Higher-order Fabry-Pérot Interferometer from Topological Hinge States
HHigher-order Fabry-Pérot Interferometer from Topological Hinge States
Chang-An Li, ∗ Song-Bo Zhang, † Jian Li,
2, 3 and Björn Trauzettel Institute for Theoretical Physics and Astrophysics,University of Würzburg, 97074 Würzburg, Germany School of Science, Westlake University, 18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China Institute of Natural Sciences, Westlake Institute for Advanced Study,18 Shilongshan Road, Hangzhou 310024, Zhejiang Province, China (Dated: January 25, 2021)We propose an intrinsic 3D Fabry-Pérot type interferometer, coined “higher-order interferometer”,that utilizes the chiral hinge states of second-order topological insulators and cannot be equivalentlymapped to 2D space because of higher-order topology. Quantum interference patterns in the two-terminal conductance of this interferometer are controllable not only by tuning the strength butalso, particularly, by rotating the direction of the magnetic field applied perpendicularly to thetransport direction. Remarkably, the conductance exhibits a characteristic beating pattern withmultiple frequencies with respect to field strength or direction. Our novel interferometer providesfeasible and robust magneto-transport signatures to probe the particular hinge states of higher-ordertopological insulators.
Introduction.—
Higher-order topological insulators(HOTIs) feature gapless excitations, similar to tra-ditional (first-order) topological insulators, that areprotected by bulk electronic topology but localized atopen boundaries at least two dimensions lower than theinsulating bulk [1–16]. For instance, 3D second-ordertopological insulators (SOTIs) host 1D chiral or helicalstates along specific hinges of the systems. In recentyears, HOTIs have triggered widespread research inter-est, owing to their discoveries in a variety of candidatesystems, promotion of our understanding of topologicalstates of matter, and potential applications [17–39].So far, most efforts have been put into the potentialrealization and electronic characterization of HOTIs.However, the transport properties of HOTIs remainlargely unexplored, despite of a few works associatedwith superconductivity [40–42]. Indeed, for 3D SOTIs,an intriguing open question is whether the emergenthinge states can exhibit any particular phenomena innormal-state transport.One appealing route towards this question involves in-terferometers built of SOTIs, which enable us to studyquantum-coherent transport of hinge states. Propagat-ing hinge states that form interference loops enclosing amagnetic flux applied to the system pick up an Aharonov-Bohm (AB) phase [43]. In presence of quantum co-herence, the AB phase will give rise to quantum oscil-lations in transport characteristics such as the chargeconductance. Quantum interference patterns in the two-terminal conductance have been employed to detect topo-logical phases of matter, for instance, surface states oftopological insulators [44–47], chiral Majorana modes[48–51], and topological Dirac semimetals [52].In this work, we propose a higher-order Fabry-Pérotinterferometer to probe hinge states of SOTIs. Our basicsetup, shown in Fig. 1(a), is composed of a rectangularchiral SOTI in contact with two leads. The chiral hinge B Energy (b)(a)
LeadLead (c)
Figure 1. (a) Schematic of the higher-order interferometer: aSOTI with four chiral hinge states (solid red and dashed bluelines) are connected to two leads (yellow). Adjacent chiralhinge states form interference loops in the presence of finitereflections at the interfaces. A magnetic field B perpendicularto z -direction is applied to the SOTI (gray). (b) The hingestates have linear dispersion and are shifted in k z -direction by B . (c) Density plot of conductance with respect to the fieldstrength B and direction θ . B = φ /S f with φ the fluxquantum and S f the area of the front surface of the SOTI. states, existing in 3D space, form interference loops dueto finite reflections (not shown) at the two interfaces,and their energy dispersions split in a non-uniform man-ner under magnetic fields, as shown in Fig. 1(b). Partic-ular quantum interference patterns in the two-terminalconductance, arising from the AB effect as exemplifiedby Fig. 1(c), can be observed either by tuning the field a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n strength B or direction θ . In addition, owing to the in-trinsic 3D nature of the interferometer, there are gener-ally two frequencies in the magneto-conductance oscilla-tions, leading to a beating pattern. These features do notdepend on the details of the junction, such as the elec-tronic spectrum of the leads, and are stable against disor-der and dephasing. Hence, they provide robust transportsignatures of hinge states in 3D SOTIs. General analysis based on scattering matrix the-ory.—
