Odd-frequency pairing mechanism of superconductivity has been investigated for several decades. Nevertheless, its properties, including the thermodynamic stability, have remained unclear. In particular, it has been argued that the odd-frequency state is thermodynamically unstable, has an unphysical (anti-) Meissner effect, and thus can not exist as a homogeneous equilibrium phase. We argue that this conclusion is incorrect because it implicitly relies on the inappropriate assumption that the odd-frequency superconductor can be described by an effective Hamiltonian that breaks the particle conservation symmetry. We demonstrate that the odd-frequency state can be properly described within the functional integral approach using non-local-in-time effective action. Within the saddle point approximation, we find that this phase is thermodynamically stable, exhibits ordinary Meissner effect, and therefore can be realized as an equilibrium homogenous state of matter.
We show that vertex corrections to Migdal's theorem in general induce an odd-frequency spin-triplet superconducting order parameter, which coexists with its more commonly known even-frequency spin-singlet counterpart. Fully self-consistent vertex-corrected Eliashberg theory calculations for a two dimensional cuprate model, isotropically coupled to an Einstein phonon, confirm that both superconducting gaps are finite over a wide range of temperatures. The subordinate $d$-wave odd-frequency superconducting gap is found to be one order of magnitude smaller than the primary even-frequency $d$-wave…
Finite-momentum superconducting states due to odd-frequency Cooper pairing correlations
Graduate School of Science, Hokkaido University（北海道大学大学院理学研究科）；RIKEN Center for Emergent Matter Science（理化学研究所新兴物质科学中心）；Department of Applied Physics, Hokkaido University（北海道大学应用物理系）
In several members of the pnictide materials, spin density wave order coexists with superconductivity over a range of dopings and temperature. In this paper we show that odd-frequency superconductivity emerges on the edges of pnictides in such a coexistence phase. In particular, the breaking of spin-rotation symmetry by SDW and translation symmetry by the edge can lead to the development of odd-frequency spin-triplet Cooper pairing on edges of superconducting pnictide samples. In this case, the odd-frequency pairing has even parity components, which are immune to disorder. Our results show tha…
We show that quasiparticle interference (QPI) due to omnipresent weak impurities and probed by Fourier transform scanning tunneling microscopy and spectroscopy acts as a direct experimental probe of bulk odd-frequency superconducting pairing. Taking the example of a conventional $s$-wave superconductor under applied magnetic field, we show that the nature of the QPI peaks can only be characterized by including the odd-frequency pairing correlations generated in this system. In particular, we identify that the defining feature of odd-frequency pairing gives rise to a bias asymmetry in the QPI, …
We study unconventional superconductivity in a two-dimensional locally noncentrosymmetric triangular lattice. The model is relevant to bilayer transition metal dichalcogenides with 2H$_b$ stacking structure, for example. The superconducting instability is analyzed by solving the linearized Eliashberg equation within the random phase approximation. We show that ferromagnetic fluctuations are dominant owing to the existence of disconnected Fermi pockets near van Hove singularity, and hence odd-parity spin-triplet superconductivity is favored. In the absence of the spin-orbit coupling, we find th…
Odd-frequency superconductivity describes a class of superconducting states where the superconducting gap is an odd function in relative time and Matsubara frequency. We present a group theoretical analysis based on the linearized gap equation in terms of Shubnikov groups of the second kind. By discussing systems with spin-orbit coupling and an interaction kernel which is symmetric under the reversal of relative time, we show that both even- and odd-frequency gaps are allowed to occur. Specific examples are discussed for the square lattice, the octahedral lattice, and the tetragonal lattice. F…
This article reviews odd-frequency (odd-w) pairing with focus on superconducting systems. Since Berezinskii introduced the concept of odd frequency order in 1974 it has been viewed as an exotic and rarely occurring in nature. Here, we present a view that the Berezinskii state is in fact a ubiquitous superconducting order that is both non-local and odd in time. It appears under quite general circumstances including in bulk materials, heterostructures and dynamically driven superconducting states. It is therefore important to understand the nature of odd-w pairing. We present the properties of o…
