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Topological Superconductors in Doubly-Coupled Nanowires with Altermagnetism
Hongfa Pan, 1 , 2 Haoyang Wang, 1 Wenguang Zhu, 1 , 2, ∗ and Zhenhua Qiao 1 , 2, 3, †
1 International Centre for Quantum Design of Functional Materials and Department of Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China 2 Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China
3 Wilczek Quantum Center, Shanghai Institute for Advanced Studies, University of Science and Technology of China, Shanghai 201315, China
nd Technology of China,Sh (Dated: August 14, 2026)
arXiv:2608.13265v1 [cond-mat.mes-hall] 13 Aug 2026
We theoretically investigate the possibility of engineering topological superconductivity in doubly- coupled nanowires integrating proximity-induced superconductivity and altermagnetism. By tuning experimentally accessible parameters, we find four distinct topological superconducting phases: class D, class BDI, and two phases hosting two Majorana zero modes per end, protected respectively by spin-group and magnetic point-group symmetries. Beyond inheriting the advantages of alternating- magnetism induced topological superconductivity, our system provides multiple tuning knobs (e.g., superconducting phase difference and inter-wire coupling) to control topological properties.
I. INTRODUCTION Topological superconductors (TSCs) hosting Majo- rana zero modes (MZMs) at their boundaries have at- tracted intense attention owing to their potential for fault-tolerant quantum computing [ 1 – 7 ]. Despite decades of progress, there still exist three core challenges: devel- oping experimentally feasible schemes for TSC realiza- tion, achieving unambiguous experimental identification of MZMs, and establishing protocols for controlled MZM manipulation [ 2 , 5 – 11 ]. Single-nanowire platforms un- der external Zeeman fields, which have been extensively studied for topological superconductivity, show great promise [ 12 – 15 ]. However, strong Zeeman fields suppress the superconducting gap, and distinguishing MZMs from trivial Andreev bound states remains a persistent exper- imental difficulty [ 16 – 18 ]. Altermagnetism, an emerging class of magnetic materials, has recently been a research focus in condensed matter physics [ 19 – 23 ]. Unlike con- ventional Zeeman-field-based schemes that suppress the superconducting gap, altermagnetism enhances the gap significantly with zero net magnetization. Moreover, it allows for the use of weak external Zeeman field to distin- guish MZMs from impurity-induced trivial states, repre- senting a substantial advantage [ 24 ]. Nevertheless, even with altermagnetism, single-nanowire platforms remain limited by the lack of MZM manipulation knobs [ 5 , 17 ]. Furthermore, the full experimental realization of TSCs with multiple symmetry-correlated boundary MZMs re- mains elusive due to pervasive technical constraints. Time-reversal-invariant TSCs hosting Majorana Kramers pairs, for instance, generally demand either unconven- tional superconducting substrates or prohibitively large values of an individual control parameter [ 25 – 28 ], while topological crystalline superconductors require higher- dimensional platforms with dedicated crystalline symme- ∗ Correspondence author: [email protected] † Correspondence author: [email protected] tries [ 29 – 31 ]. In this article, we theoretically investigate a doubly- coupled nanowire platform with proximity-induced su- perconductivity and altermagnetism from adjacent su- perconductors and altermagnets, respectively [ 24 ]. Band degeneracy lifting by altermagnetism and Rashba spin- orbit coupling leads to the emergence of both intra- band and interband pairing terms [ 32 – 37 ], whose inter- play gives rise to class D TSC with one MZM per end and class BDI TSC with one or two MZMs per end, re- spectively [ 38 – 42 ]. Notably, even within