Symmetry Breaking Solution Research Articles

Degenerate soliton solutions and their dynamics in the nonlocal Manakov system: I symmetry preserving and symmetry breaking solutions

In this paper, we construct degenerate soliton solutions (which preserve $\cal{PT}$-symmetry/break $\cal{PT}$-symmetry) to the nonlocal Manakov system through a nonstandard bilinear procedure. Here by degenerate we mean the solitons that are present in both the modes which propagate with same velocity. The degenerate nonlocal soliton solution is constructed after briefly indicating the form of nondegenerate one-soliton solution. To derive these soliton solutions, we simultaneously solve the nonlocal Manakov equation and a pair of coupled equations that arise from the zero curvature condition. The later consideration yields general soliton solution which agrees with the solutions that are already reported in the literature under certain specific parametric choice. We also discuss the salient features associated with the obtained degenerate soliton solutions.

Global elucidation of broken symmetry solutions to the independent particle model through a Lie algebraic approach.

Broken symmetry solutions-solutions to the independent particle model that do not obey all symmetries required by the Hamiltonian-have attracted significant interest for capturing multireference properties with mean-field scaling. However, identification and optimization of broken-symmetry solutions is difficult owing to the non-linear nature of the self-consistent field (SCF) equations, particularly for solutions belonging to low-symmetry subgroups and where multiple broken symmetry solutions are sought. Linearization of SCF solution space results in the Lie algebra, which this work utilizes as a framework for elucidation of the set of solutions that exist at the desired symmetry. To demonstrate that searches constructed in the Lie algebra yield the set of broken symmetry solutions, a grid-based search of real-restricted, real-unrestricted, complex-restricted, paired-unrestricted, and real-general solutions of the (nearly D 4h ) H4 molecule is performed.

Linear polarizabilities and second hyperpolarizabilities of streptocyanines: Results from broken-Symmetry DFT and new CCSD(T) benchmarks.

We present a detailed analysis of the linear polarizability (α) and second hyperpolarizability (γ) in a series of streptocyanines, as predicted with various range-separated functionals and CCSD(T)-based methods. Contrary to previous work on these systems, we find that the lowest-energy electronic states for the larger streptocyanine oligomers are not closed-shell singlets, and improved accuracy can be obtained with certain DFT methods by allowing the system to relax to a lower-energy broken-symmetry solution. Our extensive analyses are complemented by new large-scale CCSD(T) and explicitly correlated CCSD(T)-F12 calculations that comprise the most complete and accurate benchmarks of α and γ for the streptocyanine systems to date. Taken together, our CCSD(T) and broken-symmetry DFT calculations (1) show that the MP2 benchmarks used in previous studies still exhibit significant errors (~ 25% for α and ~100% for γ) and, therefore, the MP2 calculations should not be used as reliable benchmarks for polarizabilities or hyperpolarizabilities, and (2) emphasize the importance of testing for a lower-energy open-shell configuration when calculating nonlinear optical properties for these systems. © 2018 Wiley Periodicals, Inc.

Gregory-Newton problem for kissing sticky spheres

All possible non-isomorphic arrangements of 12 spheres kissing a central sphere (the Gregory-Newton problem) are obtained for the sticky-hard-sphere (SHS) model, and subsequently projected by geometry optimization onto a set of structures derived from an attractive Lennard-Jones (LJ) type of potential. It is shown that all 737 derived SHS contact graphs corresponding to the 12 outer spheres are (edge-induced) subgraphs of the icosahedral graph. The most widely used LJ(6,12) potential has only one minimum structure corresponding to the ideal icosahedron where the 12 outer spheres do not touch each other. The point of symmetry breaking away from the icosahedral symmetry towards the SHS limit is obtained for general LJ($a,b$) potentials with exponents $a,b\in \mathbb{R}_+$. Only if the potential becomes very repulsive in the short-range, determined by the LJ hard-sphere radius $\sigma$, symmetry broken solutions are observed.

