Valley-Imbalanced Quarter Metals

• Valley imbalanced quarter metals are quantum phases in multi-valley materials where conduction occurs via one spin and one valley flavor, yielding a one-fold Fermi surface.
• They arise from strong electron correlations and symmetry-breaking mechanisms, such as quantum anomalous Hall and valley-coherent charge-density wave orders, that fully lift spin-valley degeneracy.
• This phase serves as a parent state for unconventional superconductivity with pair-density-wave instabilities and Majorana Fermi pockets, exhibiting distinct transport and thermodynamic signatures.

A valley imbalanced quarter metal is a quantum itinerant metallic phase in multi-valley materials—most prominently in chirally-stacked multilayer graphene—where conduction is enabled by exactly one spin and one valley flavor, with all others gapped, resulting in a one-fold Fermi surface degeneracy. This state arises from the interplay of strong electron correlations, van Hove singularity-driven Stoner instabilities, and subtle degeneracy-lifting mechanisms such as quantum anomalous Hall or valley-coherent charge-density wave orders. The quarter-metal serves as a parent state for unconventional superconductivity, including pair-density-wave phases featuring symmetry-protected Majorana Fermi pockets and topological features, with distinct thermodynamic, spectroscopic, and transport signatures observable in high-mobility systems.

1. Band Structure and Theoretical Conditions for Quarter-Metallicity

The realization of a valley imbalanced quarter-metal requires multivalley band structures and precise control of symmetry-breaking fields and interactions. In Bernal bilayer and rhombohedral trilayer or tetralayer graphene, the key parameters are:

• Intralayer hopping $t$ (energy scale set to unity), direct interlayer hopping $t_\perp\approx 0.1$–$0.3t$, and trigonal warping $t_3\approx 0.05$–$0.2t$
• Perpendicular displacement field $D$ introducing an onsite potential ($V\approx 0.1$–$0.3t$) across the layers

Layer antiferromagnetism (driven by onsite Hubbard $U$) opens a spin-selective gap, yielding a spin-polarized but valley-degenerate half-metal. A further instability, such as quantum anomalous Hall (QAH) order from next-nearest-neighbor repulsion or a valley-coherent charge-density-wave (VC-CDW), can fully lift the four-fold (spin⊗valley) degeneracy, producing a quarter metal with only a single (spin, valley) flavor remaining at the Fermi level (Murshed et al., 2022, Szabo et al., 2021, Parra-Martinez et al., 26 Feb 2025).

Typical phase diagrams exhibit direct transitions from unpolarized metal → half-metal (spin-polarized, valley-unpolarized) → quarter metal (fully spin and valley polarized) as chemical potential and displacement field are varied, with phase boundaries tracked via negative compressibility and quantum oscillation degeneracies (Zhou et al., 2021).

2. Hamiltonian Structure, Symmetry Breaking, and Order Parameters

Following projection to low-energy bands, the effective Hamiltonian for the quarter-metal can be summarized as (for BBLG/RTLG):

$$H_{\mathrm{QM}}(\mathbf{k}) = \alpha_0\beta_1 + \left[ \alpha_1 p_1 - (\alpha_2 d_1 + \alpha_3 f_1) \right] \beta_1 + \left[ \alpha_1 p_2 + (\alpha_2 d_2 + \alpha_3 f_2) \right] \beta_2 + m(\mathbf{k})\beta_3 - \mu \beta_0$$

where the $\beta_i$ are Pauli matrices in sublattice space; $p$, $d$, $f$ represent chiral in-plane harmonics relevant to the stacking and symmetry.

Order parameters quantifying valley and spin polarization are

$$S = \langle \Psi^\dagger (\sigma_z \otimes \tau_0) \Psi \rangle, \quad \tau = \langle \Psi^\dagger (\sigma_0 \otimes \tau_z) \Psi \rangle$$

with $S=+1$ and $\tau=+1$ indicating full spin-up/valley-K polarization in the quarter-metal (Murshed et al., 2022).

