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Phase diagram of the Kitaev model on a square lattice system on the parameter μ − Δ plane (in units of t). The blue lines indicate the boundary, which separate the gapful phases (yellow) and gapless phase (green). Several points (a–g) at typical positions are indicated and the same letter represents the situations with the similar band structures. The corresponding band structures and the topology of the nodal point in the momentum space are given in Figs 2 and 3, respectively.
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We study the topological feature of gapless states in the fermionic Kitaev model on a square lattice. There are two types of gapless states which are topologically trivial and nontrivial. We show that the topological gapless phase lives in a wide two-dimensional parameter region and are characterized by two vertices of an auxiliary vector field de-...
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Topological phases of materials are characterized by topological invariants that are conventionally calculated by different means according to the dimension and symmetry class of the system. For topological materials described by Dirac models, we introduce a wrapping number as a unified approach to obtain the topological invariants in arbitrary dim...
Citations
... In contrast, the SOC induced due to the presence of noncollinear magnetic textures, characterized by g x,y , transforms into a (p x + p y )-type SC pairing with a gap p x +p y that scales with g. In the context of 2D first-order TSC, it is well known that (p x + ip y )-type pairing induces dispersive Majorana edge modes [55,116,118], while (p x + p y )-type pairing is responsible for the emergence of flat Majorana edge modes [55,63,119,120]. When the SOC strength is set to zero, λ x,y = 0, the TSC phase exclusively features the flat edge modes [63], which seems to restrict the emergence of opposite mass terms in the intersecting edge Hamiltonians. ...
We put forth a theoretical framework for engineering a two-dimensional (2D) second-order topological superconductor (SOTSC) by utilizing a heterostructure: incorporating noncollinear magnetic textures between an s-wave superconductor and a 2D quantum spin Hall insulator. It stabilizes the higher order topological superconducting phase, resulting in Majorana corner modes (MCMs) at four corners of a 2D domain. The calculated nonzero quadrupole moment characterizes the bulk topology. Subsequently, through a unitary transformation, an effective low-energy Hamiltonian reveals the effects of magnetic textures, resulting in an effective in-plane Zeeman field and spin-orbit coupling. This approach provides a qualitative depiction of the topological phase, substantiated by numerical validation within an exact real-space model. Analytically calculated effective pairings in the bulk illuminate the microscopic behavior of the SOTSC. The comprehension of MCM emergence is supported by a low-energy edge theory, which is attributed to the interplay between effective pairings of (px+py)-type and (px+ipy)-type. Our extensive study paves the way for practically attaining the SOTSC phase by integrating noncollinear magnetic textures.
... In contrast, the SOC induced due to the presence of noncollinear magnetic textures, characterized by g x,y , transforms into a (p x +p y )-type SC pairing with a gap ∆ px+py that scales with g. In the context of 2D first-order TSC, it is well-known that (p x +ip y )-type pairing induces dispersive Majorana edge modes [55,117,119], while (p x + p y )type pairing is responsible for the emergence of flat Majorana edge modes [55,64,120,121]. When the SOC strength is set to zero, λ x,y =0, the TSC phase exclusively features the flat edge modes [64], which seems to restrict the emergence of opposite mass terms in the intersecting edge Hamiltonians. ...
We put forth a theoretical framework for engineering a two-dimensional (2D) second-order topological superconductor (SOTSC) by utilizing a heterostructure: incorporating noncollinear magnetic textures between an s-wave superconductor and a 2D quantum spin Hall insulator. It stabilizes the higher order topological superconducting phase, resulting in Majorana corner modes (MCMs) at four corners of a 2D domain. The calculated non-zero quadrupole moment characterizes the bulk topology. Subsequently, through a unitary transformation, an effective low-energy Hamiltonian reveals the effects of magnetic textures, resulting in an effective in-plane Zeeman field and spin-orbit coupling. This approach provides a qualitative depiction of the topological phase, substantiated by numerical validation within exact real-space model. Analytically calculated effective pairings in the bulk illuminate the microscopic behavior of the SOTSC. The comprehension of MCM emergence is aided by a low-energy edge theory, which is attributed to the interplay between effective pairings of (px + py )-type and (px + ipy )-type. Our extensive study paves the way for practically attaining the SOTSC phase by integrating noncollinear magnetic textures.
... Other symbols in Eq. (17) indicate the same meaning as mentioned after Eq. (1). A pure 2D p-wave superconductor (in absence of any magnetic spin chain) exhibits a gapless topological superconducting phase at the bulk of the system hosting MEMs of flat type [87,88]. Afterwards, we incorporate a 1D magnetic spin chain on top of this 2D p-wave superconductor. ...
