A Numerical method

Majorana bound states in a disordered quantum dot chain

Abstract

We study Majorana bound states in a disordered chain of semiconductor quantum dots proximity-coupled to an s-wave superconductor. By calculating its topological quantum number, based on the scattering-matrix method and a tight-binding model, we can identify the topological property of such an inhomogeneous one-dimensional system. We study the robustness of Majorana bound states against disorder in both the spin-independent terms (including the chemical potential and the regular spin-conserving hopping) and the spin-dependent term, i.e., the spin-flip hopping due to the Rashba spin-orbit coupling. We find that the Majorana bound states are not completely immune to the spin-independent disorder, especially when the latter is strong. Meanwhile, the Majorana bound states are relatively robust against spin-dependent disorder, as long as the spin-flip hopping is of uniform sign (i.e., the varying spin-flip hopping term does not change its sign along the chain). Nevertheless, when the disorder induces sign-flip in spin-flip hopping, the topological-nontopological phase transition takes place in the low-chemical-potential region.

pacs:
71.10.Pm, 74.78.Na

I Introduction

Majorana bound states (MBS)(1); (2) in solid-state systems are recently attracting increasing interest, both theoretically and experimentally. Proposed by Kitaev more than ten years ago in a spinless toy model,(1) these zero-energy bound states are expected to exist in several structures with spin, including nanowires with spin-orbit coupling (SOC) in proximity to a superconductor (SC),(3); (4); (5) ferromagnetic atom chains on top of a SC,(6) topological insulator/SC hybrid structures,(7); (9); (8); (10); (11); (12) quantum dot (QD) chains with SC in adjacence,(13); (14); (15) as well as cold-atom systems.(16) Experimentally, possible signatures of MBS have been reported in nanowires,(17); (19); (18) atom chains,(20) and topological insulator/SC structures.(21)

Majorana bound states attract considerable attention partly due to their future potential applications in quantum information.(2); (22); (23); (24) One attractive possibility would be to construct Majorana qubits based on MBS.(22) Majorana qubits, among various qubit candidates,(25); (26); (27); (28); (29); (30); (31) are supposed to be robust against local perturbations and hence promising to store quantum information.(22); (32); (13) Moreover, arbitrary qubit rotations are expected to be implemented, by means of topologically-protected braiding operations(23); (33) in combination with other non-topological operations assisted by, e.g., nanomechanical resonators.(34); (35) However, recent studies reveal that the MBS are not completely robust against disorder in the Kitaev’s spinless model and in the systems with spin.(36); (41); (37); (38); (39); (40) Moreover, the Majorana qubits are not totally protected from decoherence.(43); (44); (42); (45)

Note that the studies investigating so far the effect of disorder on MBS focus solely on the spin-independent disorder, without considering the spin-dependent one. In fact, the spin-dependent disorder, e.g., the randomness in SOC, can be present inevitably in many solid-state systems and play an important role in the spin-related dynamics.(47); (46); (48) Therefore, the effect of spin-dependent disorder on the existence of MBS deserves to be investigated.

Figure 1: (Color online) Schematic diagram of a disordered chain of semiconductor quantum dots (shown in blue) in proximity to an s-wave superconductor (in red), under a transverse magnetic field . The on-site chemical potentials in the quantum dots, as well as the spin-conserving and -flip hopping terms between neighbouring quantum dots, can vary among the different sites.

In this work, we systematically study the robustness of MBS against disorder, based on a concrete structure, i.e., a QD chain in proximity to an s-wave superconductor.(14) Experimentally, such a QD chain system might have the advantage to be adaptively tuned, as suggested in Ref. (14). However, in the absence of precise control, this system is also very likely to be disordered due to, e.g., the inhomogeneity in QD sizes or QD confining potentials. Therefore, we consider a QD chain as an ideal platform to study the influence of disorder. Concretely, we calculate the topological quantum number by means of the scattering-matrix method on a tight-binding model, to identify the topological property of a disordered chain in a relatively large parameter region. Apart from the disorder in the spin-independent terms (including the chemical potential and the regular spin-conserving hopping), we also consider the disorder in the spin-dependent term, i.e., the spin-flip hopping due to the Rashba SOC. We find that the MBS are not completely immune to disorder in the spin-independent terms, especially when the disorder is strong. Meanwhile, the MBS are relatively robust against disorder in the spin-flip hopping, as long as the spin-flip hopping is of uniform sign. Nevertheless, when the disorder induces sign-flip in spin-flip hopping, a topological-nontopological phase transition in the QD chain takes place in the low-chemical-potential region.

