Session P23: Superconductivity Theory II: p-wave, chiral, and topological

Spin-orbit induced mixed-parity pairing in Sr$_2$RuO$_4$: a self-consistent quantum many-body analysis

The unusual superconducting state in Sr$_2$RuO$_4$ has long been viewed as being analogous to a superfluid state in liquid $^3$He. Nevertheless, calculations based on a pure odd-parity state are presently unable to completely reconcile the properties of Sr$_2$RuO$_4$. Using a self-consistent quantum many-body scheme that employs realistic parameters, we are able to model several signature properties of the normal and superconducting states of Sr$_2$RuO$_4$ such as the weak temperature dependence of the spin susceptibility below $T_c$. However, we find that the dominant component of the model superconducting state is of even parity and closely related to superconducting state for the high-$T_c$ cuprates although a smaller odd-parity component is induced by spin-orbit coupling. This mixed parity pairing state provides an alternative scenario for understanding the complex phenomena measured in Sr$_2$RuO$_4$.   

Spin and charge collective excitations in the multiband superconductivity of Sr2RuO4

Multiband superconductors with weak interband pair scattering can support soft collective modes. In the case of spin-singlet superconductivity, interband pair scattering leads to fluctuations of the relative phase of the order parameter on the different bands. However, when the superconductivity is spin-triplet, the interband pair scattering gives rise to fluctuations of both the relative phase and the relative spin on the different bands. One possible example of a multiband triplet superconductor is a recently proposed model [1] of the superconductivity in Sr$_2$RuO$_4$ in which the pairing occurs primarily on the two quasi-1D bands. We show that the collective excitations arising from relative spin fluctuations can lead to a double resonance peak in the presence of an oscillating magnetic field. We discuss how the presence or absence of such collective modes can yield clear information concerning the precise microscopic structure of the order parameter. \\ \\ $[1]$ S. Raghu, A. Kapitulnik, and S. Kivelson, Phys. Rev. Lett. {\bf 105}, 136401 (2010)   

Dynamics of domain walls in chiral p-wave superconductors

Two-fold degeneracy of the ground state of chiral $p$-wave superconductors or superfluids with $k_x\pm ik_y$ order parameter makes it possible for domain walls separating regions of opposite chirality to exist. In addition to affecting the scattering spectrum in the bulk, the domain walls also carry localized low-energy fermionic quasiparticles, whose energy essentially depends on the Josephson phase difference between the domains. Dynamical properties of the domain walls are determined by the transitions between different quasiparticle states induced by the domain wall motion. We present a microscopic calculation of the viscous friction coefficient and the effective mass of the domain walls.   

Stability of topological defects in chiral superconductors: London theory

We examine thermodynamic stability of chiral domain walls and vortices -- topological defects which can exist in chiral superconductors. Using London theory it is demonstrated that at sufficiently small applied and chiral fields the existence of domain walls and vortices in the sample is not favored and the sample's configuration is a single domain. The particular chirality of the single-domain configuration is neither favored nor disfavored by the applied field. Increasing the field leads to an entry of a domain wall loop or a vortex into the sample. Formation of a straight domain wall is never preferred in equilibrium. Values of the entry (critical) fields for both types of defects, as well as the equilibrium size of the domain wall loop, are calculated. The applicability of these results to $\rm Sr_2RuO_4$ -- a tentative chiral superconductor -- is discussed.   

Intrinsic Hall effect in a multiband chiral superconductor

We identify an intrinsic, dissipationless Hall effect in multiband chiral superconductors in the absence of a magnetic field (i.e., an \emph{anomalous} quantum Hall effect). Similarly to its analog in ferromagnets, this effect arises from inter-band transitions when time-reversal symmetry is spontaneously broken. We discuss the implications of this effect for the putative chiral $p$-wave superconductor, $\mathrm{Sr}_2\mathrm{RuO}_4$, and show that it can be of the right order of magnitude to contribute significantly to the polar Kerr rotation observed in experiments, depending on the structure of the order parameter across the bands.   

Topologically protected surface Majorana arcs and bulk Weyl fermions in ferromagnetic superconductors

A number of ferromagnetic superconductors have been recently discovered which are believed to be in the so-called ``equal spin pairing'' (ESP) state. In the ESP state the Cooper pairs condense forming order parameters $\Delta_{\uparrow,\uparrow}, \Delta_{\downarrow\downarrow}$, which are decoupled in the spin-sector. We show that these 3D systems generically support topologically protected surface Majorana arcs and bulk Weyl fermions as gapless excitations. Similar protected low-energy exotic quasiparticles should also appear in the recently discovered non-centrosymmteric superconductors in the presence of a Zeeman field. The protected surface arcs can be probed by angle-resolved photoemission (ARPES) as well as fourier transform scanning tunneling spectroscopty (FT-STS) experiments.   

Thermodynamic signatures of half-quantum vortices in $p+ip$ Josephson junction arrays

A very interesting type of excitation in a chiral p-wave superconductor is a half-quantum vortex. As the name suggests, they carry half of a superconducting flux quantum, and are only possible in superconductors with spin-triplet pairing. An astonishing feature of these excitations is the presence of topologically protected Majorana zero modes. Single half-quantum vortices were recently discovered (J. Jang et al, Science \textbf{331}(6014): 186-188) in superconducting mesoscopic rings made of Sr$_2$RuO$_4$, yet to this date they have not been observed in macroscopic samples. We propose a method for detecting half-quantum vortices in Josephson junction arrays, which could host a large number of these vortices. Contrary to a 3D setting, we argue that half-quantum vortices can be energetically preferable in quasi-2D chiral spin-triplet superconductors. As a result, half-quantum vortices rather than full vortices could drive a Berezinskii-Kosterlitz-Thouless transition (which manifests itself as a resistive transition). We propose to look for their signatures by comparing transition temperatures in $p+ip$ Josephson junction arrays in a transverse magnetic field in both unfrustrated and frustrated cases.   

