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Finding symmetry breaking order parameters with Euclidean neural networks
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Switching of Hybrid Improper Ferroelectricity in Oxide Double Perovskites
In ABO 3 -type perovskite oxides with Pnma symmetry, rotation (Q R+ , a 0 a 0 c + ) and tilt (Q T , a – a – c 0 ) of BO 6 octahedra are the two primary order parameters. These order parameters establish an inherent trilinear coupling with anti-ferroelectric A-site displacement (Q AFE ) to form the low-symmetry phase. The symmetry is further lowered in double perovskite oxides (DPOs) due to A/A' cation ordering. It in turn makes these systems polar via hybrid improper ferroelectric mechanism, primarily driven by Q R+ and Q T . Naturally, it has been believed that functionalities such as polarization can also be switched by tuning these primary order parameters. However, mystery around finding switching mechanism still remains. Our study based on density functional theory calculations combined with finite-temperature molecular dynamics simulations shows that the polarization switching is a two-step process, driven by out-of-phase rotation (Q R– , a 0 a 0 c – when Q T = 0 or, a – a – b – when Q T ≠ 0). A series of polar DPOs such as KLnFeOsO 6 [Ln = Sm, Gd, Dy, Tm (lanthanides) and Y (rare earth)], all belonging to P2 1 symmetry, are considered in this investigation. The polarization switching P ($\overrightarrow{P}$) occurs at a very high temperature of ~1150 K through a phase transition, from a polar (P2 1 ) phase with $\overrightarrow{P}$(+) to $\overrightarrow{P}$(-) via a non-polar P4/n phase. The switching itself is metastable in nature. The switching (both polarization and spin state) is only observed for a very short period of time (~23 ps) that poses limitation on using such a mechanism in memory device realization. We demonstrate a concurrent heating–cooling procedure to overcome such shortcoming. In conclusion, simulations conducted at 600 K further imply that long lasting switching can be achieved, at least for 1.2 ns for 600 K, and ideally for an infinite time, if the material is heated just above the T c followed by rapid cooling to a temperature below T c .
Phase equilibrium of liquid water and hexagonal ice from enhanced sampling molecular dynamics simulations
We study the phase equilibrium between liquid water and ice Ih modeled by the TIP4P/Ice interatomic potential using enhanced sampling molecular dynamics simulations. Our approach is based on the calculation of ice Ih-liquid free energy differences from simulations that visit reversibly both phases. The reversible interconversion is achieved by introducing a static bias potential as a function of an order parameter. The order parameter was tailored to crystallize the hexagonal diamond structure of oxygen in ice Ih. Furthermore, we analyze the effect of the system size on the ice Ih-liquid free energy differences, and we obtain a melting temperature of 270 K in the thermodynamic limit. This result is in agreement with estimates from thermodynamic integration (272 K) and coexistence simulations (270 K). Since the order parameter does not include information about the coordinates of the protons, the spontaneously formed solid configurations contain proton disorder as expected for ice Ih.
Theory of topological defects and textures in two-dimensional quantum orders with spontaneous symmetry breaking
In this article, we consider two-dimensional (2d) quantum many-body systems with long-range orders, where the only gapless excitations in the spectrum are Goldstone modes of spontaneously broken continuous symmetries. To understand the interplay between classical long-range order of local order parameters and quantum order of long-range entanglement in the ground states, we study the topological point defects and textures of order parameters in such systems. We show that the universal properties of point defects and textures are determined by the remnant symmetry enriched topological order in the symmetry-breaking ground states with a nonfluctuating order parameter, and provide a classification for their properties based on the inflation-restriction exact sequence. We highlight a few phenomena revealed by our theory framework. First, in the absence of intrinsic topological orders, we show a connection between the symmetry properties of point defects and textures to deconfined quantum criticality. Second, when the symmetry-breaking ground state has intrinsic topological orders, we show that the point defects can permute different anyons when braided around. They can also obey projective fusion rules in the sense that multiple vortices can fuse into an Abelian anyon, a phenomenon for which we coin “defect fractionalization.” Finally, we provide a formula to compute the fractional statistics and fractional quantum numbers carried by textures (skyrmions) in Abelian topological orders.