This paper is organized as follows. First, we describe the inhomogeneous QD chain in a tight-binding model. Then we present the scattering-matrix method used to calculate the topological quantum number. Afterwards, we numerically study the robustness of the MBS against disorder in the QD chain. Finally, we summarize our results.

Ii Model and Hamiltonian

A QD chain, as studied in Ref. (14), is schematically shown here in Fig. 1. An s-wave SC is in proximity to the QD chain and a transverse magnetic field is applied along the -axis. We assume that the QDs can be approximately treated as one dimensional along the chain-direction (-axis) due to the strong transverse confinement. By further assuming that the orbital level splitting in the QDs is much larger than both the Zeeman splitting and Rashba SOC, we consider only the Kramers doublet closest to the chemical potential energy in each QD. The general form of the tight-binding Hamiltonian describing such a chain of single-level QDs is written as(14)

(1)

Here, is the creation operator for a spin- electron in the th QD. The Pauli matrices act on the spin space. The chemical potential is labeled as . The term proportional to is the Zeeman splitting while stands for the superconducting pairing due to the proximity effect. The nearest-neighbour hopping term has two parts, i.e., the spin-conserving () and spin-flip () ones. The spin-flip hopping can be caused by the SOC which supplies an effective magnetic field during hopping. Here we only consider the Rashba type SOC, with its effective magnetic field along the -axis. Due to the inhomogeneity in the QD confining potentials and/or QD/SC sizes, as well as other disorder sources such as charged impurities, both the spin-conserving terms, and , and the spin-flip term, , can be QD-site dependent.

In the Bogoliubov-de Gennes basis , the Eq. (1) can be rewritten as(6)

(2)

where

(3)
(4)

and the Pauli matrices act on the particle-hole space.

Iii Scattering-matrix method

To identify the topological property of the QD chain, we study the scattering matrix relating the incoming and outgoing wave amplitudes at the Fermi level(49)

(5)

The subblocks {, } and {, } are the reflection and transmission matrices at the two ends of the QD chain, respectively. The topological quantum number is given by(49)

(6)

Here, sgn denotes the sign of the determinant Det. The MBS arise(49) at the ends of the QD chain only when .

The scattering matrix can be obtained by the transfer-matrix scheme. Based on Hamiltonian (2), the zero-energy Schrödinger equation gives(6)

(7)

where

(8)

Here is a four-component vector of wave amplitudes on the th site. The above recursive relation indicates that waves at the two ends ( and ) of the nanowire are related by the transfer matrix

(9)

In the basis with right-moving and left-moving waves separated in the upper and lower four components, the transfer matrix transforms as

(10)

where

(11)

In this basis, the reflection matrices () and transmission matrices () in the scattering matrix [refer to Eq. (5)] can be obtained via the relations

(12)

where

(13)

Finally, the calculation of the topological quantum number is reduced to that of the transfer matrix . In Appendix A, we present the numerical method for calculating .

Iv Results

We now numerically study(50) the topological property of the QD chain. For comparison, we first look into an ideal homogeneous QD chain and reproduce the topological phase reported in the literature, and then take into account disorder to investigate the robustness of the MBS.

iv.1 Homogeneous QD chain

For a homogeneous QD chain, we denote , and . In Fig. 2(a) we plot the phase diagram, [refer to Eqs. (5) and (6)] versus and , of a homogeneous QD chain typically with and . The blue region in this figure, with , stands for the topological phase supporting MBS. It is found that this region is nicely enclosed by the white curve plotted in the figure, which defines the topological region of a single-band homogeneous superconducting nanowire as(51); (52)

(14)

In Fig. 2(b), we further show the energy spectrum (for clarity, we present only the lowest four states close to the zero energy) of this QD chain versus when is fixed. It is clear [from the red and blue curves in Fig. 2(b)] that when the QD chain enters the topological region, the zero-energy states (localized at the two ends of the QD chain) which are separated from the higher-energy bulk states arise. Note that when varying the spin-flip hopping , the topological phase space in Fig. 2(a) remains invariant, consistent with the feature that is absent from Eq. (14).

iv.2 Inhomogeneous QD chain with disordered chemical potential and spin-conserving hopping

Figure 2: (Color online) (a) The determinant of the reflection matrix as a function of the chemical potential and the Zeeman splitting , in a homogeneous QD chain with and . The blue region with stands for the topological phase supporting MBS. (b) The energy spectrum (with only the lowest four eigenstates close to zero energy plotted) versus the chemical potential , when the Zeeman splitting is fixed as . Note that in this figure, as well as in Figs. 3-4, the chain has QDs, which is large enough for the numerical convergence.

