Influence of long-range interaction on Majorana zero modes
IInfluence of long-range interaction on Majorana zero modes
Andrzej Więckowski ∗ and Andrzej Ptok † Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,Wrocław University of Science and Technology, PL-50370 Wrocław, Poland Institute of Nuclear Physics, Polish Academy of Sciences,ul. W. E. Radzikowskiego 152, PL-31342 Kraków, Poland
Majorana bound states, with their non-Abelian properties, are candidates for the realization offault-tolerant quantum computation. Here we study the influence of long-range many-body inter-actions on Majorana zero modes present in Kitaev chains. We show that these interactions cansuppress the lifetime of the Majorana zero mode. We also discuss the role of long-range interac-tions on the Majorana state’s spatial structure, and the overlap of the Majorana states localized atopposite ends of the chain. We have determined that increasing the interaction strength leads todecreasing of the stability of the Majorana modes. Moreover, we found out that interaction betweenparticles located at more distant sites plays a more destructive role than the interaction betweennearest neighbors.
I. INTRODUCTION
Kitaev’s famous proposal [1] for the realization of Ma-jorana quasiparticles opened a period of intense studyinto these topological bound states [2–4]. Currently, re-alizations of Majorana bound states are expected in afew low-dimensional platforms. First are semiconducting-superconducting hybrid nanostructures [5–12], where theinterplay between intrinsic spin–orbit coupling, inducedsuperconductivity, and external magnetic field lead to therealization of zero-energy bound states [3]. Second areone-dimensional (1D) chains of magnetic atoms depositedon a superconducting surface [13–18], where Majoranabound states are expected to form as a consequenceof the magnetic moments in mono-atomic chains [19–22]. More recently, a realization of these zero-energybound states in two-dimensional topological supercon-ducting domains [23–25] and nanostructures with spintextures [26–28] have also been illustrated as well.An essential property of Majorana zero modes (MZMs)are their non-Abelian statistics [29]. Such peculiaritymakes them a very promising platform for the realiza-tion of fault-tolerant quantum computing [30–35]. Quan-tum computations can be realized using braiding pro-tocols [35–39], which can be practically implemented inwire-type systems [40]. Here quantum qubit registers arestored in spatially separated MZMs, which are topologi-cally protected from noise and decoherence [41, 42]. Thelocalized Majorana modes can also be manipulated bysolely acting on the quantum dots [11, 12, 43–48]. Forpractical applications, it is crucial to describe the sourceof the decoherence in the system. This is due to the factthat any decoherence can lead to additional errors inthe state’s coding [49]. From this, we should maximallyclear out the source of decoherence in the system [50, 51],which can be induced, e.g., by fluctuations [52, 53]. ∗ e-mail: [email protected] † e-mail: [email protected] In the context of the practical implementation ofquantum computers based on MZMs, the interactionintroduced in the system plays an important role inthe computation process. The stabilization of MZMcan be achieved by introducing limited interactionstrengths [54–57]. On-site repulsive interactions, in thehalf-spin fermion chain, was earlier discussed withinin the Hartree–Fock approximation [54, 58]. This typeof interaction can lead to the decreasing of the Zee-man energy minimum value needed for MZM emer-gence. Additionally, on-site interactions can stabilize theMZM [57, 59]. Long-range interactions, however, can re-duce the decoherence rate [60]. In the context of spin-less fermions, interactions between nearest sites havebeen discussed using density-matrix renormalization-group (DMRG) methods [56, 61]. Moreover, in this case,moderate repulsive interactions stabilize the topologicalorder. In the present work, we study the influence oflong-range interactions on the MZM’s lifetime and spatialstructures using exact diagonalization (ED) for the Ki-taev chain. The paper is organized as follows: In Sec. II,we introduce the microscopic model and present compu-tational details. In Sec. III, we describe the numericalresults. Finally, we summarize the results in Sec. IV.
