Thermal Magnetic Fluctuations of a Ferroelectric Quantum Critical Point

Entanglement of two different quantum orders is of an interest of the modern condensed matter physics. One of the examples is the dynamical multiferroicity, where fluctuations of electric dipoles lead to magnetization. We investigate this effect at finite temperature and demonstrate an elevated magnetic response of a ferroelectric near the ferroelectric quantum critical point (FE QCP). We calculate the magnetic susceptibility of a bulk sample on the paraelectric side of the FE QCP at finite temperature and find enhanced magnetic susceptibility near the FE QCP. We propose quantum paraelectric strontium titanate (STO) as a candidate material to search for dynamic multiferroicity. We estimate the magnitude of the magnetic susceptibility for this material and find that it is detectable experimentally.

Materials that can undergo phase transitions, especially those exhibiting an interplay of two or more quantum orders attract special interest. One of the examples featuring an entanglement of two different quantum orders is the dynamic multiferroicity (DMF), the concept reflecting the connection between electric and magnetic properties [1,2,3,4].
Ferroelectric (FE) QCPs are an important part in the discussion of FE behavior, particularly in displacive quantum paraelectrics (PE) [5,6,7]. The plethora of effects that occur near a FE QCP have been investigated in various contexts [5,6,7,8,9,10,11,12,13,14,15]. In the DMF phenomenon magnetization M (i.e. magnetic moment per unit volume) is induced by time-dependent oscillations of electrical polarization P (an electric dipole moment per unit volume): Here a 0 is the lattice spacing, and c is the speed of light. The form of magnetization in the above equation is a consequence of the relation between the magnetic moment and the angular momentum for a rotating electric dipole, see also Eq. 4 and the discussion therein. An expression for the coupling constant λ is obtained (Eq. 4) for the model of optical long wave oscillations in the material with one cation and one anyon per unit cell, with charges ±e + and masses m ± . One can induce DMF by applying external drive-like light field to ferroelectrics. An alternative approach is to rely on inherent fluctuations present in the material near phase transition. In this work we consider a bulk material in the vicinity of a ferroelectric phase transition, and investigate the enhanced magnetic susceptibility at finite temperatures upon approaching the quantum critical point (QCP). The main result of our work is the elevated magnetic response of a ferroelectric near the FE QCP, see Eq. (3) and Fig. 1. We calculate the dynamic magnetic susceptibility at finite temperature using the Kubo formula and show that it grows in the paraelectric phase upon approaching to the QCP, which is controlled by the tuning parameter δ of the ferroelectric phase. Material develops a paramagnetic susceptibility. Our estimates taken from possible candidate materials, such as SrTiO 3 (STO), show that the effect can be readily observed experimentally.
To understand the meaning of the result presented in Eq. (3), it is important to recall that the ferroelectricity is often induced by the softening of the phonon optical mode, e.g. the frequency of this ferroelectric mode can gradually decrease upon lowering the temperature [16,17,18]. In case of STO, the material we choose to illustrate the effect, FE phase transition occurs in the presence of strain or in the isotope-exchanged SrTi 18 O 3 , when the frequency of a ferroelectric mode reaches zero at finite temperature T [19,20,21,22]. Based on these observations, we assume that near the FE QCP the spectrum of a soft transverse phonon mode reads as [3,9] Here, for the physically relevant low-energy phonon modes, by virtue of the general scaling arguments [10,11], ω 0 (T, δ) = ω 0 (T )|1 − δ| νz , s ⊥ (δ) = s 0 |1 − δ| ν(z−1) , with ω 0 (T ) as a bare phonon gap, and ν (z) is the correlation length (dynamical) exponent. We emphasize that the phonon gap and the transverse sound velocity may be gradually vanishing with the distance |1 − δ| while approaching the QCP [23], implying z > 1 at the QCP located at δ = δ c = 1. Furthermore, higher order terms ∼ q α , with α ≥ 4, in the low-energy phonon dispersion are even more irrelevant in the renormalization-group sense than the leading one proportional to the velocity, and as such can be neglected. For the (static) magnetic susceptibility in the high-T limit, at T ω 0 (T, δ), based on simple physical arguments (see below), supported by a formal analysis [see Eq. (14)], we find , and set k B = 1 hereafter. Thus, a strong increase of the magnetic susceptibility may occur while approaching the QCP: the underlying reason for this enhancement is an increase of the density of states of phonons due to the softening of the ferroelectric mode near the QCP, which is consistent with z > 1.
