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Nonlocal Cooper pairs in finite topological superconductors and their relation to Majorana nonlocality
Authors:
Hiroto Mizoguchi,
Yutaro Nagae,
Yasuhiro Asano,
Satoshi Ikegaya
Abstract:
We identify two fundamental properties of the Gor'kov Green's function of finite one-dimensional topological superconductors. In the low-frequency (low-energy) regime, the normal and anomalous Green's functions, which describe single-particle and Cooper-pair correlations, respectively, become identical up to a phase factor. Moreover, they exhibit pronounced nonlocality: correlations between the tw…
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We identify two fundamental properties of the Gor'kov Green's function of finite one-dimensional topological superconductors. In the low-frequency (low-energy) regime, the normal and anomalous Green's functions, which describe single-particle and Cooper-pair correlations, respectively, become identical up to a phase factor. Moreover, they exhibit pronounced nonlocality: correlations between the two ends of the system grow exponentially with system length, whereas local correlations at either end vanish in the zero-frequency limit. These striking features signify the emergence of unconventional nonlocal Cooper pairs associated with a nonlocal fermionic mode composed of hybridized Majorana end modes. The nonlocal Cooper pairs are directly linked to fermion parity and to the nonlocal transport properties of finite topological superconductors. By focusing on pair correlations, our analysis advances the understanding of Majorana nonlocality, a key concept in topological quantum computation.
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Submitted 27 April, 2026;
originally announced April 2026.
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Robust realization of spin-polarized specular Andreev reflection in V$_2$O-based altermagnets
Authors:
Yutaro Nagae,
Andreas P. Schnyder,
Satoshi Ikegaya
Abstract:
We theoretically investigate charge transport in a junction between a conventional superconductor and a V$_2$O-based altermagnet exhibiting distinctive spin-split quasi-one-dimensional Fermi surfaces. The altermagnet is described by a microscopically motivated six-orbital model that incorporates sublattice degrees of freedom associated with both V and O sites. Based on calculations performed under…
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We theoretically investigate charge transport in a junction between a conventional superconductor and a V$_2$O-based altermagnet exhibiting distinctive spin-split quasi-one-dimensional Fermi surfaces. The altermagnet is described by a microscopically motivated six-orbital model that incorporates sublattice degrees of freedom associated with both V and O sites. Based on calculations performed under various boundary conditions, we demonstrate the robust emergence of specular Andreev reflection with a distinctive spin polarization. Furthermore, we propose an efficient multiterminal setup to detect this specular Andreev reflection through nonlocal conductance measurements. Our results establish V$_2$O-based altermagnets as a promising platform for realizing spin-resolved Cooper pair splitting, which is essential for generating energy-entangled electron pairs.
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Submitted 14 April, 2026;
originally announced April 2026.
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Reentrant Superconductivity in Zeeman Fields
Authors:
Tomoya Sano,
Kota Tabata,
Satoshi Ikegaya,
Yasuhiro Asano
Abstract:
We propose a theoretical model for a superconductor that exhibits the reentrant superconductivity in Zeeman fields. The Bogoliubov-de Gennes Hamiltonian includes three vectors in spin space: a $d$ vector of a spin-triplet superconducting state, a potential representing spin-orbit interactions, and a Zeeman field. When the three vectors are perpendicular to one another, the spin-orbit interaction s…
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We propose a theoretical model for a superconductor that exhibits the reentrant superconductivity in Zeeman fields. The Bogoliubov-de Gennes Hamiltonian includes three vectors in spin space: a $d$ vector of a spin-triplet superconducting state, a potential representing spin-orbit interactions, and a Zeeman field. When the three vectors are perpendicular to one another, the spin-orbit interaction suppresses superconductivity in weak Zeeman fields and enhances superconductivity in strong Zeeman fields. The instability (stability) of superconducting state is characterized by the appearance of odd-frequency (even-frequency) Cooper pairs.
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Submitted 12 April, 2026; v1 submitted 16 February, 2026;
originally announced February 2026.
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Exceptionally large winding number of a finite-size topological superconductor
Authors:
Satoshi Ikegaya,
Shingo Kobayashi,
Yasuhiro Asano
Abstract:
We study finite-size-induced topological phenomena in unconventional superconductors. Specifically, we focus on a thin film with a persistent spin texture, fabricated on a high-$T_{\text{c}}$ cuprate $d_{xy}$-wave superconductors. In two-dimensional $d_{xy}$-wave superconductors, flat-band Andreev bound states appear at the edges. As the system narrows, these bound states acquire an energy gap due…
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We study finite-size-induced topological phenomena in unconventional superconductors. Specifically, we focus on a thin film with a persistent spin texture, fabricated on a high-$T_{\text{c}}$ cuprate $d_{xy}$-wave superconductors. In two-dimensional $d_{xy}$-wave superconductors, flat-band Andreev bound states appear at the edges. As the system narrows, these bound states acquire an energy gap due to finite-size hybridization and spin-orbit coupling of the persistent spin texture. This induced gap gives rise to the emergence of a topological phase, characterized by an exceptionally large one-dimensional winding number that scales with the film width. We demonstrate the appearance of highly degenerate zero-energy states, leading to anomalous perfect charge transport in dirty superconducting junctions. These findings provide a promising platform for exploring fascinating topological superconducting phases driven by gapped Andreev bound states.
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Submitted 8 April, 2025;
originally announced April 2025.
