Valley-Singlet Superconducting Phase

• Valley-singlet superconductivity is a state where Cooper pairs are antisymmetric in the valley degree of freedom, enabling electrons to pair from opposite valleys.
• The phase emerges from complex interactions among spin, valley, and orbital sectors, with differing characterizations in two-valley and single-valley systems.
• Insights into valley-singlet pairing offer pathways to understand novel topological states and unconventional superconducting mechanisms in moiré and multilayer graphene.

A valley-singlet superconducting phase is a superconducting state in which the Cooper-pair wavefunction is antisymmetric in the valley sector. In two-valley systems this usually means pairing electrons from opposite valleys, but the spin assignment depends on the symmetry convention and the microscopic low-energy basis: some works identify the canonical valley-singlet state as spin-triplet and opposite-valley paired, whereas others emphasize an inter-valley spin-singlet member of a larger spin-valley multiplet selected by weak symmetry breaking. In single-valley systems the phrase is only analogical, because there is no independent valley quantum number and the antisymmetry is instead carried by orbital or pseudospin structure (Zhou et al., 2022, You et al., 2018, Pawlak et al., 2014).

1. Symmetry definition and exchange structure

In multivalley systems, the defining issue is how Fermi antisymmetry is distributed among spin, valley, and spatial sectors. A systematic statement is given in the graphene-multilayer analysis of soft-mode-mediated pairing: for pairing between electrons in opposite valleys, a valley-symmetric gap is spin-singlet, while a valley-antisymmetric gap is spin-triplet. In that convention, spin-singlet pairing is equivalent to valley-triplet, and spin-triplet pairing is equivalent to valley-singlet. The same paper identifies the valley-singlet superconducting phase near inter-valley ferromagnetism $\mathrm{FM}^+$ as the spin-triplet, valley-singlet channel selected by the singular spin fluctuation vertex (Dong et al., 2023).

A different but closely related convention appears in weak-coupling twisted bilayer graphene. There the dominant superconducting order parameter is written as

$$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$

with $\mu=0$ the spin-singlet component and $\mu=1,2,3$ the spin-triplet components. The pairing is explicitly inter-valley, and the paper treats the spin-singlet state as the experimentally relevant candidate once an anti-Hunds coupling selects it from an approximate $SO(4)\sim SU(2)_K\times SU(2)_{K'}$ multiplet. This usage makes “valley-singlet” implicit rather than a separate formal phase label, because the essential structure is opposite-valley pairing combined with a specific spin choice (You et al., 2018).

This suggests that the term does not denote a single universal irreducible representation across the literature. Instead, it labels a family of opposite-valley superconducting states whose detailed classification depends on whether the low-energy theory keeps valley, spin, and orbital indices separate, or reorganizes them into a spin-valley-locked basis.

2. Single-valley analogue in quadratic band-crossing systems

The most explicit nonliteral use of the idea occurs in the doped single-valley quadratic band-crossing problem. The low-energy theory is built from a two-component orbital or pseudospin spinor $\psi_{k\alpha}$ with real spin $\alpha=\uparrow,\downarrow$, and the interaction is written in the most general marginal symmetry-allowed form. At half filling, the weak-coupling instability is toward particle-hole order. Upon doping, however, the renormalization-group balance changes: the chemical potential scales as $\mu\to\mu e^{2\ell}$, particle-hole susceptibilities saturate near the Fermi level, and particle-particle channels continue to grow. The resulting superconductivity therefore emerges from a repulsive starting interaction because doping cuts off the tendency toward QAH, QSH, or nematic order and allows the Cooper channel to dominate (Pawlak et al., 2014).

The pairing operators are exposed by a Fierz rearrangement,

$S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$

together with the triplet operator $\vec T_2=\psi^T\sigma_2\vec s\,\psi$. In this setting the “singlet” label refers to spin-orbital structure, not to a true valley antisymmetry. That distinction is essential because the model is explicitly single-valley: the quadratic band touching is symmetry protected, there is no $$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$0 valley pair, and the superconducting order is intra-valley in the only valley present.