the BDI class, one MZM per end can be generically realized when the phase difference is not an integer multiple of π, mak- ing these phases directly applicable for topological quan- tum computing. By introducing asymmetric altermag- netism or Rashba spin-orbit coupling between the two nanowires, we further find two topological phases pro- tected respectively by spin-group symmetry [ 43 , 44 ] and magnetic point-group symmetry, hosting two symmetry- related MZMs per end. All identified topological phases occupy wide parameter ranges, with several continuously tunable through experimentally accessible controls, such as the superconducting phase difference and inter-wire coupling. This offers versatile tuning knobs for MZM manipulation and the detection of their hallmark exper- imental signatures. The remainder of the paper is organized as follows. In Sec. II , the tight-binding model Hamiltonians are intro- duced. In Sec. III , the D class and BDI class TSCs are demonstrated. In Sec. IV , two types of TSCs are real- ized with two MZMs per end, protected by a spin-group symmetry and a magnetic point-group symmetry, respec- tively. In Sec. V , a brief conclusion is provided. II. MODEL AND HAMILTONIAN Figure 1 displays our system consisting of two cou- pled semiconductor nanowires with inter-wire coupling t 0 , denoted as nanowires w n (n = 1, 2). µ, t, and α n denote respectively the chemical potential, nearest-
2 FIG. 1: (a) Side-view and (b) top-view schematics of the doubly-coupled semiconducting nanowire platform with inter-wire coupling. Two superconducting (SC) regions exhibit superconducting phases φ 1 and φ 2 , and two altermagnetic (AM) regions are labeled accordingly. . neighbor hopping amplitude and Rashba spin-orbit cou- pling in the nanowire w n . By proximity-coupled to alter- magnets and superconductors, altermagnetic exchange coupling J n with d x 2 −y 2 symmetry [ 24 ] and s-wave su- perconducting pairing potential ∆ n with phase φ n are induced in each nanowire. The model Hamiltonian of the system reads: H = H w 1 + H w 2 + H i nt , H w n = H t n + H S O n + H J n + H ∆ n , H t n = −µ j ,α c † w n , j,α c w n , j,α − t j,α ( c † w n , j+1,α c w n , j,α + H.c.), H S O n = iα n j,α,β (c † w n , j+1,α ( σ y ) α ,β c w n , j,β − H.c.), H J n = J n j ,α ( c † w n , j+1,α σ z c w n , j,α + H.c.), H ∆ n = j ( ∆ n e i φ n c † w n , j,↑ c † w n , j,↓ + H.c.), H i nt = −t 0 j,α ( c † w 1 , j,α c w 2 , j,α + H.c.), (1) where c † w n , j,α (c w n , j,α ) creates (annihilates) an electron with spin α =↑, ↓ at site j in nanowire w n . Since only the relative phase difference between two superconduc- tors is physically meaningful, we define the phases as φ n = (−1) n φ/2, where φ = φ 2 − φ 1 is the phase differ- ence across the two nanowires. Since the momentum k x is conserved, the Hamilto- FIG. 2: (a) Energy eigenvalues as a function of chemical potential µ. Point A marks a twofold-degenerate zero-energy state, with an inset showing the eigenvalue distribution near zero energy. (b) Phase diagram in the plane of phase difference φ vs altermagnetism J 1 ,2 . Parameters are set to be t = 1, α 1 ,2 = 0.35, ∆ 1 = 0.3, ∆ 2 = 0.15, t 0 = 0.25. Additional parameters: (a) J 1 ,2 = 0.2, φ = 0.6π, and nanowire length N w = 200; (b) µ = −2. . nian H can be transformed to momentum space. On the Nambu basis of {ψ † 1 , ψ 1 , ψ † 2 , ψ 2 } , where the basis is defined as {ψ † n , ψ n } = {c † n ,↑ , c † n ,↓ , c n ,↑ , c n ,↓ } , the corre- sponding Bogoliubov–de Gennes Hamiltonian takes the form H k = H k 1 s 0 + s z 2 + H k 2 s 0 − s z 2 − t 0 s x τ z , H k n = −(µ + 2t cos k x ) τ z + 2J n cos k x τ z σ z + 2α n sin k x τ z σ y + e i φ n ∆ n τ x + iτ y 2 iσ y − e − iφ n ∆ n τ x − iτ y 2 iσ y , (2) where s, τ, σ are Pauli matrices acting on the the wire, particle-hole, and spin spaces, respectively. The particle- hole symmetry of the Hamiltonian H