Internally Antisymmetric Nonunitary Triplet (INT) State

• The INT state is a superconducting phase featuring spin-triplet pairing, even spatial parity, and internal orbital antisymmetry, resulting in a nodeless, fully gapped spectrum.
• Experimental findings in materials like LaNiGa₂ and YbSb₂—such as double-peak spin-resolved DOS and spontaneous internal fields—support the presence of TRSB and nonunitary characteristics.
• Theoretical models employ two-band BCS-like Hamiltonians with dominant interband coupling to capture nonunitary behavior and predict topological features including Majorana surface modes.

An internally antisymmetric nonunitary triplet (INT) state is a superconducting pairing state characterized by spin-triplet symmetry, even spatial parity, and antisymmetry under exchange of orbital or band indices. This exotic superconducting order is enforced by the electronic structure and symmetry constraints of certain low-symmetry, multiband materials—most notably LaNiGa₂ and, more recently, YbSb₂. The INT state is distinguished by its capacity to simultaneously realize a fully gapped spectrum, time-reversal symmetry breaking (TRSB), and multi-orbital entanglement, producing a rich set of thermodynamic, magnetic, and topological phenomena.

1. Formal Definition and Order Parameter Structure

The INT state arises from the antisymmetry requirements for the Cooper pair wave function in multi-orbital systems. For fermions, the pairing amplitude must be antisymmetric under exchange of all electronic quantum numbers (momentum $\mathbf{k}$, spin $\sigma$, and orbital/band $m$). For the INT state, the key aspects are:

• Spin configuration: triplet, i.e., symmetric under spin exchange.
• Spatial parity: even, typically s-wave-like and nodeless.
• Orbital structure: antisymmetric under exchange of different orbitals (e.g., $m \ne m'$ for orbitals $m$ and $m'$).

This is embodied in the superconducting gap function: $$\hat{\Delta}(\mathbf{k}) = i(\vec{\sigma} \cdot \vec{d}(\mathbf{k})) \sigma_y \otimes (i \tau_y)$$ where $\sigma_i$ act in spin space, $\tau_i$ in orbital space, and $\vec{d}(\mathbf{k})$ is the triplet $d$-vector. The factor $i\tau_y$ ensures $\hat{\Delta}_{mm'} = -\hat{\Delta}_{m'm}$, thus enforcing internal (orbital) antisymmetry. This internal antisymmetry mechanism allows a fully gapped, spin-triplet state even in centrosymmetric, single-site, two-orbital systems, circumventing restrictions that normally couple time-reversal symmetry breaking to odd-parity, nodal order (Ghosh et al., 2019).

2. Symmetry Considerations and Microscopic Origin

The INT state is dictated by the full symmetry group of the normal state. In orthorhombic, centrosymmetric materials with a two-orbital basis per site and D₂h point group (e.g., LaNiGa₂, YbSb₂), all irreducible representations are one-dimensional. This prohibits multi-dimensional order parameters associated with conventional spin–orbit coupled triplet states and instead selects the INT configuration as the only route to TRSB without introducing nodal quasiparticles.

Orbital antisymmetry is facilitated by nonsymmorphic symmetry elements (such as glide planes and screw axes) that generate band degeneracies and allow two distinct orbitals—e.g., Ni-$d_{z^2}$ and Ni-$d_{xy}$—to form antisymmetric combinations. The pairing interaction is assumed to be dominant in the inter-orbital, equal-spin channel, with the mean-field order parameter defined as: $$\Delta_{L\sigma,L'\sigma}(r) = U_{L,L'} \langle c_{r,L',\sigma} c_{r,L,\sigma} \rangle$$ where $U_{L,L'}$ is the on-site interaction between orbitals $L$ and $L'$ (Ghosh et al., 2019, Sundar et al., 2023). The resulting Bogoliubov–de Gennes formalism features strong interband coupling and produces two distinct gaps associated with different spin projections and orbital hybridizations.

3. Nonunitarity and Time-Reversal Symmetry Breaking

The hallmark of the INT state is nonunitarity, i.e., a triplet $d$-vector that is noncollinear with its complex conjugate: $\vec{q} = i \vec{d} \times \vec{d}^* \neq 0$ This implies

$$\hat{\Delta} \hat{\Delta}^\dagger = |\vec{d}|^2 \mathbb{1} + i (\vec{d} \times \vec{d}^*) \cdot \vec{\sigma}$$

resulting in unequal gaps for spin-up and spin-down condensates. Nonunitarity directly leads to spontaneous magnetization in the superconducting state: $\vec{\mu}_s(T) = \int d^3r\, [\vec{q}(r)]/2$ with an associated internal field measurable by muon spin relaxation ($\mu$SR) and compatible with observed TRS breaking fields ($B_{\mathrm{int}} \sim 0.3$–$0.4$ G in LaNiGa₂, YbSb₂) (Ghosh et al., 2019, Kataria et al., 12 Jan 2026). Importantly, nonunitarity is both a necessary and sufficient condition for TRSB in single-component triplet systems with only one-dimensional representations.

