Spin and pair density waves in 2D altermagnetic metals

TL;DR

This work investigates how correlations induce secondary orders in 2D altermagnetic metals, where momentum-dependent spin splitting arises from altermagnetic order. By combining DFT-guided material candidates with an effective AM Hubbard model and unbiased TUFRG analysis (complemented by RPA), it uncovers competing spin-density-wave states and fluctuation-driven superconducting states, including an unconventional triplet pair-density wave with commensurate momentum $Q=(\,π,\,π)$ and an extended $s$-wave component, whose Bogoliubov spectrum can host nodes or Bogoliubov Fermi surfaces. The results reveal two distinct AM regimes (small and large AM splitting) that host different instabilities, underscoring universal mechanisms linking Fermi-surface nesting or van Hove singularities to SDW and PDW formation even in the presence of altermagnetic spin splitting. The findings provide a framework for realizing exotic many-body states in altermagnetic metals and guide experimental probes (STM, ARPES, thermal transport, NMR) and material searches, including meta-altermagnets, with potential tunability via doping or strain.

Abstract

Altermagnetism, a recently proposed and experimentally confirmed class of magnetic order, features collinear compensated magnetism with unconventional d-, g-, or i-wave spin order. Here, we show that in a metallic 2D d-wave altermagnet with combined two-fold spin and four-fold lattice rotational symmetry $[C_2||C_4]$, secondary instabilities can arise. Using an unbiased functional renormalization group approach, we analyze the weak-coupling instabilities of a 2D Hubbard model with a preexisting altermagnetic order inspired by our ab initio electronic structure calculations of realistic material candidates from V$_2$X$_2$O (X = Te, Se) family. We identify two distinct spin density wave (SDW) states that break the underlying altermagnetic $[C_2||C_4]$ symmetry. Additionally, we find spin-fluctuation-induced instabilities leading to a singlet d-wave superconducting state and an unconventional commensurate pair density wave (PDW) state with extended s-wave and spin-triplet symmetry. We establish a general criterion for the unusual exchange statistics for these pair density waves and characterize their excitation spectrum, which exhibits Bogoliubov Fermi surfaces or nodal points depending on the gap size.

Figures (10)

• Figure 1: Electronic structure of 2D AM metal. (a) The predicted band structure of $\mathrm{V}_2\mathrm{Te}_2\mathrm{O}$ with $f_{x^2-y^2}$ altermagnetic splitting and (b) its density of states. The inset corresponds to the atom arrangement in the unit cell. (c) For the effective model in our calculations we used $t=-\frac{2}{3},\,t^\prime=-\frac{1}{5}, \lambda=-1$, which provides an adequate description for the band structure for $E-E_F\in[-0.25,0.6]$. Color represents the spin quantum number of the bands.
• Figure 1: The Fermi surfaces for an $f_{x^2-y^2}$ altermagnetic metal for different chemical potentials $\mu$ and different ratios of hopping and AM splitting $t/\lambda$. For both the small and large AM regime, at $\mu=0$ there is perfect nesting for the same spin species. The location of the vH singularities as well as their spin degeneracy depends on the ratio $t/2\lambda$. In the small altermagnetic regime, they are spin split and are found at $\mathbf{X}$ and $\mathbf{Y}$. In the large AM regime they are found at $\mathbf{\Gamma}$ and $\mathbf{M}$.
• Figure 2: Effective models of an AM metal according to the bare dispersion (Eq.(\ref{['eqn:ham']})) for the two AM form factors studied. The spin splitting is tuned by $\lambda$. The relative ratio $t/\lambda$ tunes between the small (a,c) and large (b,d) altermagnetic regime and affects the location of saddle points and extrema in the BZ.
• Figure 2: The Fermi surfaces for an $f_{xy}$ altermagnetic metal for different chemical potentials $\mu$ and different ratios of hopping and AM splitting $t/\lambda$. For this case both the small and large AM regimes posses perfect nesting for the opposite spin species at $\mu=0$. Similar to the $f_{x^2-y^2}$ case, the location of the vH singularities as well as their spin degeneracy depends on the ratio $t/2\lambda$.
• Figure 3: FRG phase diagrams for small the small AM regime. (a) Critical temperatures $T$ of FRG instabilities as function of next-nearest-neighbor hopping $t'$ at $\mu_\mathrm{VH}=4t'$ for the $f_{xy}$ AM case. The results are qualitatively similar to the ones obtained in the SU(2) symmetric Hubbard model where for $t'\sim0.2t$ a transition from a SDW to d-wave pairing is observed. For larger values of $t'$ the system displays a ferromagnetic instability. In the $f_{xy}$ AM case, a transition between $\mathrm{SDW}{\perp}$ and $\mathrm{SDW}{\|}$ occurs. (b) For $f_{x^2-y^2}$ the observed instabilities $\mathrm{SDW}_{\|}$ and $\mathrm{SDW}_{\perp}$ are separated into two domes as a function of $\mu$ within our temperature resolution. The former is dominated by equal spin nesting at $\mu=0$ and the latter due to the spin split VH singularities occurring at $\mathbf{X}$ and $\mathbf{Y}$. The different symbols correspond to the instabilities identified with the vertex divergence terminating the FRG flow. The insets correspond to relevant FS for the instabilities identified.