Landau-Ginzburg theory of charge density wave formation accompanying lattice and electronic long-range ordering
Here we propose an analytical Landau-Ginzburg (LG) theory of the charge density waves coupled with lattice and electronic long-range order parameters. Examples of long-range order include the electronic wave function of superconducting Cooper pairs, structural distortions, electric polarization, and magnetization. We formulate the LG free energy density as a power expansion with respect to the charge density and other long-range order parameters as well as their spatial gradients and biquadratic coupling terms. We introduced a biquadratic coupling between the charge density gradient and long-range order parameters as well as nonlinear higher gradients of the long-range order parameters. The biquadratic gradient coupling is critical to the appearance of different spatially modulated phases in charge-ordered ferroics and high-temperature superconductors. We derived the thermodynamic conditions for the stability of the spatially modulated phases, which are the intertwined spatial waves of charge density and lattice/electronic long-range order. The analytical expressions for the energies of different phases, corresponding order parameters, charge density waves amplitudes, and modulation periods obtained in this paper can be employed to guide the comprehensive physical explanation, deconvolution, and Bayesian analysis of experimental data on quantum materials ranging from charge-ordered ferroics to high-temperature superconductors.
Kinematics and dynamics of disclination lines in three-dimensional nematics
An exact kinematic law for the motion of disclination lines in nematic liquid crystals as a function of the tensor order parameter Q is derived. Unlike other order parameter fields that become singular at their respective defect cores, the tensor order parameter remains regular. Following earlier experimental and theoretical work, the disclination core is defined to be the line where the uniaxial and biaxial order parameters are equal, or equivalently, where the two largest eigenvalues of Q cross. This allows an exact expression relating the velocity of the line to spatial and temporal derivatives of Q on the line, to be specified by a dynamical model for the evolution of the nematic. By introducing a linear core approximation for Q, analytical results are given for several prototypical configurations, including line interactions and motion, loop annihilation, and the response to external fields and shear flows. Behaviour that follows from topological constraints or defect geometry is highlighted. Finally, the analytic results are shown to be in agreement with three-dimensional numerical calculations based on a singular Maier–Saupe free energy that allows for anisotropic elasticity.
A physics-constrained neural network for multiphase flows
The present study develops a physics-constrained neural network (PCNN) to predict sequential patterns and motions of multiphase flows (MPFs), which includes strong interactions among various fluid phases. To predict the order parameters, which locate individual phases in the future time, a neural network (NN) is applied to quickly infer the dynamics of the phases by encoding observations. The multiphase consistent and conservative boundedness mapping algorithm (MCBOM) is next implemented to correct the predicted order parameters. This enforces the predicted order parameters to strictly satisfy the mass conservation, the summation of the volume fractions of the phases to be unity, the consistency of reduction, and the boundedness of the order parameters. Then, the density of the fluid mixture is updated from the corrected order parameters. Finally, the velocity in the future time is predicted by another NN with the same network structure, but the conservation of momentum is included in the loss function to shrink the parameter space. The proposed PCNN for MPFs sequentially performs (NN)-(MCBOM)-(NN), which avoids nonphysical behaviors of the order parameters, accelerates the convergence, and requires fewer data to make predictions. Numerical experiments demonstrate that the proposed PCNN is capable of predicting MPFs effectively.
Signatures of Ultrafast Reversal of Excitonic Order in Ta2NiSe 5
In the presence of electron-phonon coupling, an excitonic insulator harbors two degenerate ground states described by an Ising-type order parameter. Starting from a microscopic Hamiltonian, we derive the equations of motion for the Ising order parameter in the phonon coupled excitonic insulator Ta 2 NiSe 5 and show that it can be controllably reversed on ultrashort timescales using appropriate laser pulse sequences. Using a combination of theory and time-resolved optical reflectivity measurements, we report evidence of such order parameter reversal in Ta 2 NiSe 5 based on the anomalous behavior of its coherently excited order-parameter-coupled phonons. Finally, our Letter expands the field of ultrafast order parameter control beyond spin and charge ordered materials.