II. MODEL AND METHODS
We consider a spinless fermion chain with L sites, de-scribed by the Kitaev model [1] extended by many-bodyinteractions. The system can be represented by the fol-lowing Hamiltonian: H = L − (cid:88) i =1 (cid:16) − ta † i a i +1 + ∆ a † i a † i +1 + h.c. (cid:17) − µ L (cid:88) i =1 (cid:101) n i + L − (cid:88) r =1 V r L − r (cid:88) i =1 (cid:101) n i (cid:101) n i + r , (1)where a † i ( a i ) is fermionic creation (annihilation) opera-tor of spinless fermion at site i , while (cid:101) n i = a † i a i − / . a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t Here t is the hopping integral, ∆ is the superconductinggap, µ is the chemical potential, and V r is the r -nearestneighbor interaction strength.In the absence of the interactions, when | ∆ | > , in theKitaev model, there are two distinguished phases: topo-logical and trivial [1]. Then in the thermodynamic limitit can be shown that the topological phase is present for | µ | ≤ t and the trivial phase is present for | µ | > t .MZMs can emerge only in the topological phase. Here,one should notice that for a system with many-body in-teractions, the expression for the phase boundary canbe more complicated [61–63]. There are several methodsfor studying the presence of MZM in the system withmany-body interactions. Additionally, there exist a fewindicators, which can be used for checking if the systemis in the topological phase [56].From a theoretical point of view, studying quantumsystems with many-body interactions is a relatively diffi-cult task. For example, DMRG methods allow for study-ing systems with thousands of sites, but are limited toshort-range interactions. Here, we used ED for solvingthe chain with L sites. Unfortunately, only small systemscan be solved exactly (with L ∼ ) [64]. This methodallows us to study all possible r for selected system size.For simplification and without loss of generality we take r up to 4, due to the fact that for ED-based methods,only small system sizes are available.MZMs, as states which are indistinguishable from their anti states, should fulfill a few conditions. Each MZM isequivalent to a zero mode Γ , which is fermionic operatorsatisfying the following relations [32]: Γ = 1 , [Γ , H ] = 0 . (2)In actual physical systems with finite length, in whichone can realize a MZM, the second condition of (2) is as-sociated with the exponential suppression of energy split-ting, i.e. [Γ , H ] ∝ e − L/ξ [32], where L is system size and ξ is correlation length. MZM in the finite length systemhas nearly-zero energy and, in consequence, a finite life-time [62, 65], which is observed experimentally [66].To analyze the influence of long-range interactions intothe spatial structure of the Majorana modes, we can an-alyze the Γ modes in the Majorana basis representation.Then MZMs in the Majorana operator γ + i = a i + a † i , γ − i = i ( a i − a † i ) basis [1] can be expressed as: Γ = L (cid:88) i =1 α i γ i = L (cid:88) i =1 (cid:0) α + i γ + i + α − i γ − i (cid:1) , (3)where α ± i are real coefficients. We assume a normaliza-tion condition (cid:80) i α i = 1 when Γ = 1 [62]. Here, γ + and γ − can be understood as new orthogonal basis matrices.Moreover, this basis is a natural representation for thesystem which hosts MZM.To check if MZMs can exist in our system, we usedthe same method which was introduced for studying inte-grals of motion in the Heisenberg model [67, 68] and then was adapted for Kitaev model [62]. As the second con-dition for MZM is satisfied in the thermodynamic limitonly (except fine-tuning parameters settings), we gener-ate almost conserved MZMs by solving the optimizationproblem [62]: λ = max { α i } (cid:104) ΓΓ (cid:105) = max { α i } (cid:88) ij α i (cid:104) γ i γ j (cid:105) α j , (4)which becomes an eigenvalue problem for the matrix (cid:104) γ i γ j (cid:105) . We found the operator Γ averaged over time τ as close as possible to the operator Γ . To measure thedistance between operators Γ and Γ we used the Hilbert–Schmidt operator norm defined in the following way: (cid:104) (Γ − Γ) (cid:105) = Tr [(Γ − Γ) ] / Tr ( ) . We solved the eigen-value problem for matrix (cid:104) γ i γ j (cid:105) and in result we obtainedeigenvalues λ with corresponding eigenvectors [ α i ] . Aver-aging was done by high-oscillating terms cut-off in oper-ator energy basis: Γ = (cid:88) nm θ (cid:18) τ − | E n − E m | (cid:19) (cid:104) n | Γ | m (cid:105) | n (cid:105)(cid:104) m | , (5)where | n (cid:105) and E n are respectively eigenstate and eigenen-ergy of the Hamiltonian H and θ is the Heaviside func-tion. Such averaging, in the limit τ → ∞ is equivalentto calculating the following: lim τ →∞ τ (cid:82) τ (cid:48) d τ (cid:48) Γ( τ (cid:48) ) . Thevalue of λ carries information about the distance betweenoperators Γ and Γ . In the limit τ → ∞ , we can distinguishthree different scenarios: λ = 1 , then Γ is a strict integralof motion Γ = Γ , the distance between them is zero andthe MZM condition [ H , Γ] = 0 is satisfied exactly. For < λ < , part of the information which is stored in Γ is conserved. Finally, if λ = 0 , the information stored islost. In this work, we then look for the largest λ , and thecorresponding eigenvectors [ α i ] , which carry informationof the possible MZM realizations in the system. III. NUMERICAL RESULTS
In this Section we present the numerical results. First,we start by describing the influence of the long-rangeinteraction into Majorana modes lifetimes (Sec. III A).Next, we describe the spatial structure of the Majo-rana modes in the presence of the long-range interaction(Sec. III B).