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Noise-to-current ratio divergence as a fingerprint of dispersing Majorana edge modes
Authors:
Leo Katayama,
Andreas P. Schnyder,
Yasuhiro Asano,
Satoshi Ikegaya
Abstract:
The definitive detection of Majorana modes in topological superconductors is a key issue in condensed matter physics. Here we propose a smoking-gun experiment for the detection of one-dimensional dispersing Majorana edge modes, based on theoretical results for multi-terminal transport in a setup consisting of two normal metal leads and a topological superconductor. In the proposed device, the unpa…
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The definitive detection of Majorana modes in topological superconductors is a key issue in condensed matter physics. Here we propose a smoking-gun experiment for the detection of one-dimensional dispersing Majorana edge modes, based on theoretical results for multi-terminal transport in a setup consisting of two normal metal leads and a topological superconductor. In the proposed device, the unpaired nature of the Majorana edge modes inherently leads to the absence of the charge current in the linear response regime, while the current fluctuation remains significant. Therefore, the divergence in the noise-to-current ratio serves as unambiguous evidence for the presence of the dispersing Majorana edge modes. We reach this conclusion analytically, without relying on any specific model of topological superconductors. In addition, using tight-binding models of topological-insulator-based topological superconductors, we numerically verify the predicted divergent noise-to-current ratio. We also discuss the application of our proposal to the CoSi$_2$/TiSi$_2$ heterostructure and the iron-based superconductor FeTe$_{1-x}$Se$_x$.
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Submitted 28 February, 2025;
originally announced February 2025.
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Majorana flat bands and anomalous proximity effects in $p$-wave magnet--superconductor hybrid systems
Authors:
Yutaro Nagae,
Leo Katayama,
Satoshi Ikegaya
Abstract:
Flat-band Majorana bound states of nodal $p$-wave superconductors give rise to striking electromagnetic anomalies, reflecting their high degree of degeneracy at the Fermi level. However, experimental investigations of these states have been limited because of the scarcity of materials exhibiting intrinsic $p$-wave superconductivity. In this Letter, we demonstrate that Majorana flat bands can emerg…
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Flat-band Majorana bound states of nodal $p$-wave superconductors give rise to striking electromagnetic anomalies, reflecting their high degree of degeneracy at the Fermi level. However, experimental investigations of these states have been limited because of the scarcity of materials exhibiting intrinsic $p$-wave superconductivity. In this Letter, we demonstrate that Majorana flat bands can emerge in a hybrid system consisting of a conventional superconductor and a $p$-wave magnet, a recently proposed class of unconventional magnets that possess a unique composite symmetry, the $[C_{2\perp}||\boldsymbol{t}]$ symmetry. The degeneracy of the flat-band Majorana bound states is protected by chiral symmetry from the BDI symmetry class, which originates from the $[C_{2\perp}||\boldsymbol{t}]$ symmetry of the $p$-wave magnet. In addition, we predict the robust appearance of a zero-bias conductance peak in a dirty normal-metal--superconductor junction containing a $p$-wave magnet, which serves as an unambiguous signature of anomalous proximity effects associated with the Majorana flat bands.
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Submitted 14 February, 2025; v1 submitted 4 February, 2025;
originally announced February 2025.
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Anomalous proximity effect of a spin-singlet superconductor with a spin-orbit interaction
Authors:
Jaechul Lee,
Satoshi Ikegaya,
Yasuhiro Asano
Abstract:
The anomalous proximity effect of a spin-triplet $p$-wave superconductor has been known as a part of the Majorana physics and is explained by the penetration of zero-energy states from a surface of a superconductor to a dirty normal metal. We demonstrate that a spin-singlet $d$-wave superconductor without any surface zero-energy states exhibits the anomalous proximity effect in the presence of a s…
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The anomalous proximity effect of a spin-triplet $p$-wave superconductor has been known as a part of the Majorana physics and is explained by the penetration of zero-energy states from a surface of a superconductor to a dirty normal metal. We demonstrate that a spin-singlet $d$-wave superconductor without any surface zero-energy states exhibits the anomalous proximity effect in the presence of a specific spin-orbit interaction. The results show the quantization of the zero-bias conductance in a dirty normal-metal/superconductor junction. We also discuss a relation between our findings and results in an experiment on a CoSi$_2$/TiSi$_2$ junction.
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Submitted 21 March, 2025; v1 submitted 28 January, 2025;
originally announced January 2025.
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Thermoelectric effect in a superconductor with Bogoliubov Fermi surfaces
Authors:
Tomoya Sano,
Takumi Sato,
Akihiro Sasaki,
Satoshi Ikegaya,
Shingo Kobayashi,
Yasuhiro Asano
Abstract:
We study theoretically the thermoelectric effect in a superconducting state having the Bogoliubov-Fermi surfaces which stays in a thin superconducting layer between a conventional superconductor and an insulator. The thermoelectric coefficients calculated based on the linear response theory show the remarkable anisotropy in real space, which are explained well by the anisotropic shape of the Bogol…
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We study theoretically the thermoelectric effect in a superconducting state having the Bogoliubov-Fermi surfaces which stays in a thin superconducting layer between a conventional superconductor and an insulator. The thermoelectric coefficients calculated based on the linear response theory show the remarkable anisotropy in real space, which are explained well by the anisotropic shape of the Bogoliubov-Fermi surface in momentum space. Our results indicate a way to check the existence of the Bogoliubov-Fermi surfaces in a stable superconducting state because the anisotropy is controlled by the direction of an applied magnetic field.
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Submitted 16 March, 2025; v1 submitted 10 November, 2024;
originally announced November 2024.