Topological dynamics and excitations in lasers and condensates with saturable gain or loss.

We classify symmetry-protected and symmetry-breaking dynamical solutions for nonlinear saturable bosonic systems that display a non-hermitian charge-conjugation symmetry, as realized in a series of recent groundbreaking experiments with lasers and exciton polaritons. In particular, we show that these systems support stable symmetry-protected modes that mirror the concept of zero-modes in topological quantum systems, as well as symmetry-protected power-oscillations with no counterpart in the linear case. In analogy to topological phases in linear systems, the number and nature of symmetry-protected solutions can change. The spectral degeneracies signalling phase transitions in linear counterparts extend to bifurcations in the nonlinear context. As bifurcations relate to qualitative changes in the linear stability against changes of the initial conditions, the symmetry-protected solutions and phase transitions can also be characterized by topological excitations, which set them apart from symmetry-breaking solutions. The stipulated symmetry appears naturally when one introduces nonlinear gain or loss into spectrally symmetric bosonic systems, as we illustrate for one-dimensional topological laser arrays with saturable gain and two-dimensional flat-band polariton condensates with density-dependent loss.

Alice-Bob systems, P^-T^-Ĉ symmetry invariant and symmetry breaking soliton solutions

To describe two-place physical problems, many possible models named Alice-Bob (AB) systems are proposed. To find and to solve these systems, the parity (P^), time reversal (T^), charge conjugation (Ĉ), and their possible combinations such as P^T^, P^Ĉ, and P^T^Ĉ, etc., can be successively applied. Especially, some special types of P^-T^-Ĉ group invariant multi-soliton solutions for the KdV-KP-Toda type, mKdV-sG type, and nonlinear Schrödinger equations (NLS) type AB systems are explicitly constructed. The possible P^T^ symmetry breaking solutions of two special ABKdV systems are explicitly given. Applying the P^-T^-Ĉ symmetries to coupled Ablowitz-Kaup-Newell-Segur systems, some four-place nonlocal NLS systems are also derived.

Qualitative Views on the Polyradical Character of Long Acenes.

Spin-symmetry breaking appears in the DFT treatment of polyacenes, beyond a certain length, the critical length depending on the exchange-correlation potential. This phenomenon may be attributed to an instability with respect to HOMO-LUMO mixing, which suggests a diradical character of long acenes. However, the increase of the S2 operator with acene length questions this simple view. It is shown that this increase cannot be attributed to spin polarization of the inner MOs, and that a second symmetry breaking takes place for the pentadecacene, with four unpaired electrons centered at the first and third quarters of the chain. The spin density distributions of broken-symmetry solutions support a qualitative picture in terms of tetra-methylene hexacenes separated by Clar sextets. A strategy is proposed to identify local symmetry breaking in polyradical systems.

Coherent structure of Alice–Bob modified Korteweg de-Vries equation

To describe two-place events, Alice-Bob systems have been established by means of the shifted parity and delayed time reversal in Ref. [1]. In this paper, we mainly study exact solutions of the integrable Alice-Bob modified Korteweg de-Vries (AB-mKdV) system. The general Nth Darboux transformation for the AB-mKdV equation are constructed. By using the Darboux transformation, some types of shifted parity and time reversal symmetry breaking solutions including one-soliton, two-soliton and rogue wave solutions are explicitly obtained. In addition to the similar solutions of the mKdV equation (group invariant solutions), there are abundant new localized structures for the AB-mKdV systems.

Bifurcation analysis for a free-boundary tumor model with angiogenesis and inhibitor

This paper is concerned with the bifurcation phenomenon of a free-boundary problem modeling the tumor growth under the action of angiogenesis and inhibitor. Taking the surface tension coefficient γ as a bifurcation parameter, we prove that there exist a positive integer m^{**} and a sequence of gamma_{m} such that, for every gamma_{m} (m>m^{**}), symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solutions.

