Ferroelectric Quantum Critical Point Research Articles

Field-induced effects in the spin liquid candidate PbCuTe2O6

${\mathrm{PbCuTe}}_{2}{\mathrm{O}}_{6}$ is considered to be one of the rare candidate materials for a three-dimensional quantum spin liquid (QSL). This assessment was based on the results of various magnetic experiments, performed mainly on polycrystalline material. More recent measurements on single crystals revealed an even more exotic behavior, yielding ferroelectric order below ${T}_{\text{FE}}\ensuremath{\approx}1\phantom{\rule{0.28em}{0ex}}\text{K}$, accompanied by distinct lattice distortions, and a somewhat modified magnetic response which is still consistent with a QSL. Here we report on low-temperature measurements of various thermodynamic, magnetic, and dielectric properties of single-crystalline ${\mathrm{PbCuTe}}_{2}{\mathrm{O}}_{6}$ in magnetic fields $B\ensuremath{\le}14.5\phantom{\rule{0.28em}{0ex}}\text{T}$. The combination of these various probes allows us to construct a detailed $B\text{\ensuremath{-}}T$ phase diagram including a ferroelectric phase for $B\ensuremath{\le}8$ T and a $B$-induced magnetic phase at $B\ensuremath{\ge}11$ T. These phases are preceded by or coincide with a structural transition from a cubic high-temperature phase into a distorted noncubic low-temperature state. The phase diagram discloses a ferroelectric quantum critical point at ${B}_{c1}=7.9\phantom{\rule{0.28em}{0ex}}\text{T}$, where the second-order phase transition line associated with ferroelectric order is suppressed to zero. In addition, a magnetic quantum phase transition is revealed at ${B}_{c2}=11\phantom{\rule{0.28em}{0ex}}\text{T}$. The corresponding phase transition to a field-induced magnetic order at $B>{B}_{c2}$ is likely to be of first order. Field-induced lattice distortions, observed in the state at $T>1$ K and which are assigned to the effect of spin-orbit interaction of the ${\mathrm{Cu}}^{2+}$ ions, are considered as the key mechanism by which the magnetic field couples to the dielectric degrees of freedom in this material.

Quantum critical dynamics in the two-dimensional transverse Ising model

In the vicinity of the quantum critical point (QCP), thermodynamic properties diverge toward zero temperature governed by universal exponents. Although this fact is well known, how it is reflected in quantum dynamics has not been addressed. The QCP of the transverse Ising model on a triangular lattice is an ideal platform to test the issue, since it has an experimental realization, the dielectrics being realized in an organic dimer Mott insulator, $\ensuremath{\kappa}\text{\ensuremath{-}}{\mathrm{ET}}_{2}X$, where a quantum electric dipole represents the Ising degrees of freedom. We track the Glauber-type dynamics of the model by constructing a kinetic protocol based on the quantum Monte Carlo method. The dynamical susceptibility takes the form of the Debye function and shows a significant peak narrowing in approaching a QCP due to the divergence of the relaxation timescale. It explains the anomaly of dielectric constants observed in the organic materials, indicating that the material is very near the ferroelectric QCP. We disclose how the dynamical and other critical exponents develop near QCP beyond the simple field theory.

Superconductivity enhancement in polar metal regions of Sr0.95Ba0.05TiO3 and Sr0.985Ca0.015TiO3 revealed by systematic Nb doping

Two different ferroelectric materials, Sr0.95Ba0.05TiO3 and Sr0.985Ca0.015TiO3, can be turned into polar metals with broken centrosymmetry via electron doping. Systematic substitution of Nb5+ for Ti4+ has revealed that these polar metals both commonly show a simple superconducting dome with a single convex shape. Interestingly, the superconducting transition temperature Tc is enhanced more strongly in these polar metals when compared with the nonpolar matrix Sr(Ti, Nb)O3. The maximum Tc reaches 0.75 K, which is the highest reported value among the SrTiO3-based families to date. However, the Tc enhancement is unexpectedly lower within the vicinity of the putative ferroelectric quantum critical point. The enhancement then becomes much more prominent at locations further inside the dilute carrier-density region, where the screening is less effective. These results suggest that centrosymmetry breaking, i.e., the ferroelectric nature, does not kill the superconductivity. Instead, it enhances the superconductivity directly, despite the absence of strong quantum fluctuations.

