Selection rules for Cooper pairing in two-dimensional interfaces and sheets

Thin sheets deposited on a substrate and interfaces of correlated materials offer a plethora of routes towards the realization of exotic phases of matter. In these systems, inversion symmetry is broken which strongly affects the properties of possible instabilities -- in particular in the superconducting channel. By combining symmetry and energetic arguments, we derive general and experimentally accessible selection rules for Cooper instabilities in noncentrosymmetric systems which yield necessary and sufficient conditions for spontaneous time-reversal-symmetry breaking at the superconducting transition and constrain the orientation of the triplet vector. We discuss in detail the implications for various different materials. For instance, we conclude that the pairing state in thin layers of Sr$_2$RuO$_4$ must, as opposed to its bulk superconducting state, preserve time-reversal symmetry with its triplet vector being parallel to the plane of the system. All pairing states of this system allowed by the selection rules are predicted to display topological Majorana modes at dislocations or at the edge of the system. Applying our results to the LaAlO$_3$/SrTiO$_3$ heterostructures, we find that while the condensates of the (001) and (110) oriented interfaces must be time-reversal symmetric, spontaneous time-reversal-symmetry breaking can only occur for the less studied (111) interface. We also discuss the consequences for thin layers of URu$_2$Si$_2$ and UPt$_3$ as well as for single-layer FeSe. On a more general level, our considerations might serve as a design principle in the search for time-reversal-symmetry-breaking superconductivity in the absence of external magnetic fields.

Two-dimensional (2D) superconducting systems with spin-orbit coupling constitute a central venue for the realization of topologically non-trivial states of matter [1]. A pivotal property of these phases is their behavior under the inversion of the time direction. Not only does it determine the topological classification [2] itself but also essentially influences the electromagnetic and thermal response of these systems [3,4]. Particularly interesting electromagnetic behavior is expected for topological superconductors that break time-reversal symmetry (TRS).
The experimental realization of 2D electron liquids can be divided into two different classes. Firstly, a 2D conducting system can be realized at the interface between two different materials (one of which might be the vacuum). In this case, inversion symmetry or, more generally, any symmetry interchanging the two materials is necessarily broken. The relevant point groups [5] are C n and C nv with n = 1, 2, 3, 4, 6. Secondly, in case of 2D sheets, symmetries relating z and −z with z denoting the coordinate perpendicular to the plane do not have to be broken. Focusing on the non-centrosymmetric systems, this also makes the point groups C s , C 3h , S 4 , D n , D 3h and D 2d possible.
One particularly interesting example of the former class is given by oxide heterostructures, which show very rich electronic behavior [6]. Not only at the [001] (with C 4v symmetry) but also at the [110] (C 2v symmetry) and [111] (C 3v ) LaAlO 3 /SrTiO 3 interface, high mobility 2D electron liquids have been reported [7][8][9]. So far, superconductivity has been observed in the former two orientations of the interface [10,11]. Despite several theoretical studies of these systems [12][13][14][15][16], neither the origin of superconductivity nor its order parameter and the related topological properties have been unambiguously identified. Another exciting 2D system with C 4v symmetry is single-layer FeSe on [001] SrTiO 3 with transition temperatures significantly above 50 K [17].
Here we show that under very general assumptions, TRS-breaking superconductivity can only occur in interfaces and 2D sheets that exhibit a three-fold rotation symmetry as element of the point group of the normal phase. Thus, no TRS-breaking superconductivity is expected for the widely studied LaAlO 3 /SrTiO 3 interfaces with [001] and [110] orientation, while such a state is possible for the less studied [111]-oriented interface [18]. The assumptions under which this theorem is valid are: (i) the superconducting state is reached through a single phase transition that is, in the absence of critical fluctuations, of second order, (ii) the combination of broken inversion symmetry and spin-orbit interaction lifts the degeneracy of the Fermi surfaces in the normal state by energy scales E so that are larger than the superconducting gap ∆ 0 , and (iii) the superconducting phase does not break translation invariance of the kind that occurs, e.g., in Fulde-Ferrell-Larkin-Ovchinnikov [19,20] (FFLO) states. The first assumption can be checked experimentally rather easily and seems to be fulfilled in the oxide interfaces [10,11,21]. The second assumption, E so > ∆ 0 , also does not constitute a problem in the LaAlO 3 /SrTiO 3 interfaces, where the magnitudes of the Rashba spin-orbit interaction are known reasonably well [22][23][24]. As for the third assumption, we stress that it does not seem to be very restrictive either, since, within the microscopic calculation of [15], an FFLO state only emerges for very exotic interaction parameters.
