Introduction

Parity symmetry (\({{{{{{{\mathcal{P}}}}}}}}\)-symmetry) and time-reversal symmetry (\({{{{{{{\mathcal{T}}}}}}}}\)-symmetry) are fundamental properties of quantum materials, such as insulators, metals, magnets, and superconductors. Superconductivity is caused by the quantum condensation of either even-parity or odd-parity Cooper pairs, which correspond to spin-singlet or spin-triplet superconductivity due to the fermion antisymmetry1. The order parameter of conventional superconductors breaks neither \({{{{{{{\mathcal{P}}}}}}}}\)-symmetry nor \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry. However, competition and coexistence of multiple pairing instabilities lead to exotic superconductivity, such as chiral superconductivity with spontaneous \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking2 related to the nontrivial topology3,4 and anomalous transport5.

In particular, mixed-parity superconductivity with coexistent even- and odd-parity pairing channels has been widely discussed in noncentrosymmetric superconductors6,7, ultracold fermion systems8,9, and spin-orbit-coupled systems in the vicinity of the \({{{{{{{\mathcal{P}}}}}}}}\)-symmetry broken phase10,11,12,13,14. The \({{{{{{{\mathcal{P}}}}}}}}\)-symmetry is broken in such superconductors. Furthermore, spontaneous \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking realized by the ±π/2 phase difference between even- and odd-parity pairing potentials is energetically favored15,16,17 (Fig. 1a), when the spin-orbit coupling (SOC) due to noncentrosymmetric crystal structure is absent or weak. This class of superconducting states spontaneously breaks both \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries but maintain the combined \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetry. There have been considerable interests in studying such \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity superconductivity. The three-dimensional s + ip-wave superconductivity has attracted much theoretical attention as a superconducting analog of axion insulators18,19,20,21,22,23. The \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity pairing has also been theoretically proposed in Sr2RuO424. Furthermore, a mixed-parity superconducting state in UTe225 has been predicted to explain experimentally-observed multiple superconducting phases26,27,28,29,30,31,32,33.

Fig. 1: Schematics of \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity superconductivity (SC).
figure 1

a Schematic phase diagram in a superconductor with comparable strength of even- and odd-parity pairing interactions. The transition between the even-parity superconducting phase (Γg) and the odd-parity superconducting phase (Γu) is induced by tuning a control parameter x. For centrosymmetric systems with \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry, there is generally an intermediate mixed-parity superconducting phase (Γg + iΓu) where even- and odd-parity pairing components are coexistent with the relative phase difference ± π/2. b Illustration for a mechanism of the asymmetric Bogoliubov spectrum (BS) in \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity multiband superconductors. \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking interband pairing induces an asymmetric modulation of the BS. c Schematic figure of the anapole superconducting states. In real space, the phase of the superconducting order parameter Δ(r) becomes nonuniform along a direction parallel to the effective anapole moment T as Δ(r) ∝ eiq⋅r with q∥T, where q is the center-of-mass momentum of Cooper pairs.

Full size image

In previous works, the mixed-parity superconductivity has been theoretically studied mainly in single-band models for spin-1/2 fermions15,16,18,19,20,21,23. On the other hand, it has recently been recognized that the multiband structure of the Cooper pair’s wave function arising from internal electronic degrees of freedom (DOF) (e.g., orbital and sublattice) induces exotic superconducting phenomena. For instance, multiband superconductors have attracted much attention as a platform realizing odd-frequency pairing34. In \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking superconductors, an intrinsic anomalous Hall effect emerges owing to the multiband nature of Cooper pairs5,35,36,37,38. In particular, even-parity \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking superconductors host topologically protected Bogoliubov Fermi surfaces in the presence of interband pairing39,40.

In this work, we show that \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity superconducting states generally exhibit an asymmetric Bogoliubov spectrum (BS) in multiband systems, although it is overlooked in single-band models. We demonstrate that such asymmetric deformation of the BS is induced by a nonunitary interband pairing (see Fig. 1b), and derive the necessary conditions for generic two-band models. Although we consider two-band systems for simplicity throughout this paper, our theory is relevant for any multiband superconductors with multiple bands near the Fermi level. In addition, we show that the Bogoliubov quasiparticles with asymmetric BS stabilize the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) superconductivity41,42, which is evident from the Lifshitz invariants43, namely linear gradient terms, in the Ginzburg-Landau free energy. The Lifshitz invariants are nonzero only for the anapole superconducting states, whose order parameters are equivalent to an anapole (magnetic toroidal) moment, namely a polar and time-reversal odd multipole44, from the viewpoint of symmetry. It is shown that the phase of the superconducting order parameter is spatially modulated along the effective anapole moment of the superconducting state (see Fig. 1c). The concept of anapole order has been employed in nuclear physics45, magnetic materials science44, strongly correlated electron systems46,47, and optoelectronics48,49, and it is extended to superconductors by this work. In previous works, the FFLO superconductivity has been proposed in the presence of an external magnetic field41,42,43,50 or coexistent magnetic multipole order51,52. However, the magnetic field causes superconducting vortices, prohibiting pure FFLO states, and the proposed multipole superconducting state has not been established in condensed matters. In contrast, the anapole superconductivity realizes the FFLO state without the aid of any other perturbation or electronic order. Note that an intrinsic nonuniform superconducting state has also been discussed in the Bogoliubov Fermi surface states53, although its mechanism and symmetry are different from those of the anapole FFLO state.

Based on the obtained results, we predict the possible asymmetric BS and anapole superconductivity in UTe2, a recently-discovered candidate of a spin-triplet superconductor54. The multiple pairing instabilities26,27,28,29,30,31,32,33 and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking55,56,57,58 were recently observed there.

Results

General two-band Bogoliubov-de Gennes Hamiltonian

We begin our discussion by considering the general form of the Bogoliubov-de Gennes (BdG) Hamiltonian for two-band systems:

$${{{{{{{\mathcal{H}}}}}}}}=\frac{1}{2}\mathop{\sum}\limits_{{{{{{{{\bf{k}}}}}}}}}({\hat{c}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{\dagger}}} },{\hat{c}}_{-{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{T}}}}}}}}})\left(\begin{array}{cc}{H}_{0}({{{{{{{\bf{k}}}}}}}})&{{\Delta }}({{{{{{{\bf{k}}}}}}}})\\ {{{\Delta }}}^{{{{\dagger}}} }({{{{{{{\bf{k}}}}}}}})&-{H}_{0}^{* }(-{{{{{{{\bf{k}}}}}}}})\end{array}\right)\left(\begin{array}{c}{\hat{c}}_{{{{{{{{\bf{k}}}}}}}}}\\ {\hat{c}}_{-{{{{{{{\bf{k}}}}}}}}}^{* }\end{array}\right),$$
(1)

where \({\hat{c}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{T}}}}}}}}}=({c}_{{{{{{{{\bf{k}}}}}}}}1\uparrow },{c}_{{{{{{{{\bf{k}}}}}}}}1\downarrow },{c}_{{{{{{{{\bf{k}}}}}}}}2\uparrow },{c}_{{{{{{{{\bf{k}}}}}}}}2\downarrow })\) is a spinor encoding the four internal electronic DOF stem from spin-1/2 and extra two-valued DOF, such as orbitals and sublattices. Then, the 4 × 4 matrices H0(k) and Δ(k) can be generally expressed as a linear combination of σμ ⊗ τν matrices, where σμ and τν (μ, ν = 0, x, y, z) are the Pauli matrices for the spin and extra DOF, respectively. However, we here introduce a more convenient form of the two-band BdG Hamiltonian using the Euclidean Dirac matrices γn (n = 1, 2, 3, 4, 5), which satisfy {γm, γn} = 2δmn. See “Methods” for the correspondence between the σμ ⊗ τν and Dirac matrices. Assuming that the normal state preserves both \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries, the general form of the normal state Hamiltonian H0(k) can be expressed as