Synergetic Ferroelectricity and Superconductivity in Zero-Density Dirac Semimetals near Quantum Criticality.

We study superconductivity in a three-dimensional zero-density Dirac semimetal in proximity to a ferroelectric quantum critical point. We find that the interplay of criticality, inversion-symmetry breaking, and Dirac dispersion gives rise to a robust superconducting state at the charge-neutrality point, where no Fermi surface is present. Using Eliashberg theory, we show that the ferroelectric quantum critical point is unstable against the formation of a ferroelectric density wave (FDW), whose fluctuations, in turn, lead to a first-order superconducting transition. Surprisingly, long-range superconducting and FDW orders are found to cooperate with each other, in contrast to the more usual scenario of phase competition. Therefore, we suggest that driving charge neutral Dirac materials, e.g., Pb_{x}Sn_{1-x}Te, through a ferroelectric quantum critical point may lead to superconductivity intertwined with FDW order.

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 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.

Superconductivity mediated by polar modes in ferroelectric metals

The occurrence of superconductivity in doped SrTiO3 at low carrier densities points to the presence of an unusually strong pairing interaction that has eluded understanding for several decades. We report experimental results showing the pressure dependence of the superconducting transition temperature, Tc, near to optimal doping that sheds light on the nature of this interaction. We find that Tc increases dramatically when the energy gap of the ferroelectric critical modes is suppressed, i.e., as the ferroelectric quantum critical point is approached in a way reminiscent to behaviour observed in magnetic counterparts. However, in contrast to the latter, the coupling of the carriers to the critical modes in ferroelectrics is predicted to be small. We present a quantitative model involving the dynamical screening of the Coulomb interaction and show that an enhancement of Tc near to a ferroelectric quantum critical point can arise due to the virtual exchange of longitudinal hybrid-polar-modes, even in the absence of a strong coupling to the transverse critical modes.

Quantum critical phenomena in a compressible displacive ferroelectric

The dielectric and magnetic polarizations of quantum paraelectrics and paramagnetic materials have in many cases been found to initially increase with increasing thermal disorder and hence, exhibit peaks as a function of temperature. A quantitative description of these examples of "order-by-disorder" phenomena has remained elusive in nearly ferromagnetic metals and in dielectrics on the border of displacive ferroelectric transitions. Here, we present an experimental study of the evolution of the dielectric susceptibility peak as a function of pressure in the nearly ferroelectric material, strontium titanate, which reveals that the peak position collapses toward absolute zero as the ferroelectric quantum critical point is approached. We show that this behavior can be described in detail without the use of adjustable parameters in terms of the Larkin-Khmelnitskii-Shneerson-Rechester (LKSR) theory, first introduced nearly 50 y ago, of the hybridization of polar and acoustic modes in quantum paraelectrics, in contrast to alternative models that have been proposed. Our study allows us to construct a detailed temperature-pressure phase diagram of a material on the border of a ferroelectric quantum critical point comprising ferroelectric, quantum critical paraelectric, and hybridized polar-acoustic regimes. Furthermore, at the lowest temperatures, below the susceptibility maximum, we observe a regime characterized by a linear temperature dependence of the inverse susceptibility that differs sharply from the quartic temperature dependence predicted by the LKSR theory. We find that this non-LKSR low-temperature regime cannot be accounted for in terms of any detailed model reported in the literature, and its interpretation poses an empirical and conceptual challenge.

Anisotropic superconductivity mediated by ferroelectric fluctuations in cubic systems with spin-orbit coupling

Motivated by the experimental observation that superconductivity in bulk doped SrTiO$_{3}$ is enhanced as a putative ferroelectric quantum critical point (FE-QCP) is approached, we study the pairing instability of a cubic system in which electrons exchange low-energy ferroelectric fluctuations. Instead of the gradient coupling to the lattice distortion associated with ferroelectricity, we consider a direct coupling between the electrons and the bosonic ferroelectric field that appears in the presence of spin-orbit coupling. Working in the weak-coupling regime, we find that the pairing interaction is dominated by the soft transverse optical (TO) mode, resulting in a $T_{c}$ enhancement upon approaching the FE-QCP. Focusing on even-parity states, we find that although the $s$-wave state always wins, states with higher Cooper-pair angular momentum become close competitors as the TO mode softens. We show that the cubic anisotropy of the FE fluctuations mixes the $s$-wave and $g$-wave states, resulting in a characteristic anisotropy of the gap function. The gap anisotropy behaves non-monotonically as the FE-QCP is approached: upon decreasing the TO mode frequency, the gap anisotropy first changes sign and then increases in magnitude. We discuss the possible applications of our results to the superconducting state of SrTiO$_{3}$.