The general design principle for TRS-breaking superconductivity in interfaces and 2D sheets is therefore to look for systems with a three-fold rotation symmetry. In the remainder of this paper we give the proof of this theorem and discuss further physical implications.
In order to decide which superconducting states are symmetry allowed, we consider a 2D system with pairing Hamiltonian already taking into account assumption (iii). The fermionic creation and annihilation operators ψ † k and ψ k are N -component spinors that describe the spin and orbital degrees of freedom as well as potentially relevant subbands that result from the confinement along the direction perpendicular to the 2D plane. As long as these subbands do not significantly reduce the energetic splitting of the Fermi surfaces, they have no consequences for the validity of the design principle. In the case of oxide interfaces the relevant orbitals are the t 2g states of the Ti 3d−shell (3d xy , 3d xz , and 3d yz ). When only one subband is relevant, this yields N = 6. The normal state Hamiltonian h k and the pairing function ∆ k are N × N matrices. Due to the Fermi-Dirac statistics of electrons, the latter satisfies ∆ k = −∆ T −k . We analyze the transformation properties of h k and ∆ k under time-reversal and the elements g of the point group G of the normal state. The time-reversal operator is given by Θ = T K with unitary T and K denoting complex conjugation. Time-reversal acts on the pairing field according to Under a point group operation g ∈ G holds where R v (g) transforms 2D in-plane vectors and R ψ (g) determines the transformation of the spinor: ψ k where R s (g) denotes the representation of g on (threedimensional) pseudovectors, such as the total angular momentum J . All elements of the point group can be constructed from R ψ = exp (−iϕn · J ), where the unit vector n determines the rotation axis.
If our first assumption (i) is fulfilled we are allowed to perform a Ginzburg-Landau expansion with respect to the order parameter. Note, here we are not concerned with the treatment of critical fluctuations that might lead to a transition in the Berezinskii-Kosterlitz-Thouless universality class. Instead we investigate the symmetry of the order parameter whose existence must be settled before its fluctuations can be analyzed. To identify the order parameter, we expand the pairing field with respect to the basis of N × N matrix fields χ n kµ that transform under the n-th irreducible representation of the symmetry group. Here d n is the dimensionality of the irreducible representation n and η n µ are complex valued coefficients. Using the usual orthogonality relations of irreducible representations [5] it follows for the free energy The coefficient a n0 (T ) that first changes sign determines the irreducible representation n = n 0 , i.e. the nature of the symmetry breaking. If the representation n 0 is real, the TRS of the normal state implies that the matrix fields χ n kµ T † can be chosen to be Hermitian and from Eq. (2) follows η n µ Θ −→ η n * µ . Thus, when n 0 is onedimensional and real, TRS cannot be broken as the global phase of the order parameter can always be absorbed by a U (1) gauge transformation. Note that this is different in case of a complex one-dimensional representation, where time-reversed partners transform according to different irreducible representations. Consequently, we need to identify pairing states either in complex or in multidimensional irreducible representations to obtain a TRSbreaking superconductor.
We consider first oxide interfaces with [110] termination. The relevant point group C 2v has only real onedimensional irreducible representations, which immediately excludes TRS-breaking superconducting states. This well-known [25] mechanism of protection against spontaneous TRS-breaking is based on spin-orbit coupling which reduces the symmetry group from C 2v ⊗ SO(3) to C 2v . Note that it is unrelated to the absence of inversion symmetry and also holds when the Fermi surfaces are degenerate (see, e.g., [26,27]).
The situation is more subtle for the [001] interface with C 4v point group. To discuss this case and to make a number of important additional restrictions on the allowed pairing states for C 2v , we analyze the symmetry of pairing matrix elements on the Fermi surface in greater detail. In particular, the broken inversion symmetry will be taken into account.