$${H}_{0}({{{{{{{\bf{k}}}}}}}})=({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{0}-\mu ){{{{{{{{\bf{1}}}}}}}}}_{4}+{{{{{{{{\boldsymbol{\epsilon }}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot {{{{{{{\boldsymbol{\gamma }}}}}}}},$$
(2)

where 14 is the 4 × 4 unit matrix, γ = (γ1, γ2, γ3, γ4, γ5) is the vector of the five Dirac matrices, \({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{0}\) and \({{{{{{{{\boldsymbol{\epsilon }}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}=({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{1},{\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{2},{\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{3},{\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{4},{\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{5})\) are the real-valued coefficients of these matrices, and μ is the chemical potential. Whereas \({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{0}\) is an even function of momentum, k-parity of other coefficients \({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{n}\) (n > 0) depends on the details of the extra DOF. The superconducting state is assumed to be a mixture of even- and odd-parity pairing components. The pairing potential Δ(k) for such mixed-parity superconducting states has the general form

$$\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})={{{\Delta }}}_{1}({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{0}{{{{{{{{\bf{1}}}}}}}}}_{4}+{{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot {{{{{{{\boldsymbol{\gamma }}}}}}}})+{{{\Delta }}}_{2}\mathop{\sum}\limits_{m < n}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{mn}i{\gamma }_{m}{\gamma }_{n},$$
(3)

where \(\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})\equiv {{\Delta }}({{{{{{{\bf{k}}}}}}}}){U}_{{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }\) and UT is the unitary part of the time-reversal operator. The complex-valued constants Δ1 and Δ2 represent the superconducting order parameters for the even- and odd-parity pairing channels, respectively. As a consequence of the fermionic antisymmetry Δ(k) = − ΔT( − k), the even-parity (odd-parity) part of \(\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})\) is expressed by a linear combination of 14 and γn (iγmγn) as shown in Eq. (3) (see “Methods”). The real-valued functions \({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{0}\), \({{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}=({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{1},{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{2},{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{3},{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{4},{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{5})\), and \({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{mn}\) (1 ≤ m < n ≤ 5) determine the details of order parameters. Whereas \({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{0}\) is an even function of momentum, k-parity of others \({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{n}\) and \({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{mn}\) depends on the details of the extra DOF. Note that the k-parity of \({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{n}\) must be the same as that of \({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{n}\). Although we adopt a BCS-type description of superconductivity in this work, we consider that our argument is hardly affected by enhanced quantum fluctuations in low-dimensional systems when the long-range order occurs.

Asymmetric BS from \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity interband pairing

We here consider general BdG Hamiltonian including more than two band models, and later focus on the two-band models. In the following, we assume that each band is weakly coupled and the intraband pairing is dominant compared to the interband pairing. In such situations, spontaneous \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking with maintaining the \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetry is energetically favored in the mixed-parity superconducting states15,16, and the symmetry of the superconducting order parameter becomes equivalent to that of odd-parity magnetic multipoles44. A characteristic feature of the odd-parity magnetic multipole ordered state is the asymmetric modulation of the band structure51,52,59,60, which leads to peculiar nonequilibrium responses such as nonreciprocal transport61, magnetopiezoelectric effect62,63, and photocurrent generation48,49. Therefore, the appearance of the asymmetric BS is naturally expected in the \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity superconductors. However, the asymmetric BS is not obtained in single-band models (see later discussions).

To induce such asymmetric modulation in the BS, effects of the \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking in the particle-particle superconducting channel should be transferred into the particle-hole channel. This suggests that it is not sufficient to consider only the pairing potential Δ(k), since it is not gauge invariant. Instead of Δ(k) alone, we need to consider gauge-invariant bilinear products of Δ(k) and Δ†(k)36 in order to reveal conditions for realizing the asymmetric BS. Here, we focus on the simplest bilinear products, that is, Δ(k)Δ†(k). The parity-odd and time-reversal-odd (\({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd) part of this bilinear product is calculated as

$${M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})=\frac{1}{2}\left([{\hat{{{\Delta }}}}^{{{{{{{{\rm{g}}}}}}}}}({{{{{{{\bf{k}}}}}}}}),{\hat{{{\Delta }}}}^{{{{{{{{\rm{u{{{\dagger}}}}}}} }}}}}({{{{{{{\bf{k}}}}}}}})]+[{\hat{{{\Delta }}}}^{{{{{{{{\rm{u}}}}}}}}}({{{{{{{\bf{k}}}}}}}}),{\hat{{{\Delta }}}}^{{{{{{{{\rm{g{{{\dagger}}}}}}} }}}}}({{{{{{{\bf{k}}}}}}}})]\right),$$
(4)

where \({\hat{{{\Delta }}}}^{{{{{{{{\rm{g}}}}}}}}}({{{{{{{\bf{k}}}}}}}})\) and \({\hat{{{\Delta }}}}^{{{{{{{{\rm{u}}}}}}}}}({{{{{{{\bf{k}}}}}}}})\) are the even- and odd-parity part of \(\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})\), respectively (see Supplementary Note 1 for the derivation of Eq. (4)). Owing to the gauge invariance and \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd behavior of \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\), a nonzero \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\) can be a measure of the \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking in the particle-hole channel, which permits emergence of the asymmetric BS. Note that the pairing state must be nonunitary to induce a nonzero \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\), since \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})=0\) when Δ(k)Δ†(k) is proportional to the unit matrix. In analogy with the spin polarization of nonunitary spin-triplet superconducting states in spin-1/2 single-band models1, the \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd bilinear product \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\) can be interpreted as a polarization of an internal DOF in the superconducting state.

The emergence of a nonzero \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd bilinear product \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\) requires the interband pairing. To see this, we consider the problem in the band basis. Since H0(k) is assumed to preserve the \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries, the energy eigenvalues are doubly degenerate and labeled by a pseudospin index. Especially, we choose the so-called manifestly covariant Bloch basis10, in which the pseudospin index transforms like a true spin-1/2 under time-reversal and crystalline symmetry operations. In this basis, the intraband pairing potential is generally expressed as Δk = (ψk + dk ⋅ s)isy, where s = (sx, sy, sz) are Pauli matrices in pseudospin space. The complex-valued functions ψk and dk are even and odd functions of k, respectively. Then, in the absence of the interband pairing, the multiband BdG Hamiltonian matrix reduces to a series of decoupled blocks describing spin-1/2 single-band superconductors. The bilinear product for this intraband pairing potential is obtained as \({{{\Delta }}}_{{{{{{{{\bf{k}}}}}}}}}{{{\Delta }}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{\dagger}}} }=(| {\psi }_{{{{{{{{\bf{k}}}}}}}}}{| }^{2}+| {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}{| }^{2}){{{{{{{{\bf{1}}}}}}}}}_{2}+2{{{{{{{\rm{Re}}}}}}}}({\psi }_{{{{{{{{\bf{k}}}}}}}}}{{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{* })\cdot {{{{{{{\bf{s}}}}}}}}+i({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\times {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{* })\cdot {{{{{{{\bf{s}}}}}}}}\), and the second and third terms are nonunitary components that break \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries, respectively. Here, 12 is the 2 × 2 unit matrix. However, there appears no term breaking both \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries, and hence the \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd bilinear product for this Δk must vanish. This indicates that the interband pairing is necessary for a nonzero \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd bilinear product \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\), which is essential for realizing the asymmetric BS. This is also the reason the asymmetric BS is not obtained in single-band models.