The lattice symmetry and interaction range determine the favored gap structure. In the $$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$1 checkerboard-like case, short-ranged Hubbard interactions drive a $$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$2-wave instability, with the dominant channel typically $$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$3, whereas longer-ranged forward scattering drives fully gapped $$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$4-wave superconductivity. In the $$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$5 kagome-like case, the superconducting window is much narrower; the leading superconducting state is either $$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$6-wave or absent altogether, especially near the nearly flat-band regime where the flow can terminate before any attractive Cooper channel develops (Pawlak et al., 2014).

3. Inter-valley superconductivity in moiré graphene

In weak-coupling twisted bilayer graphene, valley-singlet superconductivity is tied to nested opposite-valley Fermi pockets near half filling. Slightly away from the magic angle, the low-energy dispersion near the $$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$7 pocket is written as

$$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$8

and the triangular distortion $$\Delta_k^\mu = c_{Kk}^{\mathsf T} i\sigma^2 s^\mu c_{K',-k},$$9 makes the $\mu=0$0 and $\mu=0$1 Fermi surfaces nearly nested. Within RPA, the dominant fluctuation is the inter-valley coherence channel $\mu=0$2, whose susceptibility peaks at the three nesting vectors $\mu=0$3. Fierz rearrangement of the renormalized interaction yields an attractive pairing kernel in the inter-valley channel, and the leading gap has mixed chiral form

$\mu=0$4

Because the Fermi surface has only $\mu=0$5 symmetry, $\mu=0$6 and $\mu=0$7 are not symmetry-distinct; the dominant state is therefore a mixed chiral inter-valley superconductor rather than a pure $\mu=0$8-wave or pure $\mu=0$9-wave state. In the spin-singlet sector, only the chiral solution is allowed, and an anti-Hunds inter-valley Heisenberg coupling favors that spin-singlet inter-valley state over the nearly degenerate triplet partners (You et al., 2018).

A different route appears in the minimal Hubbard description of twisted multilayer graphene. There the two valleys act as two orbital flavors on a triangular moiré lattice, producing an approximate SU(4) symmetry in the flat mini-band. In the Mott regime, the superexchange interaction can be Fierz-decomposed into an even-parity sector and an odd-parity sector, with the even-parity channel favored. The six-component even-parity pairing field separates into a spin-triplet, orbital-singlet sector and a spin-singlet, orbital-triplet sector. Hund’s coupling lowers the energy of the spin-triplet, orbital-singlet intermediate state, thereby selecting spin-triplet, valley-singlet pairing. On the triangular lattice, the corresponding doped state is a $\mu=1,2,3$0 superconductor with valley-singlet structure (Xu et al., 2018).

These two moiré-graphene constructions therefore realize opposite sides of the same symmetry problem. Twisted bilayer graphene emphasizes an inter-valley state whose spin-singlet component is selected from an approximate $\mu=1,2,3$1 multiplet, whereas twisted multilayer graphene emphasizes that the extra valley degree of freedom can make even-parity spin-triplet pairing possible by rendering the valley sector antisymmetric.

4. Soft modes, ordered phases, and spin-valley locking

A systematic extension of the paramagnon mechanism to two-valley graphene systems identifies valley-singlet superconductivity as a fluctuation-mediated instability near itinerant ordered phases. The model contains intravalley interaction $\mu=1,2,3$2, intervalley density interaction $\mu=1,2,3$3, and intervalley exchange or scattering $\mu=1,2,3$4. Near inter-valley ferromagnetism $\mu=1,2,3$5, defined by

$\mu=1,2,3$6

the singular pairing vertex is purely spin-like and becomes attractive only in the spin-triplet channel, which by exchange antisymmetry is the valley-singlet channel. Near inter-valley antiferromagnetism $\mu=1,2,3$7, defined by $\mu=1,2,3$8, the sign reverses and the attractive channel is spin-singlet, valley-triplet. The enhancement of $\mu=1,2,3$9 near criticality is a central conclusion, and after summing the singular series to all orders in $SO(4)\sim SU(2)_K\times SU(2)_{K'}$0, the dimensionless pairing coupling does not contain a $SO(4)\sim SU(2)_K\times SU(2)_{K'}$1 factor in the exponent (Dong et al., 2023).

