Aharonov-Bohm Effect in Three-dimensional Higher-order Topological Insulator
AAharonov-Bohm Effect in Three-dimensional Higher-order Topological Insulator
Kun Luo, Hao Geng,
1, 2
Li Sheng,
1, 2
Wei Chen,
1, 2, ∗ and D. Y. Xing
1, 2 National Laboratory of Solid State Microstructures and school of Physics, Nanjing University, Nanjing, 210093, China Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China (Dated: January 29, 2021)Hinge states are the hallmark of the 3D higher-order topological insulator(HOTI). Here, we showthat chiral hinge states can be identified by the magnetic field induced Aharonov-Bohm(AB) oscil-lation of the electron conductance in the interferometer constructed by HOTI and normal metal.Unlike AB interferometer of 3D topological insulator(TI), we find that there are different AB oscil-lation frequencies for a given direction of magnetic field in 3D HOTI. And the oscillation frequenciesare also strongly depending on the direction of magnetic field. The main conclusion in our workis that there exists a universal linear relation between different oscillation frequencies. Here, byconstructing an interference model of hinge states loops, we show both analytically and numericallythat the linear relation is fulfilled in the HOTI effective model. The four basic frequencies in thework are labeled as ω x , ω y , ω x + y , ω x − y and the main linear relations we demonstrate here are ω x ± y = ω x ± ω y . These results provide an effective way for the identification of the chiral hingestates, and the oscillation signatures are stable with different sample size and bias. I. INTRODUCTION
Over the past decade, topological phases of mattersuch as topological insulator and superconductor havebecome a new research field of condensed matter physics[1, 2]. In recent years, all kinds of higher-order topolog-ical insulator and superconductor are proposed theoret-ically [3–17]. The fingerprint of 3D HOTI is the exis-tence of topologically protected gapless hinge states atthe sample boundary. And the gapped surface and bulkstates are another characteristic feature in 3D HOTI. Al-though, the hinge states are confirmed by using scanning-tunnelling spectroscopy and Josephson interferometry[17], studies of Aharonov-Bohm(AB) interference are ab-sent.One of the expression form of the AB effect in electronsystem is the periodic oscillation of conductance for anelectron traveling around magnetic flux Φ[18, 19]. Theperiod of oscillation is equal to quantum flux Φ = h/e ,and the frequency of oscillation is equal to 2 π/ Φ . Re-cently, AB effect is used as an effective method to detectedge states in various materials, such as edge states oftopological insulators[20–23], Majorana fermions of topo-logical superconductors[24–27], surface states of topolog-ical semimetal[28, 29], and non-Abelian anyons of frac-tional quantum Hall effect[30–34].The features of 3D HOTI are propitious to use ABinterference to study these hinge states. 3D HOTI hasideal insulating surface and bulk sates, and the metallichinge states, which can realize an ideal AB interferom-eter. Because of hinge states, this interferometer couldsupport abundant interference phases and quite differ-ent from previous ones. In 3D HOTI, there are manydifferent interference loops. So, the phase shifts expe-rienced by electrons traveling along the different loops ∗ Corresponding author: [email protected] (a) (b)(c) (d) (e) B B FIG. 1. (a) The interferometer constructed by HOTI andnormal metal. The HOTI is represented by a green cuboid,the normal metal are represented by orange cuboid and leadis represented by black line. Weak magnetic field B is im-posed in x − y plane and angle to x -axis is θ . The mainly ABinterfering loop is shown in (b,c), with fluxes through themΦ x , Φ y respectively. (d,e) are shown superposition of twodifferent interference paths, with fluxes Φ x + Φ y , | Φ x − Φ y | .According to superposition principle, the two terminal trans-mission probability is affected by all fluxes, Φ x , Φ y , Φ x + Φ y , | Φ x − Φ y | . could be quite different. Specifically, an electron travelsalong hinges in a closed loop will experience a phase shift2 π Φ / Φ , where Φ is the magnetic flux through the closedloop.In our work, we investigate coherent electron transportthrough an AB interferometer composed of normal metaland HOTI; see Fig. 1(a). By applying a weak magneticfield in x − y plane in our setup , there are two basicloops that electron can travel along.They are boundariesof (1,0,0) and (0,1,0) surfaces as can be seen in Fig.1(b,c).The two loops with Φ x , Φ y magnetic fluxes through maygive electron wave function different phase shifts, andthese finally result in two main frequency components ω x , ω y respectively in the oscillations of conductance signals.In Fig. 1(d,e), there are two types of superpositions of the a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n two basic loops described above. In Fig. 1(d), electronfirstly travels around (1,0,0) surface boundary duringwhich the phase shifts 2 π Φ x / Φ , and then reflects backto travel around (0,1,0) surface boundary which givesanother 2 π Φ y / Φ phase shift. So finally, the electronexperiences 2 π (Φ x + Φ y ) / Φ phase shift and this elec-tron states inference with the reflectless states result insecondary oscillation frequency ω x + y of the conductancesignals. Similarly in Fig. 1(e), another secondary oscilla-tion frequency ω x − y appears. Therefore, the AB oscilla-tion of the transmission probability takes the form T ≈T x + y cos[2 π (Φ x + Φ y ) / Φ ] + T x − y cos[2 π (Φ x − Φ y ) / Φ ] + T x cos(2 π Φ x / Φ ) + T y cos(2 π Φ y / Φ ) + C . Remarkably,it’s interesting to find that AB oscillation frequencies sat-isfy linear relations ω x ± y = ω x ± ω y which only exist in3D HOTI.Above all, we can see the signatures of the oscillationconductance signals here are quite different from the or-dinary ones and the TI ones. In the ordinary 2D interfer-ometer, the electron experience a unique basic phase shiftwhen it goes around a closed loop. The frequency com-ponents of oscillation conductance signals only contain atypical frequency ω t and the multiples of ω t . But in ourHOTI setup, the frequency components of conductancesignals are composed of the linear combination of twobasic frequencies. For the 3D TI, the transmit probabil-ity has Φ / × π Φ / Φ because clockwise and anticlock-wise loop has opposite phase 2 π Φ / Φ . And TI naturallyhas strong spin-orbit coupling which cause weak anti-localization with maximum conductance at zero flux. Incontrast to the 3D TI with maximum or minimum trans-mission probability occurs at Φ = 0, extreme transmis-sion probability here strongly depends on the incidentenergy. Therefore, our scheme provides an unambiguousevidence to identify chiral hinge state in 3D HOTI.The rest of this paper is organized as follows. In Sec.II, we explicate the basic physical picture of the main re-sults. In Sec. III, we adopt a scattering matrix analysisbased on the effective model of the chiral hinge states tosolve the four oscillation periods of the transmit probabil-ity. Detailed numerical simulations on the lattice modelis conducted in Sec. IV to show the particular resultsfor the chiral hinge states. Finally, a brief summary andoutlook are given in Sec. V. II. PHYSICAL PICTURE
Our model is 3D chiral HOTI inturoduced bySchindler[7] and a perpendicular to z -axis weak mag-netic field is applied. Without applied magnetic field,this model is gapped in bulk and the four surface whichparallel to z -axis with open boundary condition. Andthe sigh of mass in these four surface is alternated trans-lation from one to neighbor one. Jackiw-Rebbi boundstates has arisen at the hinge between two opposite sign surface, which is called hinge states in HOTI. If thereis not time-reversal symmetry in this model, the hingestates are chiral. However, the perpendicular to z -axissurface is still gapless, because the Dirac cone is topo-logical protected by ˆ C z ˆ T symmetry. It is natural to con-sider z direction transport by imposing perpendicular toz-axis weak magnetic field. And in order to form a closedloop, the cross-section area of normal metal lead is needsmaller than HOTI.The AB interferometer constructed by the 3D HOTIand normal metal is illustrated in Fig. 1(a). The the bluearrows indicate group velocity directions of the chiralhinge states. The HOTI is represented by a cube and thenormal metal leads are represented by cylinders. Weakmagnetic field B is imposed in x − y plane. When anelectron travels along a closed loop, the effective flux canbe expressed as Φ = (cid:82) S B · dS with the area S enclosedby the loop and dS the differential area. Consider anelectron propagates coherently in the AB interferometer,which contains multiple interfering loops. Two typicalones