Casimir Stabilization of Fluctuating Electronic Nematic Order

TL;DR

This work demonstrates a nonresonant Casimir-control mechanism to stabilize fluctuating electronic nematic order, focusing on quantum Hall stripe phases near a birefringent BaTiO3 plate. Using a full electromagnetic continuum and imaginary-frequency scattering theory, the authors show F(θ) depends on the stripe orientation θ, enabling alignment of macroscopic stripe domains with energy gains up to the mK–to–K per particle range at experimentally accessible plate–sample distances. The stabilization is strongest at sub-micron separations and scales with distance roughly as ΔF ∼ d^{−3} (long distance) or ΔF ∼ d^{−2.5} in the nonretarded regime, with modest sensitivity to disorder and filling factor, and a small Hall-induced shift in the optimal angle. The results offer a viable pathway for vacuum electromagnetic control of competing correlated electronic phases, with implications for terahertz cavity experiments, metamaterial realizations, and Moiré systems, where full-continuum Casimir effects and appropriate boundary conditions can be harnessed to tailor nematic order.

Abstract

Vacuum cavity control of quantum materials is the engineering of quantum materials systems through electromagnetic zero-point fluctuations. In this work we articulate a generic mechanism for vacuum optical control of correlated electronic order: Casimir control, where the zero-point energy of the electromagnetic continuum, the Casimir energy, depends on the properties of the material system. To assess the experimental viability of this mechanism we focus on the Casimir stabilization of fluctuating nematic order. In nematic Fermi liquids, different orientations of the electronic order are often energetically degenerate. Thus, while local domains of fixed orientation may form, thermal disordering inhibits long range order. By engineering the electromagnetic environment of the electronic system, however, we show that the Casimir energy can be used as a tool to preferentially stabilize particular orientations of the nematic order. As a concrete example, we examine the interplay between a birefringent crystal -- which sources an anisotropic electromagnetic environment -- and a quantum Hall stripe system, an archetypal nematic Fermi fluid. We show that for experimentally feasible setups, the anisotropy induced by the orientation dependent Casimir energy can be $10^4$ times larger than other mechanisms known to stabilize quantum Hall stripes. This finding convincingly implies that our setting may be realized with currently available experimental technology. Having demonstrated that the Casimir energy can be used to stabilize fluctuating nematic order, we close by discussing the implications for recent terahertz cavity experiments on quantum Hall stripes, as well as pave the road towards broader Casimir control of competing correlated electronic phases.

Figures (6)

• Figure 1: Schematic picture of Casimir stabilization of the stripe phase. With no symmetry breaking field (a), stripes form thermally disordered domains, and the macroscopic sample is isotropic. In the presence of a parallel birefringent plate (optic axis $\varepsilon_\| > \varepsilon_\perp$) (b), the vacuum electromagnetic modes give stripes a twist angle $\theta$ dependent free energy, aligning them at an energy scale $\Delta F$ per unit area. Conductivity easy axis of the stripes aligned with high dielectric function $\hat{\varepsilon}_\|$ of the controlling plate (c) is energetically preferred to conductivity hard axis aligned with $\hat{\varepsilon}_\|$ (d).
• Figure 2: Free energy per particle $F(\theta)/n$, as a function of twist angle $\theta$. Values are given relative to the configuration with 2DES easy axis $\hat{e}$ parallel with the birefringent plate optic axis $\hat{\varepsilon}_\|$ ($\theta = 0$). A few anisotropy ratios of the stripe phase $\lambda = \sigma_1 /\sigma_2$ are shown. Parameters are $d = 100$ nm, $\nu = 15/2$, $\tau = 1$ ns and $n = 2.9\times 10^{11}$ cm$^{-2}$. Due to the presence of Hall conductivity, the free energy minimum is shifted away from the symmetry point $\theta = 0$ (inset).
• Figure 3: Stabilization energy per particle $\Delta F /n$ as a function of interplane distance $d$ and electron density $n$ (a), and as a function of Drude scattering time $\tau$ and filling factor $\nu$ (b). The anisotropy ratio is the highest experimental value $\lambda = 55$, remaining parameters are $d = 100$ nm, $\nu = 15/2$, $\tau = 1$ ns and $n = 2.9\times 10^{11}$ cm$^{-2}$, unless varied. The horizontal line marks the onset of relevant stabilization magnitudes $\Delta F /n = 1$ mK. Vertical lines show the window of relevant distances. Dashed slanted lines show the long-distance behavior $\Delta F \sim d^{-3}$ above the range plotted, and the non-retarded behavior in the limit of vanishing density $n$, $\Delta F \sim d^{-2.5}$.
• Figure 4: Coordinate basis for the 2DES and plate twisted at an angle $\theta$. A photon (yellow arrow) of in-plane momentum $\vb{k}_\perp$ makes an angle $\phi$ with the optic axis $\hat{\varepsilon}_\|$ of the birefringent plate. The vectors $\hat{\varepsilon}_\perp$ and $\hat{z}$ complete the basis. The same photon makes an angle $\phi - \theta$ with the easy axis $\hat{e}$ of the electron system.
• Figure 5: Free energy per particle $F(\theta) /n$ as a function of twist angle $\theta$ for increased Hall conductivities $\tilde{\sigma}_h$. Here the anisotropy ratio is at the highest experimental value $\lambda = 55$, remaining parameters are $d = 100$ nm, $\nu = 15/2$, $\tau = 1$ ns and $n = 2.9\times 10^{11}$ cm$^{-2}$. Finite Hall conductivity increases the Casimir stabilization slightly, as well as shifts the preferred twist angle (marked) away from the symmetry point $\theta =0$ (blowup).