A topological look at the quantum spin Hall state
We propose a topological understanding of the quantum spin Hall state without considering any symmetries, and it follows from the gauge invariance that either the energy gap or the spin spectrum gap needs to close on the system edges, the former scenario generally resulting in counterpropagating gapless edge states. Based upon the Kane-Mele model with a uniform exchange field and a sublattice staggered confining potential near the sample boundaries, we demonstrate the existence of such gapless edge states and their robust properties in the presence of impurities. These gapless edge states are protected by the band topology alone, rather than any symmetries.
pacs:72.25.-b, 73.20.At, 73.22.-f, 73.43.-f
Since the remarkable discovery of the quantum Hall effect (QHE) QHE (), the study of edge state physics has attracted much attention on both theoretical and experimental sides. Recently, a new class of topological states of matter has emerged, called the quantum spin Hall (QSH) states Kane1 (); Bernevig (). A QSH state of matter has a bulk energy gap separating the valence and conduction bands and a pair of spin-filtered gapless edge states on the boundary. The QSH effect was first predicted in two-dimensional (2D) models Kane1 (); Bernevig (), and was experimentally confirmed soon after in mercury telluride quantum wells Konig (). The QSH systems are 2D topological insulators TI1 (); TI2 () protected by the time-reversal symmetry (TRS), whose edge states are robust against perturbations such as nonmagnetic disorder.
A simple model of the QSH systems is the Kane-Mele model Kane1 (), defined on a honeycomb lattice, first introduced for graphene with spin-orbit couplings (SOCs). It was suggested Kane1 () that the intrinsic SOC in graphene would open a band gap in the bulk and also establish spin-filtered edge states that traverse the band gap, giving rise to the QSH effect. Even though the intrinsic SOC strength in pure graphene is too small to produce an observable effect under realistic conditions Min (), the Kane-Mele model captures the essential physics of the QSH state with nontrivial band topology Rev1 (); Rev2 (). In the presence of the Rashba SOC and an exchange field, the Kane-Mele model enters a TRS-broken QSH phase sheng () characterized by nonzero spin Chern numbers spinchern (); Prodan (). Prodan proved Prodan () that the spin-Chern numbers are topological invariants, as long as the energy gap and the spectrum gap of the projected spin operator stay open in the bulk, where is the projection operator onto the subspace of the occupied bands and the Pauli matrix for the electron spin. Unlike the invariant z2 (), the robust properties of the spin-Chern numbers remain unchanged when the TRS is broken sheng (); Prodan ().
The existence of counterpropagating edge states with opposite spin polarizations is an important characteristic of the QSH state. It is believed that the edge states can be gapless only if the TRS Kane1 () or other symmetries, such as the inversion symmetry BZhou () or charge-conjugation TRS QFSun (), are present. When the TRS is broken, it was found sheng () that a small gap appears in the spectrum of the edge states, which was obtained for a ribbon geometry under ideal boundaries, i.e., boundaries created by an infinite hard-wall confining potential. However, since the edge states are localized around the sample boundaries, they can be sensitive to the variation of on-site potentials near the boundaries niu1 ().
In this Letter, in order to reveal the general characteristics of the edge states and their connection to the bulk topological invariant in a QSH system, we present a topological argument similar to the Laughlin’s Gedanken experiment without considering any symmetries. We show that, as required by the nontrivial band topology and gauge invariance, either the energy gap or the spin spectrum gap (the gap in the spectrum of ) needs to close on the edges of a QSH system. These two scenarios will lead to gapless or gapped edge modes, respectively. In particular, it is demonstrated that gapless edge states can appear in a TRS-broken QSH system by tuning the confining potential at the boundaries. They are associated with the bulk topological invariant, and are robust against relatively smooth impurity scattering potential. Our result offers an interesting example for counterpropagating gapless edge states that are not protected by symmetries, which sheds light on the underlying mechanism of the QSH effect in a broad sense.
Let us first look back on a looped ribbon of the QHE system, with a magnetic flux (in units of flux quantum ) threading the ring adiabatically new1 (); new2 (); new3 (). The Fermi energy is assumed to lie in an energy gap. In the spirit of the Laughlin’s argument, increasing from 0 to effectively pumps one occupied state from one edge to the other, giving rise to the transfer of one charge between the edges, essentially because there is a nonzero Chern number (Hall conductivity) in the bulk. On the other hand, the system Hamiltonian is gauge invariant under integer flux changes, i.e., if is increased from to , the system will reproduce the same eigenstates as at . To assure this gauge invariance, there must be gapless edge modes on the edges (when the perimeter of the ring is large), so that the spectral flow can form a closed loop, as illustrated in Fig. 1a, along which the electron states can continuously move with changing .