Classification of electronic nematicity in three-dimensional crystals and quasicrystals
Electronic nematic order has been reported in a rich landscape of materials, encompassing not only a range of intertwined correlated and topological phenomena but also different underlying lattice symmetries. Motivated by these findings, we investigate the behavior of electronic nematicity as the spherical symmetry of three-dimensional (3D) space is systematically lowered by the lattice environment. Here, we consider all 32 crystallographic point groups as well as four major classes of quasicrystalline point groups, given the recent observations of electronic phases of interest in quasicrystalline materials and artificial twisted quasicrystals. Valuable insights are gained by establishing a mapping between the five-component charge-quadrupolar nematic order parameter of the electronic fluid and the 3D tensorial order parameter of nematic liquid crystals. We find that a uniaxial nematic state is only generically realized in polyhedral point groups (icosahedral and cubic), with the nematic director pointing along different sets of rotational symmetry axes. Interestingly, icosahedral point groups are the only ones in which the five nematic order parameter components transform as the same irreducible representation, making them the closest analog of 3D isotropic nematics. In axial point groups, one of the nematic components is always condensed, whereas the other four components decompose into an in-plane and an out-of-plane nematic doublet, resulting in biaxial nematic ground states. Because these two nematic doublets behave as Z q clock order parameters, this allows us to identify the types of crystals and quasicrystals that can host interesting electronic nematic phenomena enabled by the critical properties of the q ≥ 4 clock model, such as emergent continuous nematic fluctuations in 3D, critical phases with quasi-long-range nematic order in two dimensions (2D), and Ashkin-Teller nematicity in 2D.
Toward Automated Sampling of Polymorph Nucleation and Free Energies with the SGOOP and Metadynamics
Understanding the driving forces behind the nucleation of different polymorphs is of great importance for material sciences and the pharmaceutical industry. This includes understanding the reaction coordinate that governs the nucleation process and correctly calculating the relative free energies of different polymorphs. Here, we demonstrate, for the prototypical case of urea nucleation from the melt, how one can learn such a one-dimensional reaction coordinate as a function of prespecified order parameters and use it to perform efficient biased all-atom molecular dynamics simulations. The reaction coordinate is learnt as a function of the generic thermodynamic and structural order parameters using the “spectral gap optimization of order parameters (SGOOP)” approach [Tiwary, P. and Berne, B. J. Proc. Natl. Acad. Sci. U.S.A. (2016)] and is biased using well-tempered metadynamics simulations. The reaction coordinate gives insights into the role played by different structural and thermodynamics order parameters, and the biased simulations obtain accurate relative free energies for different polymorphs. This includes an accurate prediction of the approximate pressure at which urea undergoes a phase transition and one of the metastable polymorphs becomes the most stable conformation. Here, we believe the ideas demonstrated in this work will facilitate efficient sampling of nucleation in complex, generic systems.
Group-equivariant autoencoder for identifying spontaneously broken symmetries
We introduce the group-equivariant autoencoder (GE autoencoder), a deep neural network (DNN) method that locates phase boundaries by determining which symmetries of the Hamiltonian have spontaneously broken at each temperature. We use group theory to deduce which symmetries of the system remain intact in all phases, and then use this information to constrain the parameters of the GE autoencoder such that the encoder learns an order parameter invariant to these “never-broken” symmetries. This procedure produces a dramatic reduction in the number of free parameters such that the GE-autoencoder size is independent of the system size. We include symmetry regularization terms in the loss function of the GE autoencoder so that the learned order parameter is also equivariant to the remaining symmetries of the system. By examining the group representation by which the learned order parameter transforms, we are then able to extract information about the associated spontaneous symmetry breaking. We test the GE autoencoder on the 2D classical ferromagnetic and antiferromagnetic Ising models, finding that the GE autoencoder (1) accurately determines which symmetries have spontaneously broken at each temperature; (2) estimates the critical temperature in the thermodynamic limit with greater accuracy, robustness, and time efficiency than a symmetry-agnostic baseline autoencoder; and (3) detects the presence of an external symmetry-breaking magnetic field with greater sensitivity than the baseline method. Lastly, we describe various key implementation details, including a quadratic-programming-based method for extracting the critical temperature estimate from trained autoencoders and calculations of the DNN initialization and learning rate settings required for fair model comparisons.