among them are the main AB interfering loops, asshown in Fig. 1(b,c). Theses two AB loops are composedof the forward and backward transport channels at z -direction.And the the flux of each loop is Φ x or Φ y whichresults in ω x or ω y oscillation frequency. And these ABinterfering loops can be assembled into other interferencepaths which are shown in Figs. 1(d,e) that giving rise to ω x + y , ω x − y . Therefore, these frequencies are naturallysatisfied the universal linear relations ω x ± y = ω x ± ω y .Although rotating direction of magnetic field will changethese four frequencies, this linear relations still hold. Andthese relations are independent of the system parametersand sample size. III. SCATTERING MATRIX ANALYSIS
In this section, we study electron transport in the inter-ferometer by using the scattering matrix approach basedon the low-energy effective model of the hinge states.The scattering matrix of the whole interferometer is ob-tained by combining the scattering matrices at two nor-mal metal-HOTI interfaces and involving the phase ac-cumulation during electron propagation in HOTI. Thescattering matrix at the down interface can be parame-terized as S d = r r t (cid:48) t (cid:48) r r t (cid:48) t (cid:48) t t r (cid:48) r (cid:48) t t r (cid:48) r (cid:48) , (1)which relates the incident and outgoing waves via b d = S d a d . The four components of the wave a d , b d corre-spond to the forward and backward channels on the nor-mal metal and HOTI. Here, the unitary matrix condition S d S † d = I should be satisfied due to the current conser-vation. Here, t , t are the transmission amplitudes fromone channel in normal metal transmit to two chiral hingestates, respectively; r , r are the reflection amplitudesfrom one channel in normal metal reflect to two channelsin normal metal. The other channel in normal metal hasanalogous scattering process, and the elements of scat-tering matrix are t , t , r , r . The scatter process fromHOTI to normal metal is identified by t (cid:48) , r (cid:48) . The matrix S u for the up interface between HOTI and normal metalis defined in the similar way. S u = r u r u t u (cid:48) t u (cid:48) r u r u t u (cid:48) t u (cid:48) t u t u r u (cid:48) r u (cid:48) t u t u r u (cid:48) r u (cid:48) , (2)The phase modulation of the interferometer comes fromthe flux Φ in the different area encircled by the interferingloop, which can be described by the matrix S m = e iω
00 0 0 e − iρ e − iφ e iφ (3)where φ , φ , ω, ρ are the AB phases due to the mag-netic field, with the equation φ x = φ + ω = φ + ρ =2 π Φ x / Φ ; φ y = φ − ρ = φ − ω = 2 π Φ y /φ ; φ x + y = φ + φ = 2 π (Φ x + Φ y ) / Φ ; φ x − y = ρ + ω = 2 π (Φ x − Φ y ) / Φ being gauge invariant.Combining three matrices S d , S m , S u yields the scat-tering matrix for the whole interferometer. In the follow-ing, S d , S u is adopted as real matrix for simplicity, whichwill not change the main results. For an electron injectedfrom down terminal, its transmission probability T to upterminal is obtained after some algebra as T = F − (cid:104) C + X cos φ x + Y cos φ y + XY cos φ B + XY (cid:48) cos φ (cid:48) B (cid:105) F = M C + M B cos φ B + M (cid:48) B cos φ (cid:48) B + M X cos(2 φ x )+ M Y cos(2 φ y ) − M X cos φ x − M Y cos φ y , (4)where the specific coefficient are shown in Appendix A.Leave out the small order, M X ( Y ) , scattering prob-abilities in Eq. (4) contains the information of the ABeffect with four different oscillation frequencies. For aincident energy within surface and bulk gap, the trans-mission T is dominated by the AB oscillation with fourdifferent frequencies ω x , ω y , ω x + y , ω x − y . These four fre-quencies naturally satisfies ω x ± y = ω x ± ω y . As rotatingdirection of magnetic field will change these four frequen-cies, but this equation still holds.The previous conclusion is generally holds, althoughwe have obtained it by omitting higher-order small or-der M X ( Y ) . Now we provide numerical results of Eq.