We now provide the estimate of the magnetic susceptibility as given in Eq. (3) using rather simple physical arguments. In our case, although the system is an insulator, we use the Pauli susceptibility because the fluctuating electric dipoles, according to our mechanism, yield the magnetic response. We start with the well known expression for the static Pauli susceptibility χ µ 2 B ρ(ε), where µ B is the Bohr magneton and ρ(ε) is the density of states of the excitations at a corresponding energy. Then one needs to replace the Bohr magneton with the nuclear magneton µ N (determined by the reduced mass of two atoms in a cell). Since phonon excitations yield the fluctuating dipoles which in turn give rise to the magnetization [see Eq. (1)], we calculate the density of states of phonons at a given temperature T , using the spectrum given by Eq. (2). At T ω 0 the phonon density of states is ρ( where q T = T / s ⊥ is the thermal momentum of a phonon, yielding Eq.(3). The estimate in Eq.(3) corresponds to the result obtained below more formally within the Kubo formalism, see Eqs. (13) and (14). The case T ω 0 can be treated in a similar manner [24]. As mentioned before, one can reach the vicinity of the critical value δ c = 1 by applying strain or by O 18 isotope substitution. For example, for the value of 95% of isotope concentration, an approach to a critical point by changing temperature in a relatively narrow interval around T c ≈ 25K was achieved [21]. Moreover, the temperature dependence of the squared frequency of the ferroelectric mode near T c obeyed the Curie-Weiss law. This might imply a suppression of the quantum fluctuations and our results obtained with the use of the bare Greens's functions of optical phonons can be applied in the close proximity of the critical point.
To gain further insight into the role of the fluctuating dipoles associated with the soft phonon modes, it is instructive to derive an expression for the parameter λ in Eq. (1), which relates magnetization and electric polarization, by taking into account its microscopic origin. This analysis is based on the well-known relation between the magnetic moment µ and the orbital angular momentum L for a point-like particle with electric charge e and mass m, µ = (e/2mc)L. Considering the optical long wave oscillations in a system with two atoms per unit cell, we use m + u + + m − u − = 0. Here u + and u − are the displacements of positive and negative ions. Introducing relative displacement vector u = u + − u − , we obtain for the orbital angular momentum of the unit lattice cell, L = (m + m − /(m + + m − ))[u ×u], whereu ≡ ∂u/∂t. For the magnetic moment of the unit cell m = (e + /2c) ζ=± ζ u ζ ×u ζ , we obtain where e + is the effective charge of the cation. The expression for the coupling constant λ directly follows from Eq. (4), if one takes into account that m = a 3 0 M and P = e + u/a 3 0 . Magnetic susceptibility: finite temperature case.-We now derive our main results [including Eq.(3)] using the Kubo formalism at finite temperature. Note, that without an external magnetic field the average magnetization is zero, M = 0. We calculate the magnetic susceptibility χ ij (ω, k), as a function of (real) frequency ω and a wave vector k as given by the Kubo formula Here ω + = ω + i0, [, ] is a commutator and ... is the thermal average, with the magnetization in terms of the electric polarization as given by Eq. (1). Furthermore, the polarization created by the long wave optical phonons reads as [25] where γ ij is a piezotensor, h ξ,q is a unit vector of a phonon polarization, index ξ labels different phonon polarizations, A q = a 3 0 /2M 0 ω ξ (q), ω ξ (q) is the phonon frequency, a 0 is the lattice spacing,M 0 is the reduced mass of the two atoms in the unit cell, and b ξ,q (b † ξ,q ) are annihilation (creation) phonon operators. For simplicity, we consider a case when the tensor γ ij = γδ ij is diagonal which corresponds to a cubic crystal system. Using Eq. (5) we then find where q ≡ d 3 q/(2π) 3 , ω + = ω + i0, and Λ = (16/9)λ 2 γ 4 (a 3 0 /M 0 ) 2 . The details of the calculations are presented in the Supplemental Material (SM) [26]. Note that the result is proportional to the Bose distribution function N (ω ⊥ ). The zero temperature contribution is discussed in the footnote [27]. By using expression for λ from Eq. (1) and the estimate γ e + /a 3 0 [25], the quantity Λ can be expressed as Λ (e + /M 0 c) 2 / µ 2 N / , where we have assumed that (m − − m + )/(m + + m − ) 1. Therefore, the magnetic susceptibility in Eq. (7) scales with the nuclear magneton.
