MSH system with a magnetic skyrmion lattice. a Schematic picture of a skyrmion lattice. Spatial plot of b the magnitude of the induced Rashba spin orbit interaction, |α(r)|, c the skyrmion number density, ns(r), and d the Chern number density C(r) for skyrmion radius R = 5a0, and parameters (µ, ∆, JS) = (−5, 0.4, 0.5)t.

Contexts in source publication

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... Model. We investigate the emergence of topological superconductivity in a 2D MSH system, in which a magnetic skyrmion lattice (see Fig. 1a) is placed on the surface of a conventional s-wave superconductor, as described by the ...

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... crucial aspect for the emergence of topological superconductivity in 2D skyrmion MSH systems is that the magnetic skyrmion lattice induces an effective, spatially varying Rashba spin-orbit interaction. To demonstrate this, we apply a unitary transformation [24] to the Hamiltonian in Eq.(1) (see SI section 2) that rotates the local spin S r to thêthê z axis, yielding an out-of-plane ferromagnetic order and a spatially inhomogeneous RSO interaction, α(r) (see Fig.1b). α(r) possesses the same spatial structure as the skyrmion number density, n s (r), (see Fig.1c) -reflecting its origin in the local topological charge of the skyrmon lattice -with its largest value, α max = πa 0 t/(2R), in the center of the skyrmion, and a vanishing α(r) at the corners of the skyrmion lattice Wigner-Seitz unit cell. ...

Context 3

... demonstrate this, we apply a unitary transformation [24] to the Hamiltonian in Eq.(1) (see SI section 2) that rotates the local spin S r to thêthê z axis, yielding an out-of-plane ferromagnetic order and a spatially inhomogeneous RSO interaction, α(r) (see Fig.1b). α(r) possesses the same spatial structure as the skyrmion number density, n s (r), (see Fig.1c) -reflecting its origin in the local topological charge of the skyrmon lattice -with its largest value, α max = πa 0 t/(2R), in the center of the skyrmion, and a vanishing α(r) at the corners of the skyrmion lattice Wigner-Seitz unit cell. ...

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... inhomogeneous magnetic structure of the MSH system also allows us to reveal an intriguing connection between the Chern number density, C(r), which is a local marker for the topological nature of the system, and the Berry curvature in momentum space. In particular, the spatial structure of C(r) (see Fig. 1d) reflects that of the skyrmion lattice, but is complementary to that of the induced α(r) (see Fig. 1b), with the maximum in C(r) occurring at the corners of the skyrmion lattice unit cell. However, it is in these regions that the lowest energy states possess their largest spectral weight (see discussion below), establishing a real ...

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... allows us to reveal an intriguing connection between the Chern number density, C(r), which is a local marker for the topological nature of the system, and the Berry curvature in momentum space. In particular, the spatial structure of C(r) (see Fig. 1d) reflects that of the skyrmion lattice, but is complementary to that of the induced α(r) (see Fig. 1b), with the maximum in C(r) occurring at the corners of the skyrmion lattice unit cell. However, it is in these regions that the lowest energy states possess their largest spectral weight (see discussion below), establishing a real space analogue of the observation that the lowest energy states in momentum space in general possess the ...

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... different types of gap closings. The first one occurs via a Dirac cone, with the Dirac point being located at (a) time-reversal invariant (TRI) Γ, M, K, K points in the Brillouin zone (an example of this is shown in Fig.3 of the main text), or (b) at a non-time-reversal invariant point along the Γ − M line. An example of the latter is shown in Fig. S1a, where we present the dispersion E k of the lowest energy band at the transition between the C = 4 and C = −2 phases (see black line labelled a in Fig. S1e). Here, the multiplicity of the points in the BZ where the gap closes is m = 6 (as none of the points lie on the boundary with another BZ), implying that the Chern number changes by ...