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Spin-polarized Specular Andreev Reflections in Altermagnets
Authors:
Yutaro Nagae,
Andreas P. Schnyder,
Satoshi Ikegaya
Abstract:
We show theoretically that specular Andreev reflection occurs stably at altermagnet--superconductor interfaces, which is a phenomenon that has previously been predicted only in a limited range of materials, such as Dirac/Weyl materials with fine-tuned chemical potentials. Furthermore, the characteristic spin-split bands of the altermagnet lead to a distinctive spin polarization in the specular And…
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We show theoretically that specular Andreev reflection occurs stably at altermagnet--superconductor interfaces, which is a phenomenon that has previously been predicted only in a limited range of materials, such as Dirac/Weyl materials with fine-tuned chemical potentials. Furthermore, the characteristic spin-split bands of the altermagnet lead to a distinctive spin polarization in the specular Andreev reflections. By utilizing this feature, we propose a device that integrates the functions of both a Cooper pair splitter and a spin beam splitter, thereby creating energy-entangled electron pairs. The positive nonlocal conductance and the positive noise cross-correlation are unambiguous signatures of specular Andreev reflections in the proposed device.
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Submitted 8 April, 2025; v1 submitted 11 March, 2024;
originally announced March 2024.
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Oscillating-charged Andreev Bound States and Their Appearance in UTe$_2$
Authors:
Satoshi Ando,
Shingo Kobayashi,
Andreas P. Schnyder,
Yasuhiro Asano,
Satoshi Ikegaya
Abstract:
Surface Andreev bound states, including Majorana bound states in topological superconductors, are typically charge neutral. In this work, we demonstrate the emergence of unconventional charged Andreev bound states in a superconductor with a sublattice degree of freedom, where the sign of charge density of the Andreev bound states oscillates between the two sublattices. The oscillating-charged Andr…
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Surface Andreev bound states, including Majorana bound states in topological superconductors, are typically charge neutral. In this work, we demonstrate the emergence of unconventional charged Andreev bound states in a superconductor with a sublattice degree of freedom, where the sign of charge density of the Andreev bound states oscillates between the two sublattices. The oscillating-charged Andreev bound states lead to a complete breakdown of the proportionality among the electron part of the spectral function, the local density of states, and the tunneling conductance spectrum for energies below the superconducting gap. We also discuss the possible occurrence of these Andreev bound states in UTe$_2$ and locally noncentrosymmetric superconductors.
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Submitted 8 April, 2025; v1 submitted 3 March, 2024;
originally announced March 2024.
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Multi-locational Majorana Zero Modes
Authors:
Yutaro Nagae,
Andreas P. Schnyder,
Yukio Tanaka,
Yasuhiro Asano,
Satoshi Ikegaya
Abstract:
We show the appearance of an unconventional Majorana zero mode whose wave function splits into multiple parts located at different ends of different topological superconductors, hereinafter referred to as a multi-locational Majorana zero mode. Specifically, we discuss the multi-locational Majorana zero modes in a three-terminal Josephson junction consisting of topological superconductors, which fo…
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We show the appearance of an unconventional Majorana zero mode whose wave function splits into multiple parts located at different ends of different topological superconductors, hereinafter referred to as a multi-locational Majorana zero mode. Specifically, we discuss the multi-locational Majorana zero modes in a three-terminal Josephson junction consisting of topological superconductors, which forms an elemental qubit of fault-tolerant topological quantum computers. We also demonstrate anomalously long-ranged nonlocal resonant transport phenomena caused by the multi-locational Majorana zero mode.
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Submitted 11 March, 2024; v1 submitted 23 June, 2023;
originally announced June 2023.
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Tunable Josephson diode effect on the surface of topological insulators
Authors:
Bo Lu,
Satoshi Ikegaya,
Pablo Burset,
Yukio Tanaka,
Naoto Nagaosa
Abstract:
The Josephson rectification effect, where the resistance is finite in one direction while zero in the other, has been recently realized experimentally. The resulting Josephson diode has many potential applications on superconducting devices, including quantum computers. Here, we theoretically show that a superconductor-normal metal-superconductor Josephson junction diode on the two-dimensional sur…
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The Josephson rectification effect, where the resistance is finite in one direction while zero in the other, has been recently realized experimentally. The resulting Josephson diode has many potential applications on superconducting devices, including quantum computers. Here, we theoretically show that a superconductor-normal metal-superconductor Josephson junction diode on the two-dimensional surface of a topological insulator has large tunability. The magnitude and sign of the diode quality factor strongly depend on the external magnetic field, gate voltage, and the length of the junction. Such rich properties stem from the interplay between different current-phase relations for the multiple transverse transport channels, and can be used for designing realistic superconducting diode devices.
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Submitted 16 June, 2023; v1 submitted 18 November, 2022;
originally announced November 2022.
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Surface State of Inter-orbital Pairing State in Sr$_2$RuO$_4$ Superconductor
Authors:
Satoshi Ando,
Satoshi Ikegaya,
Shun Tamura,
Yukio Tanaka,
Keiji Yada
Abstract:
We study the (001) surface state of a recently proposed $E_g$ symmetry inter-orbital-odd spin-triplet s-wave superconducting (SC) state in Sr$_2$RuO$_4$ (SRO). We confirm that this pair potential is transformed into a chiral $d$-wave pair potential and a pseudo-Zeeman field in the band basis for a low-energy range. Due to the chiral $d$-wave pair potential, the surface states appear near zero ener…
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We study the (001) surface state of a recently proposed $E_g$ symmetry inter-orbital-odd spin-triplet s-wave superconducting (SC) state in Sr$_2$RuO$_4$ (SRO). We confirm that this pair potential is transformed into a chiral $d$-wave pair potential and a pseudo-Zeeman field in the band basis for a low-energy range. Due to the chiral $d$-wave pair potential, the surface states appear near zero energy in the momentum range enclosed by the nodal lines of the chiral d-wave pair potential for each band at the (001) surface. Nevertheless, the pseudo-Zeeman field gives band splitting of the surface states, and its splitting energy is much smaller than the SC energy gap. The local density of states (LDOS) at the (001) surface of the SC state has a pronounced peak structure at zero energy due to the surface states near zero energy when the order of the resolution is lower than the splitting energy. This peak structure is robust under perturbations, such as an orbital Rashba coupling or an $E_u$ SC pair potential at the surface.