Scattering theory of chiral Majorana fermion interferometry

Using scattering theory, we investigate interferometers composed of chiral Majorana fermion modes coupled to normal metal leads. We advance an approach in which also the basis states in the normal leads are written in terms of Majorana modes. As a consequence the scattering at the junctions of the lead to the Majorana mode couples modes effectively only pair-wise irrespective of the number of normal scattering modes. We demonstrate that the charge current can also be expressed in terms of interference between pairs of Majorana modes. These two basic facts permit a treatment and understanding of current and noise signatures of chiral Majorana fermion interferometry in an especially elegant way. As a particular example of applications, in Fabry-Perot-type interferometers where chiral Majorana modes form loops, resonances (anti-resonances) upon such loops always lead to peaked (suppressed) Andreev differential conductances and negative (positive) cross-correlations originate purely from two-Majorana-fermion exchange. These investigations are intimately related to current and noise signatures of Majorana bound states.   

Probing Majorana fermions in one-dimensional topological superconductors using non-equilibrium transport: an open-quantum-system study

We study one-dimensional topological superconductors using an open-quantum-system approach based on Langevin equations. We go beyond a low-energy effective model of Majorana fermions, to derive different non-equilibrium transport properties exactly in these systems. The role of the coupling between the superconducting wire and metallic leads in physical experiments to detect Majorana bound states is also examined.   

Majorana End States of Au Wires in Proximity to $d_{x^2-y^2}$-wave Superconductors

We propose a one dimensional DIII class Hamiltonian which respects time-reversal symmetry and supports zero-energy double Majorana end states. Single Majorana end states can appear if time-reversal symmetry is broken. Majorana fermions survive in the quasi-1D regime when multiple transverse sub-bands of the wire are occupied. More importantly, this model can be realized by inducing $d_{x^2-y^2}$-wave superconductivity on a quantum wire with strong spin-orbit coupling. We suggest Au wires deposited on doped LSCO realize this topological superconducting phase. The energy scales of this set-up, an induced proximity gap of 10meV and the Rashba energy of 60meV, are two orders of magnitude larger than the corresponding energy scales in semiconductor-based proposals.   

Topological Josephson effect for arbitrary values of the tunnel coupling in the Kitaev model

We investigate the Josephson effect for a setup with two lattice quantum wires featuring fused Majorana boundary modes at the tunnel junction. We show exactly that additional degeneracies occur when the size of the Josephson coupling attains a certain critical value, thus introducing additional energy level crossings. The physical consequences of these additional level crossings are discussed. It is shown that for this critical coupling the Andreev levels can be cast in the form $E_{m\sigma}=2\sigma\sqrt{2}w\cos(\phi/6-\pi m/3)$, where $m=-1,0,1$ and $\sigma=\pm 1$. The exact Josephson current exhibits the characteristic $4\pi$ periodicity along with additional features related to the extra crossings of Andreev levels at the critical value of the tunnel coupling.   

Single-particle off-diagonal long range order in $|Pf|$ wave function for topological superconductors

We ask what can be glimpsed about a fermion system from a boson wave function obtained as the absolute value of the fermion wave function in the particle number basis. We consider examples of spinless $p$-wave superconductors in one and two dimensions and find that thus constructed wave function shows single-boson off-diagonal long range order when the fermion system is in the weak-coupling topological phase, but only pair-boson order in the strong-coupling non-topological phase. We discuss implications when such fermion states are used as factors in trial wave functions e.g. from slave-fermion construction.   

Topological Superconductivity in Spin-Orbit Coupled Bands

Topological superconductors have nontrivial winding of their order parameter phase and are expected to support Majorana Fermions in their vortex cores. For this reason they have been sought after in the past couple of decades. Over the past few years, a new and promising route for realizing topological superconductors has opened due to recent advances in the field of topological insulators. The current proposals are based on semiconductor heterostructures. In the proposed devices, spin-orbit coupled bands are Zeeman split by a magnetic field and superconductivity is induced by proximity to a conventional superconductor. This leads to heterostructures of two or three layers. This talk will focus on the prospect of realizing a topological superconductor in materials with inherent spin-orbital coupling and an intrinsic tendency for superconductivity. The proposed device will allow simplification of recently suggested devices as the need for a superconducting layer will be eliminated.   

Theory of Superconductivity in Mesoscopic Systems

By using Bogoliubov-de Gennes (BdG) equations, we study superconducting (SC) states in a quasi 2-dimensional system of radius $R$. It is shown that no vortices exist in $s$-wave SC samples with $R < R_c\sim\xi(0)$, the $T=0$ coherence length. We predict that chiral $p$-wave states exhibit superconductivity for $R < R_c$ only in the presence of a vortex with opposite chirality. This {\it reentrant} SC phase is a consequence of non-zero chirality of the pairing order parameter and implies the presence of chiral edge currents. Our study may be applied to sharply probing the pairing symmetry of unconventional superconductors.