The presence of interband pairing can be characterized by the so-called superconducting fitness F(k), which is defined as \(F({{{{{{{\bf{k}}}}}}}}){U}_{{{{{{{{\rm{T}}}}}}}}}={H}_{0}({{{{{{{\bf{k}}}}}}}}){{\Delta }}({{{{{{{\bf{k}}}}}}}})-{{\Delta }}({{{{{{{\bf{k}}}}}}}}){H}_{0}^{* }(-{{{{{{{\bf{k}}}}}}}})\)64,65. Since a nonvanishing F(k)F†(k) quantifies the strength of interband pairing by definition64,65, its \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd part should be nonzero to realize a nonvanishing \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\). The \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd part of F(k)F†(k) is obtained as

$${M}_{-}^{(2)}({{{{{{{\bf{k}}}}}}}})=\frac{1}{2}\left([{F}^{{{{{{{{\rm{g}}}}}}}}}({{{{{{{\bf{k}}}}}}}}),{F}^{{{{{{{{\rm{u{{{\dagger}}}}}}} }}}}}({{{{{{{\bf{k}}}}}}}})]+[{F}^{{{{{{{{\rm{u}}}}}}}}}({{{{{{{\bf{k}}}}}}}}),{F}^{{{{{{{{\rm{g{{{\dagger}}}}}}} }}}}}({{{{{{{\bf{k}}}}}}}})]\right),$$
(5)

where Fg(k) and Fu(k) are the even- and odd-parity part of F(k), respectively. If the normal state preserves both \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries, they are obtained as \({F}^{{{{{{{{\rm{g,u}}}}}}}}}({{{{{{{\bf{k}}}}}}}})=[{H}_{0}({{{{{{{\bf{k}}}}}}}}),{\hat{{{\Delta }}}}^{{{{{{{{\rm{g,u}}}}}}}}}({{{{{{{\bf{k}}}}}}}})]\). Note that the \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd part of F(k)F†(k) can be extracted in the same way as Δ(k)Δ†(k) [compare Eq. (5) with Eq. (4)], since the transformation of F(k)F†(k) under space-inversion and time-reversal can be described in the same way as that of Δ(k)Δ†(k). Based on Eq. (5), not only the pair potential Δ(k) but also the normal part H0(k) must satisfy a proper condition to realize \({M}_{-}^{(2)}({{{{{{{\bf{k}}}}}}}}) \, \ne \, 0\) and asymmetric BS.

From the above discussions, we conclude that the necessary (but not sufficient) condition for the asymmetric BS can be written as \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}) \, \ne \, 0\,\cap \,{M}_{-}^{(2)}({{{{{{{\bf{k}}}}}}}}) \, \ne \, 0\), which implies the superconductivity-driven \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking in the particle-hole channel. We here write down this necessary condition for the general two-band BdG Hamiltonian. By substituting Eqs. (2) and (3) to Eqs. (4) and (5), we obtain the \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd bilinear products \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\) and \({M}_{-}^{(2)}({{{{{{{\bf{k}}}}}}}})\) as follows:

$${M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})=2{{{{{{{\rm{Im}}}}}}}}({{{\Delta }}}_{1}{{{\Delta }}}_{2}^{* })\mathop{\sum}\limits_{m < n}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{mn}({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{n}{\gamma }_{m}-{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{m}{\gamma }_{n}),$$
(6)
$${M}_{-}^{(2)}({{{{{{{\bf{k}}}}}}}})={{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})]{\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}}),$$
(7)

where \({\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})\equiv {H}_{0}({{{{{{{\bf{k}}}}}}}})-({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{0}-\mu ){{{{{{{{\bf{1}}}}}}}}}_{4}\). We see that \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\) appears inside the expression of \({M}_{-}^{(2)}({{{{{{{\bf{k}}}}}}}})\), and hence the necessary condition for the asymmetric BS can be simplified as \({{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})] \, \ne \, 0\) in two-band models. Note that it is not clear whether \({M}_{-}^{(2)}({{{{{{{\bf{k}}}}}}}})\) can be written in terms of \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})\) in more than two band models since Eq. (7) is derived for the general two-band model by using the properties of Dirac matrices. From Eqs. (6) and (7), the necessary condition (i.e., \({{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})] \, \ne \, 0\)) can be summarized as following two criteria; (i) the relative phase difference between even- and odd-parity pairing potentials must be nonzero so that \({{{{{{{\rm{Im}}}}}}}}({{{\Delta }}}_{1}{{{\Delta }}}_{2}^{* })\ne 0\), and (ii) the BdG Hamiltonian must satisfy \({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{m}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{n}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{mn} \, \ne \, 0\) or \({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{n}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{m}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{mn} \, \ne \, 0\) for 1 ≤ ∃m < ∃n ≤ 5. Interpretations of these requirements in the σμ ⊗ τν basis are shown in “Methods” section.

We now confirm that the asymmetric BS indeed appears when the above two criteria are fulfilled. A minimal two-band model satisfying the criterion (ii) can be obtained by substituting \({{{{{{{{\boldsymbol{\epsilon }}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}=r{\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{a}{{{{{{{{\bf{e}}}}}}}}}_{a}+(1-r){\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{b}{{{{{{{{\bf{e}}}}}}}}}_{b}\), \({{{{{{{{\boldsymbol{\eta }}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}=(1-r){\eta }_{{{{{{{{\bf{k}}}}}}}}}^{a}{{{{{{{{\bf{e}}}}}}}}}_{a}+r{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{b}{{{{{{{{\bf{e}}}}}}}}}_{b}\), and \({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{mn}={\delta }_{ma}{\delta }_{nb}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{ab}\) into Eqs. (2) and (3). Here, a and b are specific integers satisfying 1 ≤ a < b ≤ 5, en is the unit vector for the n-th component, and r takes the value either 0 or 1. Under this setup, we can analytically diagonalize the BdG Hamiltonian as \({{{{{{{\rm{diag}}}}}}}}({E}_{{{{{{{{\bf{k}}}}}}}}}^{+}{{{{{{{{\bf{1}}}}}}}}}_{2},{E}_{{{{{{{{\bf{k}}}}}}}}}^{-}{{{{{{{{\bf{1}}}}}}}}}_{2},-{E}_{-{{{{{{{\bf{k}}}}}}}}}^{+}{{{{{{{{\bf{1}}}}}}}}}_{2},-{E}_{-{{{{{{{\bf{k}}}}}}}}}^{-}{{{{{{{{\bf{1}}}}}}}}}_{2})\). Based on the correspondence between the Dirac matrices and σμ ⊗ τν matrices the energy spectrum \({E}_{{{{{{{{\bf{k}}}}}}}}}^{\pm }\) can be obtained as