Bistability in two simple symmetrically coupled oscillators with symmetry-broken amplitude- and phase-locking.

In the model system of two instantaneously and symmetrically coupled identical Stuart-Landau oscillators, we demonstrate that there exist stable solutions with symmetry-broken amplitude- and phase-locking. These states are characterized by a non-trivial fixed phase or amplitude relationship between both oscillators, while simultaneously maintaining perfectly harmonic oscillations of the same frequency. While some of the surrounding bifurcations have been previously described, we present the first detailed analytical and numerical description of these states and present analytically and numerically how they are embedded in the bifurcation structure of the system, arising both from the in-phase and the anti-phase solutions, as well as through a saddle-node bifurcation. The dependence of both the amplitude and the phase on parameters can be expressed explicitly with analytic formulas. As opposed to the previous reports, we find that these symmetry-broken states are stable, which can even be shown analytically. As an example of symmetry-breaking solutions in a simple and symmetric system, these states have potential applications as bistable states for switches in a wide array of coupled oscillatory systems.

Bifurcation for a free boundary problem modeling tumor growth with ECM and MDE interactions

We study a free boundary problem modeling solid tumor growth. The simplified model contains a parameter λ. Different from previous works on bifurcation analysis, a new ingredient of the present paper is that the influence of the extracellular matrix (ECM) and matrix degrading enzymes (MDE) interactions is included in the model. We first show that for each λ>0, there exists a unique radially symmetric stationary solution with radius r=RS. Then we prove that there exist a positive integer n∗ and a sequence of λn(n>n∗) for which branches of symmetry-breaking stationary solutions bifurcate from the radially symmetric one. In particular, we discover that these λn are larger than those λ˜n previously known when the effects of ECM and MDE are not considered in the model.

Stationary solutions of a free boundary problem modeling growth of angiogenesis tumor with inhibitor

We consider a free boundary problem modeling the growth of angiogenesis tumor with inhibitor, in which the tumor aggressiveness is modeled by a parameter $μ$. The existences of radially symmetric stationary solution and symmetry-breaking stationary solution are established. In addition, it is proved that there exist a positive integer $m^{**}$ and a sequence of $μ_m$, such that for each $μ_m(m > m^{**})$, the symmetry-breaking stationary solution is a bifurcation branch of the radially symmetric stationary solution.

Constraints in the variational principle for stationary states

Constraints in the variational principle for stationary states (VPSS) are classified in accordance with Dirac’s constrained classical mechanics and the time-dependent variational principle (TDVP). All of the VPSS constraints are required to belong to the first-class TDVP as constants of motion to ensure the real-valuedness of the Lagrange multipliers. The VPSS constraints are further classified as either first-class or second-class. The first-class VPSS constraints are constants of variation with symmetry-adapted wave functions. If the representation basis for the constraint operators is incomplete, however, the first-class VPSS constraints lead to broken-symmetry solutions. The nondegenerate energies of $${}^2E'$$ at the $$D_{3h}$$ geometry in the Jahn–Teller distortion of H $$_3$$ are presented as an example of a problem with broken-symmetry. An ad hoc adjustment is suggested by considering the second-class pseudo-VPSS constraints with new adiabatic parameters.

What Are the Physical Contents of Hubbard and Heisenberg Hamiltonian Interactions Extracted from Broken Symmetry DFT Calculations in Magnetic Compounds?

