Symmetry Breaking Solution Research Articles

Nonlinearity-induced symmetry breaking in a system of two parametrically driven Kerr-Duffing oscillators

Abstract We study the classical dynamics of a system comprising a pair of Kerr-Duffing nonlinear oscillators, which are coupled through a nonlinear interaction and subjected to a parametric drive. Using the rotating wave approximation (RWA), we analyze the steady-state solutions for the amplitudes of the two oscillators. For the case of almost identical oscillators, we investigate separately the cases in which only one oscillator is parametrically driven and in which both oscillators are simultaneously driven. In the latter regime, we demonstrate that even when the parametric drives acting on the two oscillators are identical, the system can transition from a stable symmetric solution to a broken-symmetry solution as the detuning is varied.

A supersymmetric SYK model with a curious low energy behavior

We consider N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2, 4 supersymmetric SYK models that have a peculiar low energy behavior, with the entropy going like S = S0 + (constant)Ta, where a ≠ 1. The large N equations for these models are a generalization of equations that have been previously studied as an unjustified truncation of the planar diagrams describing the BFSS matrix quantum mechanics or other related matrix models. Here we reanalyze these equations in order to better understand the low energy physics of these models. We find that the scalar fields develop large expectation values which explore the low energy valleys in the potential. The low energy physics is dominated by quadratic fluctuations around these values. These models were previously conjectured to have a spin glass phase. We did not find any evidence for this phase by using the usual diagnostics, such as searching for replica symmetry breaking solutions.

Symmetry breaking and multiple solutions for the Schrödinger–Poisson–Slater equation

Symmetry breaking and multiple solutions for the Schrödinger–Poisson–Slater equation

Centrosymmetry-Breaking Morphotropic Phase Boundary: A Pathway to Highly Sensitive and Strong Pressure-Responsive Nonlinear Optical Switches.

The quest for high-performance piezoelectric materials has been synonymous with the pursuit of the morphotropic phase boundary (MPB), yet the full potential of MPBs remains largely untapped outside of the realm of ferroelectrics. In this study, we reveal a new class of MPB by creating continuous molecular-based solid solutions between centro- and noncentrosymmetric compounds, exemplified by (tert-butylammonium)1-x(tert-amylammonium)xFeCl4 (0 ≤ x ≤ 1), where the MPB is formed due to disorder of molecular cations. Near the MPB, we discovered an exceptionally sensitive nonlinear optical material in the centrosymmetric phase, capable of activation at pressures as low as 0.12-0.27 GPa, and producing tunable second-harmonic generation (SHG) signals from zero to 18.8 times that of KH2PO4 (KDP). Meanwhile, synchrotron diffraction experiments have unveiled a third competing phase (P212121) appearing at low pressure, forming a triple-phase point near the MPB, thereby providing insight into the mechanism underpinning the nonlinear optical (NLO) switch behavior. These findings highlight the opportunity to harness exceptional physical properties in symmetry-breaking solid solution systems by strategically designing novel MPBs.

The Onset of Parisi's Complexity in a Mismatched Inference Problem.

We show that a statistical mechanics model where both the Sherringhton-Kirkpatrick and Hopfield Hamiltonians appear, which is equivalent to a high-dimensional mismatched inference problem, is described by a replica symmetry-breaking Parisi solution.

Systematic determination of coupling constants in spin clusters from broken-symmetry mean-field solutions.

Quantum-chemical calculations aimed at deriving magnetic coupling constants in exchange-coupled spin clusters commonly utilize a broken-symmetry (BS) approach. This involves calculating several distinct collinear spin configurations, predominantly by density-functional theory. The energies of these configurations are interpreted in terms of the Heisenberg model, H̃=∑i<jJijs̃i⋅s̃j, to determine coupling constants Jij for spin pairs. However, this energy-based procedure has inherent limitations, primarily in its inability to provide information on isotropic spin interactions beyond those included in the Heisenberg model. Biquadratic exchange or multi-center terms, for example, are usually inaccessible and hence assumed to be negligible. The present work introduces a novel approach employing BS mean-field solutions, specifically Hartree-Fock wave functions, for the construction of effective spin Hamiltonians. This expanded method facilitates the extraction of a broader range of coupling parameters by considering not only the energies, but also Hamiltonian and overlap elements between different BS states. We demonstrate how comprehensive s=12 Hamiltonians, including multi-center terms, can be straightforwardly constructed from a complete set of BS solutions. The approach is exemplified for small clusters within the context of the half-filled single-band Hubbard model. This allows to contrast the current strategy against exact results, thereby offering an enriched understanding of the spin-Hamiltonian construction from BS solutions.