We theoretically investigate the phase transition from a non-trivial topological $p$-wave superconductor to a trivial $s$-wave like superconducting phase through a gapless phase, driven by different magnetic textures as an one-dimensional spin-chain impurity, $e.g.$ Bloch-type, in-plane and out-of-plane N\'eel-type spin-chains etc. In our proposal, the chain of magnetic impurities is placed on a spin-triplet $p$-wave superconductor where we obtain numerically as well as analytically an effective $s$-wave like pairing due to spin rotation, resulting in gradual destruction of the Majorana zero modes present in the topological superconducting phase. In particular, when the impurity spins are antiferromagnetically aligned $i.e.$ spiral wave vector $G_{s}=\pi$, the system becomes an effective $s$-wave superconductor without Majorana zero modes in the local density of states. The Shiba bands, on the other hand, formed due to the overlapping of Yu-Shiba-Rusinov states play a crucial role in this topological to trivial superconductor phase transition, confirmed by the sign change in the minigap within the Shiba bands. Moreover, interference of the Shiba bands exhibiting oscillatory behavior within the superconducting gap, $-\Delta_{p}$ to $\Delta_{p}$, as a function of $G_{s}$, also reflects an important evidence for the formation of an effective $s$-wave pairing. Such oscillation is absent in the $p$-wave regime. Our study paves the way to foresee the stability and tunability of topological superconducting phase via controlled manipulation of the spin spiral configurations of the magnetic impurity chain.
... Notably, the mapping theory is applicable for H (q) instead of the core matrix H (k), where q is a set of periodic parameters instead of the momentum k. In addition to the gapped topological systems, the non-Hermitian 045141-4 generalizations are applicable for gapless topological systems [44,[141][142][143][144][145]. After introducing the non-Hermiticity in the proposed manner, the gapless degeneracy points may change into pairs of EPs, EP rings, or EP surfaces [35,47,54,75]; although the non-Hermitian phase transition occurs, the topology remains unchanged and can be characterized by winding number as indicated in Refs. ...
Non-Hermiticity can vary the topology of system, induce topological phase transitions, and even invalidate the conventional bulk-boundary correspondence. Here, we show the introduction of non-Hermiticity without affecting the topological properties of the original chiral symmetric Hermitian systems. Conventional bulk-boundary correspondence holds; the topological phase transition and the (non)existence of edge states are unchanged even though the energy bands are inseparable due to non-Hermitian phase transitions. The Chern number for energy bands of the generalized non-Hermitian system in two dimensions is proved to be unchanged and favorably coincides with the simulated topological charge pumping. Our findings provide insights into the interplay between non-Hermiticity and topology. A topological phase transition independent of the non-Hermitian phase transition is a unique feature that is beneficial for future applications of non-Hermitian topological materials.
... We focus on a concrete model to demonstrate the main idea. In the previous works 40, 41 we have demonstrated that a topologically trivial superconductor emerges as a topological gapless state, which support Majorana flat band edge modes. The new quantum state is characterized by two band-degeneracy points with opposite topological invariant. ...
A gapful topological phase is always associated with at least two isolated energy bands with nonzero Chern number, while a topological gapless phase can appear in a system with single zero-Chern number energy band. Unlike the edge state of a topological insulator, where its energy level lives in the energy gap, the edge state of a topological gapless phase hides in the bulk spectrum, cannot be identified by the energy. In both cases, a distinguishing feature of an edge state is the robustness against a disorder perturbation, and is immune to Anderson localization. We investigate the sensitivity of bulk and edge states of topological gapless phase to the disorder perturbation via a concrete two-dimensional lattice model. The topological gapless phase is characterized by two opposite vortices in BZ which correspond to edge flat band when open boundary condition is applied. We employ the Loschmidt echo (LE) for both bulk and edge states to study the dynamic effect of disorder perturbation. We show that for an initial bulk state LE decays exponentially, while converges to a constant for an initial edge state in the presence of weak disorder. Furthermore, the convergent LE can be utilized to identify the positions of vortices as well as the phase diagram. We discuss the realization of such dynamic investigations topological photonic system.
... Notably, the mapping theory is applicable for H(q) instead of the core matrix H(k), where q is a set of periodic parameters instead of the momentum k. In addition to the gapped topological systems, the non-Hermitian generalizations are applicable for gapless topological systems [68,[194][195][196][197][198]. After introducing the non-Hermiticity in the proposed manner, the gapless degeneracy points may change into pairs of EPs, EP rings, or EP surfaces [56,72,80,101]; although the non-Hermitian phase transition occurs, the topology remains unchanged and can be characterized by winding number as indicated in Refs. ...