Symmetry analysis reveals that in the quarter-metal, time reversal is broken (by QAH), spin-SU(2) is broken (by LAF), and valley-exchange symmetry is broken by the mass term $m(\mathbf{k})$ (Szabo et al., 2021). Clifford-algebraic considerations show that three commuting mass terms are needed to fully lift the fourfold degeneracy (layer polarization, antiferromagnetism, and QAH or VC-CDW), leading to a single-band metallic state (Murshed et al., 23 Oct 2025).

3. Fermiology, Topology, and Lifshitz Transitions

The valley imbalanced quarter-metal exhibits a distinctive Fermi surface consisting of a single pocket—annular or simple, depending on the material and doping. In trilayer and tetralayer graphene, phase transitions between simple (disk-like) and annular Fermi surfaces, as well as between valley-Ising ($\langle \tau_z \rangle \ne 0$) and valley-XY ($\langle \tau_x \rangle \ne 0$) orders, are generic (Das et al., 2023). These transitions manifest as annular Lifshitz (ALT) or Ising-XY transitions, tracked via quantum oscillations and magnetotransport (Zhou et al., 2021).

In systems like silicene, electrostatic control of the band gaps and exchange fields realizes a valley-polarized metal (VPM), which in terms of carrier flavor content is a nearly ideal quarter metal, with topological Chern numbers inherited from the QAH phase (Ezawa, 2012).

4. Competing Orders: QAH, VC-CDW, and Layer Parity Effects

Depending on the microscopic interactions and microscopic symmetry breaking, the flavor lifting at the quarter-metal transition may be via QAH order (producing orbital magnetization, nonzero $\sigma_{xy}$, and transport hysteresis) or via a VC-CDW (producing $2\mathbf{K}$-periodic stripe or triangular modulations in real space while retaining zero Hall response) (Murshed et al., 23 Oct 2025).

Layer parity ($n$ even vs odd) in chirally-stacked $n$-layer graphene determines the symmetry of the VC-CDW: for even $n$ ($n=2,4,\ldots$) $C_3$ symmetry is broken, producing a smectic stripe order; for odd $n$ ($n=3,5,\ldots$) $C_3$ is preserved, yielding triangular modulations. Experimental STM observations confirm these predictions, with rhombohedral trilayer ($n=3$) displaying $C_3$-symmetric VC-CDW and hexalayer ($n=6$) exhibiting stripe order (Murshed et al., 23 Oct 2025).

The coexistence of QAH and VC-CDW orders is possible, with valley polarization $$\nu_v = \Delta_1/\sqrt{\Delta_1^2 + (|d_{03}-\Delta_0|)^2}$$ interpolating between 0 (CDW-dominated) and 1 (QAH-dominated) (Murshed et al., 23 Oct 2025).

5. Superconducting Instabilities and Majorana Fermi Pockets

The valley imbalanced quarter-metal provides a parent state for unconventional superconductivity. In particular, the spin- and valley-polarized Fermi pocket is susceptible to an intra-valley, spin-triplet pair-density-wave (PDW) instability with center-of-mass momentum $2\mathbf{K}$, leading to:

• Real-space modulation of the order parameter, e.g., Kekulé (BBLG) or columnar (RTLG) patterns
• Formation of three ungapped (zero-energy) Majorana pockets at symmetry-related Fermi points, determined by $C_3$ crystal symmetry
• Nontrivial topological signatures, including Chern number $C=1$ and the appearance of Majorana zero modes at edges, vortices, and PDW dislocations (Murshed et al., 2022, Yoon et al., 24 Feb 2025)

The existence of the Majorana pockets leads to a constant low-energy density of states in the superconducting phase, producing anomalous low-$T$ thermodynamics: specific heat $C_v\sim T$, residual $\kappa/T$ in thermal conductivity, and a nonvanishing low-temperature NMR relaxation rate $1/T_1T$ (Murshed et al., 2022).

Anomalous diode effects—nonreciprocal superconducting critical current—arise when both time-reversal and inversion are broken, and are enhanced by trigonal warping and chiral $p$-wave PDW pairing, with efficiencies up to 18% in realistic parameter regimes (Yoon et al., 24 Feb 2025).