The free Hamiltonian h k is diagonalized by the unitary transformation ψ ki = a (φ ka ) i f ka that is made of its eigenfunctions φ ka satisfying h k φ ka = ε ka φ ka . Since h k is time-reversal symmetric, i.e. Θh k Θ −1 = h −k , we know that Θφ ka is an eigenstate of h −k with the same energy. The broken inversion symmetry at the interface along with spin-orbit coupling further imply that the Fermi surfaces are non-degenerate in the generic case; a statement that can for each specific system easily be verified if expressed in terms of symmetry allowed spin-orbit couplings of the Rashba or Dresselhaus type. This crucial non-degeneracy of the Fermi surface implies for the wave functions that [28] where the phase factors must satisfy the condition e iϕ a k = ∓e iϕ a −k as a consequence of Θ 2 = ∓½. The Hamiltonian can now be cast in a quadratic form using the Nambu spinor of the eigenmodes f ka and f † −ka : The crucial off-diagonal elements of the Bogoliubov-de Gennes (BdG) Hamiltonian We now make the weak pairing assumption (justified by assumption (ii) above) that implies that partners of a Cooper pair always originate within a given Fermi sheet and not between states of different sheets. If this is the case, it holds Note, this assumption does not exclude frequently discussed pairing states that are due to interband interactions [29]. It merely requires that anomalous averages are made of the same quantum numbers as the normal state. All known superconductors seem to behave like this. In this weak pairing limit, we immediately obtain the eigenvalues of the BdG Hamiltonian (8) as i.e. | ∆ ka | is the superconducting gap on the Fermi surface. Inserting D kab = ∆ ka δ ab into the Hamiltonian (7), shifting k → −k and using the behavior of the phase factors in Eq. (6) under this shift, we obtain the important property Naturally, the upper sign is most relevant for fermionic pairing, yet we include the more general behavior for two reasons: Firstly, it illustrates the importance of normal state TRS for the fact that the gap function ∆ ka = φ ka |∆ k T † |φ ka has a well-defined parity. Secondly, there are situations [30] where fermionic TRS is broken, however, the effective low-energy theory of the system has an emergent TRS that satisfies Θ 2 = ½ (see Supplemental Material for the discussion of the analogous restrictions in this case). For the remainder of the paper, we will focus on the upper signs in Eq. (11). The behavior of ∆ ka under point group operations follows from inserting Eq. (4) and using that the wave functions of non-degenerate Fermi surfaces must transform as We obtain that the basis functions ϕ µn ka = φ ka |χ n kµ T † |φ ka transform under the same, a-independent, irreducible representations as the matrix fields χ n kµ . Thus, once we have found the irreducible representation n 0 under which the pairing field ∆ k transforms, along with the associated order parameter vector η n0 1 , · · · , η n0 dn 0 , we also know the symmetry properties of the gap function as it transforms exactly the same way. Suppose that the point group contains a two-fold rotation with R v = −½, which is only allowed in even space dimensions, since det R v = 1. As ∆ ka has to be an even function of k, no solutions with finite gap can occur that transform non-trivially under this rotation. In case of C 4v , the point group of [001]-oriented oxide interfaces, this is precisely the 2D irreducible representation that is required for TRS-breaking. Thus, we can exclude a finite order parameter for the [001] interface that transforms as k x + ik y or any other superpositions of k x and k y for that matter. At the same time, one can also exclude time-reversal symmetric k 1 or k 2 pairing states for the [110] termination with k 1 and k 2 denoting the crystal momenta within the plane of the interface (along [110] and [001]). The matrix elements of the pairing field on a non-degenerate Fermi surface are too restrictive to allow for any of these pairing states.
Consider now the oxide heterostructure with [111] termination and point group C 3v [18]. In this case R v = −½ is not a point group operation. Indeed, a careful analysis of the symmetry allowed states reveales that TRSbreaking supercondictivity of the kind k 1 k 2 + i 2 k 2 1 − k 2 2 is indeed allowed. Here k 1 and k 2 represent the in-plane momenta along the [110] and [112] directions, respectively. Note that this state is nodeless. For more details on the possible superconducting phases in the oxide interfaces we refer to the Supplemental Material. It is straightforward to generalize this analysis to all possible point groups of non-centrosymmetric 2D electron systems: For analogous reasons to C 4v , TRSbreaking superconductivity is not possible for the interface point group C 4 . The same holds for the isomorphic groups D 4 , D 2d and S 4 that describe possible symmetries of 2D electronic sheets. For all other symmetry groups without any rotation symmetry or containing only a twofold rotation normal to the plane, all irreducible representations are real and one-dimensional such that, exactly as in the case of C 2v discussed above, TRS-breaking superconductivity is forbidden as well. For the remaining possible non-centrosymmetric point groups of 2D electron systems, all of which contain a three-fold rotation, one cannot exclude TRS-breaking superconductivity without further assumptions. This completes the proof of the design principle stated above.