k is characterized by the operator P = τ x K , where K denotes complex conjugation. The double-wire architecture endows the system with a significantly enlarged Hilbert space compared to single- wire counterpart, introducing additional wire-specific de- grees of freedom that expand the parameter space acces- sible to experimental manipulation. This rich parameter space enables control over the key symmetry operations: the mirror symmetry exchanging the two wires, as well as its nontrivial combinations with particle-hole symme- try and time-reversal symmetry, can be either preserved or broken in different parameter regimes. For instance, the wire-exchange mirror symmetry is broken when the
3 superconducting pairing amplitudes of the two wires are unequal (∆ 1 ̸= ∆ 2 ) , placing the system in the D class as a single TSC nanowire when their relative phase φ is not an integer multiple of π (φ ̸= 0, π, 2π). In con- trast, when φ = 0, the system realizes the BDI symme- try class, which constitutes a paradigmatic example of coupled TSC nanowire systems [ 40 ]. Remarkably, even when φ deviates from integer multiples of π, the under- lying symmetries can interconnect the two wires and re- store the BDI symmetry class. These key symmetries enable the design of distinct TSCs as shown in Table I and Table II . TABLE I: Topological classes in the Altland–Zirnbauer tenfold classification. All unlisted parameters are identical for the two nanowires in all cases. ∆ 1 = ∆ 2 φ Class Yes arbitrary BDI No φ = mπ BDI No φ ̸= mπ D TABLE II: The case of MZMs related by detailed symmetry. All unlisted identical parameters are equal for the two nanowires in all cases. J 1 = ±J 2 φ α 1 = ±α 2 Group A. D class topological superconductors When both the time-reversal symmetry is broken by the altermagnetism and additional symmetries are lifted by imposing |∆ 1 | ̸= |∆ 2 | and φ ̸= mπ for integer m, the system is driven into the D class. The conditions for topologically nontrivial states are signaled by the bulk gap closing at momenta k x = 0 or k x = π [ 3 , 24 , 42 ]. The topological phase boundary depends on a variety of experimentally tunable parameters [See Appendix. A11 ], including the phase difference φ, inter-wire coupling t 0 , chemical potential µ, and altermagnetism J 1 ,2 . As a con- sequence, the topological properties can be modulated via multiple independent knobs. Specifically, the evo- lution of energy spectrum with chemical potential µ is displayed in Fig. 2 (a). Zero-energy modes exist within a relatively broad range of µ. For example, point A corresponds to the presence of two MZMs, which is the generic case for D class TSCs. The altermagnetism J 1 ,2 and phase difference φ also provide powerful knobs. The phase diagram as a function of φ and J 1 ,2 is shown in Fig. 2 (b). The region enclosed by the two dark blue curves, which mark the gap-closing topological phase transitions, is topologically nontrivial. This demon- strates an effective approach in engineering topological superconductivity via the phase difference φ, which re- laxes the requirement of strong altermagnetism for topo- logical phase when φ becomes finite. B. BDI class topological superconductors
− 0 + T 1 − π − T 2 III. D CLASS AND BDI CLASS TOPOLOGICAL SUPERCONDUCTORS By setting J 1 = J 2 and α 1 = α 2 , the system can be reduced to a two-band superconductor with both intra- band and interband pairings [See Appendix. A2 ], whose amplitudes are respectively ∆ s and ∆ d , ∆ s = | 1 2 e − iφ 2 (∆ 1 + ∆ 2 e i φ )|, ∆ d = | 1 2 e − iφ 2 (∆ 2 e i φ − ∆ 1 ) |. (3) In this parameter regime, the topological phases are governed by the interplay between interband pairing ∆ d and intraband pairing ∆ s , where the latter corresponds to the pairing mechanism in conventional single-nanowire TSCs. This results in a rich topological phase diagram with multiple independently tunable parameters.