$${E}_{{{{{{{{\bf{k}}}}}}}}}^{\pm }= \sqrt{{\xi }_{{{{{{{{\bf{k}}}}}}}}}^{2}+\frac{1}{4}{{{{{{{\rm{Tr}}}}}}}}\left[{{\Delta }}({{{{{{{\bf{k}}}}}}}}){{{\Delta }}}^{{{{\dagger}}} }({{{{{{{\bf{k}}}}}}}})\pm \frac{{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})}{r{\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{a}+(1-r){\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{b}}\right]}\\ \pm \, [r{\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{a}+(1-r){\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{b}],$$
(8)

where \({\xi }_{{{{{{{{\bf{k}}}}}}}}}\equiv {\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{0}-\mu\). By using the transformation properties of the BdG Hamiltonian under space-inversion and time-reversal, we can confirm that Eq. (8) satisfies \({E}_{-{{{{{{{\bf{k}}}}}}}}}^{+}\ne {E}_{{{{{{{{\bf{k}}}}}}}}}^{\pm }\) and \({E}_{-{{{{{{{\bf{k}}}}}}}}}^{-}\ne {E}_{{{{{{{{\bf{k}}}}}}}}}^{\pm }\) (i.e., the BS is asymmetric) when \({{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})] \, \ne \, 0\). See “Methods” for the proof. This implies that \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}) \, \ne \, 0\,\cap \,{M}_{-}^{(2)}({{{{{{{\bf{k}}}}}}}}) \, \ne \, 0\) is indeed a necessary condition of the asymmetric BS.

Lifshitz invariants and effective anapole moment

To obtain further insight into the asymmetric BS, we now investigate the free energy of the above minimal model satisfying \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}) \, \ne \, 0\,\cap \,{M}_{-}^{(2)}({{{{{{{\bf{k}}}}}}}}) \, \ne \, 0\). By differentiating Eq. (8) with respect to Δj and \({{{\Delta }}}_{j}^{* }\) (j = 1, 2), the Ginzburg-Landau free energy for superconductivity is derived as follows (see Supplementary Note 2):

$${{{{{{{\mathcal{F}}}}}}}}= \,\, {\alpha }_{1}| {{{\Delta }}}_{1}{| }^{2}+{\alpha }_{2}| {{{\Delta }}}_{2}{| }^{2}+{\beta }_{1}| {{{\Delta }}}_{1}{| }^{4}+{\beta }_{2}| {{{\Delta }}}_{2}{| }^{4}\\ +\,4\tilde{\beta }| {{{\Delta }}}_{1}{| }^{2}| {{{\Delta }}}_{2}{| }^{2}-\tilde{\beta }({{{\Delta }}}_{1}^{2}{{{\Delta }}}_{2}^{* 2}+{{{\Delta }}}_{2}^{2}{{{\Delta }}}_{1}^{* 2})\\ +\,\mathop{\sum}\limits_{\nu =x,y,z}({\kappa }_{1,\nu }| {{{\Delta }}}_{1}{| }^{2}+{\kappa }_{2,\nu }| {{{\Delta }}}_{2}{| }^{2}){q}_{\nu }^{2}+{{{{{{{\bf{T}}}}}}}}\cdot {{{{{{{\bf{q}}}}}}}},$$
(9)

where q = (qx, qy, qz) is the center-of-mass momentum of Cooper pairs. The analytical expressions of αj, βj( > 0), \(\tilde{\beta }( > \, 0)\), and κj,ν( > 0) are shown in Supplementary Note 2. The last term is the Lifshitz invariant43 stabilizing the FFLO state with q∥T. Since the Cooper pair condensation occurs at a single q in our model, the superconducting order parameter is expressed as Δ(r) ∝ eiq⋅r in real space (Fig. 1c). The coefficient vector T = (Tx, Ty, Tz) is given by

$${{{{{{{\bf{T}}}}}}}}={\rho }_{0}{\langle {{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})]{{{{{{{{\bf{v}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\rangle }_{{{{{{{{\rm{FS}}}}}}}}}\frac{7\zeta (3)}{16{\pi }^{2}{T}^{2}},$$
(10)

where ρ0 is the density of states at the Fermi energy, 〈⋯ 〉FS denotes the average over the Fermi surface, vk ≡ ∇kξk, T is the temperature, and ζ(x) is the Riemann zeta function. T can be interpreted as the effective anapole moment of the superconducting state. To see this, we here consider conditions for T ≠ 0. Eq. (10) indicates that T is nonzero only for \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking pairing states with \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}) \, \ne \, 0\). In addition, \({\langle {{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})]{{{{{{{{\bf{v}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\rangle }_{{{{{{{{\rm{FS}}}}}}}}}\) is nonzero only when the superconducting order parameter belongs to a polar irreducible representation (IR), since the velocity vk is a polar vector and \({\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})\) is assumed to be \({{{{{{{\mathcal{P}}}}}}}}\)-symmetric. Therefore, T is a polar and time-reversal-odd vector; the symmetry is equivalent to the anapole moment44,45. Hereafter, we refer to the superconductivity with T ≠ 0 as the anapole superconductivity. The anapole superconductivity realizes a nonuniform FFLO state with q∥T (see Fig. 1c) to compensate a polar asymmetry in the BS. The \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity pairing is an origin of the anapole superconductivity. Although the stability of such pairing has been revealed15, a self-consistent calculation is desirable to justify the stability of the anapole FFLO state and clarify the properties further. Such detailed analysis is left for future work.

Finally, we comment that the anapole moment must be aligned in the conducting direction in low-dimensional systems. This restriction naturally appears through the above expression (10) of the effective anapole moment T.

Application to UTe2

We now discuss the asymmetric BS and anapole superconductivity in UTe2. Intensive studies after the discovery of superconductivity evidenced odd-parity spin-triplet superconductivity in UTe254,55,56,66,67,68,69,70,71,72,73,74,75,76,77,78,79. However, multiple superconducting phases similar to Fig. 1a have been observed under pressure26,27,28,29,30,31,32,33, and the antiferromagnetic quantum criticality implies the spin-singlet superconductivity there31. A theoretical study based on the periodic Anderson model verified this naive expectation and predicted the parity-mixed superconducting state in the intermediate pressure region25. Note that the interband pairing, which is an essential ingredient for the asymmetric BS and anapole superconductivity, may have considerable impacts on the superconductivity in UTe2 owing to multiple bands near the Fermi level25,80,81,82,83,84,85,86.

First, let us discuss the symmetry of superconductivity. Since the crystal structure of UTe2 preserves D2h point group symmetry, the superconducting order parameter is classified based on the IRs of D2h. Below we consider all the odd-parity IRs, namely, Au, B1u, B2u, and B3u, although the Au and B3u IRs may be promising candidates25,77,78,80,87. Moreover, a recent calculation has shown that the even-parity Ag superconducting state is favored by antiferromagnetic fluctuation under pressure25. Therefore, we study a mixture of the even-parity Ag and odd-parity either Au, B1u, B2u, or B3u states, while we particularly focus on the Au or B3u pairing.