(4) for a general case shown in Fig. 2. The transmissionprobability and FFT spectrum for fixed scattering matrix S u , S d and direction of magnetic field are shown in Figs. ( b ) probability f l u x [ F ] FFT w [1/F ] ( a ) FIG. 2. For fixed scattering matrix S u , S d , AB oscillationpatterns of transmission probability by scattering matrix ap-proach is shown in (a). The FFT spectrum of the transmissionprobability is shown in (b). B = ( B x , B y , B x = 0 . B , B y = 0 . B .And the flux in Fig. 2(a) is B · S , where S is the area of(1,0,0) surface. We can find that there are multiple oscil-lation frequencies in transmission probability. Accordingto FFT spectrum Fig. 2(b), the oscillation frequenciesare ω x = 0 . / Φ , ω y = 0 . / Φ , ω x − y = 0 . / Φ , ω x + y =1 / Φ , ω x = 1 . / Φ , ω y = 0 . / Φ . The oscillation fre-quencies ω x , ω y are not novel, they always appear inprevious AB effect. But ω x ± y are distinctive oscillationfrequencies which only exist in 3D HOTI. And the equa-tion ω x ± y = ω x ± ω y is true according to peaks of FFTspectrum as elucidated in Sec. II. IV. LATTICE MODEL SIMULATION
Based on the scattering matrix analysis, we can seethat there are four apparent frequencies and they satisfyuniversal relations ω x ± y = ω x ± ω y . Next, we performnumerical simulation to give rigorous results. We adoptthe Schindler’s model to describe the HOTI [7] H HOTI = (cid:32) M + t (cid:88) i cos k i (cid:33) τ z σ + ∆ (cid:88) i sin k i τ x σ i + ∆ (cos k x − cos k y ) τ y σ , (5)where σ i and τ i , i = x, y, z are Pauli matrices actingon the spin and orbital space, respectively. For 1 < | M/t | < , ∆ (cid:54) = 0, H HOTI represents a chiral3D HOTI. We map the effective model onto the squarelattice (see Appendix B for details) and calculate thescattering probabilities using Kwant package [35].For the AB transport in the z -direction, the latticeis built as shown in Fig. 1(a). The cross-section of the ( c ) G f l u x [F ] t a n (cid:1) = 2 / 3 , i e = 0 . 1 t a n (cid:1) = 1 / 4 , i e = 0 . 1 t a n (cid:1) = 2 / 3 , i e = 0 . 0 5 ( b ) FFT w [1/F ] t a n (cid:1) = 2 / 3 , i e = 0 . 1 t a n (cid:1) = 2 / 3 , i e = 0 . 0 5 ( a ) w x - y w x w y w x + y w x + y w y w x w [1/F ] t a n (cid:1) = 1 / 4 , i e = 0 . 1 w x - y FIG. 3. AB Oscillation patterns of transmission probabilityby lattice simulation are shown in (a). The cross-section ofthe HOTI in the x − y plane has the size 30 a × a , where a is the lattice constant. The black line is shown for B x =0 . B , B y = 0 . B and incident energy ie = 0 .
05, the red lineis shown for B x = 0 . B , B y = 0 . B and incident energy ie = 0 .
05, the blue line is shown for B x = 0 . B , B y = 0 . B and incident energy ie = 0 .
05. The FFT spectrums of thetransmission probability are shown in (b,c). The peaks ofFFT are the oscillation frequencies of AB effect.
HOTI in the x − y plane has the size M a × N a , where a isthe lattice constant, and M, N are integers. In our setup,we choose normal metal as lead. The magnetic field B isonly applied on the HOTI. As considering an electron in-jected from down terminal with an energy within the bulkand surface gap of the HOTI, only the hinge channelsare available for propagation. The two terminal conduc-tance G is calculated for B = ( B x , B y ,
0) with incidentenergy ie = 0 . ie = 0 .
05 as shown in Fig. 3(a), where B x = 0 . B , B y = 0 . B or B x = 0 . B , B y = 0 . B .The FFT spectrum of two terminal conductance areshown in Fig. 3(b,c). The numerical calculations havethe analogous results which resemble those from the scat-tering matrix analysis in Fig. 2(a,b). In Fig. 3(a), G is dominated by the several different oscillation pe-riod which is the particular feature in HOTI. For dif-ferent incident energy, the oscillation patterns are dif-ferent but the oscillation frequencies is not. The peakscorresponding to frequencies in FFT spectrum are al-most same in Figs. 3(b). One can see that there arefour notable peaks at 0 . / Φ , . / Φ , . / Φ , . / Φ at black line in the FFT spectrum of G for afixed magnetic field. At blue line in FFT spectrumof G for a fixed magnetic field has similar peaks0 . / Φ , . / Φ , . / Φ , . / Φ . In Fig. 3(c), for an-other direction of magnetic field, there are anotheroscillation frequencies 0 . / Φ , . / Φ , . / Φ , . / Φ .This frequencies are satisfied ω x ± y = ω x ± ω y . This nu-merical calculation is consistent with former analyticalresult. We also calculate change sample size which isshown in Appendix that do not influence our conclusion. numerical data numerical data FIG. 4. The relationship of oscillation frequencies are shownin (a,b). According to different magnetic field, we can gettwo sets of frequencies which are from analytical and numer-ical calculation. The surface and dot plot are according toanalytical and numerical calculation.
