Abstract
We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes. The degeneracy points are characterized by two vortices in momentum space, with opposite winding numbers. We show rigorously that the topological gapless phase always hosts a partial Majorana flat band edge modes in a ribbon geometry, although such a single band model has zero Chern number as a topologically trivial superconductor. The flat band disappears when the gapless phase becomes topologically trivial, associating with the mergence of two vortices. Numerical simulation indicates that the flat band is robust against the disorder.
Introduction
Topological materials have become the focus of intense research in the last years^{1,2,3,4}, since they not only exhibit new physical phenomena with potential technological applications, but also provide a fertile ground for the discovery of fermionic particles and phenomena predicted in highenergy physics, including Majorana^{5,6,7,8,9,10}, Dirac^{11,12,13,14,15,16,17} and Weyl fermions^{18,19,20,21,22,23,24,25,26}. These concepts relate to Majorana edge modes and topological gapless phases. System in the topological gapless phase exhibits band structures with bandtouching points in the momentum space, where these kinds of nodal points appear as topological defects of an auxiliary vector field. On the other hand, a gapful phase can be topologically nontrivial, commonly referred to as topological insulators and superconductors, the band structure of which is characterized by nontrivial topology. A particularly important concept is the bulkedge correspondence, which links the nontrivial topological invariant in the bulk to the localized edge modes. The number of Majorana edge modes is determined by bulk topological invariant. In general, edge states are the eigenstates of Hamiltonian that are exponentially localized at the boundary of the system. The Majorana edge modes have been actively pursued in condensed matter physics^{27,28,29,30,31,32,33} since spatially separated Majorana fermions lead to degenerate ground states, which encode qubits immune to local decoherence^{34}. This bulkedge correspondence indicates that a singleband model must have vanishing Chern number and there should be no edge modes when open boundary conditions are applied. However, the existence of topological gapless indicates that there is hidden topological feature in some single band system. A typical system is a 2D honeycomb lattice of been graphene, which is a zerobandgap semiconductor with a linear dispersion near the Dirac point. Meanwhile, there is another interesting feature lies in the appearance of partial flat band edge modes in a ribbon geometry^{35,36,37}, which exhibit robustness against disorder^{38}. Recently, it has been pointed that Majorana zero modes are not only attributed to topological superconductors. A 2D topologically trivial superconductors without chiral edge modes can host robust Majorana zero modes in topological defects^{39,40,41}.
In this paper, we investigate this issue through an exact solution of a concrete system. We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes^{42,43}. The degeneracy points are characterized by two vortices, or Dirac nodal points in momentum space, with opposite winding numbers. This work aims to shed light on the nature of topological edge modes associated with topological gapless phase, rather than gapful topological superconductor. We show rigorously that the topological gapless phase always hosts a partial Majorana flat band edge modes in a ribbon geometry. The flat band disappears when the gapless phase becomes topologically trivial, associating with the mergence of two vortices. Numerical simulation indicates that the flat band is robust against the disorder.
Results
We have demonstrated that a topologically trivial superconductor emerges as a topological gapless state, which support Majorana flat band edge modes. The new quantum state is characterized by two linear banddegeneracy points with opposite topological invariant. In sharp contrast to the conventional topological superconductor, such a system has single band, thus has zero Chern number. We prove that the appearance of this topological feature attributes to the corresponding Majorana lattice structure, which is a modified honeycomb lattice. It is natural to acquire a set of zero modes, which is robust against disorder. In the following, there are three parts: (i) We present the Kitaev Hamiltonian on a square lattice and the phase diagram for the topological gapless phase. (ii) We investigate the Majorana bound states. (iii) We perform numerical simulation to investigate the robust of the edge modes against the disorder perturbations.
Model and topological gapless phase
We consider the Kitaev model on a square lattice which is employed to depict 2D pwave superconductors. The Hamiltonian of the tightbinding model reads
where r is the coordinates of lattice sites and c_{r} is the fermion annihilation operators at site r. Vectors a = ai, aj, are the lattice vectors in the x and y directions with unitary vectors i and j. The hopping between neighboring sites is described by the hopping amplitude t. The isotropic order parameter Δ is real, which result in topologically trivial superconductor. The last term gives the chemical potential.