Our proposed interferometer involves a 3D SOTIwith four chiral hinge states attached to two leads in z -direction, as sketched in Fig. 1(a). Adjacent chiral hingestates form interference loops because of finite reflectionsat the interfaces, as will be discussed below. A magneticfield B = B (cos θ, sin θ, in x - y plane is applied in theSOTI region, where B measures the field strength and θ the field direction with respect to x -direction.Before presenting concrete results based on specificmodels, it is instructive to analyze the main transportfeatures of the interferometer using a phenomenologicalscattering matrix approach [53–55]. The transport prop-erties of the setup are encoded in a scattering matrixthat directly connects the conducting channels in the leftand right leads. The scattering processes at the twointerfaces between the leads and the SOTI can be de-scribed by two scattering matrices, respectively. Eachmatrix consists of four components: transmission fromleft to right t L/R , transmission from right to left t (cid:48) L/R ,reflection from the right r L/R and reflection from the left r (cid:48) L/R , where the subscript ( L and R ) distinguishes theleft and right surfaces. At low energies, the only con-ducting channels in the SOTI are the four hinge stateswhich have linear dispersion and are localized at thefour different hinges of the cuboid. In the presence ofa magnetic field, their propagation in the SOTI will pickup AB phases that can be described by a phase matrix U ≡ e iλ e iϕσ z ⊗ σ / e iφσ z ⊗ σ z / , where σ z is a Pauli ma-trix, σ the × identity matrix, ϕ = BLW x cos θ and φ = BLW y sin θ (1)are the magnetic fluxes threading the two surfaces, re-spectively, with L the distance between the two leads and W x/y the widths of the sample in x/y -directions. More-over, λ = k F L is the dynamical phase with k F the Fermiwave number in the absence of magnetic fields. By elim-inating the scattering amplitudes in the SOTI region, wederive analytically an effective × scattering matrixthat directly connects the two interfaces [56] S ( B, θ ) = Φ + ( e − iλ − e iλ r L (cid:48) Φ − r R Φ + ) − , (2)where the phase matrices Φ ± ≡ e i ( ϕ ± φ ) σ z / account forthe AB phase differences between the two right-movingand between the two left-moving hinge channels, respec-tively. At zero temperature, the two-terminal conduc- tance of the setup can be evaluated as G ( B, θ ) = e h Tr[ t † R t R S ( B, θ ) t L t † L S † ( B, θ )] , (3)where h is the Planck constant and e is electron charge.According to Eqs. (2) and (3), if there is no transmissionacross any of the two interfaces, i.e., t L = 0 or t R =0 , then G vanishes. In the opposite limit, where theinterfaces are completely transparent for the hinge states, r L (cid:48) = 0 and r R = 0 , we find that the matrix S as well as t † R t R and t L t † L become diagonal. As a result, G becomesquantized at e /h and is independent of the magneticfield. These results indicate the necessary condition fora successful interferometer: non-trivial transmission andreflection at the two interfaces for the hinge states.When the interfaces are partially transparent, Eq. (2)indicates the formation of Fabry-Pérot interference loops.Moreover, the matrix S contains explicitly two phases ϕ ± φ in general. This indicates the appearance ofbeating patterns with two frequencies in the magneto-conductance. Notably, the two frequencies are intimatelyconnected to the magnetic fluxes threading the differentsurfaces of the SOTI. They are solely determined by thegeometry of the sample and insensitive to the details ofthe interface barriers. The oscillation pattern of G re-mains qualitatively the same even in the presence of adynamic phase. We verified these results by properlyparametrizing the scattering matrices [56]. Model simulation and method.—