Role of electromagnetic gauge-field fluctuations in the selection between chiral and nematic superconductivity
Motivated by the observation of nematic superconductivity in several systems, we revisit the problem of the leading pairing instability of two-component unconventional superconductors on the triangular lattice—such as (p x , p y )-wave and (d x 2 –y 2 , d xy )-wave superconductors. Such a system has two possible superconducting states: the chiral state (e.g., p + ip or d + id), which breaks time-reversal symmetry, and the nematic state (e.g., p + p or d + d), which breaks the threefold rotational symmetry of the lattice. Weak-coupling calculations generally favor the chiral over the nematic superconducting state, raising the question of what mechanism can stabilize the latter. Here, we show that the electromagnetic field fluctuations can play a crucial role in selecting between these two states. Specifically, we derive and analyze the effective free energy for the two-component superconducting order parameter after integrating out the gauge-field fluctuations, which is formally justified if the spatial order parameter fluctuations can be neglected. A nonanalytic cubic term arises, as in the case of a conventional s -wave superconductor. However, unlike the latter, the cubic term depends on the relative phase and on the relative amplitudes between the two order parameter components, in such a way that it generally favors the nematic state. This result is a direct consequence of the fact that the stiffness of the superconducting order parameter is not isotropic. Competition with the quartic term, which favors the chiral state, leads to a renormalized phase diagram in which the nematic state displaces the chiral state over a wide region in the parameter space. Here, we analyze the stability of the fluctuation-induced nematic phase, generalize our results to tetragonal lattices, and discuss their applicability to candidate nematic superconductors, including twisted bilayer graphene.
Pairing symmetry of twisted bilayer graphene: A phenomenological synthesis
One of the outstanding questions in the study of twisted bilayer graphene—from both experimental and theoretical points of view—is the nature of its superconducting phase. In this paper we perform a comprehensive synthesis of existing experiments, and argue that experimental constraints are strong enough to allow the structure of the superconducting order parameter to be nearly uniquely determined. In particular, we argue that the order parameter is nodal, and is formed from an admixture of spin-singlet and spin-triplet Cooper pairs. This argument is made on phenomenological grounds, without committing to any particular microscopic model of the superconductor. Existing data is insufficient to determine the orbital parity of the order parameter, which could be either p wave or d wave. We propose a way in which the measurement of Andreev edge states can be used to distinguish between the two.
Effect of hydride vapor phase epitaxy growth conditions on the degree of atomic ordering in GaInP
We report a systematic study of CuPt-type ordering in hydride vapor phase epitaxy (HVPE)-grown Ga 0.5 In 0.5 P. Selected-area electron diffraction reveals ordering in samples grown on (001) GaAs substrates offcut toward (1$\overline{1}$1)B. The ordering is single-variant, occurring only on (1$\overline{1}$1)B planes and not on ($\overline{1}$11)B. Quantitative analysis of the order parameter by high-resolution x-ray diffraction (HRXRD) indicates that ordering increases with deposition temperature in samples grown at 600–700 °C with a constant gas-phase V/III ratio ~3. Ordering increases with V/III ratio in the range of 1.3–6.7 at a constant deposition temperature of 650°C. Photoluminescence measurements correlate the order parameter with Ga0.5In0.5P bandgap contraction, though the contraction is larger than expected based on the magnitude of order parameters measured by HRXRD. A possible reason for this discrepancy is that the photoluminescence emission occurs in the lower bandgap ordered domains, which are small and evenly dispersed throughout the material. We also show that the degree of ordering decreases with growth rate, disappearing at ~60 µm/h. The HVPE-grown material exhibits a generally weaker ordering than organometallic vapor phase epitaxy (OMVPE)-grown material, likely due to the moderate V/III ratios employed, in contrast to the V/III ratios in the 100s typical of OMVPE. However, the tendency for ordering to increase with V/III ratio suggests that the same dimer-induced stress mechanism used to explain the occurrence of ordering in OMVPE-grown material also applies to HVPE. The tendencies for ordering to increase with deposition temperature and decrease with growth rate show that kinetics limit the degree of ordering, also in agreement with OMVPE trends.
