Three-dimensional Topological Superconductors Research Articles

Topological superconductors with spin-triplet pairings and Majorana Fermi arcs

Abstract We construct a three-dimensional topological superconductor Bogoliubov-de Gennes(BdG) Hamiltonian with the normal state being a three dimensional topological insulator. By introducing inter-orbital spin-triplet pairings term △3, there are topological Majorana nodes in the bulk, and they are connected by Majorana Fermi arcs on the surface, similar to the case of Weyl semimetal. Furthermore, by adding an inversion breaking term to the normal state, momentum independent pairing terms with different parities can coexist in the BdG Hamiltonian, which creates more Majorana modes similar to Andreev bound states and a richer phase diagram.

Fully gapped pairing state in spin-triplet superconductor UTe2.

The recently discovered superconductor UTe2 is a promising candidate for spin-triplet superconductors, but the symmetry of the superconducting order parameter remains highly controversial. Here, we determine the superconducting gap structure by the thermal conductivity of ultraclean UTe2 single crystals. We find that the a-axis thermal conductivity divided by temperature κ/T in zero-temperature limit is vanishingly small for both magnetic field H‖a and H‖c axes up to H/Hc2 ∼ 0.2, demonstrating the absence of nodes around the a axis contrary to the previous belief. The present results, combined with the reduction of nuclear magnetic resonance Knight shift, indicate that the superconducting order parameter belongs to the isotropic Au representation with a fully gapped pairing state, analogous to the B phase of superfluid 3He. These findings reveal that UTe2 is likely to be a long-sought three-dimensional strong topological superconductor, hosting helical Majorana surface states on any crystal plane.

Anomalous gapped boundaries between surface topological orders in higher-order topological insulators and superconductors with inversion symmetry

We show that the gapless boundary signatures---namely, chiral/helical hinge modes or localized zero modes---of three-dimensional higher-order topological insulators and superconductors with inversion symmetry can be gapped without symmetry breaking upon the introduction of non-Abelian surface topological order. In each case, the fractionalization pattern that appears on the surface is ``anomalous'' in the sense that it can be made consistent with symmetry only on the surface of a three-dimensional higher-order insulator/superconductor. Our results show that the interacting manifestation of higher-order topology is the appearance of ``anomalous gapped boundaries'' between distinct topological orders whose quasiparticles are related by inversion, possibly in conjunction with other protecting symmetries such as time-reversal symmetry and charge conservation.

Dephasing-enhanced Majorana zero modes in two-dimensional and three-dimensional higher-order topological superconductors

The 1D Kitaev model in the topological phase, with open boundary conditions, hosts strong Majorana zero modes. These are fermion parity-odd operators that almost commute with the Hamiltonian and manifest in long coherence times for edge degrees of freedom. We obtain higher-dimensional counterparts of such Majorana operators by explicitly computing their closed form expressions in models describing 2D and 3D higher-order superconductors. Due to the existence of such strong Majorana zero modes, the coherence time of the infinite temperature autocorrelation function of the corner Majorana operators in these models diverges with the linear system size. In the presence of a certain class of orbital-selective dissipative dynamics, the coherence times of half of the corner Majorana operators is enhanced, while the time correlations corresponding to the remaining corner Majoranas decay much faster as compared with the unitary case. We numerically demonstrate robustness of the coherence times to the presence of disorder.

Unconventional Majorana fermions on the surface of topological superconductors protected by rotational symmetry

Topological superconductors are exotic gapped phases of matter hosting Majorana mid-gap states on their boundaries. In conventional three-dimensional topological superconductors, Majorana in-gap states appear in the form of spin-1/2 fermions with a quasirelativistic dispersion relation. Here, we show that unconventional Majorana states can emerge on the surface of three-dimensional topological superconductors protected by rotational symmetry. The unconventional Majorana surface states are classified into three different categories: a spin-$s$ Majorana fermion with $(2s+1)$-fold degeneracy $(s\ensuremath{\ge}3/2)$, a Majorana Fermi line carrying two distinct topological charges, and a quartet of spin-1/2 Majorana fermions related by fourfold rotational symmetry. The spectral properties of the first two types, which go beyond conventional spin-1/2 fermions, are unique to topological superconductors and have no counterparts in topological insulators. We show that unconventional Majorana surface states can be obtained in the superconducting phase of doped ${Z}_{2}$ topological insulators or Dirac semimetals with rotational symmetry.

