The study of surface states in a semi-infinite crystal

An infinite three dimensional (3D) crystal can be constructed by an infinite number of parallel 2D (hkl) crystal planes (CPs) coupled to each other. Based on lattice model Hamiltonian with the hopping between the nearest neighbor (1NN) CPs and all possible neighbor hoppings within each CP, we analytically prove that a (hkl) cut crystal will not accommodate any surface states if the original infinite crystal has the reflection symmetry which results in the forward transfer matrix F to be equal to the backward one B, named as F-B dynamical symmetry. We also study the effect of the longer range couplings among the nNN (n > 1) CPs and surface relaxation on our conclusion and find that the small perturbation from both factors has no effect on our conclusion based on the perturbation theory. Thus our model may have the potential for studying surface states in some cut crystals with low-index surfaces. Our result may be helpful to visually predict which cutting direction in some non-topological crystals is unfavorable to generate surface states.

As is well known, the strong 2NN coupling is favorable to the existence of surface states. However, for some crystal families such as some conventional insulators or semiconductors with negligible SOC as well as semi-metal (i.e., graphene), the 1NN coupling can be dominant if the Miller index (hkl) of the cutting surface is low. But the 2NN and even much longer range couplings can be compatible with (even larger than) the 1NN one for some materials with highindex (hkl) surfaces. In this paper, we will prove a conclusion (''theorem'') for the model with the hopping between the 1NN CPs. However, the nNN (n . 1) hopping among CPs is able to result in the presence of surface states, which we will discuss in the later section, and the surface decoration may be a good way to manipulate surface states. Based on the perturbation theory, we further demonstrate that the weak longer range hopping and small surface relaxation have no effect on our conclusion.

Results
Model Hamiltonian. In general, an infinite 3D crystal can be described by an infinite number of parallel 2D CPs which are periodically arranged one by one with coupling. The direction of CPs can be denoted by Miller index (hkl) where h, k and l can be arbitrary integers. A semi-infinite crystal with the (hkl) cutting surface is called as the (hkl) cut crystal. Firstly we only take into account the model with the hopping between the 1NN CPs, where all possible neighbor hoppings within each CP are included. It would be a reasonable model to some crystals with low-index surfaces where the coupling between the 1NN CPs can be dominant and the couplings among the nNN (n . 1) CPs are much weaker than that between the 1NN CPs. For a semi-infinite crystal, its each CP is still a crystal with the lower dimensionality. Thus, the Fourier transformation can be applied to each CP since the wave vector I k jj within each CP is a good quantum number. Taking the diagonal representation of the Hamiltonian for each CP, the effective Hamiltonian can be described as follows: where Y T i~( y (1) i ( I k jj ), y (2) i ( I k jj ), Á Á Á , y (ni) i ( I k jj )) and fy (a) i ( I k jj ) : a1 ,2, Á Á Á ,n i g is the second quantized Fermionic wave function of the a th electron mode in the i th crystal plane. F ni|nj (B ni|nj ) represents the n i 3 n j forward (backward) hopping matrix from the plane P i to its 1NN CP P i11 (P i21 ) and (n i ,n j ) can be any finite positive integers.
jj )g and the boundary condition is {Y i 5 0 : i , 1} From now on, we omit the symbol I k jj for simplicity. The above Hamiltonian is general and each unit cell of every CP contains many atoms (and maybe different) and each atom can also contribute many different atomic orbits. Thus the model Hamiltonian is reliable for some crystals with low-index surfaces. The conclusion is phrased as follows: Conclusion (Theorem). Based on the above model Hamiltonian with the hopping between the 1NN CPs and all possible neighbor hoppings within each CP, any low index (hkl) cut crystal with negligible SOC will not allow any surface states if the original infinite crystal has the reflection symmetry for every (hkl) CP.