From Eq. (14), one may infer that when the disorder is induced into the chemical potential or the spin-conserving hopping , the topological phase space might change in the parameter space. Now we take into account such disorder to investigate the robustness of MBS in the QD chain. We first consider disorder in the chemical potential, which is modeled to perturb the ’s independently within a uniform distribution in the interval , where is now the mean value of the chemical potential and stands for the fluctuation magnitude. Our calculations indicate that the topological phase is not completely immune to disorder. In Figs. 3(a) and (b), we present the phase diagrams of the inhomogeneous QD chain calculated with and , respectively. The comparison between these two figures indicates the effect of stronger disorder on the formation of the topological phase. To qualitatively present the effect of increasing disorder, we further study the ratio of the area of the topological region with disorder [such as the blue regions in Figs. 3(a) and (b)] to that without disorder [the region defined by Eq. (14)], labeled as , versus the fluctuation magnitude . This is a qualitative study because it is performed here in a finite parameter region, e.g., and . This result is shown by the solid curve with squares in Fig. 3(e). This curve shows that when the fluctuation magnitude of the chemical potential is larger than the superconducting gap , the topological phase can be effectively destroyed.

Figure 3: (Color online) (a) and (b) [(c) and (d)] Phase diagrams of disordered QD chains, where the chemical potentials (spin-conserving hoppings ) fluctuate in an interval [] with a uniform distribution. Note that is set as 0.5 and 1.5, respectively, in (a) and (b), and is set as 0.2 and 0.6, respectively, in (c) and (d). (e) The ratio of the area of the topological region for a disordered system [such as the blue regions in (a)-(d)] to the one for a clean system [the region defined by Eq. (14), or, enclosed by the white curves in (a)-(d)], labeled as , versus the fluctuation magnitude of the chemical potential (red curve with squares), and the fluctuation magnitude of the spin-conserving hopping (blue curve with circles). The calculations for each curve in (e) are carried out by averaging over ten disordered samples.

We then consider disorder in the spin-conserving hopping, with the other terms treated as uniform. We assume that the disorder causes the spin-conserving hopping to fluctuate in an interval with a uniform distribution (). Our calculations indicate that disorder in the spin-conserving hopping can also be detrimental to the topological phase (especially when the disorder is strong), as shown by the phase diagrams in Figs. 3(c) and (d). In Fig. 3(e), by the blue curve with circles, we also plot the ratio of the area of the topological region for a disordered system to the one for a clean system, versus the fluctuation magnitude . Also, the stronger the disorder is, the smaller the topological phase area becomes.

Figure 4: (Color online) The phase diagrams of disordered QD chains where the spin-flip hoppings fluctuate in an interval with a uniform distribution. The fluctuation magnitude increases from (a) to (d) .
Figure 5: (Color online) (a) Curves: energy spectrum (with only the lowest six eigenstates close to zero energy plotted) in an inhomogeneous QD chain with a finite length (in the calculation we set the total number of QDs to be 51), versus the variation of spin-flip hopping in one half of the QD chain . Circles: the topological quantum number [in Eq. (6)] of this inhomogeneous QD chain (with the scale on the right-hand side of the frame), versus the variation of spin-flip hopping in one half of the QD chain . The spin-flip hopping in the other half of the QD chain remains invariant as . (b) Square of the wave function of the state with its energy closest to zero. The solid curve stands for the weakly-coupled MBS in a homogeneous QD chain where , while the dashed curve stands for the state where the MBS have disappeared due to their coupling to the interface fermionic bound states in an inhomogeneous QD chain. For the homogeneous QD chain, ; while for the inhomogeneous QD chain: .

iv.3 Inhomogeneous QD chain with disordered spin-flip hopping

We now focus on the robustness of the topological phase against disorder in the spin-flip hopping. Again, for simplicity, we assume that due to disorder, the spin-flip hopping fluctuates in an interval with a uniform distribution. We find that the topological phase is relatively robust against disorder in the spin-flip hopping, as long as the spin-flip hopping is of uniform sign (i.e., ). Nevertheless, when disorder induces sign-flip in the spin-flip hopping (), a topological-nontopological phase transition in the QD chain takes place in the low-chemical-potential region. This feature can be observed from Fig. 4, which presents the phase diagrams of disordered QD chains with increasing .