Anharmonic coupling between electrons and TO phonons in the vicinity of a ferroelectric quantum critical point

We explore a novel coupling mechanism of electrons with the transverse optical (TO) phonon branch in a regime when the TO mode becomes highly anharmonic and drives the ferroelectric phase transition. We show that this anharmonicity, which leads to a collective motion of ions, is able to couple electronic and lattice displacement fields. An effective correlated electron-ion dynamics method is required to capture the effect of the onset of the local electric polarization due to this collective behavior close to the quantum critical point. We identify an intermediate temperature range where an emergent phonon drag may contribute substantially to thermoelectric conductivity in this regime. We find that, under optimal conditions, this extra contribution may be larger than values achieved so far in the benchmark material, PbTe. In the last part we make a case for the importance of our results in the generic problem of anharmonic electron-lattice dynamics.

Reply to “Comment on ‘Superconductivity at low density near a ferroelectric quantum critical point: Doped SrTiO3 ”'

In our paper (W\"olfle and Balatsky, Phys. Rev. B 98, 104505 (2018)) we presented a microscopic theory of superconductivity for doped SrTiO$_{3}$ by proposing two pairing mechanisms acting simultaneously with relative strength depending on the closeness to the ferroelectric quantum critical point. The first mechanism rests on the dynamically screened Coulomb interaction, and the second assumed a coupling to the soft transverse optical phonon. In their comment Ruhman and Lee point out an error in our estimate of the deformation potential coupling to the soft mode. We agree that this type of coupling cannot explain the gigantic isotope effect observed experimentally, so that a different coupling mechanism needs to be found. As for the first pairing mechanism, Ruhman and Lee maintain the view expressed in their paper (Ruhman and Lee, Phys. Rev. B 94, 224515 (2016)) that the energy range over which the usual longitudinal optical phonon mediated interaction operates is limited by the Fermi energy. We object to this view and in this reply present evidence that the cutoff energy is much larger. In a weak coupling system such as SrTiO$_{3}$ the cutoff is given by the energy beyond which quasiparticles cease to be well defined.

Superconductivity near a Ferroelectric Quantum Critical Point in Ultralow-Density Dirac Materials

The experimental observation of superconductivity in doped semimetals and semiconductors, where the Fermi energy is comparable to or smaller than the characteristic phonon frequencies, is not captured by the conventional theory. In this paper, we propose a mechanism for superconductivity in ultralow-density three-dimensional Dirac materials based on the proximity to a ferroelectric quantum critical point. We derive a low-energy theory that takes into account both the strong Coulomb interaction and the direct coupling between the electrons and the soft phonon modes. We show that the Coulomb repulsion is strongly screened by the lattice polarization near the critical point even in the case of vanishing carrier density. Using a renormalization group analysis, we demonstrate that the effective electron-electron interaction is dominantly mediated by the transverse phonon mode. We find that the system generically flows towards strong electron-phonon coupling. Hence, we propose a new mechanism to simultaneously produce an attractive interaction and suppress strong Coulomb repulsion, which does not require retardation. For comparison, we perform same analysis for covalent crystals, where lattice polarization is negligible. We obtain qualitatively similar results, though the screening of the Coulomb repulsion is much weaker. We then apply our results to study superconductivity in the low-density limit. We find strong enhancement of the transition temperature upon approaching the quantum critical point. Finally, we also discuss scenarios to realize a topological $p$-wave superconducting state in covalent crystals close to the critical point.

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 ∼5×10^{-7} μ_{B} per unit cell near the FE QCP.