The above conclusion also covers the case of 2D/1D crystals, in which the ''surface'' represents the atomic chain/point. In our following demonstration, the transfer matrix approach 30,31 is applied. The crystals with the reflection symmetry are only one of two types: Type I: '' ... -P-P-P-P-... '' in Fig. 1a and Type II: '' ... -P-Q-P-Q-... '' in Fig. 1b where P and Q represent CPs. The same P (Q) represents exactly the same CP while Q ? P means that P and Q are different CPs. The bar ''-'' roughly describes the distance between the 1NN CPs. The same ''-'' means the same distance. Since Type II can be transformed into Type I by simple mathematical calculations, thus we at first concentrate our attention on the proof of Type I and then turn back to Type II.
Before presenting the proof of the above conclusion, we provide a definition of ''surface state'' at first. Surface states such that they propagate along the direction of the boundary surface and their amplitudes decay exponentially in distance normal to the boundary surface. In terms of the transfer matrix language, the surface state is defined by the following decay relation: where b is a decay rate and a is the distance between the 1NN CPs. When c 5 1 (b 5 0), it corresponds to the extended mode and must associate with the bulk state. Furthermore, since our model Hamiltonian is general, it also includes some particular case that can consist of two decoupled sub-lattices A and B. Then Hamiltonian H can be decomposed into two decoupled parts: H 5 H A 1 H B Meanwhile we also assume that the sub-lattice B is not only decoupled from A H AB 5 0 but it also has no coupling among CPs. Then we have Thus fY i,B ( I k jj ) : i §1g are localized in the i th CP and have no propagation among CPs. Such states, even like fy : iw1g, should be excluded from surface states. Here we focus on surface states such that their amplitudes decay exponentially in distance from the boundary plane.
Proof for Type I. For the simplest case, each CP has only single electron mode that corresponds to one atomic orbit per unit cell. At the level of the 1NN hopping approximation, the study of surface states in this case is exactly the same as that of edge states in the semiinfinite 1D single orbit atomic chain. As is well known, no edge states exist in the semi-infinite 1D atomic chain for both Type I and II when the forward hopping constant equals to the backward one 31 . Thus, we will take into account the case that each P contains n (.1) electron modes. For a cut crystal ''P-P-P-P-... '' in Fig. 1a, it is not difficult to obtain following quantum dynamical equations (QDEs): where E n3n 5 diag{E 1 , E 2 , ... , E n } and {E a 5 E 2 v a : a 5 1,2, ... , n} E is The matrices F n 3n and E n 3n are hermitian with the dimensionality n. Here we adopt the dimensional reduction method to reduce the dimensionality n in Eq.(5) to 1. We will prove that no surface waves accommodate in such a cut crystal for any energy E. Since we are not sure that E m 3m and F m 3m (m , n) are always hermitian in the dimensional reduction process, here what we assume is that E n 3n and F n 3n are arbitrary square matrices and are not limited to be hermitian from the beginning so that the following demonstration can be used repeatly in the dimensional reduction. But it is still applicable to the hermitian F n 3n and E n 3n .
Proof for n . 2 in Type I. By means of the dimensional reduction method, we will reduce the dimensionality n in Eq.(5) into 1 or 2. Let us first consider an energy such that E: det(E n 3n ) ? 0. Since det (E n3n ) ? 0, we can obtain from Eq.(5) In the matrix theory, it is known that a square matrix (E {1 n|n F n|n ) can be decomposed into a Jordan matrix via a similarity transformation

the Jordan canonical block and its form is
where the parameter c i 5 0 or c i ? 0 depends on whether the matrix In terms of the property of the Jordan matrix J n3n and from Eq.(8), we can reach immediately for where l s describes the effective coupling of the n th eigen mode between the 1NN CPs. Eq.(9) is exactly the same as the transfer matrix equation of 1D atomic chain with the single electron mode. It has been known that there are no edge states for any energy E no matter whether l s 5 0 or l s ? 0 31 . Thus we arrive at fy (n) 0 i~0 : i §1g for surface states. After back-substituting fy (n) 0 i~0 : i §1g into Eq.(8), we find that surface states are also impermissible for the (n 2 1) th mode, yielding fy (n{1) 0 i~0 : i §1g: After step by step, we obtain fY 0 i~0 : i §1g for surface states that result in fY i~Un|n Y 0 i~0 : i §1g: Hence no surface states are allowed for det(E n3n ) ? 0.