At some special points of φ = mπ (m = 0, 1, 2), the sys- tem belongs to the BDI class, denoted by the blue dashed lines and red dashed line in Fig. 2 (b). When φ = 0, 2π (φ = π), the Hamiltonian H k possesses an anti-unitary time-reversal symmetry T 1 = K (T 2 = τ z K), which to- gether with the particle-hole symmetry P generates a unitary chiral symmetry S 1 = P T 1 (S 2 = P T 2 ) satisfying {S i , H k } = 0 for i = 1, 2. Thus, the system falls into the BDI class with Z topological invariant, which counts the number of MZMs localized per end of the nanowire. As displayed in Appendix A2 , the interplay between two pairing channels gives rise to two distinct types of BDI class topological superconductors. At φ = π, the small bulk gap is dominated by the in- traband pairing ∆ s = | 1 2 (∆ 1 −∆ 2 ) |, which is proportional to the difference between two pairings, as illustrated in Fig. 2 (a). In contrast, at φ = 0, the gap is governed by ∆ s = | 1 2 (∆ 1 + ∆ 2 ) |, leading to a significantly larger bulk gap. Hereinbelow, we focus on the φ = 0 case. With the increase of altermagnetism J 1 ,2 , two consecu- tive topological phase transitions occur, marked by the emergence of zero-energy modes corresponding to topo- logically nontrivial phases with one and two MZMs per end of the doubly coupled nanowires, respectively [see Fig. 3 (a)]. Similarly, the variation of chemical potential µ drives two transitions to zero-energy phases, followed
4 FIG. 3: (a)-(d) BDI TSCs with φ = 0: (a) Energy eigenvalues as a function of altermagnetism J 1 ,2 . (b) Energy eigenvalues as a function of the chemical potential µ. (c) The phase diagram as the function of µ and J 1 ,2 . (d) The phase diagram as the function of µ and t 0 . Parameters used in all calculations: t = 1, α 1 ,2 = 0.35, ∆ 1 = 0.3, ∆ 2 = 0.2. Additional parameters: (a) µ = −2, t 0 = 0.25 and nanowire length N w = 100. (b) J 1 ,2 = 0.3, t 0 = 0.25 and nanowire length N w = 100. (c) t 0 = 0.25; (d) J 1 ,2 = 0.3. (e)-(h) BDI TSCs with φ ̸= mπ (m = 0, 1, 2): (e) Energy eigenvalues as a function of the altermagnetism J 1 ,2 . (f) Energy eigenvalues as a function of the chemical potential µ. (g) Phase diagram as a function of φ and J 1 ,2 . (h) Phase diagram as the function of µ and J 1 ,2 . Parameters used in all calculations: t = 1, α 1 ,2 = 0.3, ∆ 1 ,2 = 0.3. Additional parameters: (e) µ = −2, t 0 = 0.25, φ = 0.85π, π and nanowire length N w = 100. (f) J 1 ,2 = 0.3, t 0 = 0.2, φ = 0.5π and nanowire length N w = 100. (g) µ = −2 and t 0 = 0.25; (h) t 0 = 0.2 and φ = 0.5π. . by two transitions back to trivial phases, corresponding sequentially to phases with one MZM, two MZMs, and no MZMs per end [Fig. 3 (b)]. To systematically explore the parameter dependence of the topological phases, we present phase diagrams as functions of chemical potential µ versus altermagnetism J 1 ,2 and inter-wire coupling t 0 in Figs. 3 (c) and 3 (d), respectively. Over a broad param- eter range, we find topologically nontrivial phases with Z = 1 and Z = 2, corresponding to one and two MZMs per end, respectively. The red line in Fig. 3 (c) represents the parameter cut shown in Fig. 3 (a), which traverses the trivial phase, Z = 1 nontrivial phase, and Z = 2 nontrivial phase. Remarkably, even when φ ̸= mπ, the system remains in the BDI class for ∆ 1 = ∆ 2 . In this parameter regime, an effective