To demonstrate thesefeatures of the interferometer explicitly, we consider aneffective model for chiral SOTIs [7] H ( k ) = (cid:16) m + b (cid:88) i = x,y,z cos k i (cid:17) τ + v (cid:88) i = x,y,z sin k i σ i τ + ∆(cos k x − cos k y ) τ , (4)where k = ( k x , k y , k z ) is the wave vector. τ = ( τ , τ , τ ) and σ = ( σ x , σ y , σ z ) are Pauli matrices acting on or-bital and spin spaces, respectively; m , b , v and ∆ aremodel parameters. Without loss of generality, we setthe lattice constant and the velocity v to unity hereafter.When < | m/b | < and ∆ = 0 , the model describes3D topological insulators with gapless surface states [57].The surface states are protected by time-reversal sym-metry T = iσ K , where K represents complex conjuga-tion. A finite ∆ (cid:54) = 0 breaks time-reversal and C rotation(with the rotation axis pointing in z -direction) symme-tries individually. It opens gaps in the surface states.However, the ∆ term preserves the combined symme-try C T , as indicated by ( C T ) H ( k x , k y , k z )( C T ) − = H ( k y , − k x , − k z ) . As a result, the gaps opened by ∆ de-pend on the surface orientation, leading to gapless chiralhinge states localized at the hinges connecting differentsurfaces.We take into account the orbital effect of the magneticfield via the Peierls replacement in the hopping interac-tion T ij → T ij exp(2 πi (cid:82) r j r i d r · A /φ ) , where T ij is thehopping amplitude from sites r i to r j , φ = h/e is fluxquantum. A is the vector potential for the magnetic fieldand it is chosen as A = B (0 , , y cos θ − x sin θ ) for con-creteness [58].For simplicity, we model the metallic leads with a con-ventional quadratic energy dispersion and assume only afew transport channels in both leads such that consider-able reflections for the hinge channels are generated atthe interfaces. Furthermore, we consider the size of thesystem to be much larger than the decay length of thehinge states in order to have a well-defined multiple-loopinterferometer based on hinge states. Under these con-siderations, we calculate the two-terminal conductancenumerically, employing the standard Landauer-Büttikerapproach [59–61] in combination with lattice Green func-tions (see the Supplemental Material [56]). We emphasizethat our main results illustrated below remain qualita-tively the same if we choose other models for SOTIs orleads. Quantum interference pattern.—
Now, we analyze thedependence of the conductance G on the magnetic field,combining general scattering theory and concrete numer-ical simulations. Equation (3) implies an oscillation pat-tern of G with respect to the field direction θ . As shownin Fig. 2(a), G ( θ ) is periodic in θ, in accordance withthe scattering theory. Explicitly, we find that for weakmagnetic fields B ≤ B , G ( θ ) is approximately a si-nusoidal function of θ and takes the maximal value at θ = π/ nπ/ , n ∈ { , , , } , when W x = W y . Here, B corresponds to the field strength at which the fluxenclosed by the front surface S f is one flux quantum for θ = 0 . Thus, G ( θ ) has a period of π/ in θ . Moreover, G ( θ ) is minimal at θ = θ c and symmetric in θ − θ c , where θ c = nπ/ . For strong magnetic fields B > B , the num-ber of conductance peaks increases with increasing B , seeFig. 1(c). When W x (cid:54) = W y , the period in θ becomes π but G ( θ ) is still symmetric in θ − θ c .Equation (3) also indicates an oscillation pattern of G with respect to the field strength B, which is again fullyconfirmed by our numerical simulations. When the mag-netic field is applied in x - or y -directions, or at the spe-cific angle θ = ± arctan ( W x /W y ) , G ( B ) exhibits simpleoscillations with a single frequency, see Fig. 2(b). Gener-ally, the oscillating conductance takes maximal or mini-mal values when the interference loop encloses half a fluxquantum. In our cases, G ( B ) takes maximal values atodd multiples of B / for θ = 0 . The oscillation ampli-tude is relatively smaller since only two of the four loopsenclose half a flux quantum at this field direction. For θ = π/ , G ( B ) takes maximal values at odd multiplesof B / √ , where the interference loop also encloses halfa flux quantum, leading to a resonance peak of G ( B ) .These features signify the interferometer formed by hingestates being of Fabry-Pérot type, as we further explainbelow.Notably, there exist beating patterns, as signified by (c) (b)(d)(a)(e)(c) (f) Figure 2. (a) Conductance G as a function of field direction θ at small field strengths B = B and B / √ , respectively.(b) G as a function of B for θ = 0 and π/ , respectively.In these cases, the oscillations have a single frequency. (c)Particular beating patterns as varying B at angle θ = 0 . π .