Quantum annealed criticality beyond the isotropic limit: anisotropic elasticity, dipolar interactions, domain physics, and quantitative tests of the generalized Larkin–Pikin framework

The generalized Larkin–Pikin scaling theory (Chandra, Coleman, Continentino and Lonzarich, Phys. Rev. Research 2, 043440 (2020)) shows that in a compressible solid whose scalar order parameter couples to isotropic elastic strain, zero-point fluctuations anneal the strain-induced first-order transition above the upper critical dimension ($d+z>4$), so that a line of classical first-order transitions terminates in a continuous quantum annealed critical point. That theory assumes isotropic elasticity, a scalar order parameter, no long-range dipolar forces, and no domain physics, and it has been confronted with experiment only qualitatively. This problem asks for the realistic-materials completion of the framework: incorporate elastic anisotropy with strain-compatibility constraints and decide whether it drives inhomogeneous or domain states in the quantum regime; develop the quantum coupling of domain dynamics to anisotropic strain; include long-range dipolar interactions and compute their modification of the dynamical kernel and the generalized instability criterion; and derive and test material-specific quantitative predictions (strain-coupling and bulk-modulus magnitudes, amplitude factors, Grüneisen-ratio and elastic anomalies) in pressure-tuning experiments. A subsidiary mechanism-level question is whether the $q=0$ boundary-induced Larkin–Pikin long-range interaction admits a topological bulk–boundary or anomaly interpretation.

Background

Phase transitions are called first-order when the order parameter (magnetization, ferroelectric polarization) jumps discontinuously and latent heat is absorbed, and continuous (critical) when it vanishes smoothly with a diverging correlation length. In a solid the order parameter couples to the elastic strain — the dimensionless deformation of the lattice — because the transition temperature depends on the sample volume. Larkin and Pikin (1969) considered a compressible solid in which the order-parameter energy density couples linearly, with strength $\lambda$, to the isotropic (volume-changing) strain. Comparing the solid clamped at fixed volume, where strain is frozen, with the free solid at fixed pressure, they showed that when the clamped system has a continuous transition with a diverging specific heat (specific-heat exponent $\alpha>0$), the strain degree of freedom mediates a long-range interaction between energy-density fluctuations that turns the free solid's transition first-order. Their criterion reads $\kappa \lesssim (\Delta C_V/T_c)(dT_c/d\ln V)^2$, where $\Delta C_V$ is the singular part of the clamped specific heat, $dT_c/d\ln V$ measures the volume-strain sensitivity of the transition, and $\kappa$ is an effective bulk modulus ($\kappa^{-1}=K^{-1}-(K+4\mu/3)^{-1}$, with $K$ the bulk modulus and $\mu$ the shear modulus). Subsequent classical work refined this picture: renormalization-group analyses showed that for weak magnetoelastic coupling the transition of a compressible magnet can remain continuous with modified exponents, and studies of anisotropic (direction-dependent) elastic coupling — Bergman and Halperin; de Moura, Lubensky, Imry and Aharony; Bruno and Sak — found behavior qualitatively different from the isotropic case, including the possibility of macroscopically inhomogeneous states.

At zero temperature a different class of fluctuations takes over. A quantum critical point (QCP) is reached by tuning a non-thermal parameter $g$ (pressure, composition, field); its scaling ties space and imaginary time together through a dynamical exponent $z$, so the effective dimensionality is $d+z$, and above the upper critical dimension ($d+z>4$) the critical fluctuations are Gaussian. Chandra, Coleman, Continentino and Lonzarich (2020) generalized the Larkin–Pikin mechanism to this quantum regime using crossover scaling that interpolates between the quantum and the finite-temperature classical critical points: temperature acts as a finite cutoff in imaginary time, limiting the quantum correlation length to $\xi_Q\sim(\hbar/k_BT)^{1/z}$, so thermal fluctuations are suppressed (annealed) at short distances by zero-point motion. As $T$ decreases, the amplitude factors $A_I(T)$ and $A_{II}(g)$ governing the classical critical behavior shrink, and for $d+z>4$ the latent work of the transition vanishes at $T=0$: the line of classical first-order transitions terminates in a continuous quantum annealed critical point at which the underlying order-parameter criticality is restored. The resulting phase diagram in temperature $T$, tuning parameter $g$, and field $h$ conjugate to the order parameter contains three multicritical points — two classical critical endpoints bounding the sheet of first-order transitions, and the quantum critical point.