Non-Hermiticity can vary the topology of system, induce topological phase transition, and even invalidate the conventional bulk-boundary correspondence. Here, we show the introducing of non-Hermiticity without affecting the topological properties of the original chiral symmetric Hermitian systems. Conventional bulk-boundary correspondence holds, topological phase transition and the (non)existence of edge states are unchanged even though the energy bands are inseparable due to non-Hermitian phase transition. Chern number for energy bands of the generalized non-Hermitian system in two dimension is proved to be unchanged and favorably coincides with the simulated topological charge pumping. Our findings provide insights into the interplay between non-Hermiticity and topology. Topological phase transition independent of non-Hermitian phase transition is a unique feature that beneficial for future applications of non-Hermitian topological materials.
... In this paper, we investigate this issue through an exact solution of a concrete system. We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes 42,43 . The degeneracy points are characterized by two vortices, or Dirac nodal points in momentum space, with opposite winding numbers. ...
... As pointed in ref. 42 , two bands touch at three types of configurations: single point, double points, and curves in the k x − k y plane, determined by the region of parameter Δ−μ plane (in units of t). We focus on the non-trivial case with nonzero Δ. ...
We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes. The degeneracy points are characterized by two vortices in momentum space, with opposite winding numbers. We show rigorously that the topological gapless phase always hosts a partial Majorana flat band edge modes in a ribbon geometry, although such a single band model has zero Chern number as a topologically trivial superconductor. The flat band disappears when the gapless phase becomes topologically trivial, associating with the mergence of two vortices. Numerical simulation indicates that the flat band is robust against the disorder.
... We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes. 40,41 The degeneracy points are characterized by two vortices, or Weyl nodal points in momentum space, with opposite winding numbers, which are not removable unless meet together. This work aims to shed light on the nature of topological edge modes associated with topological gapless phase, rather than gapful topological superconductor. ...
... As pointed in Ref., 40 two bands touch at three types of configurations: single point, double points, and curves in the k x -k y plane, determined by the region of parameter ∆-µ plane (in units of t). We focus on the non-trivial case with nonzero ∆. ...
We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes. The degeneracy points are characterized by two vortices in momentum space, with opposite winding numbers, which are not removable unless meet together. We show rigorously that the topological gapless phase always hosts a partial Majorana flat band edge modes in a ribbon geometry, although such a single band model has zero Chern number as a topologically trivial superconductor. The flat band disappears when the gapless phase becomes topologically trivial, associating with the mergence of two vortices. Numerical simulation indicates that the flat band is robust against the disorder. This finding indicates that the bulk-edge correspondence can be extended to superconductors in the topologically trivial regime as recently proposed in Ref. [PRL 118, 147003 (2017)].
We propose a theoretical framework for generating gapless topological superconductivity (GTSC) hosting Majorana flat edge modes (MFEMs) in the presence of a two-dimensional (2D) array of magnetic adatoms with noncollinear spin texture deposited on top of an unconventional superconductor. Our observations reveal two distinct topological phase transitions within the emergent Shiba band depending on the exchange coupling strength (J) between magnetic adatom spins and superconducting electrons: the first one designates transition from gapless non-topological to gapless topological phase at lower J, while the second one denotes transition from gapless topological to a trivial gapped superconducting phase at higher J. The gapless topological superconducting phase survives at intermediate values of J, hosting MFEMs. Furthermore, we investigate the nature of the bulk effective pairings, which indicate that GTSC appears due to the interplay between pseudo “s-wave” and pseudo “px+py” types of pairing. Consequently, our study opens a promising avenue for the experimental realization of GTSC in 2D Shiba lattice based on d-wave superconductors as a high-temperature platform.
We report on a realistic and rather general scheme where noncollinear magnetic textures proximitized with the most common s-wave superconductor can appear as the alternative to p-wave superconductor—the prime proposal to realize two-dimensional (2D) Kitaev model for topological superconductors (TSCs) hosting Majorana flat edge mode (MFEM). A general minimal Hamiltonian suitable for magnet/superconductor heterostructures reveals robust MFEM within the gap of Shiba bands due to the emergence of an effective “px+py”-type p-wave pairing, spatially localized at the edges of a 2D magnetic domain of spin spiral. We finally verify this concept by considering Mn (Cr) monolayer grown on an s-wave superconducting substrate Nb(110) under strain [Nb(001)]. In both 2D cases, the antiferromagnetic spin-spiral solutions exhibit robust MFEM at certain domain edges that is beyond the scope of the trivial extension of one-dimensional (1D) spin-chain model in 2D. This approach, particularly when the MFEM appears in the TSC phase for such heterostructure materials, offers a perspective to extend the realm of the TSC in 2D.