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Submitted 14 October, 2022; v1 submitted 5 October, 2022;
originally announced October 2022.
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Destruction of surface states of ($d_{zx}+id_{yz}$)-wave superconductor by surface roughness: application to Sr$_2$RuO$_4$
Authors:
Shu-Ichiro Suzuki,
Satoshi Ikegaya,
Alexander A. Golubov
Abstract:
The fragility of the chiral surface current of ($d_{zx}+id_{yz}$)-wave superconductor, a potential candidate for Sr$_2$RuO$_4$, against surface roughness is demonstrated utilizing the quasiclassical Eilenberger theory. Comparing the chiral surface currents of ($d_{zx}+id_{yz}$)-wave and ($p_{x}+ip_{y}$)-wave pairings, we conclude the chiral current for ($d_{zx}+id_{yz}$)-wave SC is much more fragi…
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The fragility of the chiral surface current of ($d_{zx}+id_{yz}$)-wave superconductor, a potential candidate for Sr$_2$RuO$_4$, against surface roughness is demonstrated utilizing the quasiclassical Eilenberger theory. Comparing the chiral surface currents of ($d_{zx}+id_{yz}$)-wave and ($p_{x}+ip_{y}$)-wave pairings, we conclude the chiral current for ($d_{zx}+id_{yz}$)-wave SC is much more fragile than that for the ($p_x+ip_y$)-wave one. The difference can be understood in terms of the orbital symmetry of the odd-frequency Cooper pairs arising at the surface. Our results show the ($d_{zx}+id_{yz}$)-wave scenario can explain the null spontaneous magnetization in Sr$_2$RuO$_4$ experiments.
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Submitted 8 July, 2022;
originally announced July 2022.
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Anomalous Fano factor as a signature of Bogoliubov Fermi surfaces
Authors:
Sayan Banerjee,
Satoshi Ikegaya,
Andreas P. Schnyder
Abstract:
Noise spectroscopy is a key technique to investigate the nature and dynamics of charge carriers in superconductors. The recently discovered superconducting hybrids with Bogoliubov Fermi surfaces exhibit a particularly intriguing and rich charge dynamics, as their charge carriers consist of both Cooper pairs and an extensive number of Bogoliubov quasiparticles. Motivated by this, we compute the noi…
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Noise spectroscopy is a key technique to investigate the nature and dynamics of charge carriers in superconductors. The recently discovered superconducting hybrids with Bogoliubov Fermi surfaces exhibit a particularly intriguing and rich charge dynamics, as their charge carriers consist of both Cooper pairs and an extensive number of Bogoliubov quasiparticles. Motivated by this, we compute the noise spectra of Bogoliubov Fermi surfaces and identify their key signatures in the differential conductance and the Fano factor. Specifically, we consider a semiconductor/superconductor hybrid device with an in-plane magnetic field, which exhibits several Bogoliubov Fermi surfaces. The number and orientation of the Bogoliubov Fermi surfaces in this device can be readily controlled by the applied magnetic field, which in turn alters the noise signal. In particular, we find that the Fano factor exhibits a reduced value, substantially lower than two, whenever the charge dynamics is governed by a large number of Bogoliubov quasiparticles. Using experimentally relevant parameters, we make a number of specific predictions for the noise spectra, that can be used as direct evidence of Bogoliubov Fermi surfaces. In particular, we find that the Fano factor as a function of magnetic field and spin-orbit coupling exhibits characteristic discontinuities at the transition lines that separate phases with different number of Bogoliubov Fermi surfaces.
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Submitted 8 July, 2022;
originally announced July 2022.
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Flat-band Majorana bound states in topological Josephson junctions
Authors:
Daisuke Oshima,
Satoshi Ikegaya,
Andreas P. Schnyder,
Yukio Tanaka
Abstract:
Nodal topological superconductors characterized by $p_x$-wave pairing symmetry host flat-band Majorana bound states causing drastic anomalies in low-energy electromagnetic responses. Nevertheless, the study of flat-band Majorana bound states has been at a standstill owing to a serious lack of candidate materials for $p_x$-wave superconductors. In this paper, by expanding a scheme of planar topolog…
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Nodal topological superconductors characterized by $p_x$-wave pairing symmetry host flat-band Majorana bound states causing drastic anomalies in low-energy electromagnetic responses. Nevertheless, the study of flat-band Majorana bound states has been at a standstill owing to a serious lack of candidate materials for $p_x$-wave superconductors. In this paper, by expanding a scheme of planar topological Josephson junctions, we propose a promising device realizing an effective $p_x$-wave superconductor. Specifically, we consider a three-dimensional Josephson junction consisting of a thin-film semiconductor hosting a persistent spin-helix state and two conventional $s$-wave superconductors. We analytically obtain a topological phase diagram and numerically demonstrate the emergence of flat-band Majorana bound states by calculating the local density of states.
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Submitted 6 November, 2021;
originally announced November 2021.
