Phase-modulated superconductivity via altermagnetism

TL;DR

This work investigates how altermagnetic spin splitting enables phase-modulated superconductivity without external fields, by developing a microscopic Ginzburg–Landau theory for three 2D models. It demonstrates that a phase-modulated Fulde–Ferrell state is stabilized by altermagnetic order, with the multisublattice degree of freedom being essential. Through GL analyses and mean-field calculations, the authors contrast phase-modulated FF states with amplitude-modulated LO states across a two-sublattice tetragonal model, a continuum model, and a conventional square-lattice model, revealing FF stabilization in the multisublattice scenarios and LO predominance otherwise. The findings highlight altermagnets as a promising platform for exotic superconductivity and suggest experimental routes, such as Josephson interference, to detect phase modulation; they also underscore the importance of spin-splitting structure in determining the superconducting ground state.

Abstract

Stimulated by recent interest in altermagnets, a novel class of antiferromagnets with macroscopic time-reversal symmetry breaking, we investigate the coexistence of altermagnetism and superconductivity. By developing a Ginzburg--Landau theory based on microscopic models, we show that a phase-modulated Fulde--Ferrell superconducting state is stabilized via altermagnetic spin splitting, in contrast to the typical amplitude-modulated states that occur under the uniform Zeeman field. We apply our framework to different models to compare the resulting phase diagrams: a two-sublattice model with altermagnetic order, a continuum model with an anisotropic Zeeman field mimicking altermagnetic spin splitting, and a conventional square-lattice model with two kinds of anisotropic Zeeman fields. We show that the multisublattice structure is crucial for realizing the phase-modulated superconductivity, and highlight spin-split altermagnets as a promising platform for exploring this exotic superconductivity without external magnetic fields.

Figures (8)

• Figure 1: Energy band structures for (a) $h = 0$ and (b) $0.4$. Energy bands of up- and down-spin electrons are represented by the red and blue lines, respectively. The green dashed lines represent the Fermi level for the electron density $n = 0.85$ per site. The inset in (a) shows the high-symmetry path in the 2D BZ. We use symbols ($\Gamma$, X, Y, S) of the primitive orthorhombic lattice since the fourfold rotation symmetry is broken in the altermagnetic state.
• Figure 2: The first BZs associated with the spin splitting structures in (a) the two-sublattice tetragonal model and (b) the conventional square lattice model with $2\bm{k}$-$d$-wave Zeeman field. In the altermagnetic state, energy dispersions have $d$-wave type spin splitting, which is illustrated by the plus and minus signs. Green lines represent the nodes, where the up- and down-spin bands are degenerate even in the presence of the magnetic order, and the dashed lines show the first BZ boundaries.
• Figure 3: Schematic phase diagram in a uniform Zeeman field $h$. The zeros of the GL coefficients $\alpha$ and $\beta$ ($\kappa$) are depicted by red and blue lines, respectively. The amplitude-modulated LO state (orange shaded area) is stabilized in the high field region, where both $\beta$ and $\kappa$ are negative.
• Figure 4: Temperature ($T$) vs altermagnetic molecular field ($h$) phase diagram for the two-sublattice tetragonal model. The zeros of the GL coefficients $\alpha$ ($\widetilde{\alpha}$), $\beta$, and $\kappa$ are depicted by red solid, blue dashed, and green dash-dotted lines, respectively. The coefficient $\beta$ is positive in the whole parameters range except in the narrow region enclosed by the blue dashed line. The filled (open) circles represent the superconducting (normal) state obtained by the MF theory, where the color of the filled circles indicates the magnitude of the optimal COM momentum $Q = |\bm{Q}|$. The effective interaction is set to $V = 0.35$.
• Figure 5: Temperature ($T$) vs field ($h$) phase diagrams for the continuum model for the (a) uniform and (b) $d$-wave Zeeman fields. Both axes are scaled by the transition temperature at zero magnetic field $T_{\mathrm{c}0}$. The $\alpha = 0$ lines are unreliable when $h / T$ is large (dark shaded areas) since we truncate the infinite series to the first 100 terms.