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... K points in the Brillouin zone (an example of this is shown in Fig.3 of the main text), or (b) at a non-time-reversal invariant point along the Γ − M line. An example of the latter is shown in Fig. S1a, where we present the dispersion E k of the lowest energy band at the transition between the C = 4 and C = −2 phases (see black line labelled a in Fig. S1e). Here, the multiplicity of the points in the BZ where the gap closes is m = 6 (as none of the points lie on the boundary with another BZ), implying that the Chern number changes by ∆C = 6 at the transition. To gain further insight into the nature of the topological phase transition, we rewrite the expression for the Chern number in ...

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... present the momentum dependence of C(k) in Fig. S1b on either side of the transition. This plot reveals that the change in the Chern number by ∆C = 6 arises from the same change in C(k) near each of the gap closing points. The second type of gap closing in the phase diagram (see black line labelled c in Fig. S1e) involves a line of momenta along which the gap vanishes (and thus not ...

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... present the momentum dependence of C(k) in Fig. S1b on either side of the transition. This plot reveals that the change in the Chern number by ∆C = 6 arises from the same change in C(k) near each of the gap closing points. The second type of gap closing in the phase diagram (see black line labelled c in Fig. S1e) involves a line of momenta along which the gap vanishes (and thus not discrete Dirac points) in the Brillouin zone, as shown in Fig. S1c. Such a gap closing does not lead to a change in the Chern number between the topological phases adjacent to it, which is also reflected in the momentum dependence of C(k) (see Fig. S1d), which does ...

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... reveals that the change in the Chern number by ∆C = 6 arises from the same change in C(k) near each of the gap closing points. The second type of gap closing in the phase diagram (see black line labelled c in Fig. S1e) involves a line of momenta along which the gap vanishes (and thus not discrete Dirac points) in the Brillouin zone, as shown in Fig. S1c. Such a gap closing does not lead to a change in the Chern number between the topological phases adjacent to it, which is also reflected in the momentum dependence of C(k) (see Fig. S1d), which does not show any qualitative change between the two ...

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... black line labelled c in Fig. S1e) involves a line of momenta along which the gap vanishes (and thus not discrete Dirac points) in the Brillouin zone, as shown in Fig. S1c. Such a gap closing does not lead to a change in the Chern number between the topological phases adjacent to it, which is also reflected in the momentum dependence of C(k) (see Fig. S1d), which does not show any qualitative change between the two ...

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... Model. We investigate the emergence of topological superconductivity in a 2D MSH system, in which a magnetic skyrmion lattice (see Fig. 1a) is placed on the surface of a conventional s-wave superconductor, as described by the ...

Context 13

... crucial aspect for the emergence of topological superconductivity in 2D skyrmion MSH systems is that the magnetic skyrmion lattice induces an effective, spatially varying Rashba spin-orbit interaction. To demonstrate this, we apply a unitary transformation [24] to the Hamiltonian in Eq.(1) (see SI section 2) that rotates the local spin S r to thêthê z axis, yielding an out-of-plane ferromagnetic order and a spatially inhomogeneous RSO interaction, α(r) (see Fig.1b). α(r) possesses the same spatial structure as the skyrmion number density, n s (r), (see Fig.1c) -reflecting its origin in the local topological charge of the skyrmon lattice -with its largest value, α max = πa 0 t/(2R), in the center of the skyrmion, and a vanishing α(r) at the corners of the skyrmion lattice Wigner-Seitz unit cell. ...

Context 14

... demonstrate this, we apply a unitary transformation [24] to the Hamiltonian in Eq.(1) (see SI section 2) that rotates the local spin S r to thêthê z axis, yielding an out-of-plane ferromagnetic order and a spatially inhomogeneous RSO interaction, α(r) (see Fig.1b). α(r) possesses the same spatial structure as the skyrmion number density, n s (r), (see Fig.1c) -reflecting its origin in the local topological charge of the skyrmon lattice -with its largest value, α max = πa 0 t/(2R), in the center of the skyrmion, and a vanishing α(r) at the corners of the skyrmion lattice Wigner-Seitz unit cell. ...