To illustrate the result in Eq. (7), we first consider the case when the system is far from the QCP deep into the paraelectric phase, and the dispersion of the transverse optical phonons is given by Eq. (2) with a fixed gap ω 0 and the transverse sound velocity s 0 . Moreover, we will assume that the temperature is low enough, T ω 0 , implying that the corresponding thermal momentum q T ω 0 /s 0 , with q T √ T ω 0 /( 1/2 s 0 ). Expanding at small momenta k q T , after performing the angular integration, we obtain [26] The denominator of the function in Eq. (8) is the difference between a square of a phase velocity of the external probe, ω/k, and a square of a group velocity of a phonon at a given momentum q, v gr (q) = s 2 0 q/ω 0 . The characteristic group velocity of a phonon is v gr (T ) s 0 T / ω 0 , obtained at q = q T . Therefore, considering the case k q T , T ω 0 , we find from Eq. (8) the imaginary part of the susceptibility in the form while for the real part, up to corresponding higher order corrections, we obtain, Note that χ(0, k) coincides with the dc result of Dzyaloshinskii and Mills, see Eq. (12c) in Ref. [3].
Vicinity of the QCP.-We now consider the magnetic susceptibility near the FE QCP where the phonon spectrum assumes the form as in Eq. (2). Furthermore, in the close proximity of the QCP, |1 − δ| 1, and the gap is small comparing to the scale set by temperature, ω 0 (T )(1 − δ) νz T . The main contribution to the integral in Eq. (7) then arises from the energies ∼ T , when the spectrum ω ⊥ (q) ≈ s ⊥ (δ)q (we here neglect anomalous dimension of ferroelectric fluctuations, i.e. η = 0). At small momenta k q T = T / s ⊥ , using Eq. (7), we then obtain where θ(x) is the step function. This result, together with the Kramers-Kronig relations, yields the following form of the real part of the susceptibility Importantly, in the limit of the zero frequency, we then obtain with Λ µ 2 N / , see Eq. (7). We therefore find a strong increase of the susceptibility in the close proximity of the FE QCP due to the vanishing phonon velocity, z > 1.
Estimates and experimental proposal.-Let us estimate the magnitude of the magnetic susceptibility, Eq. (14), in the case of bulk SrTiO 3 . We consider first the case when the system is far from the QCP, taking s ⊥ = s 0 . From the experimental data of Ref. [16] one can estimate s 0 ≈ 3 × 10 5 cm/s. Then takingM 0 ≈ 12m p , e + = 4|e|, and T = 50K we obtain χ(ω = 0) 10 −11 . If one takes 1 − δ ≈ 0.1, we obtain for the susceptibility near the QCP the value χ(ω = 0) 10 −9 . Both values are experimentally measurable.
STO may be a suitable candidate material for the observation of magnetic signatures on tuning towards the FE QCP because of its incipient ferroelectric nature [28]. Similarly, KTaO 3 , due to its incipient ferroelectric features [29], is also expected to exhibit this phenomenology. Several methods are used for tuning STO towards the FE QCP, such as strain, applied pressure [17,18] or 18 O substitution [19,20,21,22]. Biaxial strain in STO thin films, for example, can confine polarization to the plane perpendicular to the tetragonal c axis but leaves undetermined the polarization direction. This creates a favorable condition for the observation of the magnetic signatures proposed here since it allows strong fluctuations between different in-plane directions of the polarization.
As for the method of tuning STO towards the FE QCP with the use of the oxygen isotope-exchange samples, we should mention here two beautiful experiments [21,22], where the authors demonstrated the ideal softening of the ferroelectric modes at finite temperature T c . To observe the effect proposed in this work, the magnetic susceptibility should be measured as a function of temperature in the close proximity of the phase transition point. A simple experimental setup consists of a SQUID placed above an STO sample.
Conclusions. Multiferoicity as a static phenomenon is well established. On the other hand, the dynamically induced multiferroic phenonena are new and there has been no experiments that explicitly prove it. In this paper, we provide the theoretical estimates and rigorous bounds on the strength of the effect. Furthermore, we have expanded the framework of dynamic multiferroicity and predict strongly enhanced magnetic susceptibility in a paraelectric material near its FE QCP due to thermal fluctuations of the electric dipoles associated with a soft optical mode driving the transition. The predicted effect indicates an alternative way for entangled quantum orders to be induced at finite temperature. We thus further support the concept, proposed in Ref. [4], that any FE QCP is an inherent multiferroic QCP with entangled ferroelectric and ferromagnetic fluctuations. Finally, we have considered STO as a system that can be tuned towards its FE QCP, and provided an estimate of a magnetic susceptibility for this material. The predicted value could be experimentally accessible. One of the possibilities to observe magnetic signatures of fluctuating dipoles is to make SQUID measurements.
We are grateful to K. Dunnett