Taking the Fourier transformation
the Hamiltonian with periodic boundary conditions on both directions can be rewritten as
where
where the summation of k = (k_{x}, k_{y}) is \({\sum }_{{\bf{k}}}={\sum }_{{k}_{x}=\pi }^{\pi }{\sum }_{{k}_{y}=\pi }^{\pi }\). The core matrix can be expressed as
where the components of the auxiliary field B(k) = (B_{x}, B_{y}, B_{z}) are
σ are the Pauli matrices
The parameters t, Δ and μ are real number as illustrated in the phase diagram (Fig. 1(a)), which automatically requires B_{x} = 0. The Bogoliubov spectrum is
We are interested in the gapless state arising from the band touching point of the spectrum. The band degenerate point k_{0} = (k_{0x}, k_{0y}) is determined by
As pointed in ref.^{42}, two bands touch at three types of configurations: single point, double points, and curves in the k_{x} − k_{y} plane, determined by the region of parameter Δ−μ plane (in units of t). We focus on the nontrivial case with nonzero Δ. Then we have
in the region \(\mu /t\le 2\), which indicates that there are two nodal points for μ ≠ 0 and μ/t ≠ 2. The two points move along the line represented by the equation k_{0x} = −k_{0y}, and merge at k_{0} = (π, −π) when μ/t = ±2. In the case of μ = 0, the nodal points become two nodal lines represented by the equations k_{0y} = ±π + k_{0x}. The phase diagram is illustrated in Fig. 1, depending on the values of μ and Δ (compared with the hopping strength t). We plot the band structures in Figs 2 and 3 for several typical cases.
In the vicinity of the degeneracy points, we have
where q = k − k_{0}, k_{0} = (k_{0x}, k_{0y}) and (k_{0x}, k_{0y}) satisfy Eq. (10), is the momentum in another frame. Around these degeneracy points, the Hamiltonian h_{k} can be linearized as the form
which is equivalent to the Hamiltonian for twodimensional massless relativistic fermions. Here (q_{1}, q_{2}) = (q_{x}, q_{y}) and (σ_{1}, σ_{2}) = (σ_{y}, σ_{z}). The corresponding chirality for these particle is defined as
Then we have
which leads to w = ±1 for two nodal points. The chiral relativistic fermions serve as twodimensional Dirac fermions. Two Dirac nodes located at two separated degenerate points have opposite chirality. We note that for Δμ/t(μ/t − 2) = 0, we have w = 0. At this situation, two Dirac nodes merge at (0, 0) and (±π, ∓π). The topology of the nodal point becomes trivial, and a perturbation hence can open up the energy gap. We illustrate the vortex structure of the degeneracy point in k_{x} − k_{y} plane in Figs 2 and 3. As shown in figures, we find three types of topological configurations: pair of vortices with opposite chirality, single trivial vortex (or degeneracy lines), and no vortex, corresponding to topological gapless, trivial gapless and gapped phases, respectively.
Majorana flat band edge modes
Now we turn to study the feature of gapless phase in the framework of Majorana representation. The Kitaev model on a honeycomb lattice and chain provides wellknown examples of systems with such a bulkboundary correspondence^{44,45,46,47,48,49,50}. It is well known that a sufficient long chain has Majorana modes at its two ends^{51}. A number of experimental realizations of such models have found evidence for such Majorana modes^{7,52,53,54,55}. In contrast to previous studies based on a gapful system with nonzero Chern number, we focus on the Kitaev model in the topologically trivial phase. This is motivated by the desire to get a connection between the Majorana edge modes and topological nature hidden in a topologically trivial superconductor. At first, we revisit the description of the present model on a cylindrical lattice in terms of Majorana fermions.
We introduce Majorana fermion operators
which satisfy the relations
Then the Majorana representation of the Hamiltonian is
It represents a honeycomb lattice with extra hopping term a_{r+a}b_{r}, which is schematically illustrated in Fig. 1. Before a general investigation, we consider a simple case to show that a flat band Majorana modes do exist. Taking t = Δ = μ the Hamiltonian reduces to
which corresponds to a honeycomb ribbon with zigzag boundary condition. It is wellknown that there exist a partial flat band edge modes in such a lattice system^{35,36,37,38}.
In the following, we will show that this feature still remains in a wide parameter region. Consider the lattice system on a cylindrical geometry by taking the periodic boundary condition in one direction and open boundary in another direction. For a M × M Kitaev model, the Majorana Hamiltonian can be explicitly expressed as
by taking r = mi + nj → (m, n). The boundary conditions are a_{n,1} = a_{n,M+1}, b_{n,1} = b_{n,M+1}, a_{M+1,n} = 0, and b_{M+1,n} = 0.
Consider the Fourier transformations of Majorana operators
where the wave vector K = 2πl/N, l = 1, 2, …, N.