Analytical expressions of the interactions present in the Heisenberg-Dirac van Vleck and Hubbard Hamiltonians have been derived as functions of both the energy of several broken symmetry DFT solutions and their expectation value of the S2 spin operator. Then, following a strategy of decomposition of the magnetic exchange coupling into its main contributions (direct exchange, kinetic exchange, and spin polarization) and using a recently proposed method of spin decontamination, values of these interactions have been extracted. As already observed, they weakly depend on the correlation functional but strongly depend on the exchange one. In order to distinguish between the effect of the delocalization of the magnetic orbitals and that of the amount of Hartree-Fock exchange (HFX) when hybrid exchange-correlation functionals are used, we have disentangled these two contributions by either freezing the magnetic orbitals and varying the amount of HFX or varying the magnetic orbitals while keeping the same amount of HFX. As expected, increasing the amount of HFX induces a slight relocalization of the magnetic orbitals on the magnetic center which results in a weak increase of the repulsion energy U parameter and a weak decrease of both the direct exchange Kab and hopping |t| parameters. Conversely, the amount of HFX has a huge effect on all the parameters, even when some of the parameters should be exchange-independent, like U. Indeed, it is analytically demonstrated that the physical content of the U parameter extracted from several broken-symmetry solutions depends on the amount of HFX and that this pathological behavior has the same origin as the self-interaction error. This result is interesting not only to theoretical chemists working in the field of magnetic systems but also to DFT methodologists interested in using this theory for studying either excited states or strongly correlated systems. Finally, the performance of the range-separated ωB97XD functional for both ferromagnetic and antiferromagnetic transition-metal compounds and organic systems must be noted.

On a class of nonhomogeneous equations of Hénon-type: Symmetry breaking and non radial solutions

In this work we study the following Hénon-type equation −div∇up−2∇uxap=xβf(u),inB;u>0,inB;u=0,on∂B;where B≔x∈RN;x<1 is a ball centered at the origin, the parameters verify the inequalities 0≤a<N−pp, N≥4, β>0, 2≤p<Np+pβN−p(a+1), and the nonlinearity f is nonhomogeneous. By minimization on the Nehari manifold, we prove that for large values of the parameter β there is a symmetry breaking and non radial solutions appear.

QM/MM Study of the Nitrogenase MoFe Protein Resting State: Broken-Symmetry States, Protonation States, and QM Region Convergence in the FeMoco Active Site.

Nitrogenase is one of the most fascinating enzymes in nature, being responsible for all biological nitrogen reduction. Despite decades of research, it is among the enzymes in bioinorganic chemistry whose mechanism is the most poorly understood. The MoFe protein of nitrogenase contains an iron-molybdenum-sulfur cluster, FeMoco, where N2 reduction takes place. The resting state of FeMoco has been characterized by crystallography, multiple spectroscopic techniques, and theory (broken-symmetry density functional theory), and all heavy atoms are now characterized. The cofactor charge, however, has been controversial, the electronic structure has proved enigmatic, and little is known about the mechanism. While many computational studies have been performed on FeMoco, few have taken the protein environment properly into account. In this study, we put forward QM/MM models of the MoFe protein from Azotobacter vinelandii, centered on FeMoco. By a detailed analysis of the FeMoco geometry and comparison to the atomic resolution crystal structure, we conclude that only the [MoFe7S9C]1- charge is a possible resting state charge. Further, we find that of the three lowest energy broken-symmetry solutions of FeMoco, the BS7-235 spin isomer (where 235 refers to Fe atoms that are "spin-down") is the only one that can be reconciled with experiment. This is revealed by a comparison of the metal-metal distances in the experimental crystal structure, a rare case of spin-coupling phenomena being visible through the molecular structure. This could be interpreted as the enzyme deliberately stabilizing a specific electronic state of the cofactor, possibly for tuning specific reactivity on specific metal atoms. Finally, we show that the alkoxide group on the Mo-bound homocitrate must be protonated under resting state conditions, the presence of which has implications regarding the nature of FeMoco redox states as well as for potential substrate reduction mechanisms.

Bifurcation for a free boundary problem modeling the growth of tumors with a drug induced nonlinear proliferation rate

In this paper, we study a free boundary model describing growth of tumors under action of drugs. To our knowledge, in theoretical discussion for free boundary problems, the proliferation rate in tumor models discussed in previous bifurcation results is a linear function of nutrients and inhibitors. Whereas in this paper we consider the net proliferation rate as a nonlinear function depending on both nutrients and drugs. First, the existence and the uniqueness of radially symmetric stationary solutions are obtained. Second, we prove that symmetry-breaking solutions bifurcate from the radially symmetric stationary solutions when the concentration of drug on the boundary of tumor is less than one in the rescaled model.