The Nature of the Chemical Bonds of High-Valent Transition-Metal Oxo (M=O) and Peroxo (MOO) Compounds: A Historical Perspective of the Metal Oxyl-Radical Character by the Classical to Quantum Computations.

This review article describes a historical perspective of elucidation of the nature of the chemical bonds of the high-valent transition metal oxo (M=O) and peroxo (M-O-O) compounds in chemistry and biology. The basic concepts and theoretical backgrounds of the broken-symmetry (BS) method are revisited to explain orbital symmetry conservation and orbital symmetry breaking for the theoretical characterization of four different mechanisms of chemical reactions. Beyond BS methods using the natural orbitals (UNO) of the BS solutions, such as UNO CI (CC), are also revisited for the elucidation of the scope and applicability of the BS methods. Several chemical indices have been derived as the conceptual bridges between the BS and beyond BS methods. The BS molecular orbital models have been employed to explain the metal oxyl-radical character of the M=O and M-O-O bonds, which respond to their radical reactivity. The isolobal and isospin analogy between carbonyl oxide R2C-O-O and metal peroxide LFe-O-O has been applied to understand and explain the chameleonic chemical reactivity of these compounds. The isolobal and isospin analogy among Fe=O, O=O, and O have also provided the triplet atomic oxygen (3O) model for non-heme Fe(IV)=O species with strong radical reactivity. The chameleonic reactivity of the compounds I (Cpd I) and II (Cpd II) is also explained by this analogy. The early proposals obtained by these theoretical models have been examined based on recent computational results by hybrid DFT (UHDFT), DLPNO CCSD(T0), CASPT2, and UNO CI (CC) methods and quantum computing (QC).

On supercritical elliptic problems: existence, multiplicity of positive and symmetry breaking solutions

On supercritical elliptic problems: existence, multiplicity of positive and symmetry breaking solutions

A geminal theory based on the generalized electron pairing.

As in the hierarchy of the Hartree-Fock theory in terms of spin symmetry, the extension of spin functions in a theory of two-electron units or geminal, was developed in this study. A trial wave function is constructed as an antisymmetrized product of geminals, in which singlet and triplet two-electron functions are fully mixed. We present a variational optimization method for this generalized pairing wave function in the strong orthogonality condition. The present method is considered an extension of the antisymmetrized product of strongly orthogonal geminals or perfect pairing generalized valence bond methods, maintaining the compactness of the trial wave function. The obtained broken-symmetry solutions were similar to the unrestricted Hartree-Fock wave functions in terms of spin contamination while giving lower energy due to the inclusion of the electron correlation effect in geminals. The degeneracy of the obtained broken-symmetry solutions in the Sz space is reported for the tested four-electron systems.

The symmetry breaking solutions of the nonlocal Alice–Bob B-type Kadomtsev–Petviashvili system

The (2+1)-dimensional B-type Kadomtsev–Petviashvili equation is an integrable model, which can be used to describe the shallow water wave in a fluid. In this paper, the nonlocal Alice–Bob B-type Kadomtsev–Petviashvili system is induced via the principle of PˆsxPˆsyTˆd symmetry. An extended Bäcklund transformation introduced, the symmetry breaking solution, which contains the soliton, breather, lump and their hybrid structures for this system, is solved through the Hirota bilinear form.

Solutions of Schrödinger equations with symmetry in orientation preserving tetrahedral group

We consider the nonlinear Schrödinger equationΔu=(1+εV1(|y|))u−|u|p−1uinRN,N≥3,p∈(1,N+2N−2). The phenomenon of pattern formation has been a central theme in the study of nonlinear Schrödinger equations. However, the following nonexistence of O(N) symmetry breaking solution is well-known: if the potential function is radial and nondecreasing, any positive solution must be radial. Therefore, solutions of interesting patterns can only exist after violating the assumptions.O(N) symmetry breaking solutions have been presented by Wei and Yan [20]. Symmetry groups of regular polygons describe their solution patterns. Ever since work of Wei and Yan, there have been substantial generalizations but solutions with higher dimensional symmetry have not been constructed. In this study, the existence of nonradial solutions whose symmetry group is a discrete subgroup of O(3), more precisely, the orientation-preserving regular tetrahedral group is shown.