When the spin-flip hopping changes sign along the QD chain, a pair of zero-energy fermionic bound states(40) arise at the interface between the neighboring domains with different signs of the spin-flip hopping. These interface fermionic bound states can couple to other nearby bound states, including the MBS originally present at the ends of the QD chain. These couplings can destroy the zero-energy MBS. To obtain a clear view of the interface fermionic bound states and their coupling to the MBS, we further consider a simple case where a short QD chain possesses a constant spin-flip hopping on one half of the chain but a varying spin-flip hopping on the other half. Typically, we study a chain with 51 QDs connected by s-wave SCs. We set the spin-flip hopping between the neighboring QDs from the 1st to 26th sites as a constant , and adjust from to the spin-flip hopping on the remaining part. The curves in Fig. 5(a) show the energy spectrum of such an inhomogeneous system (the lowest six eigenstates close to zero are plotted) versus the parameter . It is clearly shown that with the decrease and eventually the sign-flip of , the bulk gap in the QD chain gradually closes and the zero-energy fermionic bound states located around the 26th QD arise. Accordingly, the topological quantum number changes from to 1 [as shown by the open circles in Fig. 5(a)], indicating the disappearance of the MBS due to their coupling to the fermionic bound states. In Fig. 5(b), we further present the square of the wave function of the lowest eigenstate, for the cases with and . It is found that when , i.e., the QD chain is homogeneous, two weakly-coupled MBS are present. However, when , a state resulting from the coupling between MBS and the interface bound state replaces the original MBS.

V conclusion

In this work, we have studied the MBS in a disordered QD chain in proximity to an s-wave SC. We describe this one-dimensional system by a tight-binding model. By calculating the topological quantum number based on the scattering-matrix method, we can identify the topological property of such a QD chain. In our study, we take into account disorder in both the spin-independent terms (including the chemical potential and the regular spin-conserving hopping) and the spin-independent term, i.e., the spin-flip hopping due to the Rashba SOC.

We find that the MBS are not completely immune to disorder in the spin-independent terms, especially when the disorder is strong. Meanwhile, the MBS are relatively robust against disorder in the spin-flip hopping, as long as the spin-flip hopping is of uniform sign. Nevertheless, when the disorder induces sign-flip in spin-flip hopping, a topological-nontopological phase transition in the quantum dot chain takes place in the low-chemical-potential region. This study may provide insight into the search of MBS in solid-state systems.

Acknowledgements.
The authors gratefully acknowledge E. Ya. Sherman and İ. Adagideli for valuable discussions and comments. P.Z. acknowledges the support of a JSPS Foreign Postdoctoral Fellowship under Grant No. P14330. F.N. is partially supported by the RIKEN iTHES Project, MURI Center for Dynamic Magneto-Optics via the AFOSR award number FA9550-14-1-0040, the IMPACT program of JST, and a Grant-in-Aid for Scientific Research (A).

Appendix A Numerical method

As shown in Sec. III, the topological quantum number is determined by the reflection matrix , which can be obtained by the transfer matrix via Eq. (12). However, the recursive construction [i.e., Eq. (13)] is numerically unstable.(6); (53) We stabilize it by using the method described in Ref. (53). We briefly introduce this process here.

We denote

(15)

and define

(16)

Here and are subblock matrices. In such framework, . Further, according to Eq. (12), we have and .

Based on Eqs. (8) and (10), one finds that

(17)

Therefore, one can construct a unitary matrix from the non-unitary matrix as

(18)

Now let us define

(19)

where the operation is performed as

(20)

In this way, is the unitary counterpart of , i.e.,

(21)

As a result, for numerical stability, instead of calculating by Eq. (16), one can calculate the unitary matrix based on Eq. (19).

Finally, the topological quantum number can be obtained via the relation

(22)

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