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Fingerprints of possible even-parity superconducting states in Sr$_2$RuO$_4$ detected by planar tunneling spectroscopy
Authors:
Satoshi Ikegaya,
Shu-Ichiro Suzuki,
Yukio Tanaka,
Dirk Manske
Abstract:
After more than 25 years of research, three even-parity superconducting states -- the $d+id$-wave, $d+ig$-wave, and $s+id$-wave states -- have emerged as leading candidates for the superconducting states of Sr$_2$RuO$_4$. In the present work, we propose a tunneling spectroscopy experiment for distinguishing among these three superconducting states. The key component of our proposal is that we exam…
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After more than 25 years of research, three even-parity superconducting states -- the $d+id$-wave, $d+ig$-wave, and $s+id$-wave states -- have emerged as leading candidates for the superconducting states of Sr$_2$RuO$_4$. In the present work, we propose a tunneling spectroscopy experiment for distinguishing among these three superconducting states. The key component of our proposal is that we examine the conductance spectra of normal-metal/Sr$_2$RuO$_4$ junctions with various angles between the junction interface and the crystal axis of the Sr$_2$RuO$_4$. The angle dependence of the conductance spectra shows a unique pattern in each superconducting state, which can function as a fingerprint for verifying the pairing symmetry of Sr$_2$RuO$_4$.
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Submitted 17 June, 2021;
originally announced June 2021.
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Strong anomalous proximity effect from spin-singlet superconductors
Authors:
Satoshi Ikegaya,
Jaechul Lee,
Andreas P. Schnyder,
Yasuhiro Asano
Abstract:
The proximity effect from a spin-triplet $p_x$-wave superconductor to a dirty normal-metal has been shown to result in various unusual electromagnetic properties, reflecting a cooperative relation between topologically protected zero-energy quasiparticles and odd-frequency Cooper pairs. However, because of a lack of candidate materials for spin-triplet $p_x$-wave superconductors, observing this ef…
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The proximity effect from a spin-triplet $p_x$-wave superconductor to a dirty normal-metal has been shown to result in various unusual electromagnetic properties, reflecting a cooperative relation between topologically protected zero-energy quasiparticles and odd-frequency Cooper pairs. However, because of a lack of candidate materials for spin-triplet $p_x$-wave superconductors, observing this effect has been difficult. In this paper, we demonstrate that the anomalous proximity effect, which is essentially equivalent to that of a spin-triplet $p_x$-wave superconductor, can occur in a semiconductor/high-$T_c$ cuprate superconductor hybrid device in which two potentials coexist: a spin-singlet $d$-wave pair potential and a spin--orbit coupling potential sustaining the persistent spin-helix state. As a result, we propose an alternative and promising route to observe the anomalous proximity effect related to the profound nature of topologically protected quasiparticles and odd-frequency Cooper pairs.
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Submitted 17 June, 2021; v1 submitted 9 April, 2021;
originally announced April 2021.
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Tunable Majorana corner modes in noncentrosymmetric superconductors: Tunneling spectroscopy and edge imperfections
Authors:
S. Ikegaya,
W. B. Rui,
D. Manske,
Andreas P. Schnyder
Abstract:
Majorana corner modes appearing in two-dimensional second-order topological superconductors have great potential applications for fault-tolerant topological quantum computations. We demonstrate that in the presence of an in-plane magentic field two-dimensional ($s+p$)-wave superconductors host Majorana corner modes, whose location can be manipulated by the direction of the magnetic field. In addit…
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Majorana corner modes appearing in two-dimensional second-order topological superconductors have great potential applications for fault-tolerant topological quantum computations. We demonstrate that in the presence of an in-plane magentic field two-dimensional ($s+p$)-wave superconductors host Majorana corner modes, whose location can be manipulated by the direction of the magnetic field. In addition, we discuss the effects of edge imperfections on the Majorana corner modes. We describe how different edge shapes and edge disorder affect the number and controllability of the Majorana corner modes, which is of relevance for the implementation of topological quantum computations. We also discuss tunneling spectroscopy in the presence of the Majorana corner modes, where a lead-wire is attached to the corner of the noncentrosymmetric superconductor. The zero-bias differential conductance shows a distinct periodicity with respect to the direction of the magnetic field, which demonstrates the excellent controllability of the Majorana corner modes in this setup. Our results lay down the theoretical groundwork for observing and tuning Majoran corner modes in experiments on ($s+p$)-wave superconductors.
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Submitted 1 December, 2020;
originally announced December 2020.
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Odd-parity pairing correlations in a d-wave superconductor
Authors:
Jaechul Lee,
Satoshi Ikegaya,
Yasuhiro Asano
Abstract:
We theoretically study the effects of spin-orbit interactions on symmetry of a Cooper pair in a spin-singlet d-wave superconductor in two-dimension. The pairing symmetry is analyzed in terms of the anomalous Green's function which is obtained by solving the Gor'kov equation analytically. A spin-orbit interaction induces a spin-triplet p-wave pairing correlation in a uniform superconductor. An odd-…
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We theoretically study the effects of spin-orbit interactions on symmetry of a Cooper pair in a spin-singlet d-wave superconductor in two-dimension. The pairing symmetry is analyzed in terms of the anomalous Green's function which is obtained by solving the Gor'kov equation analytically. A spin-orbit interaction induces a spin-triplet p-wave pairing correlation in a uniform superconductor. An odd-frequency spin-triplet s-wave pairing correlation appears at a surface of such superconductor as a result of breaking inversion symmetry locally. We also discuss a close relationship among the odd-frequency pairing correlation, chirality of surface bound states at the zero energy, and the anomalous proximity effect. The obtained results enable us to design a superconductor which causes the strong anomalous proximity effect.
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Submitted 4 March, 2021; v1 submitted 20 November, 2020;
originally announced November 2020.
