Neutron Scattering and Magnetism
Laboratory for Solid State Physics · ETH Zurich

Quantum multiferroics

Multiferroics are materials with coupled magnetic and ferroelectric order. The underlying microscopic mechanism is rooted in relativistic spin-orbit interactions. For obvious reasons, multiferroics are of great industrial significance. We, however, are more interested in fundamental aspects. Which exotic phases of quantum magnetic materials have electrical polarization? What can we learn about the magnetic system and magnetic quantum phase transitions by studying dielectric properties? Is it possible that dielectric degrees of freedom play an active role in quantum phase transitions and collective excitations?

Multiferroic quantum criticality

Enlarged view: Measured scaling of the inverse dielectric susceptibility at the magnetic saturation transition in Rb<sub>2</sub>Cu<sub>2</sub>Mo<sub>3</sub>O<sub>12</sub>.

Measured scaling of the inverse dielectric susceptibility along the quantum critical trajectory at the magnetic saturation transition in Rb2Cu2Mo3O12. The fitted power law gives γ = 1.64(1), in excellent agreement with the value 3/2 predicted for a three-dimensional BEC and clearly distinct from the mean-field (γ = 1) and Ising (γ = 2) alternatives [2].

Bose-Einstein condensation may be the most celebrated phase transition of all, at work in superfluids, superconductors and ultracold atomic gases. Every continuous phase transition is characterized by a divergent susceptibility: the response of the order parameter to its conjugate field. For BEC this notion runs into a problem of principle. The order parameter is the complex amplitude of the condensate wave function, and no physical field couples to it: the critical susceptibility is not merely difficult to measure, it does not correspond to any observable at all. Magnetic insulators get us halfway out of this bind. Field-induced ordering transitions of gapped quantum paramagnets are famously described as a Bose-Einstein condensation of magnons, and there the order parameter is perfectly concrete: the staggered magnetization transverse to the applied field. Its conjugate field, however, is a magnetic field that reverses direction from one atom to the next, and no laboratory magnet can produce that. The susceptibility remains as inaccessible as ever.

Quantum multiferroics provide the missing loophole. The frustrated ferro-antiferromagnetic chain compound Rb2Cu2Mo3O12 is a gapped quantum paramagnet that orders magnetically only inside a "dome" of applied fields, between roughly 2 T and the saturation field of 12.6 T [1]. The ordered phases turn out to be ferroelectric, with the uniform electrical polarization coupled linearly to the staggered magnetization, the BEC order parameter itself. A uniform electric field is precisely what a capacitor applies effortlessly. Through the magnetoelectric coupling, the ordinary dielectric susceptibility becomes, up to a constant, the critical susceptibility of the magnon condensate: for once, one can actually apply the conjugate field and measure the response.

There is one subtlety: in the zero-temperature limit the strength of the linear coupling is proportional to the uniform magnetization, which vanishes at the lower critical field but is nearly saturated at the upper one. Polarization is therefore a primary order parameter only at the saturation transition, something we verified directly: a dc electric bias, the field conjugate to a primary order parameter, visibly suppresses the transition itself. With the geometry chosen accordingly, we measured the dielectric susceptibility along the quantum critical trajectory H = Hc2. It diverges as a power law in temperature with the exponent γ = 1.64(1), in excellent agreement with the long-standing theoretical prediction of 3/2 for a three-dimensional BEC (see figure). This was the first direct measurement of the critical susceptibility at a Bose-Einstein condensation quantum critical point, in any system [2].

Dielectric relaxation via quantum-critical magnons

Enlarged view: False-color plot of the real (top) and imaginary (bottom) parts of the dielectric susceptibility of Cs2Cu2Mo3O12 as a function of magnetic field and temperature [2].

False-color plot of the real (top) and imaginary (bottom) parts of the dielectric susceptibility of Cs2Cu2Mo3O12 as a function of magnetic field and temperature [2]. The sharp ridge follows the magnetic phase boundary (dashed line), as in the previous example. The broad low-temperature feature in the dissipative part (arrows) is the new relaxation anomaly: it persists at all fields, yet dips to zero at the ~7.7 T quantum critical point.

Dielectric relaxation is textbook physics. Electric dipoles in a solid take a finite time to reorient; in an oscillating field they lag behind and dissipate energy. The relaxation time is thermally activated, τ = τ0 exp(Δ/kBT), and the "barrier" Δ is normally set by a polar lattice excitation, an optical phonon, about which one can do nothing at all. Now imagine the barrier being a magnon instead: an excitation whose energy is dialed at will by an applied magnetic field, and driven all the way to zero at a quantum critical point. This is precisely what we found in Cs2Cu2Mo3O12, the sister compound of Rb2Cu2Mo3O12 from the previous example [1,2]. The Cs system is built of the same frustrated copper chains and reaches saturation at about 7.7 T through the same kind of magnon BEC quantum critical point; unlike its sibling, it orders magnetically already in zero field. At the phase boundary its dielectric constant diverges just as described above [1].

The surprise lies elsewhere. Deep inside the ordered phase, and equally in the field-polarized regime where no phase transition exists at all, the dissipative part of the dielectric susceptibility shows a broad peak at a temperature T* (figure). There is no thermodynamic anomaly, no hysteresis, no glassiness there; the feature is pure relaxation. The entire dataset, as a function of temperature, field and frequency, is described by Debye-type relaxation of low-energy dipolar degrees of freedom residing on the crystal lattice, with a single activated relaxation time [2]. The punchline is the activation barrier: above saturation it follows the Zeeman gap of a single magnon, and on approaching the quantum critical point it collapses to zero, dragging T* down with it.

In other words, the electric dipoles in this crystal relax by exchanging energy with individual magnons. Note that a magnetic excitation with no electric dipole moment of its own would be entirely invisible to the lattice dipoles and could never mediate their relaxation. The whole mechanism works only because, as this experiment thereby proves, magnons themselves may carry an electric dipole moment. The measured "attempt" frequency, a mere 1 MHz, five orders of magnitude below the typical magnon frequency, is then a direct gauge of the weak coupling between the lattice dipoles and the spins. The magnetic field thus becomes a knob that tunes the dialog between electric and magnetic degrees of freedom, opening a rather intriguing route towards genuinely multiferroic quantum criticality.