The shifted parity and delayed time-reversal symmetry-breaking solutions for the (1+1)-dimensional Alice–Bob Boussinesq equation

The integrable Alice–Bob system with the shifted parity and delayed time reversal is presented through the Lax pair for the (1 + 1)-dimensional Boussinesq equation. After introducing an extended Bäcklund transformation, this system shows abundant exact solutions with the auxiliary functions consisting of hyperbolic functions or rational functions. The corresponding soliton structures contain line solitons, breathers, and lumps, all which satisfied the shifted parity and delayed time-reversal symmetry for the states of Alice A and Bob B. In particular, some lower-order circumstances can be expressed through their explicit solutions and their dynamic structures.

An interesting case of symmetry breaking in the absence of symmetry: the bicarbonate radical (HCO3)

As part of a computational study of the thermochemical properties of systems related to carbonic acid (HCO), an interesting phenomenon was observed in the bicarbonate radical (HCO). This species belongs to the class of formoxyl radicals, but lacks the geometric symmetry of the more heavily-studied and prototypical HCO and FCO molecules. However, it is not immune from the problems that those molecules present to electronic structure calculations. Specifically, at geometries in the vicinity of the equilibrium structure, three solutions to the unrestricted Hartree-Fock equations can be found: two in which the unpaired spin is largely localized onto one of the (inequivalent) carbonyl oxygen atoms, and one in which the spin is delocalized. While all three of these reference functions comprise orbitals that transform as pure irreducible representations of the point group, the two localized choices qualitatively correspond to the symmetry-broken solution of the formoxyl species and the delocalized solution qualitatively corresponds to the symmetry-preserving (but orbitally unstable) solution. The three solutions are described in this work, as well as the corresponding ambiguities that result in the thermochemical characterization of this species.

Solitary states in complex networks: impact of topology

The dynamical behavior of networked systems is expected to reflect the properties of their coupling structure. Yet, symmetry-broken solutions often occur in symmetrically coupled networks. An example are so-called solitary states where the dynamics of one network node is different from the synchronized rest. Here, we investigate the structural constraints of networks for the appearance of solitary states. By performing a large number of numerical simulations, we find that such states occur with high probability in asymmetric networks, among them scale-free ones. We analyze the structural properties of the networks that support solitary states. We demonstrate that the minimum neighbor node degree of a solitary node is crucial for the appearance of solitary states. Finally, we perform bifurcation analysis of dimension-reduced systems, which confirm the importance of the connectivity of the neighboring nodes.

Symmetry-breaking longitude bifurcations for a free boundary problem modeling small plaques in three dimensions.

Atherosclerosis, one of the leading causes of death in USA and worldwide, begins with a lesion in the intima of the arterial wall, allowing LDL to penetrate into the intima where they are oxidized. The immune system considers these oxidized LDL as a dangerous substance and tasks the macrophages to attack them; incapacitated macrophages become foam cells and leads to the formation of a plaque. As the plaque continues to grow, it progressively restricts the blood flow, possibly triggering heart attack or stroke. Because the blood vessels tend to be circular, two-space dimensional cross section model is a good approximation, and the two-space dimensional models are studied in Friedman et al. (J Differ Equ 259(4):1227-1255, 2015) and Zhao and Hu (J Differ Equ 288:250-287, 2021). It is interesting to see whether a true three-space dimensional stationary solution can be developed. We shall establish a three-space dimensional stationary solution for the mathematical model of the initiation and development of atherosclerosis which involves LDL and HDL cholesterols, macrophages and foam cells. The model is a highly nonlinear and coupled system of PDEs with a free boundary, the interface between the plaque and the blood flow. We establish infinite branches of symmetry-breaking stationary solutions which bifurcate from the annular stationary solution in the longitude direction.

Broken-symmetry self-consistent GW approach: Degree of spin contamination and evaluation of effective exchange couplings in solid antiferromagnets

We adopt a broken-symmetry strategy for evaluating effective magnetic constants J within the fully self-consistent GW method. To understand the degree of spin contamination present in broken-symmetry periodic solutions, we propose several extensive quantities demonstrating that the unrestricted self-consistent GW preserves the broken-symmetry character of the unrestricted Hartree-Fock solutions. The extracted J are close to the ones obtained from multireference wave-function calculations. In this paper, we establish a robust computational procedure for finding magnetic coupling constants from self-consistent GW calculations and apply it to solid antiferromagnetic nickel and manganese oxides.