(d) “Irregular” beating patterns as varying θ at a large fieldstrength B = 20 B . (e) The extracted frequencies (squareand circle dots) as a function of θ . The two frequencies canbe described by ω = S | cos θ | and ω = S | sin θ | . (f) The low-energy spectrum of the SOTI in the presence of a magneticfield B = 2 B and θ = 0 . π . Other parameters are L z = 60 a , W x = W y = 12 a , m = 2 , b = − , v = 1 , ∆ = 1 , and the Fermienergy E F = 0 . . Eq. (2), where the matrix S explicitly contains the twophases ϕ ± φ . When the magnetic field deviates away fromthe special directions at θ = nπ/ (with n ∈ { , , , } )and ± arctan ( W x /W y ) , beating oscillations of G ( B ) areclearly observed, as shown in Fig. 2(c). By performingdiscrete Fourier transformation to the beating patterns,we obtain precisely two frequencies ω and ω . These fre-quencies depend strongly on the field direction θ [dottedlines in Fig. 2(e)]. Explicitly, we find that the two fre-quencies can be well described by ω = | S cos θ | = | φ | /B and ω = | S sin θ | = | ϕ | /B [solid lines in Fig. 2(e)], re-spectively, where S is the area of the surfaces of the sys-tem (we consider the case with W x = W y for simplicity).This corresponds exactly to the two AB phases in Eq.(1), in excellent agreement with the results obtained fromscattering-matrix analysis. When θ = nπ/ , only one ofthe two frequencies survives. When θ = π/ nπ/ ,the two frequencies become identical. In both cases, thebeating behavior in the oscillations disappear. Similarly, G ( θ ) also shows beating-like patterns with respect to thefield direction θ with irregular peaks and dips for largemagnetic fields B (cid:29) B , as shown in Fig. 2(d). Thisdirection-induced beating behavior is another manifesta-tion of the two AB phases. Higher-order Fabry-Pérot interference.—
Next, weclarify, in which sense our quantum interference pat-tern is a higher-order Fabry-Pérot type interference. Thetwo frequencies in the beating patterns correspond phys-ically to two areas of interference loops. As rotating themagnetic field, the two frequencies match the effectiveareas of front surface | S cos θ | and top surface | S sin θ | quite well, see Fig. 2(e). This fact indicates: (i) theadjacent hinge states with opposite chirality form effec-tive interference loops and the interference is typically ofFabry-Pérot type; and (ii) there are totally four interfer-ence loops but any two opposite surfaces of the sample(namely, the front and back surfaces, or the top and bot-tom surfaces) have the same effective area because of thechosen symmetry of the system [62]. The interferenceloops are made of chiral hinge modes located in 3D space,protected by higher-order topology. When rotating themagnetic field, one of the frequencies increases, whereasthe other one decreases. Moreover, the ratio between thetwo frequencies depends on θ as S f /S t = W y | cot θ | /W x .Thus, the two frequencies coincide at the critical field di-rections θ c = arctan ( W y /W x ) and π − arctan ( W y /W x ) ,as shown in Fig. 2(e). These features indicate the 3Dnature of the interferometer.The mechanism of the interferometer can be better un-derstood by analyzing the splitting of hinge states un-der magnetic fields. In the absence of magnetic fields,the four chiral hinge states have a double degeneratelinear spectrum in k z -direction, i.e., ± vk z . The mag-netic field gives rise to a spatially varying vector po-tential. Note that the hinge states are localized at dif-ferent hinges of the system. The local vector poten-tial splits the linear spectrum of the hinge states. Un-der the chosen gauge, the spectra of hinge states aresplit as + v ( k z ± δk z ) and − v ( k z ± δk z ) , where the split-tings are determined by δk z = BW x | sin( θ − π/ | / and δk z = BW y | cos( θ + π/ | /2 [56]. Thus, the hinge statesacquire finite momenta even for vanishing Fermi energy[Fig. 2(f)]. When propagating across the SOTI region,the hinge channels pick up extra phases, ± δk z / L z . Suchphases turn out to be exactly the AB phases φ and ϕ ,stemming from the magnetic flux enclosed by each loop.Explicitly, the flux enclosed by front surface S f and topsurface S t of the central SOTI region in Fig. 1(a) aregiven by ( δk z + δk z ) L z and δk z L z , respectively. Plug-ging φ, ϕ = δk z L z into Eq. (2), this indicates that thedependence of G on B can be attributed to the higher- (b)(a) Figure 3. (a) Low-energy spectrum of the SOTI in the pres-ence of a magnetic field. (b) The three frequencies in the caseof C symmetric SOTIs as functions of θ . The lattice modeland related parameters can be found in the Supplemental Ma-terial [56]. order Fabry-Pérot interference of the four hinge states.At special values of θ , say θ = π/ or π/ , one kind ofsplitting vanishes whereas the other one remains, δk z = 0 and δk z (cid:54) = 0 (similar results occur for θ = +3 π/ , − π/ ).In these cases, we have only one frequency. Generalization to C symmetric SOTIs.