We now propose a topological understanding of the QSH system in terms of the same looped ribbon geometry. The occupied valence band can be decomposed into two spin sectors by using the projected spin operator Prodan (); sheng (). (The unoccupied conduction band can be divided similarly.) The two spin sectors are separated by a nonzero spin spectrum gap in the bulk. They carry opposite spin Chern numbers, so that increasing pumps a state of the spin up sector in the occupied band from one edge to the other, and pumps another state of the spin down sector in the opposite direction. In order for the system to recover the initial eigenstates as changes from to , the spectral flow needs to form closed loops, similarly to the QHE system. However, for the QSH system, if not enforced by any symmetry, two different scenarios can occur on the edges. One is that gapless edge modes appear on the edges, so that states can move between the conduction and valence bands with changing to form closed loops in the spin-up and spin-down sectors separately, as shown in Fig. 1b. In this case, the states in the two loops cannot evolve into each other due to the nonvanishing spin spectrum gap both in the bulk and on the edges. The other scenario is that a closed loop of spectral flow is formed between the two spin sectors within the valence (or conduction) band, as shown in Fig. 1c. In this case, the spin spectrum gap must vanish on the edges, but the energy gap may remain open in the edge state spectrum.
The above topological discussion on the QSH system is very general, independent of any symmetries. To demonstrate the two scenarios in Figs. 1b and 1c, in what follows we take the Kane-Mele model Kane1 () for a honeycomb lattice ribbon as an example, by taking into account different confining potentials near the edges of the ribbon. It was shown that in a suitable parameter range, the Kane-Mele system is in the QSH phase protected by the TRS, and it can become a TRS-broken QSH phase sheng (), when a spin-splitting exchange field is applied. Consider an armchair ribbon along the direction, as shown in Fig. 2a, including dimer lines across the ribbon. (Results for a zigzag ribbon are similar.) The boundaries are at and , where the distance between nearest-neighbor sites is chosen as the unit of length. The Hamiltonian can be written as with
as the Hamiltonian of the Kane-Mele model. Here, the first term is the nearest-neighbor hopping term with as the electron creation operator on site and the angular bracket in standing for nearest-neighbor sites. The second term is the intrinsic SOC with coupling strength , where are the Pauli matrices, and are two next nearest neighbor sites, is their unique common nearest neighbor, and vector points from to . The third term is the Rashba SOC with coupling strength . stands for a uniform exchange field of strength . represents a sublattice staggered confining potential, which is given by with
where is taken to be positive for sites on sublattice (solid dots) and negative on sublattice (hollow dots), as shown in the Fig. 2a. In Eq. (2), is strongly dependent across the ribbon, equal to at the edges ( and ). It decays exponentially away from the edges, with a characteristic length , as shown in Fig. 2b. When the ribbon width is much greater than , essentially vanishes in the middle region of the ribbon. Here, we note that in the case of a uniform staggered potential ( being independent of ), it was shown sheng () that with increasing , there is a transition from the TRS-broken QSH phase to an ordinary insulator state, where the middle band gap closes and then reopens. Therefore, for the confining potential given by Eq. (2) with large , the ribbon in Fig. 2 can be regarded as a TRS-broken QSH ribbon sandwiched in between two trivial band insulators.
In order to assure the system in the TRS-broken QSH state, we set the parameters , , and . The length of the armchair ribbon is taken to be infinite. The energy spectrum of the ribbon, together with the corresponding eigenfunctions , can be numerically obtained by diagonalizing the Hamiltonian for each momentum in the direction. The calculated energy spectrum of the armchair ribbon with width , for the confining potential with fixed and two different decay lengths, is plotted in Fig. 3a and Fig. 3b. One can see easily that the edge states appear as thin lines in the middle bulk band gap of the energy spectrum. In Figs. 3a and 3b, the spin polarization of the edge states is labeled with and , indicating that two spin-filtered channels on each edge flow along opposite directions. For the nearly hard-wall confining potential of , the sublattice potential is nonzero only on the outermost armchair lines, similar to the assumption in Ref. niu1 (). In this case, two small energy gaps are observed in the edge state spectrum shown in Fig. 3a, in agreement with the previous observation sheng (), as a consequence of the broken TRS. With increasing the decay length of the confining potential, the energy gaps of the edge states become smaller and smaller. As is large enough, e.g., in Fig. 3b, interestingly, the edge states become gapless.
We now calculate the -dependent spectrum of projected spin operator , whose matrix elements are given by with and running over all the occupied states. By diagonalizing this matrix, the spectrum of the projected spin can be obtained. For the Kane-Mele model Eq. (1), if , commutes with the Hamiltonian . Therefore, is a good quantum number. It follows that the spectrum of consists of just two values , which are highly degenerate. When the Rashba term is turned on, and no longer commute, and the degeneracy is lifted. In this case, the spectra of between and spread towards the origin, but a gap remains for a bulk sample if the amplitude of the Rashba term dose not exceed a threshold Prodan ().