Based on the above facts, we introduce a minimal model for UTe2 as follows:

$${H}_{0}({{{{{{{\bf{k}}}}}}}})=({\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}-\mu ){\sigma }_{0}\otimes {\tau }_{0}+{{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}\otimes {\tau }_{z},$$
(11)
$$\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})=\, {{{\Delta }}}_{1}({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}{\sigma }_{0}\otimes {\tau }_{0}+{{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}\otimes {\tau }_{z})\\ +\,{{{\Delta }}}_{2}({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}\otimes {\tau }_{0}+{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}}{\sigma }_{0}\otimes {\tau }_{z}),$$
(12)

where τν represent the Pauli matrices for a sublattice DOF originating from a ladder structure of U atoms (Fig. 2). We assume a simple form of the single-particle kinetic energy as \({\varepsilon }_{{{{{{{{\bf{k}}}}}}}}}=-2{\sum }_{\nu = x,y,z}{t}_{\nu }\cos {k}_{\nu }\). The second term of Eq. (11) is a sublattice-dependent staggered form of Rashba SOC with \({{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}=\alpha (\sin {k}_{y}\hat{{{{{{{{\bf{x}}}}}}}}}-\sin {k}_{x}\hat{{{{{{{{\bf{y}}}}}}}}})\), arising from the local \({{{{{{{\mathcal{P}}}}}}}}\)-symmetry breaking at U sites25,88. Since the local site symmetry descends to C2v from D2h owing to the ladder structure of U atoms, the existence of the Rashba-type SOC with opposite coupling constants ± α at each sublattice is naturally expected (see Fig. 2). The local \({{{{{{{\mathcal{P}}}}}}}}\)-symmetry breaking also leads to a sublattice-dependent parity mixing of the pair potential89. Then, the even-parity (odd-parity) pair potential is assumed to be a mixture of intrasublattice spin-singlet (spin-triplet) and staggered spin-triplet (spin-singlet) components as shown in Eq. (12). We assume the form of the k-dependent coefficients \({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}\) and \({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}\) (\({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}}\) and \({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}}\)) so as to be consistent with the basis functions of the Ag IR (Au, B1u, B2u, or B3u IRs).

Fig. 2: Schematic of local \({{{{{{{\mathcal{P}}}}}}}}\)-symmetry breaking in UTe2.
figure 2

U atoms form a ladder structure along the a axis. The ladder structure generates two non-equivalent sublattices U1 (blue circle) and U2 (orange circle). Since the \({{{{{{{\mathcal{P}}}}}}}}\)-symmetry is locally broken in this crystal structure, a sublattice-dependent Rashba-type spin-orbit coupling (SOC) appears. To maintain the global \({{{{{{{\mathcal{P}}}}}}}}\)-symmetry, the magnitude of Rashba SOC α changes sign at each sublattice (i.e., +α for U1 and −α for U2).

Full size image

We now consider the necessary conditions for an asymmetric BS in UTe2. As discussed in the above sections, a nonzero \({{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})]\) is necessary for the asymmetric BS in a two-band model. For Eqs. (11) and (12), this quantity is obtained as \({{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})]=-8{{{{{{{\rm{Im}}}}}}}}({{{\Delta }}}_{1}{{{\Delta }}}_{2}^{* })[{{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot ({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}\times {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}})]\). Therefore, \({{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot ({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}\times {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}}) \, \ne \, 0\) must be satisfied to realize the asymmetric BS. This indicates that the sublattice-dependent SOC and spin-triplet pairing components \({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g,u}}}}}}}}}\) are essential for the asymmetric BS. In contrast, the spin-singlet pairing components \({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g,u}}}}}}}}}\) do not play an important role for realizing the asymmetric BS in this model. Hereafter, we assume \({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}=1\) and \({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}}=0\) for simplicity. The basis functions of \({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g,u}}}}}}}}}\) and corresponding \({{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot ({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}\times {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}})\) for possible mixed-parity superconducting states in UTe2 are summarized in Table 1. As shown in Table 1, \({{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot ({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}\times {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}})\propto \alpha ({\phi }_{x}^{{{{{{{{\rm{g}}}}}}}}}+{\phi }_{y}^{{{{{{{{\rm{g}}}}}}}}}){\phi }_{z}^{{{{{{{{\rm{u}}}}}}}}}\) for all patterns of the superconducting state, where \({\phi }_{\nu }^{{{{{{{{\rm{g,u}}}}}}}}}\) is a real-valued coefficient of the ν-th component of \({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g,u}}}}}}}}}\). Therefore, \({\phi }_{x}^{{{{{{{{\rm{g}}}}}}}}}+{\phi }_{y}^{{{{{{{{\rm{g}}}}}}}}} \, \ne \, 0\) and \({\phi }_{z}^{{{{{{{{\rm{u}}}}}}}}} \, \ne \, 0\) are necessary for the asymmetric BS. According to a recent numerical calculation25, the magnetic anisotropy of UTe2 leads to \(| {\phi }_{y}^{{{{{{{{\rm{g}}}}}}}}}| \gg | {\phi }_{x}^{{{{{{{{\rm{g}}}}}}}}}|\) for the Ag state. Then, we assume \({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}=\sin {k}_{x}\hat{{{{{{{{\bf{y}}}}}}}}}\) (i.e., \({\phi }_{x}^{{{{{{{{\rm{g}}}}}}}}}=0\) and \({\phi }_{y}^{{{{{{{{\rm{g}}}}}}}}}=1\)) in the following calculations. On the other hand, we assume \({\phi }_{\nu }^{{{{{{{{\rm{u}}}}}}}}}={\delta }_{\nu z}\) for the odd-parity pairing component to extract only the essential ingredient for the asymmetric BS and make a clear discussion.

Table 1 Symmetry analysis of Bogoliubov spectrum asymmetry for UTe2.
Full size table

The numerical results of the BS for this UTe2 model are shown in Fig. 3. We here consider only the Ag + iAu (\({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}}=\sin {k}_{z}\hat{{{{{{{{\bf{z}}}}}}}}}\)) and Ag + iB3u (\({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}}=\sin {k}_{y}\hat{{{{{{{{\bf{z}}}}}}}}}\)) states as promising candidates of the \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity superconductivity in UTe2. It is shown that the BS of both Ag + iAu and Ag + iB3u states are indeed asymmetric along some directions in the Brillouin zone (see Fig. 3a, b). The BS in the Ag + iAu state exhibits a kxkykz-type tetrahedral asymmetry as depicted in Fig. 3c, while the BS in the Ag + iB3u state shows a \({k}_{x}{k}_{y}^{2}\)-type unidirectional asymmetry as depicted in Fig. 3d. Consistent with these numerical results, Table 1 reveals that \({{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot ({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}\times {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}})\) of the Ag + iAu and Ag + iB3u states are proportional to kxkykz and \({k}_{x}{k}_{y}^{2}\), respectively. This implies that the type of asymmetry in the BS is determined by the symmetry of \({{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})]\), which is an essential ingredient for realizing the asymmetric BS.