How spectrum-wide quantum criticality protects surface states of topological superconductors from Anderson localization: Quantum Hall plateau transitions (almost) all the way down

We review recent numerical studies of two-dimensional (2D) Dirac fermion theories that exhibit an unusual mechanism of topological protection against Anderson localization. These describe surface-state quasiparticles of time-reversal invariant, three-dimensional (3D) topological superconductors (TSCs), subject to the effects of quenched disorder. Numerics reveal a surprising connection between 3D TSCs in classes AIII, CI, and DIII, and 2D quantum Hall effects in classes A, C, and D. Conventional arguments derived from the non-linear σ-model picture imply that most TSC surface states should Anderson localize for arbitrarily weak disorder (CI, AIII), or exhibit weak antilocalizing behavior (DIII). The numerical studies reviewed here instead indicate spectrum-wide surface quantum criticality, characterized by robust eigenstate multifractality throughout the surface-state energy spectrum. In other words, there is an “energy stack” of critical wave functions. For class AIII, multifractal eigenstate and conductance analysis reveals identical statistics for states throughout the stack, consistent with the class A integer quantum-Hall plateau transition (QHPT). Class CI TSCs exhibit surface stacks of class C spin QHPT states. Critical stacking of a third kind, possibly associated to the class D thermal QHPT, is identified for nematic velocity disorder of a single Majorana cone in class DIII. The Dirac theories studied here can be represented as perturbed 2D Wess–Zumino–Novikov–Witten sigma models; the numerical results link these to Pruisken models with the topological angle ϑ=π. Beyond applications to TSCs, all three stacked Dirac theories (CI, AIII, DIII) naturally arise in the effective description of dirty d-wave quasiparticles, relevant to the high-Tc cuprates.

Higher-order topological superconductivity of spin-polarized fermions

We study the superconductivity of spin-polarized electrons in centrosymmetric ferromagnetic metals. Due to the spin-polarization and the Fermi statistics of electrons, the superconducting pairing function naturally has odd parity. According to the parity formula proposed by Fu, Berg, and Sato, odd-parity pairing leads to conventional first-order topological superconductivity when a normal metal has an odd number of Fermi surfaces. Here, we derive generalized parity formulae for the topological invariants characterizing higher-order topology of centrosymmetric superconductors. Based on the formulae, we systematically classify all possible band structures of ferromagnetic metals that can induce inversion-protected higher-order topological superconductivity. Among them, doped ferromagnetic nodal semimetals are identified as the most promising normal state platform for higher-order topological superconductivity. In two dimensions, we show that odd-parity pairing of doped Dirac semimetals induces a second-order topological superconductor. In three dimensions, odd-parity pairing of doped nodal line semimetals generates a nodal line topological superconductor with monopole charges. On the other hand, odd-parity pairing of doped monopole nodal line semimetals induces a three-dimensional third-order topological superconductor. Our theory shows that the combination of superconductivity and ferromagnetic nodal semimetals opens up a new avenue for future topological quantum computations using Majorana zero modes.

Unhinging the Surfaces of Higher-Order Topological Insulators and Superconductors.

We show that the chiral Dirac and Majorana hinge modes in three-dimensional higher-order topological insulators (HOTIs) and superconductors (HOTSCs) can be gapped while preserving the protecting C_{2n}T symmetry upon the introduction of non-Abelian surface topological order. In both cases, the topological order on a single side surface breaks time-reversal symmetry, but appears with its time-reversal conjugate on alternating sides in a C_{2n}T preserving pattern. In the absence of the HOTI/HOTSC bulk, such a pattern necessarily involves gapless chiral modes on hinges between C_{2n}T-conjugate domains. However, using a combination of K-matrix and anyon condensation arguments, we show that on the boundary of a 3D HOTI/HOTSC these topological orders are fully gapped and hence "anomalous." Our results suggest that new patterns of surface and hinge states can be engineered by selectively introducing topological order only on specific surfaces.

Wiedemann-Franz-type relation between shot noise and thermal conduction of Majorana surface states in a three-dimensional topological superconductor

We compare the thermal conductance $G_{\rm thermal}$ (at temperature $T$) and the electrical shot noise power $P_{\rm shot}$ (at bias voltage $V\gg k_{\rm B}T/e$) of Majorana fermions on the two-dimensional surface of a three-dimensional topological superconductor. We present analytical and numerical calculations to demonstrate that, for a local coupling between the superconductor and metal contacts, $G_{\rm thermal}/P_{\rm shot}= {\cal L}T/eV$ (with ${\cal L}$ the Lorenz number). This relation is ensured by the combination of electron-hole and time-reversal symmetries, irrespective of the microscopics of the surface Hamiltonian, and provides for a purely electrical way to detect the charge-neutral Majorana surface states. A surface of aspect ratio $W/L\gg 1$ has the universal shot-noise power $P_{\rm shot}=(W/L)\times (e^2/h)\times (eV/2\pi)$