6. Experimental Diagnostics and Signatures

Distinctive observable features of valley imbalanced quarter metals include:

• Quantum oscillation degeneracy reduction: A sequential decrease from fourfold (unpolarized), twofold (half-metal), to single-flavor (quarter-metal), as revealed by Shubnikov–de Haas frequencies $f_\nu = 1, \frac12, \frac14$ (Zhou et al., 2021)
• Negative electronic compressibility: Strongly negative spikes in compressibility mark the first-order transitions into the quarter-metal phase (Zhou et al., 2021)
• Anomalous Hall conductivity and hysteresis: In QAH-driven quarter metals, observation of quantized $\sigma_{xy} = e^2/h$ at $B=0$, robust hysteresis in Hall resistivity, and Kerr or magnetic circular dichroism signatures (Szabo et al., 2021, Murshed et al., 23 Oct 2025)
• STM and Josephson STM: Direct imaging of $2\mathbf{K}$ periodic charge order (VC-CDW) and real-space modulations of superconducting order parameter in the PDW phase (Murshed et al., 2022)
• Thermodynamics and transport: Linear-in-$T$ specific heat and residual thermal conductivity as diagnostic of Majorana Fermi pockets in the superconducting state (Murshed et al., 2022)

A combination of gate-tuned Fermiology, broken symmetries (valley/spin), superconducting spatial modulation, and thermodynamic scaling is necessary to conclusively demonstrate the quarter-metal and its descendant phases experimentally.

7. Quarter Metals in Other Multi-Valley and Anisotropic Systems

While much of the quarter-metal phenomenology has been developed in multilayer graphene, similar interaction-driven spin and valley flavor-selective metallic phases emerge in other multi-valley materials:

• In AlAs quantum wells, variational Monte Carlo establishes a sequence of isotropic (unpolarized), valley-polarized (half-metal), and spin+valley-polarized (quarter-metal) phases as the dimensionless $r_s$ increases. Critical values $r_s^{c1}\approx 12$ (unpolarized $\to$ valley) and $r_s^{c2}\approx 27$ (valley $\to$ quarter-metal) are found for realistic mass anisotropy $\eta\simeq5.2$. Distinctive nematic transport is diagnostic of the quarter-metal (Valenti et al., 2023).
• In monolayer and buckled honeycomb materials (e.g., silicene), tunable Dirac mass terms and exchange fields yield valley-polarized metallic and marginal metallic regions, with phase boundaries analytically classified in $(E_z, M)$ space. The VPM region is equivalent to an almost perfect quarter-metal at the Fermi level and sits adjacent to QAH and QSH insulating phases with quantized edge modes (Ezawa, 2012).

These findings suggest that valley imbalanced quarter-metallicity is a general emergent phase of strongly interacting multi-valley electron systems with tunable symmetry-breaking perturbations.

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Key references:

• "Nodal pair-density-waves from quarter-metal in crystalline graphene multilayers" (Murshed et al., 2022)
• "Metals, fractional metals, and superconductivity in rhombohedral trilayer graphene" (Szabo et al., 2021)
• "Charge-density waves and stripes in quarter metals of graphene heterostructures" (Murshed et al., 23 Oct 2025)
• "Quarter Metal Superconductivity" (Yoon et al., 24 Feb 2025)
• "Band Renormalization, Quarter Metals, and Chiral Superconductivity in Rhombohedral Tetralayer Graphene" (Parra-Martinez et al., 26 Feb 2025)
• "Quarter-Metal Phases in Multilayer Graphene: Ising-XY and Annular Lifshitz Transitions" (Das et al., 2023)
• "Half and quarter metals in rhombohedral trilayer graphene" (Zhou et al., 2021)
• "Valley-Polarized Metals and Quantum Anomalous Hall Effect in Silicene" (Ezawa, 2012)
• "Nematic metal in a multi-valley electron gas: Variational Monte Carlo analysis and application to AlAs" (Valenti et al., 2023)