We investigate emergent odd-frequency pairs and proximity effect in nematic and chiral states of superconducting topological insulators (STIs), such as $M_x$Bi$_2$Se$_3$ ($M=$ Cu, Sr, Nb). The interplay of superconducting gap symmetry, the orbital degrees of freedom, and strong spin-orbit interaction generates a variety of odd-frequency pairs in the bulk and surface of STIs. The nematic and chiral states are the prototypes of topological superconductors with and without time-reversal symmetry, respectively. We find that the Fermi surface evolution from a closed spheroidal to an open cylindrica…
We introduce the concept of Floquet odd-frequency superconducting pairs and establish their emergence in time-periodic conventional superconductors, where the continuous time-translation invariance is broken. We show that these exotic Cooper pairs are possible because the Floquet modes in time-periodic systems provide an additional index (a Floquet index) that broadens the classification of pair symmetries, with no analog in the static regime. Our results thus put forward a different route for odd-frequency superconducting pairs and pave the way for Floquet engineered dynamical superconducting…
We first show the realization of exceptional points in a non-Hermitian superconducting system based on a conventional superconductor and then demonstrate that, surprisingly, the system hosts odd-frequency pairing, solely generated by the non-Hermiticity. While there is a coexistence of even- and odd-frequency pairs under general conditions, we find that the even-frequency term vanishes at the exceptional degeneracies, leaving only odd-frequency pairing. This exceptional odd-frequency pairing is directly given by the imaginary part of the eigenvalues at the exceptional points and can be measure…
Odd-frequency superconductivity represents a truly unconventional ordered state which, in contrast to conventional superconductivity, exhibits pair correlations which are odd in relative time and, hence, inherently dynamical. In this review article we provide an overview of recent advances in the study of odd-frequency superconducting correlations in one-dimensional systems. In particular, we focus on recent developments in the study of nanowires with Rashba spin-orbit coupling and metallic edges of two-dimensional topological insulators in proximity to conventional superconductors. These syst…
We study the emergence of odd-frequency superconducting pairing in UPt$_3$. Starting from a tight-binding model accounting for the nonsymmorphic crystal symmetry of UPt$_3$ and assuming an order parameter in the $E_{2u}$ representation, we demonstrate that odd-frequency pairing arises very generally, as soon as inter-sublattice hopping or spin-orbit coupling is present. We also show that in the low temperature superconducting $B$ phase, the presence of a chiral order parameter together with spin-orbit coupling, leads to additional odd-frequency pair amplitudes not present in the $A$ or $C$ pha…
Majorana Fermions and Odd-frequency Cooper Pairs in a Nano Wire
我们讨论了马约拉纳费米子和奇频库珀对之间的强关系，出现在一个无序的正常（N）纳米线连接到一个拓扑非平凡的超导（S）的。超导纳米线结中的输运性质表现出与无序度无关的普遍行为：NS结中的量子化零偏压微分电导（2 e ^{2}/h）和SNS结中约瑟夫森效应的分数电流-相位（J-φ）关系J \propto \sin（φ/2）。这种行为与奇宇称自旋三重态超导体的反常邻近效应完全相同。我们发现，奇频库珀对支持普遍的输运性质。奇频率对存在于马约拉纳费米子所在的地方。
We investigate odd-frequency superconducting correlations in normal-superconductor (NS) and short superconductor-normal-superconductor (SNS) junctions with the S region described by the Kitaev model of spinless fermions in one dimension. We demonstrate that, in both the trivial and topological phases, Andreev reflection is responsible for the coexistence of even- and odd-frequency pair amplitudes at interfaces, while normal reflections solely contribute to odd-frequency pairing. At NS interfaces we find that the odd-frequency pair amplitude exhibits large, but finite, values in the topological…
Quantum transport in a normal metal/odd-frequency superconductor junction
最近的实验结果表明，在化合物CeCu $_2$ Si $_2$和CeRhIn $_5$中可能实现体奇频超导态。出于这一动机，我们提出了一个正常的金属/奇数频率超导体结的量子输运性质的研究，在寻找探针，以揭开奇频对称性。从Eliashberg方程，我们执行一个准经典近似占奇频超导体的Keldysh形式主义的运输形式主义。具体地说，我们考虑了隧穿电荷电导和隧穿热导。我们发现定性不同的行为在奇频率的情况下相比，传统的偶频率的情况下，在电流和热电流。这是一个有用的工具，以确定可能存在的体奇频超导态。
We consider normal-superconductor (NS) and superconductor-normal-superconductor (SNS) junctions based on one-dimensional nanowires with Rashba spin-orbit coupling and proximity-induced $s$-wave spin-singlet superconductivity and analytically demonstrate how both even- and odd-frequency and spin-singlet and -triplet superconducting pair correlations are always present. In particular, by using a fully quantum mechanical scattering approach, we show that Andreev reflection induces mixing of spatial parities at interfaces, thus being the unique process which generates odd-frequency pairing; on the…
The Su-Schrieffer-Heeger model describes fermions that hop on a one-dimensional chain with staggered hopping amplitude, where the unit cell contains two sites, or two sublattices. In this work we consider the Su-Schrieffer-Heeger model with superconducting pairing and show that the sublattice index acts as an additional quantum number in the classification of Cooper pairs, giving rise to inter- and intra-sublattice odd-frequency pair correlations in the bulk. Interestingly, this system behaves as a two band superconductor where the bulk odd-frequency correlations depend solely on the intrinsic…