Next let us think over some energy such that E: det(E n 3n ) 5 0. Now we apply a Jordan transformation V n 3n to the matrix E n|n : where we have arranged such that the sub-matrix J 1 contains l 1 5 0. Without loss of generality, here we can assume the first block is a two-order Jordan sub-matrix at first. For other cases where the order of J 1 (l 1 5 0)is one or greater than two, we can do similar demonstrations as we do for a two-order Jordan block. The derivation can proceed by considering two scenarios: We can obtain from the first row of Eq.(10) Substituting Eq. (11) into Eq. (10), we can arrive at www.nature.com/scientificreports : a,b~f2,3, Á Á Á ,ngg. Thus, we have reduced the dimensionality n in Eq.(10) into n21.
2) Next suppose F 0 11~0 . Now we focus on the first column matrix elements of the matrix F 0 n|n . If they are all zero, the reduction of the dimensionality in Eq.(10) is already reached. Thus what we assume here is that there exists some b such that F 0 b1 =0 and b ? 1 then we have After substituting Eq.(13) into Eq.(10), we can get H (n21)3 (n21) is zero, we will continue to reduce the dimensionality n21 in Eq.(12) (Eq. (14)) to n22 by means of the similar steps from Eq.(10) to Eq. (14). If necessary, we can do more reductions similar to above and eventually reduce the dimensionality in Eq.(10) to 1 or 2. Meanwhile, we can see that other modes fDy (l) i : l~2,3, Á Á Á ,n(l~3,4, Á Á Á ,n)g are either the linear combinations of fy i : j~1 (j~1,2)g or can be decoupled as local modes when the dimensionality in Eq.(10) is reduced to 1 (2). No surface states exist for the dimensionality 1 (as well known) when the forward hopping constant is equal to the backward one, neither for the dimensionality 2, as will be proved in the following.
Proof for n 5 2 in Type I. When n 5 2, Eq.(5) is rewritten as where Y T i~( y (1) i ,y (2) i ) and assuming E 232 and F 232 are general matrices in order to cover the previous case where the dimensionality in Eq.(10) is reduced to 2 when n . 2 and det (E n 3n ) 5 0. To ensure the validity of the proof for any energy E and any crystal structures, we must discuss all possible matrix structures of E 232 and F 232. At first, note that when det(E 232 ) ? 0 or det(F 232 ) ? 0 we obtain fy (1) i~y (2) i~0 : i §1g for surface states by the use of the similar steps from Eq.(6) to Eq.(9). Next, think over the special case where det(E 232 ) 5 0 and det(F 232 ) 5 0. We apply a Jordan similarity transformation U 232 for E 232 then Eq.(15) can be written as where Proof for Type II. In Type II, the crystal has two different CPs: P and Q. We just discuss the Q cut crystal ''Q-P-Q-P-... '' in Fig. 1b since the discussion for the P cut crystal will be similar. Now the QDEs for the Q cut crystal are where {W i 5 0 m31 ,Y i 5 0 n31 : i , 1} and the CP P has n modes and Q has m ones. n and m can be equal or unequal. fE (P) n|n , E (Q) m|m g are defined as E (a) la|la~d iagfE a 1 ,E a 2 , Á Á Á ,E (a) la g, fE a i~E {v a i : i~1,2, Á Á Á , l a g and l p(Q) 5 n(m) when a 5 P(Q). After simple calculations, Eq. (19) can be written as where W 0 i~W i z W iz1 and DP i 5 P i11 1 P i21 . Now the Q cut crystal structure ''Q-P-Q-P-... '' in Type II is equivalent to ''P9-P9-P9-... '' in Type I with the dimensionality m 1 n. We can find fP i~0 ,W 0 i~0 ,Y i~0 : i §1g for surface waves. fW 0 i~0 : i §1g yield W i 1 W i11 5 0 : i $ 1 that further lead to {W i 5 0 : i $ 1} for surface states. Hence the conclusion is also valid for Type II. So far, we have completed the proof of the conclusion for Type I and II. From the above demonstration, we clearly know that F n3n 5 B n3n is the key to the conclusion. Our conclusion is only valid at the level of the hopping approximation between the 1NN CPs. In some real materials, although the longer range hoppings among the nNN (n . 1) CPs may be weak for some materials with low-index surfaces, they always exist. Meanwhile, the surface of real cut materials is imperfect and the surface relaxation is always unavoidable near the surface, even surface reconstruction. Thus here we study the effect of both factors on our conclusion.