Chiral gravitomagnetic effect in topological superconductors and superfluids

We theoretically search for dynamical cross-correlated responses of three-dimensional topological superconductors and superfluids. It has been suggested that a gravitational topological term, which is analogous to the theta term in topological insulators, can be derived in three-dimensional time-reversal invariant topological superconductors and superfluids, and that the dynamical gravitational axion field can be realized by the fluctuation of the relative phase, i.e., by the Leggett mode between topological $p$-wave pairing and conventional $s$-wave pairing. In the presence of the dynamical gravitational axion field, we propose the emergence of the "chiral gravitomagnetic effect", a thermal current generation by gravitomagnetic fields, i.e., by mechanical rotations. This effect can be regarded as a thermal counterpart of the chiral magnetic effect which has been studied mainly in Weyl semimetals. We also show the occurrence of the anomalous thermal Hall effect in the bulk. We discuss a possible application of our study to the thermal responses of Weyl superconductors.

Higgs mechanism, phase transitions, and anomalous Hall effect in three-dimensional topological superconductors

We demonstrate that the Higgs mechanism in three-dimensional topological superconductors exhibits unique features with experimentally observable consequences. The Higgs model we discuss has two superconducting components and an axion-like magnetoelectric term with the phase difference of the superconducting order parameters playing the role of the axion field. Due to this additional term, quantum electromagnetic and phase fluctuations lead to a robust topologically non-trivial state that holds also in the presence of interactions. In this sense, we show that the renormalization flow of the topologically nontrivial phase cannot be continuously deformed into a topologically non-trivial one. One consequence of our analysis of quantum critical fluctuations, is the possibility of having a first-order phase transition in the bulk and a second-order phase transition on the surface. We also explore another consequence of the axionic Higgs electrodynamics, namely, the anomalous Hall effect. In the low frequency London regime an anomalous Hall effect is induced in the presence of an applied electric field parallel to the surface. This anomalous Hall current is induced by a Lorentz-like force arising from the axion term, and it involves the relative superfluid velocity of the superconducting components. The anomalous Hall current has a negative sign, a situation reminiscent of, but quite distinct in physical origin from the anomalous Hall effect observed in high-$T_c$ superconductors. In contrast to the latter, the anomalous Hall effect in topological superconductors is non-dissipative and occurs in the absence of vortices.

Tri-Dirac surface modes in topological superconductors

We propose a new type of topological surface modes having cubic dispersion in three-dimensional topological superconductors. Lower order dispersions are prohibited by the threefold rotational symmetry and time-reversal symmetry. Cooper pairing in the bulk changes sign under improper rotations, akin to$^{3}$He-B. The surface manifestations are a divergent surface density of states at the Fermi level and isospins that rotate three times as they circle the origin in momentum space. We propose that Heusler alloys with band inversion are candidate materials to harbor the novel topological superconductivity.

Surface States around a Vortex in Topological Superconductors: Intersection of a Surface and a Vortex

We numerically show that the zero-energy Majorana surface states are suppressed around a vortex in the three-dimensional topological superconductors such as Cu$_{x}$Bi$_{2}$Se$_{3}$ and Sn$_{1-x}$In$_{x}$Te. On the other hand, the zero-energy Majorana bound states along the vortex line are robust against cut by the surface. The suppression of the surface bound states is similar to that with a magnetic impurity on the surface of the topological insulator. The suppression of the surface-bound states around a vortex can be observed as the unconventional energy dependence of imaging of the scanning tunneling microscopy/spectroscopy.

Surface transport coefficients for three-dimensional topological superconductors

We argue that surface spin and thermal conductivities of three-dimensional topological superconductors are universal and topologically quantized at low temperature. For a bulk winding number $\nu$, there are $|\nu|$ "colors" of surface Majorana fermions. Localization corrections to surface transport coefficients vanish due to time-reversal symmetry (TRS). We argue that Altshuler-Aronov interaction corrections vanish because TRS forbids color or spin Friedel oscillations. We confirm this within a perturbative expansion in the interactions, and to lowest order in a large-$|\nu|$ expansion. In both cases, we employ an asymptotically exact treatment of quenched disorder effects that exploits the chiral character unique to two-dimensional, time-reversal-invariant Majorana surface states.