Let us next go beyond the weak pairing limit and clarify under which conditions a (translation invariant) superconducting phase with a vanishing intra-Fermi surface order parameter can occur. Similar to the order parameter in Eq. (4), the effective electron-electron interaction can always be expanded according to the irreducible representations of the point group G. Suppose that, at low energies, it is dominated by the Cooper channel (coupling constant g) transforming under the representation n 0 that is non-trivial under the two-fold rotation C 2 ∈ G. From the analysis above, we know that the matrix ele-ments D kab of the associated order parameter ∆ k with respect to the eigenfunctions of the normal state Hamiltonian vanish for a = b. This type of superconducting order can only occur if the zero-temperature condensation energy is positive for some finite ∆ 0 = g|η n0 |. Here ǫ ka and E ka denote the different bands of the normal state and superconducting mean-field Hamiltonian, respectively. To focus on the essential part of the physics, let us consider only N = 2 singly degenerate bands. Replacing |D k12 | by its maximum value ∆ 0 m yields an upper bound E max c (∆ 0 ) on the condensation energy. Physically, it corresponds to the situation of "optimal basis functions" with |D k12 | being constant except for negligibly small regions where it has to vanish as dictated by symmetry. One finds that the condensation energy can only be positive when the spin-orbit splitting E so on the Fermi surface satisfies E 2 so < 4Λ 2 e −1/λ / sinh(1/λ), where the dimensionless coupling constant is defined by λ = 2ρ F m 2 g. Here Λ and ρ F denote the energetic cutoff of the attractive interaction and the density of states at the Fermi level, respectively. This means that, in the weak coupling limit, λ ≪ 1, superconductivity can only emerge when the spin-orbit coupling is exponentially small. Put differently, in any system, such as the oxide heterostructures, where spin-orbit coupling is comparable to the Fermi energy [22][23][24], superconductivity of this type is a genuine strong coupling phenomenon. Furthermore, one finds that ∆ 0 > E so /2 at the positive maximum of E max c (∆ 0 ) which shows the relation between the weak pairing approximation and assumption (ii). It now becomes obvious that our design principle is not only based on symmetry considerations but also relies on energetic arguments.
Let us finally discuss further physical implications for oxide heterostructures. The design principle derived above indicates that the superconducting states observed in the [001] and [110] interfaces of LaAlO 3 /SrTiO 3 are most likely TRS-preserving states. The condensate then belongs to class DIII [2] and, when fully gapped, can be classified by the 2 index ν = a sign( ∆ k a a ) [31] that distinguishes between topologically trivial (ν = 1) and non-trivial (ν = −1) [1]. Here k a denotes an arbitrary point on the Fermi surface a of the normal phase and the product involves all Fermi surfaces. The symmetry analysis cannot yield information about the topological properties as the relative sign between the ∆ k a a depends on details of the basis functions. Our microscopic calculation performed in [15] shows that the topological properties are rather related to the origin (electronphonon/purely electronic) of the interaction driving the superconducting instability.
Alternatively, the superconductivity observed in the [001] heterostructure could be in the strong coupling regime rendering possible one of the superconducting phases transforming under a 2D representation of C 4v . As applying a magnetic field will break the TRS in the normal phase and alter the structure of the wavefunctions φ ka , the band-diagonal matrix elements ∆ ka of the order parameter will become finite. Therefore, very unusual behavior of the superconducting phase is expected, such as a maximum of the critical temperature T c at a finite strength of an in-plane magnetic field. We emphasize that the suppression of such a strong coupling superconducting state with increasing spin-orbit splitting disagrees with the observed gate-voltage dependence of T c and E so [22,23,32].
So far we have assumed that the normal state of the system is time-reversal symmetric. At first sight, this seems to be inconsistent with the observation of magnetic order in [001] LaAlO 3 /SrTiO 3 heterostructures [33][34][35]. However, one has to recall that the magnetism is predominantly due to the 3d xy orbitals that are closest to the interface and localized [36][37][38][39]. As indicated by experiment [40,41], superconductivity is mostly associated with the itinerant bands derived from the 3d xz and 3d yz orbitals. The magnetic order can only affect the analysis presented above if the resulting effective Zeeman splitting E Z in these bands is comparable to or larger than the spin-orbit splitting E so on the Fermi surface. As discussed in [38], the opposite limit E Z ≪ E so seems to be realized as a consequence of the small orbital admixture of 3d xy in the bands relevant for superconductivity. This justifies the application of our design principle.
