---
title: Valley-Singlet Superconducting Phase
url: https://www.emergentmind.com/topics/valley-singlet-superconducting-phase
type: topic
---

# Valley-Singlet Superconducting Phase

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 [2206.05599, 1805.06867, 1411.3633].

## 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 [2306.11005].

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 [1805.06867].

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 [1411.3633].

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 \(\pm K\) 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 \(C_{4v}\) checkerboard-like case, short-ranged Hubbard interactions drive a \(d\)-wave instability, with the dominant channel typically \(d_{xy}\), whereas longer-ranged forward scattering drives fully gapped \(s\)-wave superconductivity. In the \(C_{6v}\) kagome-like case, the superconducting window is much narrower; the leading superconducting state is either \(s\)-wave or absent altogether, especially near the nearly flat-band regime where the flow can terminate before any attractive Cooper channel develops [1411.3633].

## 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 \(\Gamma_M\) pocket is written as
\[
\epsilon_k = k^2-\mu+\alpha \,\mathrm{Re}\,k_+^3,
\]
and the triangular distortion \(\alpha\) makes the \(K\) and \(K'\) Fermi surfaces nearly nested. Within RPA, the dominant fluctuation is the inter-valley coherence channel \(I_q^\mu\), whose susceptibility peaks at the three nesting vectors \(Q_{1,2,3}\). Fierz rearrangement of the renormalized interaction yields an attractive pairing kernel in the inter-valley channel, and the leading gap has mixed chiral form
\[
\Delta_k^\mu = u^\mu w_k + v^\mu w_k^*, \qquad
w_k = w_d k_+^2 + w_p k_-.
\]
Because the Fermi surface has only \(C_3\) symmetry, \(d+id\) and \(p-ip\) are not symmetry-distinct; the dominant state is therefore a mixed chiral inter-valley superconductor rather than a pure \(d\)-wave or pure \(p\)-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 [1805.06867].

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 \(d+id\) superconductor with valley-singlet structure [1803.08057].

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 \(SO(4)\) 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 \(U_1\), intervalley density interaction \(U_2\), and intervalley exchange or scattering \(U_3\). Near inter-valley ferromagnetism \( \mathrm{FM}^+ \), defined by
\[
(U_1 + U_3)\Pi(0)=1,
\]
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 \( \mathrm{FM}^- \), defined by \((U_1-U_3)\Pi(0)=1\), the sign reverses and the attractive channel is spin-singlet, valley-triplet. The enhancement of \(T_c\) near criticality is a central conclusion, and after summing the singular series to all orders in \(U_3\), the dimensionless pairing coupling does not contain a \(1/U_3\) factor in the exponent [2306.11005].

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
\[
P_-^{(\uparrow)}=-P_-^{(\downarrow)},
\]
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 \(\Delta\sim 1~\mathrm{meV}\) and an estimated in-plane critical field \(B_c \approx \Delta/(2\mu_B)\sim 8~\mathrm{T}\). The symmetry reduction from \(C_6\) to \(C_3\) increases Fermi-line anisotropy and enables a Kohn-Luttinger instability; the paper estimates \(T_c\sim 1~\mathrm{K}\) at \(\nu=-2.4\), still around \(\sim 1~\mathrm{K}\) but weaker at \(\nu=-2.8\), and only \(\sim 0.1~\mathrm{K}\) near \(\nu=-3.6\) [2110.11294].

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 \(f\)-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 \(30^\circ\), however, the moiré reconstruction violates simple valley conservation, the layer-resolved \(f\)-wave components combine with relative phase \(\pm\pi/2\), and a chiral \(f\pm if'\) state emerges. When the reconstructed pockets coalesce into a single connected Fermi surface, the BdG problem is in class D with bulk Chern number \(C=\pm 3\), yielding three chiral Majorana edge modes, a single Majorana zero mode in the vortex core, and a \(\pi\)-periodic inter-layer Josephson effect [2206.05599].

In twisted multilayer graphene, the two mean-field valley-singlet \(d+id\) 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 \(hc/(4e)\), because a \(\pi\) 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-\(4e\) order parameter survives, and the Kosterlitz-Thouless transition is governed by half-vortex unbinding with universal stiffness jump \(8T_c/\pi\) [1803.08057].

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 \(\vec q=\pm 2\vec K\), 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 [2112.03278]. 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 [2402.02747].

## 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 \(A_1\) state whose order parameter condenses at \(K\) and \(K'\), carries finite center-of-mass momentum, and forms a helical pair-density-wave with a \(p\)-Kekulé pattern. The conventional inter-valley spin-singlet \(s\)-wave state appears for on-site attraction, but it is not the exotic phase emphasized in that work [1606.00986].

Likewise, the triangular-lattice \(t\)-\(J\) model with valley-contrasting flux does not isolate a separate valley-singlet superconducting phase. Turning on flux \(\Phi\) breaks SU(2) spin symmetry down to U(1) and produces a mixed state in which the familiar chiral singlet \(d-id\) component is admixed with a spin-triplet \(p+ip\) component; the resulting superconductor undergoes topological transitions between phases with \(|C|=2\) and \(|C|=1\), and a small \(C_3\)-breaking anisotropy can instead stabilize a nodal nematic state [2209.10023]. A closely related conclusion holds in the moiré \(t\)-\(J\)-\(U\) model for twisted WSe\(_2\) bilayer: the stable paired state is a mixed \(d+id\) singlet and \(p-ip\) 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 [2302.13841].

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 [2505.22915]. The spin-ARPES study of \(2H\)-NbSe\(_2\) 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 [1603.05207]. The single-band study of mixed \(s\)- and \(d\)-wave singlet superconductivity on the square lattice is only indirectly relevant, because it contains no valley degree of freedom at all [1804.09513].

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.

Source: https://www.emergentmind.com/topics/valley-singlet-superconducting-phase