Impurity-Induced Anomalous Thermal Hall Effect in Chiral Superconductors
Chiral superconductors exhibit novel transport properties that rely on the topology of the order parameter, topology of the Fermi surface, the spectrum of bulk and edge Fermionic excitations, and the structure of the impurity potential. In the case of electronic heat transport, impurities induce an anomalous (zero-field) thermal Hall conductivity that is easily orders of magnitude larger than the quantized edge contribution. The effect originates from branch-conversion scattering of Bogoliubov quasiparticles by the chiral order parameter, induced by potential scattering. The former transfers angular momentum between the condensate and the excitations that transport heat. The anomalous thermal Hall conductivity is shown to depend to the structure of the electron-impurity potential, as well as the winding number ν of the chiral order parameter $Δ(p)=|Δ(p)|e^{iν\phi_p}$. The results provide quantitative formulas for interpreting heat transport experiments seeking to identify broken T and P symmetries, as well as the topology of the order parameter for chiral superconductors.
Probing configurational disorder in ZnGeN 2 using cluster-based Monte Carlo
ZnGeN 2 is sought as a semiconductor with comparable lattice constant to GaN and tunable band gap for integration in optoelectronic devices. Configurational disorder on the cation sublattice of ZnGeN 2 can strongly modify the electronic structure compared to the ordered material, and both ordered and disordered forms of ZnGeN 2 are candidates for light-emitting diodes and other emitters. The nonisovalent character of the disordered species (Zn 2+ and Ge 4+ ) subjects the cation ordering to strong short-range order effects. To model these effects, we use Monte Carlo (MC) simulations utilizing a cluster expansion to approximate formation enthalpy. Representative disordered configurations in 1024-atom supercells are relaxed in density functional theory calculations. From the MC structures, we extract a short-range order parameter (the N-cation coordination motif), and two long-range order parameters (Bragg-Williams and stretching parameters), and examine their correlations. We perform a thermodynamic integration to determine the mixing entropy and free energy. ZnGeN 2 exhibits a first-order phase transition with pronounced discontinuities in enthalpy and entropy, as well as in the structural order parameters. We discuss the relationship between the effective temperature used in the MC simulation and the growth temperatures in experiment in relation to the crossover from the nonequilibrium to the equilibrium growth regime. This work expands on current models of site disorder in ZnGeN 2 and provides atomic structure models with a systematic variation of the degree of cation disorder.
Hopper flows of mixtures of spherical and rod‐like particles via the multisphere method
Abstract The discharge of granular media from hoppers plays a vital role in many solids handling operations. The dynamic flow of poly‐disperse nonspherical particles differs markedly from mono‐disperse spheres and highlights the need to rigorously account for particle shape effects. The multisphere discrete element method (MS‐DEM) was utilized here to examine the bulk packing fraction, discharge rate, collisional stress tensor, and order parameter of a bi‐disperse sphere‐rod mixture evacuating a hopper. A modified form of the Myers and Sellers correlation with variable shape factor is capable of replicating the MS‐DEM discharge data. The collisional stress tensor is observed to be highly symmetric for mono‐disperse spheres but becomes asymmetric when rod‐like particles are introduced. The order parameter within the hopper increases significantly during discharge and the rod‐like particles undergo shear induced alignment with their long dimensions orthogonal to the outflow plane.