Fig. 3: Asymmetric Bogoliubov spectrum (BS) for the UTe2 model.
figure 3

a, b Numerical results of BS in the UTe2 model for the Ag + iAu state and the Ag + iB3u state. c Schematic of kxkykz-type asymmetric modulation in the Brillouin zone that is induced in the BS of Ag + iAu state in a. d Schematic of kx-type asymmetric modulation that is induced in the BS of Ag + iB3u state in b. The symbols of the horizontal axis in a and b denote the k-points in the Brillouin zone of a primitive orthorhombic lattice; Γ = (0, 0, 0), \({{{{{{{\rm{X}}}}}}}}=-\bar{{{{{{{{\rm{X}}}}}}}}}=(\pi ,0,0)\), \({{{{{{{\rm{S}}}}}}}}=-\bar{{{{{{{{\rm{S}}}}}}}}}=(\pi ,\pi ,0)\), \({{{{{{{\rm{T}}}}}}}}=-\bar{{{{{{{{\rm{T}}}}}}}}}=(0,\pi ,\pi )\), \({{{{{{{\rm{R}}}}}}}}=-\bar{{{{{{{{\rm{R}}}}}}}}}=(\pi ,\pi ,\pi )\), Γy = (0, π/4, 0), and \({{{{{{{{\rm{X}}}}}}}}}_{{{{{{{{\rm{y}}}}}}}}}=-{\bar{{{{{{{{\rm{X}}}}}}}}}}_{{{{{{{{\rm{y}}}}}}}}}=(\pi ,\pi /4,0)\). In the numerical calculations, parameters are set to be tν = 1.0, μ = − 4.0, α = 0.4, and (Δ1, Δ2) = (0.2, 0.2i). The BS asymmetry appears in a and b consistent with the symmetry analysis of \({{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})]\propto {{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot ({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{g}}}}}}}}}\times {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{{{{{{{\rm{u}}}}}}}}})\).

Full size image

Finally, we discuss the possible anapole superconductivity in UTe2. The Ag + iAu state belongs to the nonpolar \({A}_{{{{{{{{\rm{u}}}}}}}}}^{-}\) IR (IRs with odd time-reversal parity are denoted by Γ−), which corresponds to nonpolar odd-parity magnetic multipoles such as magnetic monopole, quadrupole, and hexadecapole from the viewpoint of symmetry. On the other hand, the Ag + iB3u state belongs to the polar \({B}_{{{{{{{{\rm{3u}}}}}}}}}^{-}\) IR with the polar x axis, which is symmetrically equivalent to the anapole moment Tx. Since the anapole superconducting states are allowed only when the superconducting order parameter belongs to a polar IR, the Ag + iB3u state is a possible candidate of the anapole superconductivity. Indeed, as discussed above, the BS of the Ag + iB3u state exhibits a polar \({k}_{x}{k}_{y}^{2}\)-type asymmetry, while the BS of the Ag + iAu state exhibits a nonpolar kxkykz-type asymmetry (see Fig. 3). It should also be noted that the BS in the Ag + iB3u state possesses the polarity along the kx axis, which coincides with the polar axis of the \({B}_{{{{{{{{\rm{3u}}}}}}}}}^{-}\) IR.

Based on the above classification and the Ginzburg-Landau free energy (9), the anapole FFLO state with \({{{{{{{\bf{q}}}}}}}}\propto {{{{{{{\bf{T}}}}}}}}\parallel \hat{{{{{{{{\bf{x}}}}}}}}}\) should be naturally realized in the Ag + iB3u state. In the same manner, we expect the anapole superconducting states with \({{{{{{{\bf{T}}}}}}}}\parallel \hat{{{{{{{{\bf{y}}}}}}}}}\) and \({{{{{{{\bf{T}}}}}}}}\parallel \hat{{{{{{{{\bf{z}}}}}}}}}\) in the Ag + iB2u and Ag + iB1u states, respectively (see Supplementary Note 3 for possible anapole superconductivity in UTe2).

Discussion

From the analogy with magnetic states, we can predict various exotic superconducting phenomena closely related to the asymmetric BS. For instance, the asymmetry of the BS will significantly affect the superconducting piezoelectric effect90, nonreciprocal optical responses91, and nonreciprocal Meissner effect92. All of these phenomena are caused by the absence of \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries in the superconducting state. Hence, they will be useful probes to offer conclusive evidence for the \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking and the BS asymmetry in superconductors. Studies for the interplay of these exotic superconducting phenomena and asymmetric BS will be presented elsewhere.

Experimental detection of the anapole superconductivity should be possible by observing its domain structure. The anapole superconducting state effectively carries a supercurrent along the anapole moment T, since the order parameter is spatially modulated with eiq⋅r ~ eiT⋅r. This indicates the emergence of superconducting vortices at the anapole domain boundaries (see Fig. 4a) even though an external magnetic field is absent. The anapole domains can be generally formed owing to the degeneracy between the Γg + iΓu pairing and Γg − iΓu pairing states, which have the opposite anapole moment. Therefore, the observation of vortices at a zero magnetic field can be solid evidence of the anapole superconductivity. In addition, the anapole domain can be switched by the supercurrent in a similar way to the electrical switching of antiferromagnets93,94. In an anapole superconductor, the effective anapole moment T couples to the applied electric current j, which is a symmetry-adapted field of the anapole moment. Then, the anapole superconducting domain should be switched to align the effective anapole moments along the injected supercurrent j (see Fig. 4b). It should also be noticed that the anapole domain switching eliminates the internal magnetic field from the vortices at the domain boundaries, since the domain structure disappears by applying the supercurrent. Therefore, the anapole superconducting domain switching can be regarded as a process of erasing magnetic information. These properties indicate potential applications of anapole superconductivity as a novel quantum device for magnetic information storage and processing.

Fig. 4: Anapole domain and domain switching through supercurrent.
figure 4

a Vortices at the boundary of anapole superconducting domains. b The proposed domain switching in anapole superconductors. The effective anapole moment T is aligned along the injected supercurrent j.

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Conclusions

In summary, we have established that the \({{{{{{{\mathcal{PT}}}}}}}}\)-symmetric mixed-parity superconductors generally exhibit asymmetry in the BS. The essential ingredient for the asymmetric BS is the \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd nonunitary part of the bilinear product ΔΔ† arising from the interband pairing. Therefore, the multiband nature of superconductivity is essential. Especially, we have shown that an FFLO state is stabilized in the absence of an external magnetic field when the superconducting state belongs to a polar and time-reversal-odd IR. The stabilization of the FFLO state is evidenced by the emergence of Lifshitz invariants in the free energy due to the effective anapole moment. The physics of asymmetric BS appears in any multiband superconductors when even- and odd-parity pairing interactions are comparable in strength. As a specific example, we considered the mixed-parity pairing states in UTe2, which is caused by an accidental competition of ferromagnetic and antiferromagnetic spin fluctuations under pressure25. We have shown that the mixed-parity superconductivity in UTe2 realizes the asymmetric BS and anapole superconductivity owing to the locally noncentrosymmetric crystal structure. The asymmetric BS may be linked to the asymmetric spectrum in the scanning tunneling microscope measurement55, although it was interpreted based on the chiral superconductivity. The vortex structure near the anapole domain boundary (Fig. 4a) may also cause the polar Kerr effect, reported for UTe258. Spontaneous ordering of strongly parity-mixed pairing state and resulting asymmetric BS can also be expected in superconductivity mediated by parity-breaking fluctuations11. To further broaden the scope of application of our theory, it is important to find microscopic electronic interactions that induce competing even- and odd-parity pairing instabilities.

We predicted various superconducting phenomena induced by the asymmetric BS, such as the superconducting piezoelectric effect, nonlinear optical responses, nonreciprocal Meissner effect, and anapole domain switching from the analogy with magnetic materials. Topological properties and collective modes associated with the asymmetric BS may also be intriguing issues. Exploration of such exotic phenomena will be a promising route for future research.

Methods

Correspondence between Pauli matrices and Dirac matrices

In this section, we show that the general form of the BdG Hamiltonian with spin-1/2 and a two-valued extra DOF can be expressed by using the Euclidean Dirac matrices.