A. Majorana modes lifetimes
Results in Figs. 1–4 show the most stable λ , whichwe found by solving Eq. (4). Note that the biggest λ isdoubly degenerate – for each MZM: Γ + and Γ − , whichare defined later. To study the influence of long-rangeinteraction, we study each V r independently, consideringonly one non-zero r -nearest neighbor interaction V r forgiven r . In Fig. 1 we compare influence of V r and µ on V r / t (a) V (b) V V r / t µ / t (c) V µ / t (d) V λ (d) V Figure 1. MZM correlation function λ as a function of chem-ical potential µ and r -nearest neighbor interaction strength V r for ∆ /t = 0 . , τ = 50 and L = 10 . Panels correspondto different long-range interaction V r , as labeled. Black andwhite contour marks λ = 0 . and 0.1, respectively. λ . It is known that moderate interaction V can lead tothe broadening of the topological regime [54]. The samefeature can be seen in our result, note the contour λ = 0 . in Fig. 1(a). However, this topological phase broadeningis much smaller, when longer range interactions V , V and V are present in the system [Fig. 1(b)–Fig. 1(d)].Moreover, increasing the interaction range r decreases thearea of strong MZM [see yellow area under the λ = 0 . contour in Fig 1(a)–Fig 1(d)]. Here we can notice that V r / t (a) V (b) V V r / t ∆ / t (c) V ∆ / t (d) V λ (d) V Figure 2. The same as in Fig. 1, but as a function of ∆ and V r . Black and red contour marks λ = 0 . and 0.5, respectively.( µ = 0 ) V r / t (a) τ = (b) τ = V r / t µ / t (c) τ = µ / t (d) τ = λ (d) τ = Figure 3. Finite time τ scaling. The same as in Fig. 1(c),but for different times τ = 1 , , , . Black and whitecontour marks λ = 0 . and . , respectively. the transition from trivial to topological regime does notoccur exactly at | µ | = 2 t in the case without interactions( V r = 0 ), due to a finite-size effect [1].In Fig. 2 we present the same as in Fig. 1, but as afunction of ∆ , instead of µ . Again one can see the topo-logical phase decreasing as the interaction range r grows.One can see here characteristic line along ∆ /t = 1 . Thisfading line is related to the fact that Kitaev model for thenon-interacting case ∆ = | t | and µ = 0 (special param-eter tweak) contains MZM, which are exactly integralsof motion even for finite system size [1]. It seems that V r / t (a) L = (b) L = V r / t ∆ / t (c) L = ∆ / t (d) L =
12 0.00.20.40.60.81.0 λ (d) L = Figure 4. Finite size L scaling. The same as in Fig. 2(d), butfor different system sizes L = 6 , , , and time τ = 100 .Black contour marks λ = 0 . . V r / t (a) V (b) V V r / t µ / t (c) V µ / t (d) V − − − δ E / t (d) V Figure 5. Ground-state degeneracy δE as a function of inter-action V r and chemical potential µ . Panels correspond to thedifferent long-range interaction V r , as labeled. System param-eters the same as in Fig. 1. for large τ and ∆ /t (cid:29) MZM are absent in the sys-tem. However, this is only a finite size effect, which weexplained in detail in Fig. 4.To study the topological phase in the thermodynamiclimit L → ∞ and τ → ∞ one should be extremelycareful doing the size/time scaling. Except for specialparameter tweak, following limit always tends to zero: lim L →∞ lim τ →∞ λ = 0 . In Fig. 3 we show how λ vanishesover time τ for the selected case. However, there is a non-zero topological regime even for a large τ = 1000 and for V r / t (a) V (b) V V r / t µ / t (c) V µ / t (d) V ∆ E / t (d) V Figure 6. Energy gap ∆ E as a function of interaction V r and chemical potential µ . Panels correspond to different long-range interaction V r , as labeled. System parameters same asin Fig. 1. a relatively small system with L = 10 . In a contrast, limit Λ = lim L →∞ lim τ →∞ λ in general can be different than0. The order of these limits is essential, for an almoststrong MZM value of Λ (cid:39) [62]. In Fig. 4, a finite-sizescaling is presented. The procedure of extrapolation of Λ can be found