Context 15

... inhomogeneous magnetic structure of the MSH system also allows us to reveal an intriguing connection between the Chern number density, C(r), which is a local marker for the topological nature of the system, and the Berry curvature in momentum space. In particular, the spatial structure of C(r) (see Fig. 1d) reflects that of the skyrmion lattice, but is complementary to that of the induced α(r) (see Fig. 1b), with the maximum in C(r) occurring at the corners of the skyrmion lattice unit cell. However, it is in these regions that the lowest energy states possess their largest spectral weight (see discussion below), establishing a real ...

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... allows us to reveal an intriguing connection between the Chern number density, C(r), which is a local marker for the topological nature of the system, and the Berry curvature in momentum space. In particular, the spatial structure of C(r) (see Fig. 1d) reflects that of the skyrmion lattice, but is complementary to that of the induced α(r) (see Fig. 1b), with the maximum in C(r) occurring at the corners of the skyrmion lattice unit cell. However, it is in these regions that the lowest energy states possess their largest spectral weight (see discussion below), establishing a real space analogue of the observation that the lowest energy states in momentum space in general possess the ...

Context 17

... different types of gap closings. The first one occurs via a Dirac cone, with the Dirac point being located at (a) time-reversal invariant (TRI) Γ, M, K, K points in the Brillouin zone (an example of this is shown in Fig.3 of the main text), or (b) at a non-time-reversal invariant point along the Γ − M line. An example of the latter is shown in Fig. S1a, where we present the dispersion E k of the lowest energy band at the transition between the C = 4 and C = −2 phases (see black line labelled a in Fig. S1e). Here, the multiplicity of the points in the BZ where the gap closes is m = 6 (as none of the points lie on the boundary with another BZ), implying that the Chern number changes by ...

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... K points in the Brillouin zone (an example of this is shown in Fig.3 of the main text), or (b) at a non-time-reversal invariant point along the Γ − M line. An example of the latter is shown in Fig. S1a, where we present the dispersion E k of the lowest energy band at the transition between the C = 4 and C = −2 phases (see black line labelled a in Fig. S1e). Here, the multiplicity of the points in the BZ where the gap closes is m = 6 (as none of the points lie on the boundary with another BZ), implying that the Chern number changes by ∆C = 6 at the transition. To gain further insight into the nature of the topological phase transition, we rewrite the expression for the Chern number in ...

Context 19

... present the momentum dependence of C(k) in Fig. S1b on either side of the transition. This plot reveals that the change in the Chern number by ∆C = 6 arises from the same change in C(k) near each of the gap closing points. The second type of gap closing in the phase diagram (see black line labelled c in Fig. S1e) involves a line of momenta along which the gap vanishes (and thus not ...

Context 20

... present the momentum dependence of C(k) in Fig. S1b on either side of the transition. This plot reveals that the change in the Chern number by ∆C = 6 arises from the same change in C(k) near each of the gap closing points. The second type of gap closing in the phase diagram (see black line labelled c in Fig. S1e) involves a line of momenta along which the gap vanishes (and thus not discrete Dirac points) in the Brillouin zone, as shown in Fig. S1c. Such a gap closing does not lead to a change in the Chern number between the topological phases adjacent to it, which is also reflected in the momentum dependence of C(k) (see Fig. S1d), which does ...

Context 21

... reveals that the change in the Chern number by ∆C = 6 arises from the same change in C(k) near each of the gap closing points. The second type of gap closing in the phase diagram (see black line labelled c in Fig. S1e) involves a line of momenta along which the gap vanishes (and thus not discrete Dirac points) in the Brillouin zone, as shown in Fig. S1c. Such a gap closing does not lead to a change in the Chern number between the topological phases adjacent to it, which is also reflected in the momentum dependence of C(k) (see Fig. S1d), which does not show any qualitative change between the two ...

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... black line labelled c in Fig. S1e) involves a line of momenta along which the gap vanishes (and thus not discrete Dirac points) in the Brillouin zone, as shown in Fig. S1c. Such a gap closing does not lead to a change in the Chern number between the topological phases adjacent to it, which is also reflected in the momentum dependence of C(k) (see Fig. S1d), which does not show any qualitative change between the two ...