Furthermore, it is additionally confirmed by the experimental observation [34] that there is no significant spatial correlation between the isolated ferromagnetic patches and the superconducting order parameter. Irrespective of how small E Z is, the absence of correlation also seems not to be consistent with an FFLO state [38,[42][43][44] that results from the interplay of the spin-orbit coupling and the exchange coupling to the ferromagnetic moments.
In conclusion, we have shown that, in 2D, the lack of degeneracy of Fermi surfaces leads to severe symmetry restrictions for spontaneous TRS-breaking at a superconducting phase transition. In case of oxide interfaces, only the [111] interface is a natural candidate system for a superconductor that breaks TRS in the absence of an external magnetic field.
We thank E. J. König for discussions. .

The [001] interface
As mentioned in the main part, the relevant point group is C 4v in this case. The resulting superconducting phases transforming under the associated irreducible representations are shown in Table I. To find the three allowed order parameter vectors (η E 1 , η E 2 ) in case of the 2D representation E, one has to consider higher order terms in the Ginzburg-Landau expansion (5).
In Table I, X(k) and Y (k) are continuous real valued scalar basis functions transforming as k x and k y under C 4v , e.g. X(k) = sin(k x ) and Y (k) = sin(k y ). We list both the weak pairing gap function ∆ ka as well as the microscopic order parameter ∆ k . For the latter, it is assumed that the orbitals relevant for superconductivity are the Ti 3d xz and 3d yz states as explained in the main text. We use σ j and τ j to denote Pauli matrices in spin and orbital space, respectively. Note that the generalization to a three-orbital order parameter that takes into account the 3d xy band is straightforward. Although the system is not inversion symmetric, we split the order parameter into a spin-trivial (∆ s k ) and a spin-nontrivial (∆ t k ) term according to where From the symmetry of the gap function ∆ ka one readily obtains the (minimal) number of nodes assuming that all Fermi surfaces enclose the Γ point and do not cross the boundaries of the Brillouin zone.
As dictated by symmetry, all order parameters transforming under the 2D irreducible representation have vanishing intra-Fermi surface matrix elements ∆ ka . Using the microscopic expressions for ∆ k in the two-orbital model as given in Table I, this can also be verified explicitly.
TABLE I. Summary of the possible superconducting phases of the [001] oxide interface as derived from the C4v point symmetry of the high-temperature phase. We show both the microscopic two-orbital order parameter ∆ k as well as the superconducting gap function ∆ ka . It is indicated whether the superconductor preserves (y) or breaks (n) TRS and the number of nodes on a given Fermi surface is shown. Ungapped means that D kab has vanishing diagonal elements and, at weak coupling, there is no stable superconducting state of this type.

The [110] interface
In this case, the analysis is simpler as the point group C 2v only admits one-dimensional irreducible representations. The results are summarized in Table II. Here we only show the order parameter in the weak pairing description which is independent of the details of the microscopic model of the normal phase. In obvious analogy to the notation introduced above, X 1,2 (k) are continuous real valued scalar basis functions transforming as k 1,2 under C 2v . Here k 1 and k 2 denote in-plane momenta along the [110] and [001] directions, respectively.
In this case, the symmetries allow only for two distinct superconducting phases with finite intra-Fermi surface order parameters. For this orientation, the relevant point group is C 3v leading to the possible superconducting phases listed in Table III. Here the basis functions X 1 (k) and X 2 (k) refer to the transformation properties of the in-plane crystal momenta along the [110] and [112] directions, respectively.
In accordance with the design principle of the main text, only for this orientation of the interface a TRS-breaking superconducting state with a non-vanishing intra-Fermi surface order parameter is possible.