Helical Majorana surface states of strongly disordered topological superconductors with time-reversal symmetry

Noncentrosymmetric superconductors with strong spin-orbit coupling and the B phase of $^{3}\mathrm{He}$ are possible realizations of topological superconductors with time-reversal symmetry. The nontrivial topology of these time-reversal invariant superconductors manifests itself at the material surface in the form of helical Majorana modes. In this paper, using extensive numerical simulations, we investigate the stability and properties of these Majorana states under strong surface disorder, which influences both bulk and surface states. To characterize the effects of strong disorder, we compute the level spacing statistics and the local density of states of both two- and three-dimensional topological superconductors. The Majorana surface states, which are located in the outermost layers of the superconductor, are protected against weak disorder due to their topological characteristic. Sufficiently strong disorder, on the other hand, partially localizes the surface layers, with a more pronounced effect on states with energies close to the gap than on those with energies close to zero. In particular, we observe that for all disorder strengths and configurations there always exist two extended states at zero energy that can carry thermal current. At the crossover from weak to strong disorder the surface state wave functions and the local density of states show signs of critical delocalization. We find that at this crossover the edge density of states of two-dimensional topological superconductors exhibits a zero-energy divergence, reminiscent of the Dyson singularity of quasi-one-dimensional dirty superconductors.

Robustness against non-magnetic impurities in topological superconductors

We study the robustness against non-magnetic impurities in a three-dimensional topological superconductor, focusing on an effective model (massive Dirac Bogoliubov-de Gennes (BdG) Hamiltonian with s-wave on-site pairing) of CuxBi2Se3 with the parameter set determined by the first-principles calculation. With the use of the self-consistent T- matrix approximation for impurity scattering, we discuss the impurity-concentration dependence of the zero-energy density of states. We show that a single material variable, measuring relativistic effects in the Dirac-BdG Hamiltonian, well characterizes the numerical results. In the nonrelativistic limit, the odd-parity fully-gapped topological superconductivity is fragile against non-magnetic impurities, since this superconductivity can be mapped onto the p-wave superconductivity. On the other hand, in the ultrarelativistic limit, the superconductivity is robust against the non-magnetic impurities, since the effective model has the s-wave superconductivity. We derive the effective Hamiltonian in the both limit.

Nonmagnetic impurity effects in a three-dimensional topological superconductor: Fromp- tos-wave behaviors

Unconventional features in superconductivity are revealed by responses to impurity scattering. We study nonmagnetic impurity effects in a three-dimensional topological superconductor, focusing on an effective model (massive Dirac Hamiltonian with $s$-wave on-site pairing) of copper-doped bismuth-selenium compounds. Using a self-consistent $T$-matrix approach for impurity scattering, we examine in-gap states in density of states. We find that the results are well characterized by a single material variable, which measures relativistic effects in the Dirac Hamiltonian. In the nonrelativistic regime, an odd-parity superconducting state is fragile against nonmagnetic impurities. We show that this behavior is caused by a $p$-wave character involved in the topological superconducting state. In contrast, we show that in the relativistic regime the superconductivity is robust against nonmagnetic impurities, owing to an $s$-wave character. To summarize, the system has two aspects, $p$- and $s$-wave features, depending on the weight of relativistic effects.

Topological protection, disorder, and interactions: Survival at the surface of three-dimensional topological superconductors

We consider the interplay of disorder and interactions upon the gapless surface states of 3D topological superconductors. The combination of topology and superconducting order inverts the action of time-reversal symmetry, so that extrinsic time-reversal invariant surface perturbations appear only as "pseudomagnetic" fields (abelian and non-abelian vector potentials, which couple to spin and valley currents). The main effect of disorder is to induce multifractal scaling in surface state wavefunctions. These critically delocalized, yet strongly inhomogeneous states renormalize interaction matrix elements relative to the clean system. We compute the enhancement or suppression of interaction scaling dimensions due to the disorder exactly, using conformal field theory. We determine the conditions under which interactions remain irrelevant in the presence of disorder for symmetry classes AIII and DIII. In the limit of large topological winding numbers (many surface valleys), we show that the effective field theory takes the form of a Finkel'stein non-linear sigma model, augmented by the Wess-Zumino-Novikov-Witten term. The sigma model incorporates interaction effects to all orders, and provides a framework for a controlled perturbative expansion; the inverse spin or thermal conductance is the small parameter. For class DIII we show that interactions are always irrelevant, while in class AIII there is a finite window of stability, controlled by the disorder. Outside of this window we identify new interaction-stabilized fixed points.

New Class of Topological Superconductors Protected by Magnetic Group Symmetries

We study a new type of three-dimensional topological superconductor that exhibits Majorana zero modes (MZM) protected by a magnetic group symmetry, a combined antiunitary symmetry composed of a mirror reflection and time reversal. This new symmetry enhances the noninteracting topological classification of a superconducting vortex from Z2 to Z, indicating that multiple MZMs can coexist at the end of one magnetic vortex of unit flux. Especially, we show that a vortex binding two MZMs can be realized on the (001) surface of a topological crystalline insulator SnTe with proximity induced BCS Cooper pairing, or in bulk superconductor InxSn1-xTe.

Hunting Majorana Fermions in Topological Superconductors

Theoretical research shows a vortex core in three-dimensional topological superconductors traps a spin-polarized Majorana fermion.