† Corresponding author. E-mail:
‡ Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant No. 41174116), the Graduate Student Education Teaching Reform Project, China (Grant No. JG201512), and the Young Teachers’ Research Project of Yanshan University, China (Grant No. 13LGB028).
In a recent paper, Lang et al. proposed that edge states and topological phases can be observed in one-dimensional optical superlattices. They showed that the topological phases can be revealed by observing the density profile of a trapped fermion system, which displays plateaus with their positions. However, disorders are not considered in their model. To study the effect of disorders on the topological phases, we introduce random potentials to the model for optical superlattcies. Our calculations show that edge states are robust against the disorders. We find the edge states are very sensitive to the number of the sites in the optical superlattice and we propose a simple rule to describe the relationship between the edge states and the number of sites. The density plateaus are also robust against weak disorders provided that the average density is calculated over a long interval. The widths of the plateaus are proportional to the widths of the bulk energy gaps when there are disorders. The disorders can diminish the bulk energy gaps. So the widths of the plateaus decrease with the increase of disorders and the density plateaus disappear when disorders are too strong. The results in our paper can be used to guide the experimental detection of topological phases in one-dimensional systems.
Topological properties have long been used to characterize the two-dimensional (2D) electron systems.[1–11] For example, the integer quantum Hall (IQH) effect and the quantum spin Hall (QSH) effect in the 2D system with periodic potentials can both be related to the topological properties of occupied energy bands.[2–7] A topological invariant called the Chern number can be used to characterize the integer quantum Hall effect and the Chern number is directly related to the quantum Hall conductance. The topology of the occupied bands of the 2D QSH system is characterized by a Z2 topological invariant.[6] The systems exhibit QSH states are topological insulators. Both of the IQH and the QSH systems have bulk energy gaps and topologically protected gapless edge states.[12]
Recently, the ultra cold atomic systems have been used to simulate the topological insulators. Optical superlattices populated with cold atoms are proven to be a powerful tool to study the topological properties of the 2D systems.[13–15] Lang et al. proposed that the one-dimensional (1D) quasiperiodic optical lattice populated with trapped fermions can display nontrivial topological properties.[16] The localized edge states appear in the energy gaps of 1D optical lattices with the open boundary condition, which indicates that the bulk 1D system is topologically nontrivial. They also showed that the topological property of the 1D system can be detected by observing the density profile of the trapped fermions, which displays plateaus with positions. Hence, their work opens the possibility to observe the topological phase in the 1D system.
It is generally believed that the topological properties are robust against weak disorders.[17,18] For example, the numerical results show that the edge states remain gapless with weak time reversal symmetric disorders in the QSH state.[18] The edge states become gapped only when the disorders are strong enough to make the bulk gap collapse. In this paper, we check whether the 1D topological phases are robust against disorders. For this purpose, we introduce random potentials to the 1D system. We study the effect of the disorders on both of the bulk and edge properties of the 1D system. The edge states and the density plateaus are simulated with different strengths of disorders. We hope our work may provide some insight into the 1D topological phases.
We consider the tight-binding model for the 1D polarized Fermi gas loaded in a bichromatic optical lattice as in Refs. [16] and [19]–[21]
![](cpb_25_7_077303/cpb160295ieqn1.gif)
![](cpb_25_7_077303/cpb160295ieqn2.gif)
This 1D model can be easily mapped to a 2D model when we identify δ with one component of the wave vector ky of the 2D system,[16] so the topology of the 1D model can be studied by using the topological concepts for the 2D system. For example the Chern number can still be defined for the 1D system if we consider δ as a variable parameter. In this sense we still use the 2D topological properties to study the topology of our 1D system.
Without disorders, that is Vr = 0, and if the periodic boundary condition is used the eigenstates are Bloch waves. In this case the energies and the eigenstates are functions of the crystal momentum k and they form energy bands.[16] When the random potentials are turned on, the translational symmetry will be broken and the crystal momentum will no longer be a good quantum number to describe the eigenstate.
