Quantum Metric Senses A Persistent Spin Helix

Abstract

Persistent spin helices are a manifestation of symmetry-protected spin textures in systems with balanced spin-orbit coupling. They enable long-lived spin structures that are of interest for spintronics and coherent spin manipulation. The quantum metric has recently emerged as a promising tool for characterizing the geometric structure of quantum states. Here, we demonstrate that the quantum metric provides a sensitive geometric probe of the persistent spin helix. Within the Rashba-Dresselhaus Hamiltonian, we analytically evaluate the quantum metric components and uncover a divergent geometric contribution that emerges precisely at the persistent spin helix condition. We reveal that this divergence originates from a hidden line degeneracy that forms when the strengths of Rashba and Dresselhaus spin-orbit coupling become equal. We further study the role of higher-order cubic spin-orbit interactions and determine how these corrections regularize the geometric response and control the scaling behavior of the quantum metric. Our results establish quantum geometry as a powerful framework for identifying and characterizing persistent spin helices and related symmetry-protected spin textures.

Figures (3)

• Figure 1: Spin texture and variation of the quantum metric. Schematic of the spin texture with (a) Rashba ($\alpha \neq 0, \beta=0$), (b) persistent spin helix ($\alpha=\beta$), and (c) Dresselhaus ($\alpha=0, \beta \neq 0$) terms plotted in the $k_x-k_y$ plane. (d) Schematic of the quantum metric components, $g_{\mu\nu}$, with varying Dresselhaus to Rashba strength ratio, $\beta/\alpha$. We discover a sharp enhancement of all components of the metric when the Rashba and Dresselhaus strengths become equal, $\beta=\alpha$. This is a striking signature of the persistent spin helix.
• Figure 2: Distribution of quantum metric components. The momentum space distribution of the quantum metric components (a1)-(a4) $g_{xx}(\mathbf{k})$, (b1)-(b4) $g_{yy}(\mathbf{k})$, and (c1)-(c4) $g_{xy}(\mathbf{k})$ for different Dresselhaus to Rashba strength ratios, $\beta/\alpha$, noted at the top of each column. In the absence of the Dresselhaus term, the diagonal components $g_{xx}(\mathbf{k})$ and $g_{yy}(\mathbf{k})$ present mirror symmetries along $k_y=0$ and $k_x=0$, respectively, as shown in panels (a1) and (b1). The off-diagonal component, $g_{xy}(\mathbf{k})$, exhibits a quadrupolar petal-like pattern with alternating equal-sized positive and negative lobes, as shown in panel (c1). As the $\beta/\alpha$ ratio increases, the mirror symmetry in the diagonal elements is lost [(a2)-(b2)]. The positive and negative lobes of the off-diagonal component become unequal [(c2)]. The distributions become skewed along the $k_x=-k_y$ direction. As the $\beta/\alpha$ ratio approaches one, a singular ridge forms along the $k_x=-k_y$ direction, where all components of the quantum metric take large values, as shown in panels (a3)-(c3). This enhancement signals the formation of the persistent spin helix. At larger values of $\beta/\alpha$, beyond unity, the ridge disappears giving way to asymmetrically distributed quantum metric components [(a4)-(c4)].
• Figure 3: Integrated quantum metric and nature of band structures. Quantum metric components (a) $g_{xx}$, (b) $g_{yy}$, and (c) $g_{xy}$ integrated over the momentum space, plotted as a function of the Dresselhaus to Rashba strength ratio, $\beta/\alpha$. We find a sharp enhancement in the quantum metric components near the persistent spin helix condition ($\alpha=\beta$). The band structure for the Rashba-Dresselhaus model for (d) $\beta/\alpha=0.5$, (e) $\beta/\alpha=1.0$, and (f) $\beta/\alpha=1.5$. The corresponding band gaps are presented in panels (g)-(i). Notably a hidden line degeneracy, with a vanishing band gap, appears along the $k_x=-k_y$ direction when the persistent spin helix forms, as shown in panels (e) and (h).