Since we assume that the normal state preserves both \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries, H0(k) transforms under the space-inversion \({{{{{{{\mathcal{P}}}}}}}}\) and the time-reversal \({{{{{{{\mathcal{T}}}}}}}}\) as

$${H}_{0}({{{{{{{\bf{k}}}}}}}})\mathop{\to }\limits^{{{{{{{{\mathcal{P}}}}}}}}}{U}_{{{{{{{{\rm{P}}}}}}}}}^{{{{\dagger}}} }{H}_{0}(-{{{{{{{\bf{k}}}}}}}}){U}_{{{{{{{{\rm{P}}}}}}}}}={H}_{0}({{{{{{{\bf{k}}}}}}}}),$$
(13)
$${H}_{0}({{{{{{{\bf{k}}}}}}}})\mathop{\to }\limits^{{{{{{{{\mathcal{T}}}}}}}}}{U}_{{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }{H}_{0}^{* }(-{{{{{{{\bf{k}}}}}}}}){U}_{{{{{{{{\rm{T}}}}}}}}}={H}_{0}({{{{{{{\bf{k}}}}}}}}),$$
(14)

where UP and UT are unitary matrices. In this paper, we consider a spin-1/2 system satisfying \({U}_{{{{{{{{\rm{T}}}}}}}}}{U}_{{{{{{{{\rm{T}}}}}}}}}^{* }=-{{{{{{{{\bf{1}}}}}}}}}_{4}\). In addition, we require that the time-reversal commute with the space-inversion (i.e., \({U}_{{{{{{{{\rm{P}}}}}}}}}{U}_{{{{{{{{\rm{T}}}}}}}}}={U}_{{{{{{{{\rm{T}}}}}}}}}{U}_{{{{{{{{\rm{P}}}}}}}}}^{* }\)), and the space-inversion operator is its own inverse (i.e., \({U}_{{{{{{{{\rm{P}}}}}}}}}^{2}={{{{{{{{\bf{1}}}}}}}}}_{4}\)). Under the above assumptions, H0(k) can be generally expressed as

$${H}_{0}({{{{{{{\bf{k}}}}}}}})=({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{0}-\mu ){\sigma }_{0}\otimes {\tau }_{0}+{f}_{{{{{{{{\bf{k}}}}}}}}}{\sigma }_{0}\otimes {\tau }_{{x}_{i}}\\ +\,{{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}\otimes {\tau }_{{y}_{i}}+{h}_{{{{{{{{\bf{k}}}}}}}}}{\sigma }_{0}\otimes {\tau }_{{z}_{i}},$$
(15)

where σ = (σx, σy, σz) and σ0 ⊗ τ0 = 14. Hermiticity requires all coefficients in Eq (15) are real. The index i specifies the extra DOF and (xi, yi, zi) is a permutation of (x, y, z). Since UP and UT vary depending on the extra DOF, the general models (15) are classified by the index i. In this paper, we consider three representative examples shown in Table 2. For i = 1 (i = 2), the extra DOF is orbitals with the same (opposite) parity, and UP = σ0 ⊗ τ0 (UP = σ0 ⊗ τz). For i = 3, the extra DOF is sublattices in a locally noncentrosymmetric crystal structure, and UP = σ0 ⊗ τx. In these cases, UT = iσy ⊗ τ0. Although the extra DOF can be other than the above three cases, Eq. (15) holds for all the cases unless \({U}_{{{{{{{{\rm{P}}}}}}}}}{U}_{{{{{{{{\rm{T}}}}}}}}}\ne {U}_{{{{{{{{\rm{T}}}}}}}}}{U}_{{{{{{{{\rm{P}}}}}}}}}^{* }\), UPUP ≠ 14, or \({U}_{{{{{{{{\rm{T}}}}}}}}}{U}_{{{{{{{{\rm{T}}}}}}}}}^{* }\ne -{{{{{{{{\bf{1}}}}}}}}}_{4}\)37. Since the set of σμ ⊗ τν matrices is completely anticommuting in Eq. (15), we can substitute them by the five anticommuting Euclidean Dirac matrices. Then, we can rewrite Eq. (15) as Eq. (2).

Table 2 Classification of two-band models based on the extra degrees of freedom (DOF).
Full size table

The pairing potential Δ(k) transforms under the space-inversion and the time-reversal as Δ(k) \({{{{{{\mathcal{P}}}}}}}\atop{\longrightarrow}\) \({U}_{{{{{{{{\rm{P}}}}}}}}}^{{{{\dagger}}} }{{\Delta }}(-{{{{{{{\bf{k}}}}}}}}){U}_{{{{{{{{\rm{P}}}}}}}}}^{* }\) and Δ(k) \({{{{{{\mathcal{T}}}}}}}\atop{\longrightarrow }\) \({U}_{{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }{{{\Delta }}}^{* }(-{{{{{{{\bf{k}}}}}}}}){U}_{{{{{{{{\rm{T}}}}}}}}}^{* }\), respectively. In terms of \(\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})={{\Delta }}({{{{{{{\bf{k}}}}}}}}){U}_{{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }\), these relations can be rewritten as

$$\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})\mathop{\to }\limits^{{{{{{{{\mathcal{P}}}}}}}}}{U}_{{{{{{{{\rm{P}}}}}}}}}^{{{{\dagger}}} }\hat{{{\Delta }}}(-{{{{{{{\bf{k}}}}}}}}){U}_{{{{{{{{\rm{P}}}}}}}}},$$
(16)
$$\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})\mathop{\to }\limits^{{{{{{{{\mathcal{T}}}}}}}}}{\hat{{{\Delta }}}}^{{{{\dagger}}} }({{{{{{{\bf{k}}}}}}}}).$$
(17)

We note that Eq. (16) is equivalent to the transformation of H0(k) under the space-inversion [see Eq. (13)], while Eq. (17) corresponds to the Hermiticity condition. Whereas H0(k) is assumed to preserve both \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries, we admit that Δ(k) spontaneously breaks the \({{{{{{{\mathcal{P}}}}}}}}\)- and \({{{{{{{\mathcal{T}}}}}}}}\)-symmetries. The only requirements for the pairing potential is satisfying the fermionic antisymmetry Δ(k) = − ΔT( − k), which can be rewritten as

$$\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})={U}_{{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }{\hat{{{\Delta }}}}^{{{{{{{{\rm{T}}}}}}}}}(-{{{{{{{\bf{k}}}}}}}}){U}_{{{{{{{{\rm{T}}}}}}}}},$$
(18)

where we used the fact that \({U}_{{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }={U}_{{{{{{{{\rm{T}}}}}}}}}^{{{{{{{{\rm{T}}}}}}}}}=-{U}_{{{{{{{{\rm{T}}}}}}}}}\) by choosing UT as real (i.e., \({U}_{{{{{{{{\rm{T}}}}}}}}}={U}_{{{{{{{{\rm{T}}}}}}}}}^{* }\)). It should be noticed that Eq. (18) is formally equivalent to the time-reversal symmetry for H0(k) [see Eq. (14)]. Since the even-parity part of \(\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})\) obeys transformation properties completely equivalent to those of H0(k) under the time-reversal and the space-inversion, it can be expressed as a linear combination of six σμ ⊗ τν matrices allowed to appear in H0(k). On the other hand, the other ten σμ ⊗ τν matrices, which correspond to iγmγn (1 ≤ m < n ≤ 5), constitute the odd-parity pairing potential. Then, we obtain a general form of Δ(k) as