in Ref. [62]. However, in this work, tocompare the influence of interaction range r , finite timeresults are sufficient for the discussion.Next, we check the necessary condition for soft MZM,i.e. degeneracy of the ground-state energies δE = | E o0 − E e0 | and spectral gap ∆ E = min { E e1 − E e0 , E o1 − E o0 } , where E e n ( E o n ) is n -eigenenergy from even (odd)parity regime [60]. Ground-state degeneracy δE and en-ergy gap ∆ E results for different interaction V r range r can be found in Figs. 5 and 6, respectively. Surpris-ingly, one may conclude from the results presented inFig. 5 that increasing the interaction range r topologicalphase increases as a consequence of increasing the yellowregime, where δE is small. Simultaneously, in Fig. 6, thearea with a bigger energy gap ∆ E grows with the inter-action range r . It should be stressed that the δE con-dition is necessary, but it is not sufficient. In Fig. 5(a)one can identify a few yellow stripes. These lines sep-arate regions where the ground-state average particlenumber (cid:104) N (cid:105) = (cid:104) (cid:80) i a † i a i (cid:105) is close to an integer value: , , . . . , L (see Supplementary Material for Ref. [62]).These lines are the consequence of energy level crossingsand are not related to MZM presence in the system. B. Spatial structure of Majorana modes
To study the spatial profile of the MZM, we can expressthe Γ state in the Majorana basis, which was includedearlier (cf. Sec. II). Then, we can find a pair of orthogonaloperators Γ + and Γ − : Γ ± = L (cid:88) i =1 α ± i γ ± i , (6)which describe a projection of the Γ states into the pure–Majorana γ states. Because of this, every Γ ± state con-tains only α ± i (cid:54) = 0 (describing contribution of the γ ± state), while in the same time α ∓ i = 0 . Examples of nu-merical results are shown in Fig. 7, where | α + i | + | α − i | ispresented. This quantity corresponds to the local densityof states [69] or differential conductance [70]. Addition-ally, in the case of a uniform chain, due to symmetry inthe chain midpoint, the coefficients for Γ + and Γ − mustbe swapped in space, i.e., α + i = α − L +1 − i . Using such con-straint, one can generate coefficients only for one of Γ ± to study the spatial structures. As we can see, increasinginteraction range r leads to decrease of the MZM local-ization, i.e. when r grows, the sum | α + i | + | α − i | at thecenter of the chain increases [cf. Fig. 7(b)–Fig. 7(d)]. Atthe same time, the value of this expression decreases atthe ends of the chain. Such behavior can be explained bydecreasing the overlap between MZM located at both left | α + i | + | α − i | site iV V V V Figure 7. Spatial structure of the Majorana states Γ + and Γ − . Results for L = 10 , ∆ /t = 0 . , V r /t = 1 , and µ/t = 0 . . and right ends of the chain. In contrast, the interactionbetween nearest-neighbor sites leads to the stabilizationincrement of the MZM [57, 62]. Moreover, this empha-sizes the importance of many-body interactions on theMZM lifetime.As a degree of non-locality of the two Majorana states,we can define their overlap [12, 46]: Ω = (cid:107) Γ + (cid:101) Γ − (cid:107) = L (cid:88) i =1 | α + i α − i | , (7)where (cid:101) Γ − = U Γ − U † is reflection in space of Γ + and U is unitary operator described by transformation of ba-sis matrices U γ ± i U † = γ ∓ i . From the definition, Ω takesranges from 0 (no overlap) to 1 (perfect overlap). Here,we note that the Ω strongly depends on L [71]. More-over, this quantity can be associated with the resilienceof the Majorana qubit to local environmental noise, withcomplete non-locality Ω = 0 denotes topological qubitprotection [46].In general, the overlap Ω can be controlled by some pa-rameter modification, like electrostatic potential [72–75]or inter-site interactions [57, 62]. In our case, we control | α + i α − i | ( × − ) site iV V V V Figure 8. Local overlap | α + i α − i | between MZM Γ + and Γ − .System parameters are the same as in Fig. 7. V r / t (a) V (b) V V r / t µ / t (c) V µ / t (d) V − − − − Ω (d) V Figure 9. Overlap Ω between left, and right Majorana states,as a function of interaction V r and chemical potential µ . Pan-els correspond to different long-range interaction V r , as la-beled. System parameters are the same as in Fig. 1. Ω by modification of the long-range interaction V r andthe chemical potential µ in the whole system, for whichthe result is presented in Fig. 9. For weak V r and doping µ overlapping is exponentially small. When interaction V r increases, Ω decreases – this effect seems to be indepen-dent of interaction V r range. As one can see, the MZMoverlap Ω is more sensitive to controlling the chemicalpotential µ than interaction V r modifications.Similar behaviour can be observed in Fig. 10, wherewe show site index i for which the “local” overlapping V r / t (a) V (b) V V r / t µ / t (c) V µ / t (d) V s it e i (d) V Figure 10. The site number i for which | a + i a − i | is maximal asa function of interaction V r and chemical potential µ . Panelscorrespond to different long-range interaction V r , as labeled.System parameters are the same as in Fig. 1. | a + i a − i | reaches maximal value. As one can see, increasingchemical potential µ leads to a stronger overlap betweenMajorana states (at the center of the chain, i.e., i = 5 ).In contrast, increasing of the long-range interaction V r leads to increasing overlapping near the edge of the chain– the maximal value of overlap is more visible outsidethan in the center of the chain. Note that, fast changesof site index i for µ/t ∼ are associated with numericalaccuracy, i.e. local overlap | α + i α − i | is relatively small andcomparable for all i (small variance). IV. SUMMARY
We have studied the influence of interaction range onthe Majorana zero mode lifetime and spatial structure inthe Kitaev chain. The Majorana zero mode’s lifetime isan important quantity from a practical point of view andcan be related to the topological qubit decoherence time.For the practical application of Majorana zero modes,one needs to extend the decoherence time, which will aidthe realization of a quantum computers based on theirnon-Abelian properties.From previous theoretical calculations based onDMRG methods, moderate repulsive interactions be-tween the nearest sites can lead to the stabilization ofthe topological order [56, 61]. It should be emphasizedthat the dissipation and dephasing of the Majorana zeromodes have also been studied in the presence of near-est neighbor interactions [60]. In this case, the dissipa- tion and dephasing noises can induce parity- and non-parity preserving transitions. Moreover, the dissipationand dephasing rates can be reduced by increasing the in-teraction strength at sufficiently low temperature, whichcan lead to extended coherence times for the Majoranamode [60].In this paper, we have shown that long-range inter-action strongly modifies the lifetime of the Majoranazero mode. These interactions decrease the lifetime of theMZM. Moreover, we have discovered that interaction be-tween particles located at distant sites is more significantthan the interaction between nearest neighbors. This be-havior can have a crucial role from the practical pointof view in real materials, where interaction decays withdistance. This destructive character can be crucial forthe practical implementation of Majorana zero modes astopological qubits. This type of interaction leads to theoverlap between two Majorana bound states localized atthe opposite end of the chain. Naturally, it can be a sourceof decoherence of these states. In summary, to guaranteethe efficiency of quantum computers based on Majoranazero modes, the suppression of the long-range interactionis required.
ACKNOWLEDGMENTS
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