A Fermi Surface Driven Spiral Spin Liquid

Abstract

EuAg$_4$Sb$_2$ is a model material to study the interplay of electronic and spin texture degrees of freedom, exhibiting numerous multi-$q$ magnetic textures coupled with the electronic properties. It is generally understood that some combination of conduction-electron mediated interactions, frustration, and higher order interactions are responsible for complex incommensurate spin textures in centrosymmetric lanthanide materials. Here, we refine an effective model of the magnetic interactions in EuAg$_4$Sb$_2$ through measurements of diffuse magnetic neutron scattering above the ordering temperature. These diffuse measurements reveal a ring of fluctuating spin modulations that reflects a manifold of nearly degenerate propagation vectors known as a spiral spin liquid (SSL). We further identify that this approximate $U$(1) symmetric SSL emerges from magnetic interactions mediated by a quasi-2D hole pocket and exhibits critical scaling of the spatial correlations. Further, Monte Carlo simulations reveal excellent agreement with experiment and provide a comprehensive understanding of the phase diagram. This study emphasizes the connection between the rich spin textures in this material, the electronic structure, and spin liquidity$\unicode{x2014}$uncovering new insights into design principles for nano-scale spin texture materials with advantageous intertwined electronic, magnetic, and topological properties, and new mechanisms for generating the physics of spiral spin liquids.

Figures (5)

• Figure 1: A Fermi Surface Mediated Spiral Spin Liquida Schematic 2D Fermi surface in a hexagonal Brillouin zone. b RKKY-mediated momentum dependent susceptibility for a parabolic band in one, two, and three dimensions (after kittel1969indirect). The general concept that more refined approximations can favor $q=2k_F$ order in 2D is depicted schematically with a dashed black line. b Magnetic susceptibility of a 2D electron gas treated to second order (dashed line) and fourth order (solid blue line), after wang2020skyrmion. c Schematic illustration of a case proximate to the 2D RKKY case where the ferromagnetic ($\bm{q}=0$) spin susceptibility is slightly less favored than a 2D line of $|q|=2k_F$ in momentum space. d-f Model momentum-space resolved susceptibility and g-i schematic of 2D SSL at three representative temperatures depicting the high temperature paramagnetic, intermediate temperature spiral spin liquid, and low temperature cycloidal ordered behavior, respectively.
• Figure 2: Preferred Propagation Vectors and Critical SSL Fluctuationsa-c Temperature dependence of the ICM phases, with red hexagons emphasizing the line on which the peaks of all three phases lie, collected at zero field and 2, 8, and 10 K, respectively. d The magnitude of the magnetic propagation vector $q$ in ICM 1-3 as a function of azimuthal angle $\psi$ (see inset of b for definition), where 0$^\circ$ is along the $a^*$ direction rotating counterclockwise about $c$. The expectation for a circular (hexagonal) nodal line is the dashed black (green) line. e The width $\sigma_q$, in blue, of the radially averaged diffuse scattering peak, with the instrument resolution subtracted (estimated from the width of the peaks in the ordered phase), and the correlation length $\xi=1/\sigma_q$, in orange, as a function of temperature above the ordering temperature (vertical dashed gray line), see Eq. \ref{['crit']} for definitions of $\xi$ and $\sigma_q$. f The integrated intensity of the radially averaged diffuse scattering as a function of temperature. d-f Were obtained by fitting to a Gaussian. The errorbars are the uncertainty in the fit values. g The diffuse scattering at 11 K just above the transition temperature. h The radial integration of the SANS intensity at 11 K. $q_r=\sqrt{q_x^2+q_y^2}$. i The $q_z$ dependence of the integrated SANS intensity at 11 K (the FWHM of the ground state peak is indicated with a horizontal bar as an estimate of the instrument resolution).
• Figure 3: Model of Diffuse Scattering and Susceptibilitya-h Fit of diffuse single crystal SANS intensity to Heisenberg model as a function of temperature (see text, table \ref{['tab:Jr']}, and Supplemental Information for details). The data is shown on the left half of each figure, while the model is depicted on the right half of each figure. i Model of the $L$-dependence of the single crystal SANS scattering intensity. The data is fit in a 3D cube, and plotted here as a integral over the $H$ and $K$ directions. j Model fit of susceptibility data. k Model fit of the diffuse powder diffraction data. The SANS data is binned, smoothed, inversion symmetrized, and a high temperature (20 K) background is subtracted. The powder data has a high temperature (100 K) background subtracted. Single crystal susceptibility data is reproduced from kurumaji2025electronic.
• Figure 4: Momentum Space Pairing $J(\bm{Q})$ The maximum eigenvalue of the interaction matrix in reciprocal space, $J(\bm{Q})$ plotted a-b in the $HK0$ plane, c-d along the $(H,0,0)$ and $(-H/2,H,0)$ directions, and e along (0.102, 0, $L$) in blue and (0.058, 0.058, $L$) in orange. The line cuts plotted in c are indicated in b by lines, and the intersections of the line cuts plotted in d are indicated with crosses in b. In b, the location of the maximum as a function of in-plane angle is indicated with a red-white-blue line, with the color of the line indicating the component along $L$. e The radial in-plane distance of the maximum (along the red line in b) of $J(\bm{Q})$ from the origin in Å$^{-1}$ as a function of in-plane angle $\theta$, where $\theta=0$ corresponds to the $a^*$ direction. f The value of $J(\bm{Q})$ at the maximum (along the red line in b) as a function of in-plane angle $\theta$. See supplementary information for additional cuts of $J(\bm{Q})$.
• Figure 5: Monte Carlo Simulationa The MC simulated (circles) and experimental (lines) magnetization as a function of applied field along the $a$ (red) and $c$ (blue) directions at 2 K. The temperature-dependent phase diagrams are compared in supplementary Fig. S9. b-e The MC simulated magnetic textures at 1 K for various applied fields (indicated by green letters in a). The field directions are depicted with black arrows, and the magnetic propagation vectors are depicted with green arrows. One $ab$-plane layer of magnetic spins is depicted, with the in-plane magnetization shown with black arrows, and the out-of-plane magnetization depicted with the colorscale. The spin size is normalized to one. b is simulated in zero field, c is simulated with 2 T of field along $c$, and d,e are simulated with 0.4 T and 0.8 T of field along $a$, respectively. These represent the four states observed in our MC modeling. The diffraction patterns corresponding to b-e are depicted in supplementary Fig. S10. Each 10x10 simulation was periodically tiled for visual clarity. Single crystal magnetization data is reproduced from kurumaji2025electronic.