— So far, wehave focused on the case of chiral SOTIs with four hingestates and a sample with (effective) C symmetry. How-ever, our scattering theory can be generalized and ap-plied to SOTIs with more pairs of hinge states. As anexample, we consider a C symmetric SOTI with threepairs of chiral hinge states [63] and show the spectrum inFig. 3(a). The geometry considered here is a hexagonalprism with C symmetry in x - y plane. The hinge statessplit generally with different amounts of momenta undera magnetic field. If we consider an interferometer similarto the setup in Fig. 1(a), we can also observe characteris-tic oscillations and beating patterns in the conductancewhich depend sensitively on the field direction θ . In thiscase, the conductance G ( θ ) is π/ periodic in θ . Sincethere are three pairs of counter-propagating hinge states,the oscillations can exhibit three frequencies in general[56]. Figure 3(b) illustrates the three frequencies as afunction of field direction θ . Particularly, the oscillationsare described by a single and two frequencies for θ = 0 and θ = π/ , respectively. Discussion and summary.—
In realistic samples, disor-der and dephasing [64] due to environmental noises maybe detrimental to the interference pattern of hinge states.However, we show numerically that the oscillation pat-terns of the conductance in our setups persist under weakdisorder and dephasing [56]. This indicates the robust-ness of our proposal. Our results based on chiral SO-TIs can also be applied to helical SOTIs, which can beregarded as two copies of chiral SOTIs related by time-reversal symmetry. Recently, SOTIs have been proposedin many candidate materials. Among these candidates,bismuth [17] and axion insulators including EuIn As and MnBi Te [65, 66] provide potential platforms to testour predictions.In summary, we have proposed a higher-order Fabry-Pérot interferometer and revealed unique Aharonov-Bohm oscillations arising from topological hinge states bytuning either strength or direction of an applied magneticfield. 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Appendix S1: General scattering matrix analysis
In this section, we present the details for the scattering matrix analysis of the setup in the main text. Suppose thereare p L/R conducting modes at the Fermi level in the left/right leads. We can define generally ( p L/R + 2) × ( p L/R + 2) scattering matrices, S L and S R , to describe the scattering at the left and right interfaces, respectively, (cid:18) b L b L (cid:48) (cid:19) = S L (cid:18) a L a L (cid:48) (cid:19) , S L = (cid:18) r L t L (cid:48) t L r L (cid:48) (cid:19) , (S1.1) (cid:18) b R (cid:48) b R (cid:19) = S R (cid:18) a R (cid:48) a R (cid:19) , S R = (cid:18) r R t R (cid:48) t R r R (cid:48) (cid:19) . (S1.2)Here, a L/R and b L/R indicate the incoming and outgoing modes that propagate in the leads and scatter at the left/rightinterface, respectively; a L (cid:48) /R (cid:48) and b L (cid:48) /R (cid:48) indicate the incoming and outgoing hinge modes that propagate in the SOTIand scatter at the left/right interface, respectively. The scattering matrix S L/R consists of four components t L/R ,t (cid:48) L/R , r
L/R , and r (cid:48) L/R , corresponding to the transmission from left to right, transmission from right to left, reflectionfrom the right, and reflection from the left, respectively. In the center SOTI region, the conducting chiral hinge statespick up an AB phases when applying an external magnetic field. Thus, the incoming and outgoing modes in the SOTIcan be connected by a phase matrix as a R (cid:48) a R (cid:48) a L (cid:48) a L (cid:48) = e iλ/ e i ( ϕ + φ ) / e − i ( ϕ + φ ) / e i ( ϕ − φ ) /