To further clear present our conclusion, we plot thefunction of frequencies as shown in Fig. 4 which areoriginated from FFT peaks. The red and blue plane arethe equation that we suggested, and the black and orangesolid circle are the numerical data from peaks of FFTspectrum. We can see that solid circle are near or atthe frequency plane, that shows our conclusion ω x ± y = ω x ± ω y is right. V. SUMMARY AND OUTLOOK
In summary, we investigated the AB effect in 3D chi-ral HOTI. We found that there are several interferencepaths which leads to some different AB oscillation peri-ods. We focused on the relation between those oscillationfrequencies. The oscillation frequencies are satisfied with ω x ± y = ω x ± ω y , which is the particular feature thatcan be used to confirm the chiral hinge states of 3D chi-ral HOTI. Our work provides a promising and effectiveapproach to confirm the 3D chiral HOTI. Although ourstudy is based on 3D chiral HOTI, the idea of detect-ing hinge states can be hopefully extended to 3D helicalHOTI. ACKNOWLEDGMENTS
This work was supported by the National Natural Sci-ence Foundation of China under Grant No. 12074172(W.C.), No. 11674160 and No. 11974168 (L.S.), thestartup grant at Nanjing University (W.C.), the StateKey Program for Basic Researches of China under GrantsNo. 2017YFA0303203 (D.Y.X.) and the Excellent Pro-gramme at Nanjing University.
Note added .—Just as we finished our paper, we find arelated preprint[36].
Appendix A: specific forms of coefficient for analysiscalculation
In equation(4), the specific forms of coefficient are C = ( t + t )( W + W + W + X + X + X ) + ( t + t )( Y + Y + Y + Z + Z + Z ), X = 2( W W + X X )( t + t ) + 2( Y Y + Z Z )( t + t ) + 2( W Y + W Y + X Z + X Z )( t t + t t ), Y = 2( W W + X X )( t + t ) + 2( Y Y + Z Z )( t + t ) + 2( W Y + W Y + X Z + X Z )( t t + t t ), XY = 2( W Y + W Y )( t t + t t ), XY (cid:48) = 2( W W + X X )( t + t ) +2( Y Y + Z Z )( t + t ) + 2( W Y + W Y + X Z + X Z )( t t + t t ), M B = 2 M M + 2 M M , M (cid:48) B =2 M M + 2 M M , M X = 2 M M , M Y = 2 M M , M X = 2 M M + 2 M M , M Y = 2 M M + 2 M M , M C = M + M + M + M + M with W = t u , W = t u r (cid:48) r u − t u r (cid:48) r u , W = t u r (cid:48) r u − t u r (cid:48) r u , X = t u , X = t u r (cid:48) r u − t u r (cid:48) r u , X = t u r (cid:48) r u − t u r (cid:48) r u , Y = t u , Y = t u r (cid:48) r u − t u r (cid:48) r u , Y = t u r (cid:48) r u − t u r (cid:48) r u , Z = t u , Z = t u r (cid:48) r u − t u r (cid:48) r u , Z = t u r (cid:48) r u − t u r (cid:48) r u , M =1+ r (cid:48) r u r (cid:48) r u + r (cid:48) r u r (cid:48) r u − r (cid:48) r u r (cid:48) r u − r (cid:48) r u r (cid:48) r u , M = r (cid:48) r u , M = r (cid:48) r u , M = r (cid:48) r u , M = r (cid:48) r u .And Fig. 2 is obtained by taking the following pa-rameters, t = √ . t = √ . t = √ . t = √ . r = √ . r = √ . r = √ . r = √ . r (cid:48) = √ . r (cid:48) = √ . r (cid:48) = √ . r (cid:48) = −√ . t u = √ . t u = √ . t u = √ . t u = −√ . r u = √ . r u = √ . r u = −√ . r u = −√ . Appendix B: Lattice model for numerical calculation
In order to run numerical calculations, we use a 3Dsquare lattice model for whole system by discretizing thecontinuous effective hamiltonian H . The lattice Hamil- probability f l u x [ F ] FFT w [ ] FIG. 5. AB Oscillation patterns of transmission probabilityby lattice simulation are shown in (a). the blue line is shownfor B x = 0 . B , B y = 0 . B , incident energy ie = 0 . a × a . The FFT spectrumof the transmission probability is shown in (b). The peaks ofFFT are the oscillation frequencies of AB effect. tonian is H = (cid:88) i c † i M σ τ z c i + (cid:40) (cid:88) i c † i + x (cid:20) e iφ x σ τ y + tσ τ z + i ∆ σ x τ x ) (cid:21) c i − (cid:88) i c † i + y (∆ σ τ y + tσ τ z + i ∆ σ y τ x ) c i + (cid:88) i c † i + z e iφ z tσ τ z + i ∆ σ z τ x ) c i + h.c. (cid:41) , (B1)where c i = ( c a, ↑ ,i , c b, ↑ ,i , c a, ↓ ,i , c b, ↓ ,i ) are the annihilate op-erators of electron with spin up and spin down in a andb orbits at site i. The Peierls phase φ r = x,z is equal to e (cid:126) (cid:82) r j r i A ( r ) · dr , where A ( r ) is the vector potential. Appendix C: numerical result for different size
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