Dynamic Multiferroicity of a Ferroelectric Quantum Critical Point

Quantum matter hosts a large variety of phases, some coexisting, some competing; when two or more orders occur together, they are often entangled and cannot be separated. Dynamical multiferroicity, where fluctuations of electric dipoles lead to magnetization, is an example where the two orders are impossible to disentangle. Here we demonstrate an elevated magnetic response of a ferroelectric near the ferroelectric quantum critical point (FE QCP), since magnetic fluctuations are entangled with ferroelectric fluctuations. We thus suggest that any ferroelectric quantum critical point is an inherent multiferroic quantum critical point. We calculate the magnetic susceptibility near the FE QCP and find a region with enhanced magnetic signatures near the FE QCP and controlled by the tuning parameter of the ferroelectric phase. The effect is small but observable—we propose quantum paraelectric strontium titanate as a candidate material where the magnitude of the induced magnetic moments can be $\sim 5\times {10}^{-7}{\mu }_B$ per unit cell near the FE QCP.

Figure 1

(a) Phase diagram near a ferroelectric QCP (where $x=x_{cr}$) with the magnetic susceptibility (red line, dashed in the FE phase) at $\omega =0.5{\omega }_0$ which diverges at the vertical dashed line, leading to a new “multiferroic PE” phase. The ferroelectric quantum critical region (pale green) is now a dynamical multiferroic quantum critical region. In both the PE (white background) and FE (yellow background) phases, qualitatively similar behaviors of ${\chi }_m$ are expected, despite the different underlying orders. The blue shading indicates the main regions where induced magnetic signatures are expected: a dome around the FE QCP due to quantum fluctuations; a narrow band around the finite temperature phase transition line due to thermally induced fluctuations. (b) A simple experiment using a SQUID could detect magnetic signatures resulting from rotating electric dipoles in a system towards its ferroelectric phase transition. Here, the electric dipoles are constrained to the horizontal plane and lead to an out-of-plane magnetic moment $\mathbf{m}$ and susceptibility.

Figure 2

The total magnetic susceptibility in units of the common prefactor $C^2={\lambda }^2{V}^4{P}_0^4$, and with ${\mathrm{\Lambda }}^3/{\omega }_0=1\times {10}^5$, as a function of ${\delta }_x$ at several energies. (a) The real part; (b) the imaginary part. The behavior in the FE phase ${\delta }_x<0$ is expected to share the main qualitative features despite the underlying order. The effects of changing the ${\chi }^{(2)}$ prefactor ${\mathrm{\Lambda }}^3/{\omega }_0$ and the individual contributions of ${\chi }^{(1)}$ and ${\chi }^{(2)}$ are discussed in Supplemental Material, Sec. IV [30].

Figure 3

Total magnetic susceptibility, in units of $C^2={\lambda }^2{V}^4{P}_0^4$, and with ${\mathrm{\Lambda }}^3/{\omega }_0=1\times {10}^5$, as a function of $\omega /{\omega }_0$ at several distances from the FE QCP. (a) Real part; (b) imaginary part. The peaks in the imaginary part ($\delta$ functions plotted as Lorentzian functions) corresponding to the divergence and sign change of the real part are weak for ${\mathrm{\Lambda }}^3/{\omega }_0=1\times {10}^5$ used; their presence is more easily seen at smaller values of ${\mathrm{\Lambda }}^3/{\omega }_0$ (Supplemental Material, Sec. IV [30]).