Twisted trilayer graphene provides a distinct mechanism in which valley-singlet superconductivity emerges only after the normal state has already broken valley symmetry in a spin-selective way. In self-consistent Hartree-Fock, the valley-breaking order parameter satisfies

$SO(4)\sim SU(2)_K\times SU(2)_{K'}$2

so the two spin projections are attached to opposite valleys. This spin-valley locking forces a singlet Cooper pair to occupy different valleys and different Fermi lines, yielding what the paper interprets as a valley-singlet, spin-singlet Ising superconductor. The same spin-selective valley symmetry breaking generates an effective intrinsic spin-orbit coupling through an imaginary next-nearest-neighbor hopping, with an energy scale $SO(4)\sim SU(2)_K\times SU(2)_{K'}$3 and an estimated in-plane critical field $SO(4)\sim SU(2)_K\times SU(2)_{K'}$4. The symmetry reduction from $SO(4)\sim SU(2)_K\times SU(2)_{K'}$5 to $SO(4)\sim SU(2)_K\times SU(2)_{K'}$6 increases Fermi-line anisotropy and enables a Kohn-Luttinger instability; the paper estimates $SO(4)\sim SU(2)_K\times SU(2)_{K'}$7 at $SO(4)\sim SU(2)_K\times SU(2)_{K'}$8, still around $SO(4)\sim SU(2)_K\times SU(2)_{K'}$9 but weaker at $\psi_{k\alpha}$0, and only $\psi_{k\alpha}$1 near $\psi_{k\alpha}$2 (Gonzalez et al., 2021).

Taken together, these works place valley-singlet superconductivity near soft collective modes of spin and valley order, rather than treating it only as a weak-coupling band-structure instability.

5. Topological, finite-momentum, and large-angle manifestations

The spin-triplet valley-singlet state can be topologically trivial in one setting and non-Abelian in another. In isolated layers of the maximally twisted double-layer construction, the basic state is a fully gapped $\psi_{k\alpha}$3-wave spin-triplet valley-singlet superconductor built from same-spin electrons in opposite valleys. Its gap function changes sign between valleys but avoids nodal intersections with the disconnected Fermi pockets, so the isolated-layer state is fully gapped and nevertheless topologically trivial because it belongs to class BDI in two dimensions. Near the maximal twist angle of $\psi_{k\alpha}$4, however, the moiré reconstruction violates simple valley conservation, the layer-resolved $\psi_{k\alpha}$5-wave components combine with relative phase $\psi_{k\alpha}$6, and a chiral $\psi_{k\alpha}$7 state emerges. When the reconstructed pockets coalesce into a single connected Fermi surface, the BdG problem is in class D with bulk Chern number $\psi_{k\alpha}$8, yielding three chiral Majorana edge modes, a single Majorana zero mode in the vortex core, and a $\psi_{k\alpha}$9-periodic inter-layer Josephson effect (Zhou et al., 2022).

In twisted multilayer graphene, the two mean-field valley-singlet $\alpha=\uparrow,\downarrow$0 states are already topological. One breaks time-reversal symmetry and supports gapless chiral edge modes; the other preserves time reversal and supports counterpropagating Majorana edge modes protected by symmetry. Both admit half-vortices carrying flux $\alpha=\uparrow,\downarrow$1, because a $\alpha=\uparrow,\downarrow$2 winding of the superconducting phase can be compensated by the internal order-parameter structure. At nonzero temperature in strictly two dimensions, the spin-vector part of the order cannot maintain true long-range order, but an algebraic charge-$\alpha=\uparrow,\downarrow$3 order parameter survives, and the Kosterlitz-Thouless transition is governed by half-vortex unbinding with universal stiffness jump $\alpha=\uparrow,\downarrow$4 (Xu et al., 2018).