Using Brillouin's theorem for obtaining symmetry-preserving and symmetry-breaking solutions. Application to graphene quantum dots

Using Brillouin's theorem for obtaining symmetry-preserving and symmetry-breaking solutions. Application to graphene quantum dots

Bifurcation for a free-boundary problem modeling small plaques with reverse cholesterol transport

In this paper, we rigorously analyze a free boundary model of plaque growth under the influence of reverse cholesterol transport (RCT). We will analyze a model with a highly nonlinear and mutually coupled PDE system of plaque growth dynamics, including LDL and HDL cholesterol, M1 inflammatory and M2 anti-inflammatory macrophages, and foam cells. This free boundary problem is associated with two different types of surfaces: a vascular surface with the boundary set to r=1 and the interface Γ(t) between the plaque and blood flow. For this model, the explicit stationary solutions are nonexistent due to the highly nonlinear nature of the problem. We shall first prove the existence and uniqueness of the radially symmetric stationary solution. Then, we shall combine parameters into a single bifurcation parameter μ, and establish the existence of a sequence μn, such that for each μn(n≥2), the symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solutions for sufficiently small plaques.

Spin Alternation Rule in USCF for Through-Bond Magnetic Coupling─A New Look: Why and When Does It Arise and How To See It.

This work takes a new look at the spin alternation rule in unrestricted self-consistent-field (USCF) calculations in terms of structural characteristics such as periodicity, impurity location, and Coulomb exchange. For clarity, the systems considered are biradicals produced from linear conjugated hydrocarbons. Both site-parametrized Hamiltonian models for theoretical analysis and spin unrestricted density functional theory (DFT) calculations are used. Theoretical analysis leads to the following conclusions: (1) The diradical state is an excited state of a linear chain of N conjugated carbon atoms (when N is about ≤ 10). Spin alternation is a consequence of the (truncated) periodic symmetry combined with filling each closed-shell pi orbital with two electrons and each singly occupied molecular orbital (SOMO) with one electron. Spin polarization is evident in triplet (T) and broken symmetry (BS) solutions for an odd N and only in the T solution for an even N. Spin alternation is visible in the BS for an odd N and always remains muted in the calculated T. (2) For a doped chain with two radical centers, spin alternation is generally visible in the BS for an odd N. The sign of spin population on the radical centers in the BS indicates the stable spin. For radical centers separated by an odd (even) number of pz electrons, spin alternation favors T (S) state with FM (AFM) interaction. Spin oscillation remains less transparent for an even N without exchange. (3) In an unrestricted treatment with exchange, spin alternation becomes observable. Without SCF iterations, the more stable state can be identified from a clear spin oscillation in the BS. An irregular oscillation indicates a possible singlet ground state. These observations are supported by density functional calculations using the B3LYP functional and the 6-311+g(d,p) basis set on linear decapentaene diradicals with nitronyl nitroxide moieties substituted on two sets of conjugated atoms, (3,9) and (3,10). Because of the SCF procedure, one finds spin alternation in the T (BS) solution and erratic oscillation in the BS (T) solution of the 3,9 (3,10) diradical in respective equilibrium geometries. The ground state is T (S). DFT adiabatic coupling constants, SOMO energies, spin population plots, and SOMO lobe diagrams compare well with molecular electronic characteristics from theoretical analysis using Hamiltonian parameters.

Local analysis of a two phase free boundary problem concerning mean curvature

We consider an overdetermined problem for a two phase elliptic operator in divergence form with piecewise constant coefficients. We look for domains such that the solution $u$ of a Dirichlet boundary value problem also satisfies the additional property that its normal derivative $\partial_n u$ is a multiple of the radius of curvature at each point on the boundary. When the coefficients satisfy some non-criticality condition, we construct nontrivial solutions to this overdetermined problem employing a perturbation argument relying on shape derivatives and the implicit function theorem. Moreover, in the critical case, we employ the use of the Crandall-Rabinowitz theorem to show the existence of a branch of symmetry breaking solutions bifurcating from trivial ones. Finally, some remarks on the one phase case and a similar overdetermined problem of Serrin type are given.