$$\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}}) = \, {{{\Delta }}}_{1}\left[\mathop{\sum}\limits_{\nu =0,{x}_{i},{z}_{i}}{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{\nu }{\sigma }_{0}\otimes {\tau }_{\nu }+{{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{y}_{i}}\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}\otimes {\tau }_{{y}_{i}}\right]\\ +\,{{{\Delta }}}_{2}\left[\mathop{\sum}\limits_{\nu =0,{x}_{i},{z}_{i}}{{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{\nu }\cdot {{{{{{{\boldsymbol{\sigma }}}}}}}}\otimes {\tau }_{\nu }+{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{y}_{i}}{\sigma }_{0}\otimes {\tau }_{{y}_{i}}\right],$$
(19)

where \({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{\nu }\) and \({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{\nu }\) are real-valued coefficients. Note that Δ1 and Δ2 are complex valued since \(\hat{{{\Delta }}}({{{{{{{\bf{k}}}}}}}})\ne {\hat{{{\Delta }}}}^{{{{\dagger}}} }({{{{{{{\bf{k}}}}}}}})\) in \({{{{{{{\mathcal{T}}}}}}}}\)-symmetry breaking superconducting phases. From Eq. (19), we obtain Eq. (3) as a general form of Δ(k) in two-band models.

From Eqs. (15) and (19), we obtain

$${{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})]= \,\, 8{{{{{{{\rm{Im}}}}}}}}({{{\Delta }}}_{1}{{{\Delta }}}_{2}^{* })\times \left[({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{z}_{i}}{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{y}_{i}}-{{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{y}_{i}}\cdot {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{z}_{i}}){f}_{{{{{{{{\bf{k}}}}}}}}}\right.\\ +\,({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{x}_{i}}{{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{z}_{i}}-{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{z}_{i}}{{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{x}_{i}}-{{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{y}_{i}}\times {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{0})\cdot {{{{{{{{\bf{g}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}\\ +\left.\,({{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{y}_{i}}\cdot {{{{{{{{\bf{d}}}}}}}}}_{{{{{{{{\bf{k}}}}}}}}}^{{x}_{i}}-{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{x}_{i}}{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{y}_{i}}){h}_{{{{{{{{\bf{k}}}}}}}}}\right].$$
(20)

Then, in the σμ ⊗ τν basis, the necessary conditions for the asymmetric BS (i.e., \({{{{{{{\rm{Tr}}}}}}}}[{M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}}){\tilde{H}}_{0}({{{{{{{\bf{k}}}}}}}})] \, \ne \, 0\)) can be summarized as shown in Table 3. For example, the condition (I) means that the asymmetric BS appears when \({{{{{{{\rm{Im}}}}}}}}({{{\Delta }}}_{1}{{{\Delta }}}_{2}^{* }) \, \ne \, 0\) and \({\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{z}_{i}}{\psi }_{{{{{{{{\bf{k}}}}}}}}}^{{y}_{i}}{f}_{{{{{{{{\bf{k}}}}}}}}} \, \ne \, 0\).

Table 3 Necessary conditions for the asymmetric Bogoliubov spectrum in two-band models (σμ ⊗ τν basis).
Full size table

Asymmetry of BS in the minimal two-band model

We here prove that Eq. (8) indeed expresses the asymmetric BS. For r = 1, Eq. (8) leads to

$${E}_{{{{{{{{\bf{k}}}}}}}}}^{\pm } = \sqrt{{\xi }_{{{{{{{{\bf{k}}}}}}}}}^{2}+| {{{\Delta }}}_{1}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{b}{| }^{2}+| {{{\Delta }}}_{2}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{ab}{| }^{2}\pm 2{{{{{{{\rm{Im}}}}}}}}({{{\Delta }}}_{1}{{{\Delta }}}_{2}^{* }){\eta }_{{{{{{{{\bf{k}}}}}}}}}^{b}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{ab}} \pm {\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{a}.$$
(21)

Then, we need to specify the k-parity of \({\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{a}\), \({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{b}\), and \({\eta }_{{{{{{{{\bf{k}}}}}}}}}^{ab}\), which depend on the details of the extra DOF, to investigate the property of the BS \({E}_{-{{{{{{{\bf{k}}}}}}}}}^{\pm }\). We here denote \({\epsilon }_{-{{{{{{{\bf{k}}}}}}}}}^{a}={p}_{a}{\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{a}\), \({\eta }_{-{{{{{{{\bf{k}}}}}}}}}^{b}={p}_{b}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{b}\), and \({\eta }_{-{{{{{{{\bf{k}}}}}}}}}^{ab}={p}_{ab}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{ab}\) (pa, pb, pab = ± 1). From Eqs. (13) and (14), we obtain \({p}_{a}{\gamma }_{a}={U}_{{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }{\gamma }_{a}^{* }{U}_{{{{{{{{\rm{T}}}}}}}}}={U}_{{{{{{{{\rm{P}}}}}}}}}^{{{{\dagger}}} }{\gamma }_{a}{U}_{{{{{{{{\rm{P}}}}}}}}}\). On the other hand, the \({{{{{{{\mathcal{P}}}}}}}},{{{{{{{\mathcal{T}}}}}}}}\)-odd behavior of \({M}_{-}^{(1)}({{{{{{{\bf{k}}}}}}}})=2{{{{{{{\rm{Im}}}}}}}}({{{\Delta }}}_{1}{{{\Delta }}}_{2}^{* }){\eta }_{{{{{{{{\bf{k}}}}}}}}}^{b}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{ab}{\gamma }_{a}\) leads to \(-{p}_{b}{p}_{ab}{\gamma }_{a}={U}_{{{{{{{{\rm{T}}}}}}}}}^{{{{\dagger}}} }{\gamma }_{a}^{* }{U}_{{{{{{{{\rm{T}}}}}}}}}={U}_{{{{{{{{\rm{P}}}}}}}}}^{{{{\dagger}}} }{\gamma }_{a}{U}_{{{{{{{{\rm{P}}}}}}}}}\). Thus, pa = − pbpab holds in general. Using this relation, we obtain

$${E}_{-{{{{{{{\bf{k}}}}}}}}}^{\pm } = \sqrt{{\xi }_{{{{{{{{\bf{k}}}}}}}}}^{2}+| {{{\Delta }}}_{1}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{b}{| }^{2}+| {{{\Delta }}}_{2}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{ab}{| }^{2}\mp {p}_{a}2{{{{{{{\rm{Im}}}}}}}}({{{\Delta }}}_{1}{{{\Delta }}}_{2}^{* }){\eta }_{{{{{{{{\bf{k}}}}}}}}}^{b}{\eta }_{{{{{{{{\bf{k}}}}}}}}}^{ab}} \pm {p}_{a}{\epsilon }_{{{{{{{{\bf{k}}}}}}}}}^{a},$$
(22)

where \({p}_{a}=\pm 1\). Comparing Eq. (22) with (21), we can safely say that \({E}_{-{{{{{{{\bf{k}}}}}}}}}^{\pm }\ne {E}_{{{{{{{{\bf{k}}}}}}}}}^{+},{E}_{{{{{{{{\bf{k}}}}}}}}}^{-}\) and the BS is asymmetric. In the same manner, we can prove the asymmetry of Eq. (8) for r = 0.