electron waves propagating in cut crystals. In Ref.[38],it has also been shown that edge states are absent in the presence of LR hopping when only one electron mode near the Fermi surface is important in the cut crystal with type I, resulting from the transfer matrix in the bulk identical to that in the boundary layers. Consequently, an interesting question is as follows: Are there edge states whenm ≥2 and do LR terms exist? If yes,what could the condition for the existence of edge states be?

Fig. 1. The schematic illustration of the lattice structure of a semiinfinite 2D crystal in(a). The dispersion relation of edge states reached based on the exact diagonalization method(blue dashed line)and TMM(red dashed line)in(b)with t′1=t′′1 =0.9t1,t2=0.7t1,t3=t′3=0.4t1,t4=t′4=t′′4 =0,t5=t′5=0,εA=?0.6t1 and εB=0.9t1.

3. Edge states in type II

In this section,we analyze such model crystals with type II,consisting of two different CLs: P and Q.Here we are only looking to the Q cut crystal Q–P–Q–P–···, since the discussion for thePcut crystal will be similar. When the LR coupling is taken into account,QDEs for theQcut crystal are

Fig. 2. The schematic illustration of the lattice structure of a semiinfinite 2D crystal in(a).The energy spectrum of edge states obtained in terms of the exact diagonalization method(blue dashed line)and TMM(red dashed line)in(b). The parameters are t2=t′2=0.4t1,t3=t′3=0,t4=0.2t1,εA=?1.0t1 and εB=1.5t1.

Here, owing to its infinity iny-direction, the Fourier transformation can be applied to wave functions forA(B)sublattices. When we investigate the hopping from 1NN to 3NN CLs, one can adopt the{n,ky}representation and introduce QDEs based on a set of Fermion operators{ψn,A(ky),ψn+1,B(ky):n=1,3,5,...,∞}

Now we focus on the general case in type II wheremandlare finite positive integers and LR couplings are introduced.Here we are still talking about electron modes within the CL Q because the similar result for the CLPcan be obtained. Equation (8) can be rewritten as the following matrix form after some derivation:

4. Edge states in type III

In the section, we investigate such cut crystals with type III in detail. Since the original infinite crystals with type III have no reflection symmetry,the forward transfer matrix fromnNN(n=2i?1:i=1,2,...)CLs is not equal to the backward one,and QDEs read

Fig. 3. The schematic illustration of the lattice structure of a semiinfinite 2D crystal in(a)and the dispersion relation of edge states shown via the exact diagonalization method(blue dashed line)and TMM(red dashed line) in (b) with t1 =0.7t, t2 =0.4, t3 =0.0, t4 =0.15,=0.18and εφ =0.

5. Conclusion and perspectives

In conclusion, based on the lattice model Hamiltonian with the LR coupling among CLs and all possible neighbor hoppings within each CL,we have analytically determined the condition of the appearance of edge states in three model crystals. The expressions we obtain are general and hold for any choice of the LR hopping. In fact, the key in our demonstration is that the transfer matrix in the bulk is different from that in the boundary layers. It is this reason so that edge states can arise in such cut crystals. The study of edge states has become a central focus for physical investigation in recent years,such as extended Kitaev chain and Su–Schrieffer–Heeger chain. Our approach could be generalized to oneand three-dimensional systems.Meanwhile our demonstration can also apply to other model crystals, such as···–P=P≡P–P=P≡P–···and···–Q=P–Q=P–···. We believe that this work can provide new proposals to solve novel edge states in extended Kitaev chain and Su–Schrieffer–Heeger chain. From a more general perspective, our study highlights the effects of the LR coupling in strongly correlated systems.

For(2)electron mode,S(2)has the similar matrix form toS(1),not given in detail.