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Anomalous Proximity Effect of Planer Topological Josephson Junctions
Authors:
Satoshi Ikegaya,
Shun Tamura,
Dirk Manske,
Yukio Tanaka
Abstract:
The anomalous proximity effect in dirty superconducting junctions is one of most striking phenomena highlighting the profound nature of Majorana bound states and odd-frequency Cooper pairs in topological superconductors. Motivated by the recent experimental realization of planar topological Josephson junctions, we describe the anomalous proximity effect in a superconductor/semiconductor hybrid, wh…
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The anomalous proximity effect in dirty superconducting junctions is one of most striking phenomena highlighting the profound nature of Majorana bound states and odd-frequency Cooper pairs in topological superconductors. Motivated by the recent experimental realization of planar topological Josephson junctions, we describe the anomalous proximity effect in a superconductor/semiconductor hybrid, where an additional dirty normal-metal segment is extended from a topological Josephson junction. The topological phase transition in the topological Josephson junction is accompanied by a drastic change in the low-energy transport properties of the attached dirty normal-metal. The quantization of the zero-bias differential conductance, which appears only in the topologically nontrivial phase, is caused by the penetration of the Majorana bound states and odd-frequency Cooper pairs into a dirty normal-metal segment. As a consequence, we propose a practical experiment for observing the anomalous proximity effect.
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Submitted 16 October, 2020; v1 submitted 25 July, 2020;
originally announced July 2020.
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Nodal Andreev Spectra in Multi-Majorana Three-Terminal Josephson Junctions
Authors:
Keimei Sakurai,
Maria Teresa Mercaldo,
Shingo Kobayashi,
Ai Yamakage,
Satoshi Ikegaya,
Tetsuro Habe,
Panagiotis Kotetes,
Mario Cuoco,
Yasuhiro Asano
Abstract:
We investigate the Andreev-bound-state (ABS) spectra of three-terminal Josephson junctions which consist of 1D topological superconductors (TSCs) harboring multiple zero-energy edge Majorana bound states (MBSs) protected by chiral symmetry. Our theoretical analysis relies on the exact numerical diagonalization of the Bogoliubov-de Gennes (BdG) Hamiltonian describing the three interfaced TSCs, comp…
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We investigate the Andreev-bound-state (ABS) spectra of three-terminal Josephson junctions which consist of 1D topological superconductors (TSCs) harboring multiple zero-energy edge Majorana bound states (MBSs) protected by chiral symmetry. Our theoretical analysis relies on the exact numerical diagonalization of the Bogoliubov-de Gennes (BdG) Hamiltonian describing the three interfaced TSCs, complemented by an effective low-energy description solely based on the coupling of the interfacial MBSs arising before the leads get contacted. Considering the 2D synthetic space spanned by the two independent superconducting phase differences, we demonstrate that the ABS spectra may contain either point or line nodes, and identify $\mathbb{Z}_2$ topological invariants to classify them. We show that the resulting type of nodes depends on the number of preexisting interfacial MBSs, with nodal lines necessarily appearing when two TSCs harbor an unequal number of MBSs. Specifically, the precise number of interfacial MBSs determines the periodicity of the spectrum under $2π$-slidings of the phase differences and, as a result, also controls the shape of the nodal lines in synthetic space. When chiral symmetry is preserved, the lines are open and coincide with high-symmetry lines of synthetic space, while when it is violated the lines can also transform into loops and chains. The nodal spectra are robust by virtue of the inherent particle-hole symmetry of the BdG Hamiltonian, and give rise to distinctive experimental signatures that we identify.
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Submitted 30 March, 2020;
originally announced March 2020.
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Identification of Spin-Triplet Superconductivity through a Helical-Chiral Phase Transition in Sr$_2$RuO$_4$ Thin-Films
Authors:
S. Ikegaya,
K. Yada,
Y. Tanaka,
S. Kashiwaya,
Y. Asano,
D. Manske
Abstract:
Despite much effort for over the two decades, the paring symmetry of a Sr$_2$RuO$_4$ superconductor has been still unclear. In this Rapid Communication, motivated by the recent rapid progress in fabrication techniques for Sr$_2$RuO$_4$ thin-films, we propose a promising strategy for identifying the spin-triplet superconductivity in the thin-film geometry by employing an antisymmetric spin-orbit co…
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Despite much effort for over the two decades, the paring symmetry of a Sr$_2$RuO$_4$ superconductor has been still unclear. In this Rapid Communication, motivated by the recent rapid progress in fabrication techniques for Sr$_2$RuO$_4$ thin-films, we propose a promising strategy for identifying the spin-triplet superconductivity in the thin-film geometry by employing an antisymmetric spin-orbit coupling potential and a Zeeman potential due to an external magnetic field. We demonstrate that a spin-triplet superconducting thin-film undergoes a phase transition from a helical state to a chiral state by increasing the applied magnetic field. This phase transition is accompanied by a drastic change in the property of surface Andreev bound states. As a consequence, the helical-chiral phase transition, which is unique to the spin-triplet superconductors, can be detected through a sudden change in a tunneling conductance spectrum of a normal-metal/superconductor junction. Importantly, our proposal is constructed by combining fundamental and rigid concepts regarding physics of spin-triplet superconductivity.
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Submitted 31 December, 2019;
originally announced December 2019.