Effects of the longer range hopping and surface relaxation. At first, we focus on the effect of the longer range hoppings among the nNN (n . 1) CPs in the absence of surface relaxations. When they are much weaker than that between the 1NN CPs for the crystals with low-index surfaces, their effect can be estimated in terms of the perturbation theory. In the case, QDEs for a cut crystal with Type I can be written as where {Y i 5 0 : i , 0} and l characterizes the order of the small perturbation. The boundary surface is set at the zeroth layer. The terms fF a ð Þ n|n : a~2,3, Á Á Á ,mg result from the longer range hopping among CPs. F a ð Þ n|n represents the n 3 n hopping matrix from one CP to its a th neighbor CP. According to the gradual development spirit of the perturbation theory, we can set Inputting Eq.(22) into Eq. (21), we obtain where {b, c 5 0, 1, 2 }. From Eq.(23), we can obtain fY (0) i~0 : i §0g for surface states from the above discussion. After back-substituting As demonstrated above, we can reach fY (1) i~0 : i §0g. After step by step, we also can get fY (l) i~0 : i §0, l~2,3, Á Á Ág and further have {Y i 5 0 : i $ 0} for surface states. Thus, surface states in crystals with the reflection symmetry are not allowable when they are much weaker than that between the 1NN CPs. When l becomes large, the perturbation theory will be invalid and as a result surface states can emerge even if the crystal has the reflection symmetry (see the example of armchair edged graphene shown in the following section). In practice, the longer range hoppings within the CP may be large, but they are usually small among CPs for some crystals with low-index surfaces and can be regarded as a perturbation.
It is similar to consider small surface relaxations. Here we assume there are surface relaxations from 0 th to (j 2 1) th layer CP and energies of eigen modes renormalized at each CP are unchanged at the level of the hopping approximation between the 1NN CPs, then we obtain QDEs where L (i,k) n|n is from the small surface relaxation between the 1NN CPs and non-zero matrix when 0 # i # j. We input Eq.(22) into Eq.(24) and receive where {b 5 0, 1, 2, … } and fY c ð Þ k~0 : cv0g. As it has been done in the absence of surface relaxations, we can obtain fY (l) i~0 : i §0, l~0, 1, 2, Á Á Ág step by step. Thus, the small surface relaxation has no impact on our conclusion. So far, it is known that the weak longer range hopping and small surface relaxation cannot influence our conclusion, but they have the effect on the bulk states.