00 0 0 e − i ( ϕ − φ ) / b L (cid:48) b L (cid:48) b R (cid:48) b R (cid:48) , (S1.3)where λ = k F L is the dynamic phase with k F the Fermi wave number in k z -direction and L the length of the SOTI,and the two phases are given by φ = BS sin θ, ϕ = BS cos θ, (S1.4)with θ the angle between the magnetic field direction and x -axis.Substituting Eq. (S1.3) into Eq. (S1.2), we obtain (cid:18) e − iλ/ Φ †− a L (cid:48) b R (cid:19) = S R (cid:18) e iλ/ Φ + b L (cid:48) a R (cid:19) , (S1.5)where Φ ± ≡ e i ( ϕ ± φ ) σ z / and the Pauli matrix σ z acts on (pseudo-)spin space for two left- or right-moving hinge states.Writing Eqs. (S1.5) explicitly, we have e − iλ/ Φ †− a L (cid:48) = e iλ/ r R Φ + b L (cid:48) + t R (cid:48) a R , (S1.6) b R = e iλ/ t R Φ + b L (cid:48) + r R (cid:48) a R . (S1.7)From Eq. (S1.6), we find a L (cid:48) = e iλ Φ − r R Φ + b L (cid:48) + e iλ/ Φ − t R (cid:48) a R . (S1.8)Writing Eq. (S1.1) explicitly, we have b L = r L a L + t L (cid:48) a L (cid:48) , (S1.9) b L (cid:48) = t L a L + r L (cid:48) a L (cid:48) , (S1.10)Plugging Eq. (S1.8) into Eq. (S1.10), we obtain b L (cid:48) = t L a L + r L (cid:48) ( e iλ Φ − r R Φ + b L (cid:48) + e iλ/ Φ − t R (cid:48) a R ) , (S1.11)and hence, b L (cid:48) = (1 − e iλ r L (cid:48) Φ − r R Φ + ) − ( t L a L + e iλ/ r L (cid:48) Φ − t R (cid:48) a R ) . (S1.12)Plugging this result into Eq. (S1.7), we find b R as b R = t R e iλ/ Φ + (1 − e iλ r L (cid:48) Φ − r R Φ + ) − ( t L a L + e iλ/ r L (cid:48) Φ − t R (cid:48) a R ) + r R (cid:48) a R = e iλ/ t R Φ + (1 − e iλ r L (cid:48) Φ − r R Φ + ) − t L a L + [ e iλ t R Φ + (1 − e iλ r L (cid:48) Φ − r R Φ + ) − r L (cid:48) Φ − t R (cid:48) + r R (cid:48) ] a R . (S1.13)Plugging Eq. (S1.10) into Eq. (S1.6), we obtain e − iλ/ Φ †− a L (cid:48) = e iλ/ r R Φ + ( t L a L + r L (cid:48) a L (cid:48) ) + t R (cid:48) a R , (S1.14)and hence, a L (cid:48) = (1 − e iλ Φ − r L (cid:48) ) − ( e iλ Φ − r R Φ + t L a L + e iλ/ Φ − t R (cid:48) a R ) . (S1.15)Plugging Eq. (S1.15) into Eq. (S1.9), we find b L as b L = r L a L + t L (cid:48) (1 − e iλ Φ − r L (cid:48) ) − ( e iλ Φ − r R Φ + t L a L + e iλ/ Φ − t R (cid:48) a R )= [ r L + e iλ t L (cid:48) (1 − e iλ Φ − r L (cid:48) ) − Φ − r R Φ + t L ] a L + e iλ/ t L (cid:48) (1 − e iλ Φ − r L (cid:48) ) − Φ − t R (cid:48) a R . (S1.16)Rewriting Eqs. (S1.13) and (S1.16), we obtain the effective scattering matrix of the junction (cid:18) b L b R (cid:19) = S (cid:18) a L a R (cid:19) , S = (cid:18) r t (cid:48) t r (cid:48) (cid:19) , (S1.17)where t = t R S t L , r (cid:48) = r R (cid:48) + e iλ/ t R S r L (cid:48) Φ − t R (cid:48) ,r = r L + e iλ/ t L (cid:48) S (cid:48) r R Φ + t L , t (cid:48) = t L (cid:48) S (cid:48) t R (cid:48) , S = e iλ/ Φ + (1 − e iλ r L (cid:48) Φ − r R Φ + ) − , S (cid:48) = e iλ/ (1 − e iλ Φ − r L (cid:48) ) − Φ − . (S1.18)With this general scattering matrix, the two-terminal conductance can be written as G ( B, θ ) = e h tr( tt † ) = e h tr( t † R t R S t L t † L S † ) . (S1.19) Appendix S2: Chiral hinge states under magnetic fields
In this section, we demonstrate the splitting behavior of the chiral hinge states when rotating the magnetic field. Inthe absence of magnetic fields, the four chiral hinge states have a double degenerate linear spectrum in k z -direction.The left-moving hinge states cross with the right-moving ones at k z = 0 . The magnetic field gives rise to a spatiallyvarying vector potential. Remember that the hinge states are localized at different hinges of the system. Thelocal vector potential splits the linear spectrum of hinge states. Under the chosen gauge for the vector potential, A = (0 , , B [ y cos θ − x sin θ ]) , the spectrum of hinge states is split as + v ( k z ± δk z ) and − v ( k z ± δk z ) , where thesplitting strengths are determined by δk z = BW x | sin( θ − π/ / | and δk z = BW y | cos( θ + π/ / | . Thus, the splittingof hinge state spectrum strongly depends on the field direction θ .We focus on the splitting of the chiral hinge states in the lower-energy spectrum presented in Fig. S1. At θ = 0 π, two pairs of the hinge states split by equal value. At θ = π/ , only one pair of the hinge states can be split, whereasthe other one remain unaltered, i.e., δk z = 0 . The spectrum at θ = π/ looks the same as for θ = 0 . Another specialfield direction is at θ = 3 π/ at which we have instead δk z = 0 . At θ = π , the spectrum is the same as that at θ = 0 .This evolution with rotating the magnetic field is consistent with the analytical results δk z = BW x | sin( θ − π/ / | and δk z = BW y | cos( θ + π/ / | . Figure S1. Evolution of the hinge states spectrum when rotating the field direction from θ = 0 to π . Here, we choose parameters: L z = 60 a , W x = W y = 12 a , m = 2 , b = − , v = 1 , ∆ = 1 . The field strength is fixed at B = 2 B . Appendix S3: Numerical simulation details