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Josephson effect in two-band superconductors
Authors:
Akihiro Sasaki,
Satoshi Ikegaya,
Tetsuro Habe,
Alexander A. Golubov,
Yasuhiro Asano
Abstract:
We study theoretically the Josephson effect between two time-reversal two-band superconductors, where we assume the equal-time spin-singlet $s$-wave pair potential in each conduction band. %as well as the band asymmetry and the band hybridization in the normal state. The superconducting phase at the first band $\varphi_1$ and that at the second band $\varphi_2$ characterize a two-band superconduct…
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We study theoretically the Josephson effect between two time-reversal two-band superconductors, where we assume the equal-time spin-singlet $s$-wave pair potential in each conduction band. %as well as the band asymmetry and the band hybridization in the normal state. The superconducting phase at the first band $\varphi_1$ and that at the second band $\varphi_2$ characterize a two-band superconducting state. We consider a Josephson junction where an insulating barrier separates two such two-band superconductors. By applying the tunnel Hamiltonian description, the Josephson current is calculated in terms of the anomalous Green's function on either side of the junction. We find that the Josephson current consists of three components which depend on three types of phase differences across the junction: the phase difference at the first band $δ\varphi_1$, the phase difference at the second band $δ\varphi_2$, and the difference at the center-of-mass phase $δ(\varphi_1+\varphi_2)/2$. A Cooper pairs generated by the band hybridization carries the last current component. In some cases, the current-phase relationship deviates from the sinusoidal function as a result of time-reversal symmetry breaking down.
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Submitted 12 December, 2019;
originally announced December 2019.
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Anomalous Nonlocal Conductance as a Fingerprint of Chiral Majorana Edge States
Authors:
Satoshi Ikegaya,
Yasuhiro Asano,
Dirk Manske
Abstract:
Chiral $p$-wave superconductor is the primary example of topological systems hosting chiral Majorana edge states. Although candidate materials exist, the conclusive signature of chiral Majorana edge states has not yet been observed in experiments. Here we propose a smoking-gun experiment to detect the chiral Majorana edge states on the basis of theoretical results for the nonlocal conductance in a…
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Chiral $p$-wave superconductor is the primary example of topological systems hosting chiral Majorana edge states. Although candidate materials exist, the conclusive signature of chiral Majorana edge states has not yet been observed in experiments. Here we propose a smoking-gun experiment to detect the chiral Majorana edge states on the basis of theoretical results for the nonlocal conductance in a device consisting of a chiral $p$-wave superconductor and two ferromagnetic leads. The chiral nature of Majorana edge states causes an anomalously long-range and chirality-sensitive nonlocal transport in these junctions. These two drastic features enable us to identify the moving direction of chiral Majorana edge states in the single experimental setup.
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Submitted 22 May, 2019; v1 submitted 22 January, 2019;
originally announced January 2019.
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Symmetry conditions of a nodal superconductor for generating robust flat-band Andreev bound states at its dirty surface
Authors:
Satoshi Ikegaya,
Shingo Kobayashi,
Yasuhiro Asano
Abstract:
We discuss the symmetry property of a nodal superconductor that hosts robust flat-band zero-energy states at its surface under potential disorder. Such robust zero-energy states are known to induce the anomalous proximity effect in a dirty normal metal attached to a superconductor. A recent study has shown that a topological index ${\cal N}_\mathrm{ZES}$ describes the number of zero-energy states…
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We discuss the symmetry property of a nodal superconductor that hosts robust flat-band zero-energy states at its surface under potential disorder. Such robust zero-energy states are known to induce the anomalous proximity effect in a dirty normal metal attached to a superconductor. A recent study has shown that a topological index ${\cal N}_\mathrm{ZES}$ describes the number of zero-energy states at the dirty surface of a $p$-wave superconductor. We generalize the theory to clarify the conditions required for a superconductor that enables ${\cal N}_\mathrm{ZES}\neq 0$. Our results show that ${\cal N}_\mathrm{ZES}\neq 0$ is realized in a topological material that belongs to either the BDI or CII class. We also present two realistic Hamiltonians that result in ${\cal N}_\mathrm{ZES}\neq 0$.
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Submitted 17 January, 2018;
originally announced January 2018.
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Tunable $\varphi$-Josephson junction with a quantum anomalous Hall insulator
Authors:
Keimei Sakurai,
Satoshi Ikegaya,
Yasuhiro Asano
Abstract:
We theoretically study the Josephson current in a superconductor/quantum anomalous Hall insulator/superconductor junction by using the lattice Green function technique. When an in-plane external Zeeman field is applied to the quantum anomalous Hall insulator, the Josephson current $J$ flows without a phase difference across the junction $θ$. The phase shift $\varphi$ appealing in the current-phase…
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We theoretically study the Josephson current in a superconductor/quantum anomalous Hall insulator/superconductor junction by using the lattice Green function technique. When an in-plane external Zeeman field is applied to the quantum anomalous Hall insulator, the Josephson current $J$ flows without a phase difference across the junction $θ$. The phase shift $\varphi$ appealing in the current-phase relationship $J\propto \sin(θ-\varphi$) is proportional to the amplitude of Zeeman fields and depends on the direction of Zeeman fields. A phenomenological analysis of the Andreev reflection processes explains the physical origin of $\varphi$. A quantum anomalous Hall insulator breaks time-reversal symmetry and mirror reflection symmetry simultaneously. However it preserves magnetic mirror reflection symmetry. Such characteristic symmetry property enable us to have a tunable $\varphi$-junction with a quantum Hall insulator.
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Submitted 8 December, 2017; v1 submitted 7 September, 2017;
originally announced September 2017.
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Stability of flat zero-energy states at the dirty surface of a nodal superconductor
Authors:
Satoshi Ikegaya,
Yasuhiro Asano
Abstract:
We discuss the stability of highly degenerate zero-energy states tha appear at the surface of a nodal superconductor preserving time-reversal symmetry. The existence of such surface states is a direct consequence of the nontrivial topological numbers defined in the restricted Brillouin zones in the clean limit. In experiments, however, potential disorder is inevitable near the surface of a real su…
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We discuss the stability of highly degenerate zero-energy states tha appear at the surface of a nodal superconductor preserving time-reversal symmetry. The existence of such surface states is a direct consequence of the nontrivial topological numbers defined in the restricted Brillouin zones in the clean limit. In experiments, however, potential disorder is inevitable near the surface of a real superconductor, which may lift the high degeneracy at zero energy. We show that an index defined in terms of the chiral eigenvalues of the zero-energy states can be used to measure the degree of degeneracy at zero energy in the presence of potential disorder. We also discuss the relationship between the index and the topological numbers.