anti-unitary time-reversal symmetry T 3 = M K emerges, where M = s x denotes the operator exchang- ing the two nanowires. This effective time-reversal sym- metry combines with the particle-hole symmetry P to generate a unitary chiral symmetry S 3 = P T 3 , satisfying {S 3 , H k } = 0. The evolution of the energy spectrum with altermagnetism J 1 ,2 and chemical potential µ is presented in Figs. 3 (e) and 3 (f), respectively, revealing topologi- cally nontrivial phases over a broad parameter range. As displayed in Fig. 3 (e), at φ = 0.85π, zero-energy modes corresponding to MZMs persist over a wide range of J 1 ,2 . Near φ = π, the bulk gap tends to close because the in- traband pairing amplitude ∆ s = | 1 2 (∆ 1 − ∆ 2 ) | vanishes when ∆ 1 = ∆ 2 . These two cases are respectively marked by the purple and blue lines in Fig. 3 (g), which shows the phase diagram as a function of phase difference φ and al- termagnetism J 1 ,2 . Away from φ = π point, the system exhibits a topologically nontrivial phase with one MZM per end over a wide range of J 1 ,2 and φ, with a strong dependence on the phase difference φ, which also relaxes the requirement for strong altermagnetism. Topologi- cally nontrivial phases over a broad parameter range is also observed in the phase diagram as a function of µ and J 1 ,2 in Fig. 3 (h). IV. SPIN-GROUP SYMMETRY AND MAGNETIC POINT-GROUP SYMMETRY PROTECTED TOPOLOGICAL SUPERCONDUCTORS TSCs protected by magnetic point-group symmetry or time-reversal symmetry have been extensively stud- ied both theoretically and experimentally, where multiple MZMs can emerge at system boundaries. However, most of these realizations are based on higher-dimensional sys- tems or unconventional pairings [ 29 – 31 ]. In this Section, we demonstrate that two distinct types of TSCs can be realized in our quasi-one-dimensional system by tuning either the exchange couplings J 1 ,2 or the Rashba spin- orbit couplings α 1 ,2 . These two phases are protected by
5 FIG. 4: (a)-(d) spin-group symmetry protected TSCs: (a) Energy eigenvalues as a function of the chemical potential µ. (b) Energy eigenvalues as a function of the phase difference φ. (c) Phase diagram as a function of µ and J 1 ,2 . (d) Phase diagram as a function of t 0 and J 1 ,2 . Parameters used in all calculations: t = 1 and ∆ 1 ,2 = 0.3. Additional parameters: (a) α 1 ,2 = 0.35, J 1 ,2 = ±0.4, t 0 = 0.2, φ = 0 and nanowire length N w = 100; (b) µ = −2 and the other fixed parameters are same as those in (a); (c) J 1 = −J 2 and the fixed parameters are same as those in (a); (d) J 1 = −J 2 , α 1 ,2 = 0.3, µ = −2 and φ = 0. (e)-(h) Magnetic point-group symmetry protected TSCs: (e) Energy eigenvalues as a function of the chemical potential µ. (f) Energy eigenvalues as a function of the phase difference φ. (g) Phase diagram as a function of µ and J 1 ,2 . (h) Phase diagram as a function of t 0 and J 1 ,2 . Parameters used in all calculations: t = 1, α 1 ,2 = ±0.3, and ∆ 1 ,2 = 0.3. Additional parameters: (e) J 1 ,2 = ±0.4, t 0 = 0.25, φ = π and nanowire length N w = 100; (f) µ = −2 and the other fixed parameters are same as those in (e); (g) J 1 = −J 2 and the fixed parameters are same as those in (e); (h) J 1 = −J 2 , µ = −2.3 and the other fixed parameters are same as those in (e). . spin-group symmetry [ 43 , 44 ] and magnetic point-group symmetry, respectively. In