With the periodic boundary condition the numerical results show that the random potentials affect the system in two ways. First, the random potentials broaden the energy bands and thus shrink the energy gaps. The energy spectra for p = 1, q = 3, and L = 1000q with different random potentials are shown in Fig.
As long as the random potentials do not become too large there are energy gaps between the energy bands. These results can be understood if we treat the weak random potentials as perturbations. In Fig.
The second effect of random potentials is the eigenstates become localized. Anderson localization is a ubiquitous phenomenon in the one-dimensional system when disorders present.[19,22] The numerical results show that the bulk states become localized and the localization length decreases when the strength of the disorder increases as is shown in Fig.
If the open boundary condition is used there will be edge states at the ends of the optical superlattices. When there is no disorder the spectrum of the edge states as functions of δ are gapless, that is they continuously connect the upper and lower energy bands. The gapless edge states are related to the topology of the system which can be characterized by the Chern numbers of the occupied energy bands.[2,3,12,23,24]
We find that the edge states remain gapless even when disorders are relatively strong as is shown in Fig.
From another point of view, when the energy of the edge state is located at the gap of the bulk energy band, it is energy costly to scatter fermions from the edge states to the bulk. However, we must be cautious when the disorders are strong enough and some edge states are merged to the bulk bands. In this case it is difficult to discriminate the edge states and the bulk states as in Figs.
In Ref. [16], it is shown that the 1D system can be mapped to a 2D Hofstadter system. So the robustness of gapless edge states can be explained by the theory for 2D quantum Hall systems. In Ref. [25] it is shown that the Chern number can still be defined for the many-particle ground states when disorders are present as long as there is finite energy gap. So the gaplessness of the edge states can be related to this nontrivial topological invariant. We conclude that the edge states will remain gapless as long as the bulk energy gap is not closed by the disorders.
![]() | Fig. 4. (a)–(c) Open boundary energy spectra as functions of δ with 102, 101, and 100 sites respectively. Solid and dashed black curves denote edge states at left and right ends, respectively. |
Though the topology of the edge states is not affected by weak disorders, we find that the edge states are very sensitive to the number of sites in the optical superlattice. When the number of sites is not a multiple of q, the missing sites can be considered as point disorders. As is shown in Fig.
As in Ref. [16], when there is confining potential, the Vi in Eq. (
The density distribution is given by
When there is no disorder, we choose M = q. The local density is depicted as a function of sites and the number of Fermions N in Fig.
To study the effect of the disorders on the density plateaus we plot the local density with different strengths of disorders. As is shown in Figs.
As in the edge state case, the Anderson localization seems to not have much effect on the density plateaus. Figure
To study the relationship between the width of the plateaus and the bulk energy gap we first consider the clean case. For a given fermion number N = 600 we plot the local average density as a function of sites and δ with Vr = 0. It can be seen from Fig.
Now we check whether the conclusion on the relationship between the widths of the density plateaus and the bulk energy gap still works when the disorders are present. As is discussed above the bulk energy gaps decrease with disorders increasing. So if the widths of density plateaus still are approximately in proportion to the sizes of the bulk energy gap, the decrease of widths of the plateaus can be explained by the shrinking of energy gaps with disorders. The numerical results confirm our speculation: the widths of the density plateaus approximately are in proportion to the sizes of the bulk energy gaps as is shown in Fig.
In conclusion, by studying the edge states and the density plateaus of the fermions in 1D optical lattices with random potentials, we find that the edge states and density plateaus are robust with weak disorders, which confirms that the edge states and the density plateaus are topologically protected. We qualitatively explain why the Anderson localization has little effect on the topological properties. The edge states are found to be sensitive to the number of sites and a simple rule that describes the relationship between the edge state and the number of the sites is proposed. We find that the relationship between the sizes of energy gaps and the widths of the density plateaus is the same for both the clean and the disordered cases: the widths of the density plateaus are approximately in proportion to the sizes of the bulk energy gaps and they both decrease with the increase of disorders. The density plateaus and the bulk energy gap both disappear when the disorders become too strong.
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