Finite-momentum and nonstandard intervalley condensates broaden the landscape further. In a dilute two-valley triangular-lattice electron fluid, spontaneous valley polarization can favor singlet superconducting tendencies that are not standard zero-momentum inter-valley states: at zero field the leading instability can be an intra-valley singlet pair-density wave with $\alpha=\uparrow,\downarrow$5, and at high in-plane field a distinct reentrant singlet superconducting regime can appear because the field reshapes the flavor occupancies and restores an almost nested pairing condition. The same work argues for a reduced magnetic response in the valley-polarized state and therefore a finite violation of the Pauli limit (Han et al., 2021). In hybrid exciton/superconductor systems, a different intervalley condensate appears: a finite-energy Cooper-pair condensate composed of conduction- and valence-band electrons from different valleys, with both even- and odd-frequency components. That state is explicitly intervalley but is not formally classified as valley-singlet or valley-triplet in the paper (Kornich, 2024).

6. Terminological boundaries and related but distinct states

Several nearby superconducting phases should not be conflated with valley-singlet superconductivity. In the Kane-Mele honeycomb system with attractive interactions, nearest-neighbor attraction drives an intra-valley spin-triplet, valley-triplet $\alpha=\uparrow,\downarrow$6 state whose order parameter condenses at $\alpha=\uparrow,\downarrow$7 and $\alpha=\uparrow,\downarrow$8, carries finite center-of-mass momentum, and forms a helical pair-density-wave with a $\alpha=\uparrow,\downarrow$9-Kekulé pattern. The conventional inter-valley spin-singlet $\mu\to\mu e^{2\ell}$0-wave state appears for on-site attraction, but it is not the exotic phase emphasized in that work (Tsuchiya et al., 2016).

Likewise, the triangular-lattice $\mu\to\mu e^{2\ell}$1-$\mu\to\mu e^{2\ell}$2 model with valley-contrasting flux does not isolate a separate valley-singlet superconducting phase. Turning on flux $\mu\to\mu e^{2\ell}$3 breaks SU(2) spin symmetry down to U(1) and produces a mixed state in which the familiar chiral singlet $\mu\to\mu e^{2\ell}$4 component is admixed with a spin-triplet $\mu\to\mu e^{2\ell}$5 component; the resulting superconductor undergoes topological transitions between phases with $\mu\to\mu e^{2\ell}$6 and $\mu\to\mu e^{2\ell}$7, and a small $\mu\to\mu e^{2\ell}$8-breaking anisotropy can instead stabilize a nodal nematic state (Zhou et al., 2022). A closely related conclusion holds in the moiré $\mu\to\mu e^{2\ell}$9-$S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$0-$S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$1 model for twisted WSe$S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$2 bilayer: the stable paired state is a mixed $S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$3 singlet and $S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$4 triplet superconductor with two domes around a Mott-like state, and gate tuning can drive a transition between pure singlet and pure triplet, but the paper does not define a separate valley-singlet phase (Zegrodnik et al., 2023).

Other works are relevant mainly as constraints or contrasts. The spontaneous-vortex-lattice mechanism due to orbital magnetization is formulated for valley-polarized superconductors with broken time-reversal symmetry and does not derive an analogous instability for a valley-singlet condensate (Jahin et al., 28 May 2025). The spin-ARPES study of $S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$5-NbSe$S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$6 establishes a normal state with strong spin-valley locking and hidden layer-dependent spin polarization, thereby constraining any future valley-singlet pairing proposal without directly identifying one (Bawden et al., 2016). The single-band study of mixed $S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$7- and $S_j=\psi^T\sigma_j s_2\psi,\qquad j=0,1,3,$8-wave singlet superconductivity on the square lattice is only indirectly relevant, because it contains no valley degree of freedom at all (Timirgazin et al., 2018).

A persistent misconception is therefore that “valley-singlet” is synonymous with either “inter-valley,” “spin-singlet,” or “unconventional.” The literature shows all three identifications can fail. Inter-valley pairing may be spin-singlet or spin-triplet; single-valley systems may realize only an analogue of the concept; and several unconventional superconductors with pronounced valley physics are mixed-parity, valley-triplet, or valley-polarized rather than valley-singlet. The most precise usage reserves the term for states in which the exchange antisymmetry of the Cooper pair is carried by the valley sector itself.