the first case, By setting J 1 = −J 2 , φ = 0, and all other parameters to be identical in the two nanowires, the anti-unitary spin-group symmetry T 1 = M T is pre- served, satisfying T 1 H k (k x ) T −1 1 = H k ( −k x ) . T = iσ y K is the time-reversal operator. In the topologically non- trivial phase, two MZMs reside per end of the coupled nanowire system, related by the T 1 symmetry. The non- trivial phase is denoted by the red lines in Fig. 4 (a). Since this symmetry is composed of M and T , and cannot be broken by a phase difference, a nonzero φ does not de- stroy the topologically nontrivial phase, in contrast to the time-reversal-invariant TSCs protected solely by T , as illustrated in Fig. 4 (b). To characterize this topo- logical invariant, we introduce a new chiral symmetry S T 1 = P T 1 under these parameter settings. The com- bination with the chiral symmetry S 1 = P T 1 yields a unitary symmetry U 1 = S 1 S T 1 that commutes with the Hamiltonian, [U 1 , H k ] = 0. By block-diagonalizing the Hamiltonian using U 1 symmetry, the integer topological invariant Z of the BDI class can be computed for each block via the S 1 chiral symmetry. The phase diagrams as functions of µ vs J 1 ,2 and t 0 vs J 1 ,2 are presented in Figs. 4 (c) and 4 (d), respectively. A broad topologically nontrivial region hosting two MZMs per end, correspond- ing to Z = 2, is clearly present in Fig. 4 (c). The white line in Fig. 4 (c) corresponds to the parameter trajectory shown in Fig. 4 (a). Figure 4 (d) further shows that the critical altermagnetism J 1 ,2 required for the topologically nontrivial phase increases with increasing t 0 . The second case is characterized by the parameter re- lations J 1 = −J 2 , α 1 = −α 2 , and φ = π, with all other parameters being identical in the two nanowires. A dis- tinct anti-unitary symmetry T 2 = M T of magnetic point- group symmetry is preserved, satisfying T 2 H k (k x ) T −1 2 = H k (−k x ) , where M = s x τ z σ z and T = iσ y K. As shown in Fig. 4 (e), the nontrivial phase is denoted by the blue lines. The energy spectrum exhibits a distinct 4π peri- odicity as a function of phase difference φ (see Fig. 4 (f)), which provides a clear experimental signature for this TSC phase. A unitary symmetry U 2 = S T 2 S 2 that com- mutes with the Hamiltonian, [U 2 , H k ] = 0, can also be constructed, where S T 2 = P T 2 and S 2 = P T 2 are chi- ral symmetries with T 2 = τ z K. The integer topological invariant Z can be computed similarly via the S 2 chiral symmetry. The phase diagrams as functions of µ vs J 1 ,2 and t 0 vs J 1 ,2 are displayed in Figs. 4 (g) and 4 (h), respec- tively. A broad region hosting two MZMs, corresponding to Z = 2, is shown in Figs. 4 (g) and 4 (h). The red line in Fig. 4 (g) corresponds to the parameter trajectory shown
6 in Fig. 4(e). It also relaxes the requirement for alter- ence Foundation of China (Grants No. 380 12474158, No. magnetism or the inter-wire coupling when the other is 12234017, and No. 12488101). We also thank the Super- non-zero as shown in Fig. 4(h). computing Center of University of Science and Technol-
V.
ence Foundation of China (Grants No. 380 12474158, No. 12234017, and No. 12488101). We also thank the Super- computing Center of University of Science and Technol- ogy of China for providing high-performance computing resources.