High-pressure synthesis techniques have allowed for the growth of Sn$_{1-x}$In$_x$Te samples beyond the ambient In-saturation limit of $x$ = 0.5 (T$_c \sim$ 4.5 K). In this study, we present measurements of the temperature dependence of the London penetration depth $Δλ(T)$ in this superconducting doped topological insulator for $x$ = 0.7, where T$_{c,onset}\approx 5$ K. The results indicate fully gapped BCS-like behavior, ruling out odd-parity $A_{2u}$ pairing; however, odd-parity $A_{1u}$ pairing is still possible. Critical field values measured below 1 K and other superconducting parameters …
High-pressure synthesis techniques have allowed for the growth of samples on the indium-rich side of (Pb,In)Te, which have increased superconducting transition temperatures compared to lead-rich compounds. In this study we present measurements of the temperature dependence of the London penetration depth $Δλ(T)$ in the compound In$_{0.8}$Pb$_{0.2}$Te, which shows a bulk $T_{c,onset}$ of $\sim4.75$ K. The results indicate fully gapped BCS-like behavior, ruling out odd-parity, topologically nontrivial $A_{2u}$ pairing; however, odd-parity $A_{1u}$ pairing is still possible. Critical field values…
CeRh$_2$As$_2$ is an unconventional superconductor with multiple superconducting phases and $T_\mathrm{c} = 0.26$ K. When $H\parallel c$, it shows a field-induced transition at $μ_0H^* = 4$ T from a low-field superconducting state SC1 to a high-field state SC2 with a large critical field of $μ_0H_\mathrm{c2} = 14$ T. In contrast, for $H\perp c$, only the SC1 with $μ_0H_\mathrm{c2} = 2$ T is observed. A simple model based on the crystal symmetry was able to reproduce the phase-diagrams and their anisotropy, identifying SC1 and SC2 with even and odd parity superconducting states, respectively. H…
The superconducting pairing in Sr$_2$RuO$_4$ is widely considered to be chiral $p$-wave with $\vec{d}_{\boldsymbol k} \sim (k_x + ik_y)\hat{z}$, which belongs to the $E_u$ representation of the crystalline $D_{4h}$ group. However, this superconducting order appears hard to reconcile with a number of key experiments. In this paper, based on symmetry analysis we discuss the possibility of odd-parity pairing with inherent three-dimensional (3D) character enforced by the inter-orbital interlayer coupling and the sizable spin-orbit coupling (SOC) in the material. We focus on a yet unexplored $E_u$ …
Topological superconductors give rise to unconventional superconductivity, which is mainly characterized by the symmetry of the superconducting pairing amplitude. However, since the symmetry of the superconducting pairing amplitude is not directly observable, its experimental identification is rather difficult. In our work, we propose a system, composed of a quantum point contact and proximity induced s-wave superconductivity at the helical edge of a two dimensional topological insulator, for which we demonstrate the presence of odd-frequency pairing and its intimate connection to unambiguous …
We demonstrate that SrTiO$_3$ can be a platform for observing the bulk odd-frequency superconducting state owing to the multiorbital/multiband nature. We consider a three-orbital tight-binding model for SrTiO$_3$ in the vicinity of a ferroelectric critical point. Assuming an intraorbital spin-singlet $s$-wave superconducting order parameter, it is shown that the odd-frequency pair correlations are generated due to the intrinsic LS coupling which leads to the local orbital mixing. Furthermore, we show the existence of additional odd-frequency pair correlations in the ferroelectric phase, which …
Odd-frequency superconductivity is an exotic phase of matter in which Cooper pairing between electrons is entirely dynamical in nature. Majorana zero modes exhibit pure odd-frequency superconducting correlations due to their specific properties. Thus, by tunnel-coupling an array of Majorana zero modes to a spin-polarized wire, it is in principle possible to engineer a bulk one-dimensional odd-frequency spinless $s$-wave superconductor. We here point out that each tunnel coupling element, being dependent on a large number of material-specific parameters, is generically complex with sample varia…
The Yu-Shiba-Rusinov (YSR) state appears as a bound state of a quasiparticle at a magnetic atom embedded in a superconductor. We discuss why the YSR state has energy below the superconducting gap and why the pair potential changes the sign at the magnetic atom. Although a magnetic atom in a superconductor has been considered as a pair breaker since 1960s, we propose an alternative physical picture to explain these reasons. We show that a magnetic atom converts a spin-singlet s-wave Cooper pair into an odd-frequency pair rather than breaking it. The odd-frequency pairing correlations always coe…