To calculate the conductance, we employ the Landauer-Büttiker formalism [59–61] in combination with latticeGreen functions. The two-terminal conductance is evaluated as G = e h Tr[Γ L G r Γ R G a ] , (S3.1)where the line width function Γ β = i [Σ β − Σ † β ] (S3.2)with the Σ β being the self-energy due to coupling of the lead β ∈ { L, R } to the central region of interest. The retardedand advanced Green function, G r and G a , are obtained as G r = ( G a ) † = ( E F − H c − Σ rL − Σ rR ) − . (S3.3)Here, both the self-energy Σ β and the Green function G r/a can be calculated by using the recursive method [68].In the numerical simulations, we choose the parameters L z = 60 a , W x = W y = 12 a , m = 2 ,b = − , v = 1 and ∆ = 1 for the SOHI, and m = 3 , v = 0 , b = − , and ∆ = 0 , and chemical potential µ = − . for the two leads.Without loss of generality, we set the Fermi energy in the SOHI at E F = 0 . . Appendix S4: Trivial cases of perfect transmission
In this section, we demonstrate the behaviors when the interfaces of the proposed setup are totally transparent.Under this condition, the chiral hinge states do not talk to each other and thus no interference loop is forming. As aresult, the two-terminal conductance G is quantized at e /h and independent of the magnetic field. Let us considertwo scenarios responsible for such transparent interfaces:0 Figure S2. For the two trivial cases 1 (a) and 2 (b) as discussed in this section, the conductance is quantized at e /h and hasno dependence on either the field strength B or field direction θ . • Case 1: the leads are also made of the same SOTIs. Chiral hinge states exist in all regions of space and passfrom one lead to the other lead directly;• Case 2: the leads are made of conventional semiconductors or topological insulators but highly doped. In thiscase, there are too many channels in the leads such that chiral hinge states loose quantum coherence onceentering the leads.The results for these two cases are presented in Fig. S2. The two-terminal conductance is fixed at e /h and has nodependence on neither the field strength B nor the direction θ . Appendix S5: Parametrizing the scattering matrix
In this section, we parameterize the scattering matrix with the help of the numerical method. There are twointerfaces in our proposed setup. Each interface is described by a × scattering matrix, i.e., S L and S R as listedabove, respectively. Due to time-reversal symmetry breaking, S L and S R are unitary matrices.Parametrizing thesescattering matrices is cumbersome because of the choice of at least 16 free parameters. Instead, we obtain thescattering matrices directly from numerical simulations as explained below.Let us consider a simpler junction with two semi-infinite regions in z direction: one region is made of a trivialinsulators in the region z < , and the other regions made of the SOTI in the region z > . The parameters for leadand SOTI are taken the same as those in our interference setup. Then, the interface of this simpler setup mimicsthe left interface of our interference setup. The scattering matrix at this interface can be obtained numerically bycalculating the retarded Green functions for the two regions and then employing the Fisher-Lee relation, or directlyusing the Kwant algorithm [67]. A similar procedure applies for the right interface.Known from the conductance, described by Eqs. (2) and (3) in the main text, the relevant four matrices are t L , t R , r L (cid:48) and r R (or another group t L , t R , r L and r R (cid:48) ). Under the same parameter setting with the original setup,we obtain the four matrices as t L = (cid:18) − . . i, − . − . i . . i, . − . i (cid:19) ,t R = (cid:18) . . i, − . . i − . − . i, − . . i (cid:19) ,r L (cid:48) = (cid:18) . − . i, . . i . − . i, − . − . i (cid:19) ,r R = (cid:18) − . − . i, − . − . i . . i, − . − . i (cid:19) . (S5.1)Figure S3 presents the transport properties obtained using the analytical formula Eq. (3) in the main text after weparameterize the relevant scattering matrix according to the corresponding parameter settings. We see that the mainfeatures of the conductance are qualitatively the same as the numerical results in Fig. 2 of the main text. As shownin Fig. S3 (a), each pattern has single frequency; the oscillation amplitude is larger at θ = π/ ; and the period of red1 Figure S3. Transport properties