CONCLUSION
Appendix A
To conclude, our work extends the frontier of altermagnetism-based topological superconductivity be- yond single nanowires, introducing a versatile platform with unparalleled experimental tunability. This system hosts the D class and BDI class topological supercon- ducting phases, and phases protected by spin-group and magnetic point-group symmetries. The rich set of inde- pendent tuning knobs inherent to this architecture may enable the detection of topological superconducting sig- natures and implementation of non-Abelian braiding and fusion of MZMs.
yond single nanowires, introducing a versatile platform with unparalleled experimental tunability. This system hosts the D class and BDI class topological supercon- ducting phases, and phases protected by spin-group and magnetic point-group symmetries. The rich set of inde- pendent tuning knobs inherent to this architecture may enable the detection of topological superconducting sig- natures and implementation of non-Abelian braiding and fusion of MZMs. ACKNOWLEDGMENTS This work was financially supported by Innovation Program for Quantum Science and Technology (Grant No. 382 2021ZD0302800) and the National Natural Sci- For both D class and BDI class TSCs, the Hamilto- nian H k can be transformed to a new Hamiltonian H d expressed in the basis {ψ † d 1 , ψ d 1 , ψ † d 2 , ψ d 2 } , where ψ † d 1 = 1 √ 2 {c † 2 ,↑ − c † 1 ,↑ , c † 2 ,↓ − c † 1 ,↓ } , ψ † d 2 = 1 √ 2 {c † 2 ,↑ + c † 1 ,↑ , c † 2 ,↓ + c † 1 ,↓ } (A1) The Hamiltonian H d can be denoted by H d ( k) = H 1 H ∆ 1 2 H † ∆ 1 2 H 2 (A2) where the sub-matrices H i with i = 1, 2 and H ∆ 1 2 are denoted respectively by H i = 2 (J 1 − t) cos(k) − µ i −2iα 1 0 1 2 e − iφ 2 (∆ 1 + ∆ 2 e i φ ) 2iα 1 2(−J 1 − t) cos(k) − µ i − 1 2 e − iφ 2 (∆ 1 + ∆ 2 e i φ ) 0 0 − 1 2 e − iφ 2 (∆ 2 + ∆ 1 e i φ ) 2(−J 1 + t) cos(k) + µ i 2iα 1 1 2 e − iφ 2 (∆ 2 + ∆ 1 e i φ ) 0 −2iα 1 2(J 1 + t) cos(k) + µ i (A3) and H ∆ 12 = 0 0 0 1 2 e − iφ 2 (−∆ 1 + ∆ 2 e i φ ) 0 0 1 2 e − iφ 2 (∆ 1 − ∆ 2 e i φ ) 0 0 1 2 e − iφ 2 (∆ 1 e i φ − ∆ 2 ) 0 0 1 2 e − iφ 2 (−∆ 1 e i φ + ∆ 2 ) 0 0 0 (A4) where µ 1 = −t 0 + µ and µ 2 = t 0 + µ. Thus, the Hamiltonian describes a two-band super- conductor with both intraband and interband pairing terms [ 32 – 37 ], characterized by an effective chemical po- tential difference 2t 0 . Intraband pairing favors the forma- tion of a TSC state, whereas interband pairing tends to suppress it [ 3 , 40 ]. The magnitudes of these two pairing components are given by ∆ s = | 1 2 e − iφ 2 (∆ 1 + ∆ 2 e i φ )| (A5) ∆ d = | 1 2 e − iφ 2 (∆ 2 e i φ − ∆ 1 ) | (A6) which can be controlled by tuning the phase difference φ and the pairing strengths ∆ 1 and ∆ 2 . When ∆ 1 = ∆ 2 and φ = 0, 2π, The magnitudes of these two pairing components reduce to ∆ s = |∆ 1 | , ∆ d = 0 (A7) Only intraband pairing dominates, allowing the model to be mapped onto two decoupled Kitaev p-wave supercon- ducting chains with effective chemical potentials t 0 − µ and −t 0 − µ, respectively. As a special case, this model belongs to the BDI symmetry class [ 40 ]. When ∆ 1 = ∆ 2 and φ = π, the magnitudes of these