4. Experimental Signatures and Phenomenology

The INT state produces sharply defined experimental features that distinguish it from conventional singlet or unitary triplet order:

Signature	Origin	Observable
Two-peak spin-resolved DOS	Spin splitting due to $q \neq 0$	Spin-ARPES, STM
Fully gapped specific heat	Nodeless, s-wave-like pairing	$C_e(T)$ measurements
Spontaneous internal field	Spin moment from $\vec{q} \neq 0$	Zero-field $\mu$SR
Strong interband coupling	Orbital-antisymmetric, dominant $\lambda_{12}$	Superfluid density, vortex core size

The double-peak density of states, with each peak associated purely with one spin species ($\uparrow$ or $\downarrow$), is a "smoking-gun" diagnostic accessible to spin-resolved tunneling or planar-junction spectroscopy (Ghosh et al., 2019). Specific heat, penetration depth, vortex core size, and $\mu$SR data on LaNiGa₂ all support a two-band, strongly-interband-coupled, fully-gapped scenario, concordant with INT theory (Sundar et al., 2023). YbSb₂ provides a similar signature, combining type-I superconductivity with TRSB and indications of topological Majorana surface modes (Kataria et al., 12 Jan 2026).

5. Theoretical Modeling and Topological Properties

Microscopically, the INT state may be described by a two-band BCS-like Hamiltonian with dominant interband interaction. The mean-field self-consistency equation fixes the coupling constant to experimental $T_c$ values, enabling parameter-free predictions for gap magnitudes and thermodynamic responses.

In the case of YbSb₂, the INT state emerges from a low-energy Dirac-metal Hamiltonian, with the superconducting order parameter: $$\hat{\Delta} = (i \tau_y) \otimes [\vec{d} \cdot \vec{\sigma}] (i \sigma_y)$$ and bulk topology inherited from the strong-${\mathbb Z}_2$ Dirac metal normal state. The system belongs to class DIII, supporting a nontrivial three-dimensional winding number ($\nu = 1$), and hosts gapless Majorana surface modes, whose spectral function displays characteristic Dirac-cone and "twisting" arc-states (Kataria et al., 12 Jan 2026).

6. Controversies and Challenges: NQR Constraints and Hebel–Slichter Peak

Recent nuclear quadrupole resonance (NQR) data on LaNiGa₂ challenge the ubiquity of nonunitary INT pairing as the universal explanation for TRSB in this compound. While $\mu$SR, thermodynamics, and strong interband coupling support the INT scenario, the observation of a pronounced Hebel–Slichter coherence peak in the spin-lattice relaxation rate ($1/T_1$) is inconsistent with even small degrees of nonunitarity. Detailed modeling shows that for the triplet INT case, a finite gap splitting ($\Delta_1 \ne \Delta_2$) rapidly suppresses the peak, while unitary two-gap singlet models preserve it despite similar gap ratios. This observation suggests that either true nonunitary triplet order is absent in the measured samples, or that unitary triplet or two-gap singlet states dominate (Sherpa et al., 27 Oct 2025).

Order Parameter	Gap Structure	Hebel–Slichter Peak	TRSB Evidence
INT, unitary	$\Delta_1 = \Delta_2$	present	no
INT, nonunitary	$\Delta_1 \ne \Delta_2$	absent/diminished	yes
Two-gap singlet	$\Delta_1 \ne \Delta_2$	present	no

A plausible implication is the need to reconcile distinct probes (NQR vs. $\mu$SR/thermodynamics) and revisit the precise degree of nonunitarity and its relationship to time-reversal-symmetry breaking in candidate INT materials.

7. Material Realizations, Outlook, and Topological Implications

The INT state has been conclusively discussed in LaNiGa₂ and YbSb₂. In LaNiGa₂, the state is compatible with observed TRSB, fully gapped spectra, strong interband coupling, and a double-peak spin-resolved DOS. YbSb₂ provides an unprecedented example of a type-I superconductor hosting both INT order and topological surface Majorana modes, with first-principles calculations revealing a ${\mathbb Z}_2$ Dirac nodal line and a DIII bulk topological index ($\nu=1$) (Kataria et al., 12 Jan 2026). Experimental evidence includes:

• Internal fields $B_\mathrm{int} \sim 0.3$–0.44 G
• $T_c \sim$ 0.95–2.1 K
• Spin-resolved DOS splitting $\Delta_\uparrow - \Delta_\downarrow \sim 0.2$ meV Validated signatures include the double-peak DOS (per spin), spontaneous magnetization, and fully gapped specific heat.

Further experimental discrimination—particularly via spin-resolved tunneling, ARPES, and NQR—is required to solidify the relationship between TRSB, nonunitarity, and the topological aspects of internally antisymmetric nonunitary triplet superconductivity. The INT paradigm provides a minimal platform for realizing multiband, TRS-breaking superconductivity with potential relevance to topological quantum computing and the study of Majorana modes in crystalline superconductors (Ghosh et al., 2019, Sundar et al., 2023, Sherpa et al., 27 Oct 2025, Kataria et al., 12 Jan 2026).