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Submitted 19 January, 2017;
originally announced January 2017.
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Quantization of Conductance Minimum and Index Theorem
Authors:
Satoshi Ikegaya,
Shu-Ichiro Suzuki,
Yukio Tanaka,
Yasuhiro Asano
Abstract:
We discuss the minimum value of the zero-bias differential conductance $G_{\textrm{min}}$ in a junction consisting of a normal metal and a nodal superconductor preserving time-reversal symmetry. Using the quasiclassical Green function method, we show that $G_{\textrm{min}}$ is quantized at $ (4e^2/h) N_{\mathrm{ZES}}$ in the limit of strong impurity scatterings in the normal metal. The integer…
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We discuss the minimum value of the zero-bias differential conductance $G_{\textrm{min}}$ in a junction consisting of a normal metal and a nodal superconductor preserving time-reversal symmetry. Using the quasiclassical Green function method, we show that $G_{\textrm{min}}$ is quantized at $ (4e^2/h) N_{\mathrm{ZES}}$ in the limit of strong impurity scatterings in the normal metal. The integer $N_{\mathrm{ZES}}$ represents the number of perfect transmission channels through the junction. An analysis of the chiral symmetry of the Hamiltonian indicates that $N_{\mathrm{ZES}}$ corresponds to the Atiyah-Singer index in mathematics.
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Submitted 20 April, 2016;
originally announced April 2016.
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Degeneracy of Majorana bound states and fractional Josephson effect in a dirty SNS junction
Authors:
S. Ikegaya,
Y. Asano
Abstract:
We theoretically study the stability of more than one Majorana Fermion appearing in a $p$-wave superconductor/dirty normal metal/$p$-wave superconductor junction in two-dimension by using chiral symmetry of Hamiltonian. At the phase difference across the junction $\varphi$ being $π$, we will show that all of the Majorana bound states in the normal metal belong to the same chirality. Due to this pu…
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We theoretically study the stability of more than one Majorana Fermion appearing in a $p$-wave superconductor/dirty normal metal/$p$-wave superconductor junction in two-dimension by using chiral symmetry of Hamiltonian. At the phase difference across the junction $\varphi$ being $π$, we will show that all of the Majorana bound states in the normal metal belong to the same chirality. Due to this pure chiral feature, the Majorana bound states retain their high degree of degeneracy at the zero energy even in the presence of random potential. As a consequence, the resonant transmission of a Cooper pair via the degenerate MBSs carries the Josephson current at $\varphi=π-0^+$, which explains the fractional current-phase relationship discussed in a number of previous papers.
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Submitted 7 April, 2016;
originally announced April 2016.
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Anomalous Proximity Effect and Theoretical Design for its Realization
Authors:
Satoshi Ikegaya,
Yasuhiro Asano,
Yukio Tanaka
Abstract:
We discuss the stability of zero-energy states appearing in a dirty normal metal attached to a superconducting thin film with Dresselhaus [110] spin-orbit coupling under the in-plane Zeeman field. The Dresselhaus superconductor preserves an additional chiral symmetry and traps more than one zero-energy state at its edges. All the zero-energy states at an edge belong to the same chirality in large…
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We discuss the stability of zero-energy states appearing in a dirty normal metal attached to a superconducting thin film with Dresselhaus [110] spin-orbit coupling under the in-plane Zeeman field. The Dresselhaus superconductor preserves an additional chiral symmetry and traps more than one zero-energy state at its edges. All the zero-energy states at an edge belong to the same chirality in large Zeeman field due to the effective $p$-wave pairing symmetry. The pure chiral nature in the wave function enables the penetration of the zero-energy states into the dirty normal metal with keeping their high degree of degeneracy. By applying a theorem, we prove the the perfect Andreev reflection into the dirty normal metal at the zero-energy. This paper gives a microscopic understanding of the anomalous proximity effect.
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Submitted 1 May, 2015; v1 submitted 12 November, 2014;
originally announced November 2014.
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Majorana Bound States in Proximity Junctions of Superconducting Nanowires with Dresselhaus Spin-orbit Coupling
Authors:
Satoshi Ikegaya,
Yasuhiro Asano,
Yukio Tanaka
Abstract:
We theoretically study transport properties of nanowires with the Dresselhaus [110] spin-orbit coupling under the in-plane Zeeman potential and the proximity-induced s-wave pair potential. In the topologically nontrivial phase, the nanowire hosts the Majorana fermions at its edges and the number of the Majorana bound states is equal to the number propagating channels (Nc). When we attach a normal…
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We theoretically study transport properties of nanowires with the Dresselhaus [110] spin-orbit coupling under the in-plane Zeeman potential and the proximity-induced s-wave pair potential. In the topologically nontrivial phase, the nanowire hosts the Majorana fermions at its edges and the number of the Majorana bound states is equal to the number propagating channels (Nc). When we attach a normal metal to the superconductor, such Majorana bound states penetrate into the dirty normal segment and form the Nc resonant transmission channels there. We show that chiral symmetry of the electronic states protects the Majorana bound states at the zero energy even in the presence of impurities. As a result, we find that the zero-bias conductance of normal-nanowire/superconducting-nanowire junctions is quantized at 2e2Nc/h independent of the random potentials.
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Submitted 14 October, 2014;
originally announced October 2014.