Comparison of nitrogen-vacancy complexes in diamond and cubic SiC: dose dependencies and spin-Hamiltonian parameters

At present the nitrogen-vacancy (NV) complex in diamond is the most promising defect for application in the area of quantum computing. This provides a stimulus for an extensive search of other defects in semiconductors with similar properties. Recently it was shown that the NCVSi defect complex in SiC is perspectively appropriate for this goal as well. In the present work we perform comparative ab initio studies of NV complexes in diamond and 3C-SiC. We focus both on radiospectroscopic characterization of these defects and on the calculation of the equilibrium concentration of complexes in irradiated crystals. In particular a full set of spin-Hamiltonian parameters including g-tensors, hyperfine tensors and the spin–spin part of zero-field splitting constant Dss were calculated for both negative and neutral charge states as well as for excited quartet states of neutral complexes. Comparison of calculated values with the available experimental data and results of other calculations show good agreement, especially in the case when hybrid and meta-hybrid functionals were used. This makes the unambiguous identification of negative NV complexes in both materials possible. Our calculations reveal that the ground states of neutral complexes are a difficult case for both DFT calculations and experimental observations. This is caused by multi-determinantal behavior of wave function for such complexes, which leads to a large amount of spin contamination and to the broken symmetry solution which appeared for single Slater determinant DFT calculations. Based on the calculated minimum of free energy of neutral and negative complexes in SiC and diamond we obtained the equilibrium concentrations of these complexes depending on the vacancy concentration produced by irradiation. We show that in some dose regions both negative and neutral complexes coexist, while in other regions only one charge state prevails. Comparison of the calculated and experimental dose dependencies for diamond shows good qualitative agreement.

Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes

ABSTRACTWe describe the implementation of orbital optimisation for the models in the perfect pairing hierarchy. Orbital optimisation, which is generally necessary to obtain reliable results, is pursued at perfect pairing (PP) and perfect quadruples (PQ) levels of theory for applications on linear polyacenes, which are believed to exhibit strong correlation in the π space. While local minima and σ-π symmetry breaking solutions were found for PP orbitals, no such problems were encountered for PQ orbitals. The PQ orbitals are used for single-point calculations at PP, PQ and perfect hextuples (PH) levels of theory, both only in the π subspace, as well as in the full σπ valence space. It is numerically demonstrated that the inclusion of single excitations is necessary also when optimised orbitals are used. PH is found to yield good agreement with previously published density matrix renormalisation group data in the π space, capturing over 95% of the correlation energy. Full-valence calculations made possible by our novel, efficient code reveal that strong correlations are weaker when larger basis sets or active spaces are employed than in previous calculations. The largest full-valence PH calculations presented correspond to a (192e,192o) problem.

Approximating the XY model on a random graph with aq-state clock model

Numerical simulations of spin glass models with continuous variables set the problem of a reliable but efficient discretization of such variables. In particular, the main question is how fast physical observables computed in the discretized model converge toward the ones of the continuous model when the number of states of the discretized model increases. We answer this question for the XY model and its discretization, the $q$-state clock model, in the mean-field setting provided by random graphs. It is found that the convergence of physical observables is exponentially fast in the number $q$ of states of the clock model, so allowing a very reliable approximation of the XY model by using a rather small number of states. Furthermore, such an exponential convergence is found to be independent from the disorder distribution used. Only at $T=0$ the convergence is slightly slower (stretched exponential). Thanks to the analytical solution to the $q$-state clock model, we compute accurate phase diagrams in the temperature versus disorder strength plane. We find that, at zero temperature, spontaneous replica symmetry breaking takes place for any amount of disorder, even an infinitesimal one. We also study the one step of replica symmetry breaking (1RSB) solution in the low-temperature spin glass phase.