Discussion
In application of the conclusion, we can give some examples to support our demonstration and see its availability for qualitatively predicting the presence or absence of surface states in some materials. Firstly, we can easily check armchair edged graphene with the 1NN hopping t 1 only has no edge states since it has ''P-P-P-P-... '' type structure, consistent with the previous theoretical analysis 32 . We also study the effect of the 2NN hopping t 2 on surface states in terms of the exact diagonalization method where t 2 is taken in the region 0.02t 1 # t 2 # 0.2t 1 33,34 and find that surface states can appear for the ratio t 2 =t 1 [ 0:16, 0:18, 0:2 ð Þin Fig. 2a-c, but not for t 2 =t 1 [ 0:02, 0:04, 0:06 ð Þ in Fig. 2d-f. Thus the longer range hopping among CPs is able to induce surface states if it is strong enough and not regarded as a perturbation, compatible with our demonstration. Furthermore, the type structure of zigzag edged graphene is ''P 5 P-P 5 P-... '' where the F-B symmetry is broken, thus it is in favor of the existence of edge states, shown in the zero mode result 31 . Secondly, let us further address ABO 3 Perovskites since they are very much important in device applications. Ab initio calculations of ABO 3 Perovskite (001) surfaces have shown no surface states 24,25 . The SOC and/or correlation effect in these materials do not play an important role and as a result they are suitable for our model. From the structure symmetry analysis of ABO 3 Perovskites such as PbTiO 3 , we predict that the semi-infinite c-cut ABO 3 in the para- electric phase have no surface states at the level of the 1NN hopping approximation since its type structure is ''P-Q-P-Q-... '' in Fig. 3a, consistent with ab initio calculations 24,25 . But we can find that surface states may appear in the above c-cut ABO 3 with the polarization along the c-axis since it breaks the F-B symmetry in Fig. 3b. Another example is the c-cut ferroelectric YMnO 3 in Fig. 3c which has the hexagonal structure. It has surface states due to the F-B asymmetry, compatible with the previous study 21 .
In conclusion, based on lattice model Hamiltonian with the hopping between the 1NN CPs and all possible neighbor hoppings within each CP, we have proved that there will not be surface states in a (hkl) cut crystal if the original infinite crystal has the reflection symmetry about each (hkl) CP. Meanwhile we also consider the effect of the longer range hoppings among the nNN (n . 1) CPs and surface relaxations on our conclusion. For some types of crystals (not like ionic crystals) with low-index surfaces, the longer range hoppings are weak enough and can be regarded as a perturbation, then they have no effect on our conclusion. It is also shown that small surface relaxations have no impact on our conclusion. In fact, the F-B dynamical symmetry (F n3n 5 B n3n ) is the key in our demonstration. For the crystals without SOC and/or strong correlation effect, we find that different cutting surfaces of the same crystals may have the different behavior for the existence of surface states. Moreover, our proof can be extended to F n3n 5 e id B n3n where d is a k-dependent or zero. Our model in the above demonstration, in some sense, is much closer to real materials than previous simple ones. Thus our conclusion may be helpful to visually predict which cutting direction of the crystals is unfavorable for generating surface states in future research. Finally, we would also mention that our model is limited. In practice, surface states can exist on low index metal surfaces of FCC and BCC elemental metals such as Ag, Nb and Fe where SOC and/or strong correlation effect are/is dominant [35][36][37] . Thus our model is invalid for the crystal with strong SOC or/and strong correlation effect, such as TI and TSC. Although our conclusion may have the potential for predicting the absence of surface states in some cut crystals, a criterion for the presence of surface states still remains to be investigated.

Methods
We started from a lattice model Hamiltonian at the level of effective single-particle approximation and analytically proved that there should not be surface states in a cut crystal with F-B symmetry. The Jordan matrix property and transfer matrix approach were also used to reduce the dimensionality of QDEs. Furthermore, we also extend our method to present the more general argument for the effect of the coexistence of the weak longer range hopping and small surface relaxation. What we assume is that there exist small relaxations from 0 th to (j 2 1) th layer CP and the weak longer range hoppings among the nNN CPs (2 # n # m) and energies of eigen modes renormalized at each CP are slightly changed due to atomic relative movement at each CP, then we can obtain QDEs n|n Y (c{a) iza g, i:e:, l 0 : E n|n Y (0) i~F (1) where fY (c) i~0 : cv0g. As demonstrated in the absence of both factors, we can arrive at fY (l) i~0 : i §0,l~0,1,2, Á Á Ág step by step and find that the small perturbation from both factors has no impact on our conclusion.