The experimental motivation comes from ferroelectric and magnetic materials that display quantum criticality although their ambient-temperature transitions are first-order. Quantum paraelectrics such as SrTiO3 sit at the border of ferroelectricity, and pressure or substitution drives them toward ferroelectric QCPs whose dielectric susceptibility is described by self-consistent Gaussian scaling (Rowley et al. 2014; Chandra et al. 2017; Coak et al. 2020). In ferroelectrics the long-range dipolar forces between polarization charges (falling off as $1/r^3$) are known to gap the longitudinal polarization fluctuations while transverse fluctuations remain critical, without destroying the quantum criticality (Roussev and Millis 2003, building on earlier Soviet-era work). Sharp experimental markers include the Grüneisen ratio — thermal expansion divided by specific heat — predicted to diverge as $1/T^2$ at a three-dimensional ferroelectric QCP, together with anomalies in the bulk modulus and the sound velocity.

The 2020 theory is, however, deliberately idealized: isotropic elasticity, a scalar (Ising-like) order parameter, no dipolar forces, no domains. The authors themselves list what remains open. Elastic anisotropy may drive the system into an inhomogeneous state classically, and its quantum consequences — including possible new instabilities at the quantum annealed critical point — were left unexamined; the coupling of domain dynamics to anisotropic strain has been studied only classically for ferroelectrics (Brierley and Littlewood 2014). Long-range dipolar interactions are not included at all, beyond the qualitative gapping statement. Quantitative, material-specific predictions and their confrontation with pressure experiments remain to be worked out. And at the mechanism level, the authors note that the long-range Larkin–Pikin interaction is carried by the $q=0$ (uniform, boundary-dominated) component of the strain: the integral of the strain around a closed loop on a torus equals the Burgers vector — the net lattice displacement accumulated around the enclosed dislocations — of those defects, a topological invariant connected with torsion (the geometric measure of dislocation density in the theory of elastic defects; Katanaev). This suggests a possible bulk–boundary correspondence, or an interpretation of the interaction, which breaks the Lorentz invariance of the underlying short-range critical theory, as a symmetry-breaking anomaly (a symmetry of the short-distance theory that fails to survive in the presence of the long-range strain coupling).

Problem Statement

Establish the quantum-annealing theory of strain-induced first-order transitions — the generalized Larkin–Pikin framework of Chandra, Coleman, Continentino and Lonzarich (2020) — for realistic compressible solids, and thereby determine whether the quantum annealed critical point survives or is preempted. The problem comprises four linked deliverables that jointly support this single objective:

(1) Anisotropic elasticity. Incorporate elastic anisotropy — with the Saint-Venant compatibility constraints (the differential conditions ensuring that the strain field derives from a single-valued displacement field) — into the quantum generalized Larkin–Pikin action, and determine the fate of the quantum annealed critical point: does anisotropic strain coupling drive inhomogeneous (domain or tweed-like, i.e. regular cross-hatched precursor-pattern) states or new instabilities in the quantum regime, generalizing the classical results on anisotropic elastic coupling?

(2) Quantum domain–strain coupling. Develop the low-temperature counterpart of the classical coupling between domain dynamics and anisotropic strain: how do domain walls and their fluctuations interact with anisotropic strain near the quantum annealed critical point, within or beyond the crossover-scaling treatment?

(3) Long-range dipolar interactions. Include long-range dipolar forces in the generalized Larkin–Pikin framework and compute quantitatively how they modify the dynamical kernel $V_{dyn}(q)$ of the strain-mediated long-range interaction and the generalized instability criterion, going beyond the qualitative statement that longitudinal fluctuations are gapped while transverse ones remain critical.

(4) Quantitative experimental confrontation. Derive material-specific predictions — the magnitudes of the strain coupling $\lambda$ and the effective bulk modulus $\kappa$, the amplitude factors $A_I(T)$ and $A_{II}(g)$ of the crossover scaling, the Grüneisen-ratio divergence, and bulk-modulus and sound-velocity anomalies — and test them against pressure-tuning measurements on candidate magnetic or ferroelectric materials whose ambient-pressure transitions are first-order.

Deliverables (1)–(3) jointly fix the phase diagram and stability of quantum annealed criticality in real solids; deliverable (4) certifies the framework experimentally. A complete answer requires all four deliverables (1)–(4) and states whether the quantum annealed critical point survives anisotropic and dipolar realism, is preempted by a weak first-order transition, or gives way to inhomogeneous states. Deliverable (4) is discharged by at least one concrete, quantitative confrontation that names the candidate material and its pressure-tuning data set and reports agreement within the measurement uncertainties or a documented discrepancy; the choice of candidate material is otherwise left open. A mechanism-level sub-question that may be discharged alongside these: whether the $q=0$ boundary-induced Larkin–Pikin long-range interaction admits a topological bulk–boundary interpretation or a symmetry-breaking-anomaly formulation.

References

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