obtained using the analytical formula, Eq. (3) in the main text, after we parametrize therelevant scattering matrix according to the corresponding parameter settings. (a) Conductance oscillation pattern as functionof field strength B for different field directions θ = 0 and θ = 0 . π , respectively. (b) Conductance oscillation pattern asfunction of field direction θ for different field strengths B = B / √ ≈ . B and B = 1 B , respectively. (c) Beating patternof conductance as function of field strength B . (d) Beating pattern of conductance as function of field direction θ . line is about √ times that of the blue line. In Fig. S3 (b), there are two peaks and the oscillation amplitude is morepronounced when B = B / √ ≈ . B . In Fig. S3 (c), the conductance shows a beating pattern of B at θ = 0 . π .Finally, in Fig. S3 (c), the conductance shows an “irregular” beating pattern as a function of θ for large B . Appendix S6: Model of C symmetric SOTIs The effective model for a C symmetric SOTI on a stacked hexagonal lattice can be written as [63] H hex = (cid:18) h + ms z σ h AB h † AB h + ms z σ (cid:19) , (S6.1)where h = ˜ C − C (cos k + cos k + cos k ) + v k + sin k + sin k )Γ + v √ k − sin k )Γ + w [ − sin k + sin k + sin k ]Γ + (cid:104) M − M (cos k + cos k + cos k ) (cid:105) Γ h AB = − C cos k z + 2 v z sin k z Γ − M cos k z Γ , (S6.2)and k = k x , k = ( k x + √ k y ) / and k = k − k . h AB describes the hopping between neighboring layers. The Γ matrices are defined as Γ i = s i σ with i ∈ { , , } , Γ = s σ and Γ = s σ with s and σ the Pauli matricesand s and σ the corresponding identity matrices. The model parameters are defined as ˜ C = C + 2 C + 4 C , and ˜ M = M + 2 M + 4 M . In the numerical calculations, we set the parameters C = 0 , C = C = 0 . , M = − . , M = M = 1 , v = v z = 1 , m = 0 . and w = 2 . The perimeter of the hexagonal prisms is × n s with side length n s = 25 a and L z = 50 a .2 Figure S4. Left panel: influence of disorder on the interference pattern. We average over 100 disorder configurations. Rightpanel: influence of dephasing on the interference pattern. We choose parameters: L z = 60 a , W x = W y = 12 a , m = 2 , b = − , v = 1 , ∆ = 1 . The magnetic field strength is fixed at B = B / √ ≈ . B .Figure S5. (a) Cross section of the SOTI in a trapezoid geometry. (b) Four frequencies as functions of field direction θ . Here,S1 (S2, S3, S4) indicates the bottom (right-side, top, left-side) surface of the trapezoid. Solid lines are analytical results, andthe dotted lines are obtained by Fourier transformation from the conductance beating pattern. Discrepancy between themmaybe due to the finite-size effects. We choose parameters for the SOTI as: L z = 200 a , m = 2 , b = − , v = 1 , and ∆ = 1 . Appendix S7: Disorder and dephasing
In this section, we show that the interference pattern of our interferometer is robust against disorder and dephasing.To mimic disorder, we consider the onsite type V dis = V ( r ) I × with random function V ( r ) distributed uniformlywithin the interval [ − U / , U / and U being the disorder strength. It is shown in Fig. S4 that as increasingthe disorder strength U , the oscillation amplitude decreases gradually. However, the interference pattern of theconductance remains even when the disorder strength is quite strong (comparable with the bulk gap).We also consider dephasing in the SOTI region in our setup by attaching each site in the discretized lattice modelwith a virtual lead [64]. These virtual leads are coupled to the system via the self-energy − i Γ / with Γ measuringthe dephasing strength ( / Γ signifies the quasiparticle life time). It is shown in Fig. S4 that the interference patternof the conductance remains under weak dephasing strength. As increasing dephasing strength Γ , the electrons loose3their phase memory quickly and thus the oscillation amplitudes decrease accordingly.The above results shows the oscillation pattern basically remains under weak disorder and dephasing, which indicatesthe robustness of our proposal to show quantum interference of hinge states. Appendix S8: Multiple frequencies when the cross section is a trapezoid
In this section, we present the multiple-frequency case when the cross section of SOTI is a trapezoid, as shown inFig. S5(a). In this case, there are generally four frequencies in the conductance oscillation as function of field strength B . Figure S5(b) shows the four frequencies as varying the field direction θθ