7 two pairing components reduce to ∆ s = 0, ∆ d = |i∆ 1 | (A8) Only interband pairing dominates in this regime. When the t 0 and −t 0 chemical potential shifts are omitted, the model can similarly be mapped onto two decoupled Ki- taev p-wave superconducting chains. In the general case, these shifts globally displace the energies of the bands originating from the pairing terms. When t 0 is compara- ble to the pairing amplitude, as shown in Fig. 3 (c), the superconducting gap typically closes. When ∆ 1 ̸= ∆ 2 and φ = 0, 2π, the magnitudes of these two pairing components reduce to ∆ s = 1 2 |∆ 1 + ∆ 2 | , ∆ d = 1 2 |∆ 1 − ∆ 2 | (A9) It is obvious that the intraband pairing dominates when the difference between ∆ 1 and ∆ 2 is small. For ∆ 1 ̸= ∆ 2 with φ = π, the magnitudes of these two pairing components reduce to ∆ s = | i 2 (∆ 1 − ∆ 2 ) |, ∆ d = | i 2 (∆ 1 + ∆ 2 ) | (A10) The intraband pairing is dependent on the difference be- tween ∆ 1 and ∆ 2 . 2. Conditions of D class TSCs The conditions for the topologically nontrivial phase are determined by the closing of the bulk energy gap in the Hamiltonian H k at k x = 0 or π [ 3 , 24 ]. At k x = 0, the critical exchange coupling for gap clo- sure is given by J c 1 = m 0 ± √ n 0 2 √ 2 , (A11) with m 0 = 2(2t + µ) 2 + ∆ 2 1 + ∆ 2 2 + 2t 2 0 and n 0 = 16t 2 0 ( 2t + µ) 2 + (∆ 2 1 − ∆ 2 2 ) 2 + 4t 2 0 ( ∆ 2 1 + ∆ 2 2 − 2∆ 1 ∆ 2 cos φ). At k x = π, the corresponding critical exchange cou- pling reads J c 2 = m π ± √ n π 2 √ 2 , (A12) with m π = 2(−2t + µ) 2 + ∆ 2 1 + ∆ 2 2 + 2t 2 0 and n π = 16t 2 0 ( −2t+µ) 2 +(∆ 2 1 −∆ 2 2 ) 2 +4t 2 0 ( ∆ 2 1 +∆ 2 2 −2∆ 1 ∆ 2 cos φ). The topological phase boundary is thus defined by Eqs. ( A11 ) and ( A12 ), which is dependent on µ. For example with µ × t < 0, the system resides in the topologically nontrivial phase with √ m 0 − √ n 0 2 √ 2 < J 1 ,2 < √ m 0 + √ n 0 2 √ 2 . These critical conditions depend on a vari- ety of experimentally adjustable parameters. In particu- lar, the topologically nontrivial phase can be effectively controlled by the phase difference φ when |t| is compara- ble to |µ|, since all other parameters are generally small in realistic experimental settings. In this regime, J c 1 is dominated by ∆ 1 ,2 and t 0 of comparable magnitude, can be continuously tuned via φ, as shown in Fig. 1 (d). 3. Calculation of the topological invariant A chiral symmetry S 0 is defined as {S 0 , H 0 } = 0. In the basis where S 0 is diagonal, H 0 takes the block-off- diagonal form: H(k) = 0 A(k) A T (−k) 0 . (A13) The winding number W is given by W = −i 2π π − π dz(k) z(k) , (A14) where the complex function z(k) = det[A(k)]/| det[A(k)]|. For TSCs protected by spin-group symmetry and mag- netic point-group symmetry in our system, the Hamil- tonian H k can be block-diagonalized using the unitary symmetries U 1 and U 2 , respectively, yielding H(k) = H a 0 0 H b . (A15) The total topological invariant for these phases is ob- tained by summing the invariants of the two diagonal blocks, each of which is calculated using the chiral sym- metry acting on the corresponding submatrix. [1] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. M. P. A. Fisher, Non-Abelian statistics and topological
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