The Hamiltonian H_{M} can be rewritten as
where η = [(t − Δ)e^{iK} + (t + Δ)e^{−iK} − 2μ]/4, δ_{1} = (t − Δ)/4 and δ_{2} = (t + Δ)/4, and \({h}_{{\rm{M}}}^{K}\) obeys
i.e., H_{M} has been block diagonalized. We would like to point that operators α_{m,K} and β_{m,K} are not Majorana fermion operators except the cases with K = 0 or π. We refer such operators as to auxiliary operators. We note that each \({h}_{{\rm{M}}}^{K}\) represents a modified SSH chain about auxiliary operators α_{m,K} and β_{m,K} with η, δ_{1}, and δ_{2} hopping terms. One can always get a diagonalized \({h}_{{\rm{M}}}^{K}\) through the diagonalization of the matrix of the corresponding singleparticle modified SSH chain. For simplicity, we only consider the case with positive parameters t, Δ, and μ. In large M limit, there are two zero modes for \({h}_{{\rm{M}}}^{K}\) under the condition 0 < μ < 2 (in units of t). Actually, it can be checked that \({h}_{{\rm{M}}}^{K}\) can contribute a term
where
Here \(A={(2\sum _{j=1}^{M}{{p}_{+}^{j}{p}_{}^{j}}^{2})}^{\frac{1}{2}}\) is normalization constant, and
The term in Eq. (24) exists under the convergence condition
As expected, we note that operators γ_{K} satisfy the fermion commutation relations
representing edge modes. The sufficient condition for Eq. (27) is
which leads to
or more explicitly form
where \(R(K,{\rm{\Delta }},\mu ,t)=({t}^{2}{{\rm{\Delta }}}^{2})\,{\cos }^{{\rm{2}}}(K)2\mu t\,\cos (K)+{\mu }^{2}{t}^{2}\). To demonstrate this result, considering a simple case with Δ = μ = t, we find that R reduces to −2Δ^{2}cos(K) and satisfies above equations when take −π/3 < K < π/3. Furthermore the parameters become η = Δ(e^{−iK} − 1)/2, δ_{1} = 0 and δ_{2} = Δ/2, and \({h}_{{\rm{M}}}^{K}\) corresponds to a simple SSH chain with η < δ_{2}. The edge mode wave functions can be obtained from p_{+} = 0 and \({p}_{}=\,\sqrt{2(1\,\cos \,K)}\). Then the existence of edge modes is well reasonable. The zero modes in the plot of energy band in Fig. 2(i) corresponds this flat band of edge mode.
Disorder perturbation
One of the most striking features of topologically protected edge states is the robustness against to certain types of disorder perturbation to the original Hamiltonian. In this section, we investigate the robustness of the Majorana edge flat band in the presence of disorder. The disorder we discuss here arises from the parameters {t, Δ, μ} in the Hamiltonian H_{M}. More precisely, one can rewrite the Hamiltonian in the form
where h represents a 2M^{2} × 2M^{2} matrix in the basis
We introduce the disorder perturbation to H_{M} by preserving the time reversal symmetry, i.e., keeping the reality of the parameters {t, Δ, μ}. We take the randomized matrix elements in h by the replacement
to get the disorder matrix h_{DP}. Here r^{a,b,c} are three M × M matrices which consist of random numbers in the interval of (1 − ξ, 1 + ξ), influencing each matrix elements. Real factor ξ plays the role of the disorder strength.
We investigate the influence of nonzero ξ by comparing two sets of eigenvalues obtained by numerical diagonalization of finitedimensional matrices h and h_{DP}, respectively. The plots in Fig. 4. indicate that the zero modes remain unchanged in the presence of random perturbations with not too large ξ. The numerical result support our conclusion that the topological gapless states correspond to the presence of topologically protected edge modes.
Discussion
According to the bulkedge correspondence, it seems that the existence of edge states requires a gapped topological phase. This may not include the case with a single band which contains topological gapless states. The topological character of a gapless state does not require the existence of the gap. This arises the question: What is the essential reason for the edge state, energy gap or topology of the energy band? Obviously, energy gap is not since many gapped systems do not support the edge states. Then it is possible that a special single band system supports the edge states. In the case of the lack of an exact proof, concrete example is desirable. As such an example we have considered a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes. The degeneracy points are characterized by two vortices, or Dirac nodal points in momentum space, with opposite winding numbers. We demonstrated that a topologically trivial superconductor emerges as a topological gapless state, which support Majorana flat band edge modes. The new quantum state is characterized by two linear banddegeneracy points with opposite topological invariant. In sharp contrast to the conventional topological superconductor, such a system has single band, thus has zero Chern number. We prove that the appearance of this topological feature attributes to the corresponding Majorana lattice structure, which is a modified honeycomb lattice. The topological feature of an edge state is the robustness against disorder. The numerical results indicate that such a criteria is met for this concrete example. We also note that the topological gapless state and the edge state have the same energy level, which is also an open question in the future.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (under Grant No. 11874225).
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Z.S. conceived the idea and carried out the study. K.L.Z., P.W. and Z.S. discussed the results. Z.S. and K.L.Z. wrote the manuscript with inputs from all the other authors.
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Correspondence to Z. Song.
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Zhang, K.L., Wang, P. & Song, Z. Majorana flat band edge modes of topological gapless phase in 2D Kitaev square lattice. Sci Rep 9, 4978 (2019) doi:10.1038/s4159801941529y
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