Van-Hove singularities and competing instabilities in an altermagnetic metal

TL;DR

The paper investigates how spin-polarized Van Hove singularities in altermagnetic metals on a square lattice drive interaction-induced instabilities. Using a two-patch model and parquet renormalization group, it identifies two stable fixed points for both $d_{xy}$ and $d_{x^2-y^2}$ AM forms, with instabilities predominantly in spin-density-wave channels and no onset of spin-singlet superconductivity at VH filling. The AM-induced spin splitting fundamentally alters the RG flow away from the spin-degenerate limit, making SU(2) symmetry breaking crucial for the instability structure. The work also connects to experimental AM candidates and proposes a microscopic realization via the Emery model, highlighting the potential for triplet pairing beyond the patch approximation and encouraging further momentum-dependent analyses.

Abstract

Van-Hove (VH) singularities in the single-particle band spectrum are important for interaction-driven quantum phases. Whereas VH points are usually spin-degenerate, in newly proposed altermagnets VH singularities can become spin-dependent, due to momentum-dependent spin polarization of the Fermi surfaces arising from combined rotation and time-reversal symmetry. We consider two altermagnetic models ($d_{x^2-y^2}$- and $d_{xy}$-wave) on a square lattice with spin-polarized VH points, and study their stable fixed-point solutions indicating interaction-induced instabilities using parquet renormalization group. For both models, we find new stable fixed-point solutions of the renormalization group equations which are not connected to the solution in the spin-degenerate limit. This implies that on the square lattice, the system with VH singularities is unstable with respect to altermagnetic perturbations. The leading instability for the $d_{x^2-y^2}$-model is real transverse spin density wave. For the $d_{xy}$-wave model, it is found to be real transverse spin density wave at large altermagnetic splitting. At small altermagnetic splitting both imaginary charge density wave and real longitudinal spin density waves are dominant.

Figures (12)

• Figure 1: (a)-(b): The effective models for Van Hove points of a square-lattice altermagnet [Eqs. \ref{['eq:VHXxy']},\ref{['eq:VHYxy']},\ref{['eq:VHYx2-y2']}, \ref{['eq:VHXx2-y2']}] can have different microscopic realizations since they only rely on the symmetry-induced spin splitting of bands. A simple example is a single-site unit cell with altermagnetism-induced spin-dependent hoppings that realize (a) the $d_{xy}$-wave form factor \ref{['eq:form-factor-dxy']}; (b) the $d_{x^2-y^2}$-wave form factor \ref{['eq:form-factor-d-wave']} with altermagnetic vector $\mathbf{n}$ chosen along the z-direction. The system is invariant under $C_4\mathcal{T}$ transformation. (c) Schematic phase diagram of the two models as a function of the AM strength $\lambda$. The instabilities at $\lambda=0$ (not shown here) are not adiabatically connected to the phases shown. (d)-(e) schematic diagrams of the leading instabilities. (d) Real longitudinal spin-density-wave (l-SDW) corresponds to having AFM electronic spins collinear with $\mathbf{n}$. Besides translation, the l-SDW does not break any pure spin symmetry, but the $C_4\mathcal{T}$ (with on-site rotation center) of the AM phase. (e) Real transverse spin-density-wave (t-SDW) has AFM electronic spins in the plane perpendicular to $\mathbf{n}$. It breaks spin U(1), AM $C_4\mathcal{T}$, and translation symmetry. (f) The imaginary charge density wave gives different phases to the hopping of electrons and breaks translation symmetry nayak2000density.
• Figure 2: Fermi surface of the $d_{xy}$-wave model at $\mu=0$ and $\lambda = 0.5 t$: (a) on the entire Brillouin zone; (b) near the patch $X$. Red and blue represents spin up and down respectively. The Van Hove points are located at the centers of patch $X,Y$, e.g. the origin in (b). The areas bounded by dashed lines of size $\Lambda$
• Figure 3: Fermi surface of the $d_{x^2-y^2}$-wave model at $\lambda = 0.1 t$ and $\mu=2\lambda$: (a) on the entire BZ; (b) near the patch $X$. Red and blue represents spin up and down respectively. Only the spin up (down) VH point exists at patch $X$ ($Y$). The areas bounded by dashed lines of size $\Lambda$ are the patches.
• Figure 4: The scattering channels for coupling constants in systems with two VH points. (a) couplings without external spin indices. Solid lines correspond to electrons at patch $X$ and dashed lines to electrons at patch $Y$. Putting in parallel or anti-parallel electron spins gives the coupling constants in Eq. \ref{['eq:coupling-constants']}. In (b) we show the $g_{p4}, \widetilde{g}_{p4}$ and $g_{a2}, \widetilde{g}_{a2}$ couplings which are in the same channels but are not related by $C_4 \mathcal{T}$ symmetry.
• Figure 5: One-loop diagrams for the coupling constants Eq. \ref{['eq:coupling-constants']}. (a)-(d) correspond to $g_{i1}, g_{i2},g_{i3}, g_{i4}$ where $i = p,a$; $\widetilde{g}_{a2}$ and $\widetilde{g}_{p4}$ have the same diagrams as $g_{a2}$ and $g_{p4}$. Solid lines correspond to patch $X$ while dashed lines to patch $Y$. The spin indices are not shown explicitly. The pRG equations \ref{['eq:RG-dxy']} and \ref{['eq:RG-d-wave-full']} are obtained by setting the incoming electron spin indices to be parallel and anti-parallel respectively and using the one-loop bare susceptibilities in Sec. \ref{['sec:bandstructure']}.