For the ribbon geometry, the situation is more complicated due to the existence of the edges, and numerical calculations are performed to obtain the spin spectrum. The calculated spectrum of for the same parameters as those in Figs. 3a and 3b is shown in Figs. 3c and 3d, which exhibits very interesting behavior. For the hard-wall confining potential with , while the energy spectrum of edge states are slightly gapped, with increasing the spectrum of continuously change between to without showing any gap, corresponding to the second scenario shown in Fig. 1c. On the other hand, for a relatively smooth confining potential with , the energy spectrum of the edge states are gapless, but the spectrum of displays a big gap, and the sudden changes happen at the cross points in the energy spectrum of the edge states, corresponding to the first scenario shown in Fig. 1b. From Fig. 3, it follows that as long as the system is in the QSH state, a gapless characteristic always appears either in the energy spectrum of edge states or in the spectrum of , leading to the two types of closed loops for the continuous flow of the electron states illustrated in Figs. (1b, 1c).
The result shown in Figs. 3b and 3d, for the relatively smooth confining potential, is of particular interest. It indicates that gapless edge states can exist in the TRS-broken QSH system, accompanied with a gapped spectrum of . Such an interesting behavior can be further understood by the following argument. As long as the bulk energy gap does not close, the projected spin operator is exponentially localized in real space with a characteristic length about , where is the Fermi velocity and is the magnitude of the energy gap. Prodan () For the parameter set used in Fig. 3, is estimated to be between to lattice constants. When the confining potential is varying relatively slowly in space, i.e., , one can find that roughly commutes with the confining potential. In this case, the confining potential is of no influence on the spectrum of . Since the spin spectrum has a gap in the bulk sheng (), this gap remains to open on the smooth edges, as seen from Fig. 3d. As a result, the energy gap has to close due to the topological requirement, resulting in gapless edge modes, as observed in Fig. 3b.
Finally, we wish to discuss the robustness of the gapless edge states found in the present TRS-broken QSH system. We consider a sample forming a looped geometry as that shown in Fig. 1. nonmagnetic impurities are assumed to be randomly distributed in the sample at positions with . An extra term is added to the total Hamiltonian to describe the effect of the impurity scattering, where with as the position of the -th atom site. The impurity scattering potential is taken to be with as the correlation length and the strength of the scattering potential. By inclusion of in the prefactor, the area integral of the impurity potential is set to be independent of . Figure 4 shows the evolution of the calculated eigenenergies of a system in the band gap upon adiabatic insertion of a magnetic flux into the ring, for three different impurity scattering potentials. The number concentration of the impurities is fixed at . For a very short correlation length , for which the impurity potential is nearly uncorrelated from one site to another, we see from Fig. 4a that at , the energy levels of the edge states avoid to cross each other as they move close, resulting in small energy gaps in the spectrum, as indicated by the arrows. This level repulsion behavior is a signature of the onset of backward scattering spinchern (). When the characteristic length is increased to with fixed, corresponding to a relatively smooth impurity scattering potential, all the energy gaps vanish, as shown in Fig. 4b. The energy levels move in straight lines and continue to cross each other, a clear indication of quenching of the backward scattering spinchern (). Such a level crossing feature is intact when is increased up to for fixed , as shown in Fig. 4c. We thus conclude that the edge states remain to be robust in the presence of relatively smooth impurity scattering potential of intermediate strength. This result can be understood based upon an argument similar to that in the pure case. When is greater than the characteristic length of the projected spin operator , the impurity scattering potential nearly commutes with , and hence does not affect much the spin spectrum gap, so that the energy gap needs to close on the edges, which explains the level crossing behavior of the edge modes.
In summary, based upon a general topological argument without relying on the TRS or other symmetries, we show that in a QSH system either the energy gap or the gap in the spectrum of needs to close on the edges. We find that a TRS-broken QSH system can have either gapless or gapped edge states, depending on the properties of the confining potential near the boundaries. The gapless edge states are protected by the bulk topological invariant rather than any symmetries, which can remain to be robust in the presence of impurities.
Acknowledgment This work is supported by the State Key Program for Basic Researches of China under Grant Nos. 2009CB929504 (LS), 2011CB922103 and 2010CB923400 (DYX), by the National Natural Science Foundation of China under Grant Nos. 11074110 (LS), 11174125, 11074109, and 91021003 (DYX), and by a project funded by the PAPD of Jiangsu Higher Education Institutions.
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