We see that even when the point group does not contain a two-fold rotation, the non-degeneracy of the Fermi surfaces can influence the symmetry properties of the superconducting phases: In case of A 2 , the leading basis function transforms as X 1 (3X 2 2 − X 2 1 ) which, however, has odd parity. Consequently, it cannot open up a gap in the weak pairing limit. As C 3v does not contain an operation with (X 1 , X 2 ) → (−X 1 , −X 2 ), the basis function of a given representation can have distinct parity. The leading contribution to ∆ ka thus reads X 1 X 2 (3X 2 1 − X 2 2 )(3X 2 2 − X 2 1 ) yielding 12 (and not 6) as the minimal number of nodes per Fermi surface.

Spinless fermions
For completeness we also discuss the scenario of a TRS satisfying Θ 2 = ½. Physically, this corresponds to spinless fermions. In the following we will first analyze the general implications for possible pairing states and then illustrate how a TRS of this form can emerge in an effective low-energy theory.

Restrictions on possible pairing states
As compared to the design principle derived in the main part, there are only two crucial differences for spinless fermions: Firstly, the weak pairing gap function ∆ ka has to be odd under k → −k as stated in Eq. (11). Secondly, Θ 2 = ½ allows for non-degenerate Fermi surfaces even when inversion symmetry is not broken. This means that the weak pairing assumption is also meaningful in case of centrosymmetric systems. In particular, one can immediately conclude that the following statements hold in the weak pairing limit: 1. In case of the centrosymmetric point groups (C i , C 2h , C 4h , C 6h , S 6 , D 2h , D 4h , D 6h , D 3d , T h , O h ), the order parameter must necessarily transform according to one of the ungerade representations. Note that this constitutes a strong restriction as it rules out half of the possible superconducting states. Furthermore, we emphasize that this statement holds in arbitrary spatial dimension as opposed to the other restrictions resulting from the non-degeneracy of the Fermi surfaces.
2. For the non-centrosymmetric point groups of both interfaces and 2D sheets that contain a two-fold rotation (C 2 , C 4 , C 6 , C 2v , C 4v , C 6v and S 4 , D 2 , D 4 , D 6 , D 2d ), the order parameter must transform according to a representation that is non-trivial in this rotation.
3. There are much less restrictions for obtaining a superconducting state that spontaneously breaks the effective TRS. Of course, as before, when all representations (e.g. in case of C 2v ) are one-dimensional and real, TRS cannot be broken spontaneously. However, when the symmetry group has either complex or multi-dimensional representations (e.g. in case of C 4 or C 4v ) we cannot exclude TRS-breaking. 4. In case of the point groups C 4 and S 4 , TRS must necessarily be broken as the only representations which are odd under the two-fold rotation are complex.

Strong in-plane magnetic field
Next we will discuss an example of how an emergent TRS satisfying Θ 2 = ½ can be realized. Consider a system of spin-1/2 fermions with no spin-orbit coupling and physical TRS described by Θ 1/2 = ½ o ⊗ iσ 2 K. Here ½ o is the unit matrix in orbital space and σ j refers to spin. The Hamiltonian then has the form h o (k) ⊗ σ 0 . Suppose that h o (k) is invariant under space symmetries forming the group G o . Due to the lack of spin-orbit coupling, the full symmetry group is G = G o ⊗ SO(3). Applying an in-plane magnetic field, say along the x direction, leads to an additional Zeeman term. The resulting Hamiltonian now only has G B = G o ⊗ SO(2) symmetry. The physical spin-1/2 TRS Θ 1/2 is broken, however, it still holds Θh(−k) Θ −1 = h(k), For the analysis of the symmetry properties of ∆ ka , it is essential that the representation R Ψ of the point group can be chosen to commute with Θ. More physically, it means that the associated generators, i.e. the angular momenta of the theory, have to anticommute with the time-reversal operator Θ.
For the SO(2)-part of G B this is not the case as Θ does not anticommute with σ 1 . Let us thus assume that the Zeeman energy E Z is larger than the bandwidth of h o (k) and, hence, we can project onto one of the two Zeeman-split multiplets to obtain an effective low-energy theory. Without loss of generality, let us choose the multiplet that is energetically lower. The effective Hamiltonian then reads which now only has the point symmetry group G o . All point symmetry operations in the effective theory are generated by the orbital angular momentum l satisfying {Θ, l} = 0 with Θ = ½ o K. It is therefore always possible to choose the representation R Ψ eff of the symmetry group in the effective theory to commute with Θ.
We have thus managed to construct a physical example where a TRS with Θ 2 = ½ emerges and the restrictions on possible pairing states derived above can be applied.