Dielectric Response of Quantum Critical Ferroelectric as a Function of Pressure

In this work we report for the first time measurements of the dielectric loss of single-crystal SrTiO3 under the application of hydrostatic pressure up to 20 kbar and temperatures down to 200 mK which allow us to comment on the evolution of new fundamental material properties and their relationship with the recently discovered quantum critical phenomena in this material. The well known 18 K peak or shoulder was no longer observed after pressure was applied, even after subsequently removing it, suggesting it is associated with the twin walls formed at the 110 K cubic-tetragonal transition. The family of familiar peaks were all seen to increase in temperature linearly with pressure and the height of the 9.4 K peak was drastically suppressed by even the smallest pressures. This peak is discussed in the context of a postulated ferroelectric quantum critical point in SrTiO3 and the behaviour of its size linked to the position of this point on the recently established phase diagram.

Quantum Phenomena Emerging Near a Ferroelectric Critical Point in a Donor–Acceptor Organic Charge-Transfer Complex

When a second-order transition point is decreased to zero temperature, a continuous quantum phase transition between different ground states is realized at a quantum critical point (QCP). A recently synthesized organic charge-transfer complex, TTF-2,5-QBr 2 I 2 , provides a platform for the exploration of the quantum phenomena that accompany a ferroelectric QCP. Here, we summarize the recent results showing the quantum phenomena associated with the ferroelectric QCP in TTF-2,5-QBr 2 I 2 . Whereas the enhanced quantum fluctuations lead to quantitative changes in the critical exponents of the critical phenomena, they qualitatively change the nature of the domain-wall kinetics from thermally activated motion to temperature-independent tunneling motion. The present findings highlight the great influence of quantum fluctuations on the low-temperature physical properties and suggest that TTF-2,5-QBr 2 I 2 is a model system for the uniaxial ferroelectric QCP.

Athermal domain-wall creep near a ferroelectric quantum critical point.

Ferroelectric domain walls are typically stationary because of the presence of a pinning potential. Nevertheless, thermally activated, irreversible creep motion can occur under a moderate electric field, thereby underlying rewritable and non-volatile memory applications. Conversely, as the temperature decreases, the occurrence of creep motion becomes less likely and eventually impossible under realistic electric-field magnitudes. Here we show that such frozen ferroelectric domain walls recover their mobility under the influence of quantum fluctuations. Nonlinear permittivity and polarization-retention measurements of an organic charge-transfer complex reveal that ferroelectric domain-wall creep occurs via an athermal process when the system is tuned close to a pressure-driven ferroelectric quantum critical point. Despite the heavy masses of material building blocks such as molecules, the estimated effective mass of the domain wall is comparable to the proton mass, indicating the realization of a ferroelectric domain wall with a quantum-particle nature near the quantum critical point.

Quantum Critical Origin of the Superconducting Dome in SrTiO_{3}.

We expand the well-known notion that quantum criticality can induce superconductivity by proposing a concrete mechanism for superconductivity due to quantum ferroelectric fluctuations. To this end, we investigate the origin of superconductivity in doped SrTiO_{3} using a combination of density functional and strong coupling theories within the framework of quantum criticality. Our density functional calculations of the ferroelectric soft mode frequency as a function of doping reveal a crossover related to quantum paraelectricity at a doping level coincident with the experimentally observed top of the superconducting dome. Thus, we suggest a model in which the soft mode fluctuations provide the pairing interaction for superconductivity carriers. Within our model, the low doping limit of the superconducting dome is explained by the emergence of the Fermi surface, and the high doping limit by departure from the quantum critical regime. We predict that the highest critical temperature will increase and shift to lower carrier doping with increasing ^{18}O isotope substitution, a scenario that is experimentally verifiable. Our model is applicable to other quantum paraelectrics, such as KTaO_{3}.

Quantum critical behavior of a magnetic quantum paraelectric

We discuss the interplay between anti-ferromagnetic order and polarization fluctuations in a magnetic quantum paraelectric. Using an action where anti-ferromagnetic order parameter couples to the polarization fluctuations and as well as magnetic field, we derive a set of self consistent equations to study both the temperature and the magnetic field dependence of the static dielectric susceptibility. The temperature dependence of dielectric susceptibility near both the anti-ferromagnetic quantum critical point and ferroelectric quantum critical point are described using scaling arguments. Discussions on achieving various quantum critical points in experiments are also made.