Problem Of Symmetry Breaking Research Articles

Physical systems in a space with noncommutativity of coordinates

We consider a space with canonical noncommutativity of coordinates. The problem of rotational symmetry breaking is studied in this space. To preserve the rotational symmetry we consider the generalization of constant matrix of noncommutativity to a tensor defined with the help of additional coordinates governed by a rotationally symmetric system. The properties of physical systems are examined in the rotationally invariant space with noncommutativity of coordinates. Namely, we consider an effect of coordinate noncommutativity on the energy levels of the hydrogen atom in the rotationally invariant noncommutative space. The motion of a particle in the uniform field is also studied in the noncommutative space with preserved rotational symmetry. On the basis of exact calculations we show that there is an effect of coordinate noncommutativity on the mass of a particle and conclude that noncommutativity causes the anisotropy of mass.

Semiclassical treatment of symmetry breaking and bifurcations in a non-integrable potential

We have derived an analytical trace formula for the level density of the Hénon–Heiles potential using the improved stationary phase method, based on extensions of Gutzwiller's semiclassical path integral approach. This trace formula has the correct limit to the standard Gutzwiller trace formula for the isolated periodic orbits far from all (critical) symmetry-breaking points. It continuously joins all critical points at which an enhancement of the semiclassical amplitudes occurs. We found a good agreement between the semiclassical and the quantum oscillating level densities for the gross shell structures and for the energy shell corrections, solving the symmetry breaking problem at small energies.

A distributed low tree-depth decomposition algorithm for bounded expansion classes

We study the distributed low tree-depth decomposition problem for graphs restricted to a bounded expansion class. Low tree-depth decomposition have been introduced in 2006 and have found quite a few applications. For example it yields a linear-time model checking algorithm for graphs in a bounded expansion class. Recall that bounded expansion classes cover classes of graphs of bounded degree, of planar graphs, of graphs of bounded genus, of graphs of bounded treewidth, of graphs that exclude a fixed minor, and many other graphs. There is a sequential algorithm to compute low tree-depth decomposition (with bounded number of colors) in linear time. In this paper, we give the first efficient distributed algorithm for this problem. As it is usual for a symmetry breaking problem, we consider a synchronous model, and as we are interested in a deterministic algorithm, we use the usual assumption that each vertex has a distinct identity number. We consider the distributed message-passing $$\mathcal {CONGEST}_\mathrm{BC}$$CONGESTBC model, in which messages have logarithmic length and only local broadcast are allowed. In this model, we present a logarithmic time distributed algorithm for computing a low tree-depth decomposition of graphs in a fixed bounded expansion class. In the sequential centralized case low tree-depth decomposition linear time algorithm are used as a core procedure in several non-trivial linear time algorithms. We believe that, similarly, low tree-depth decomposition could be at the heart of several non-trivial logarithmic time algorithms.

Anonymous Meeting in Networks

A team consisting of an unknown number of mobile agents, starting from different nodes of an unknown network, possibly at different times, have to meet at the same node. Agents are anonymous (identical), execute the same deterministic algorithm and move in synchronous rounds along links of the network. An initial configuration of agents is called gatherable if there exists a deterministic algorithm (even dedicated to this particular configuration) that achieves meeting of all agents in one node. Which configurations are gatherable and how to gather all of them deterministically by the same algorithm? We give a complete solution of this gathering problem in arbitrary networks. We characterize all gatherable configurations and give two universal deterministic gathering algorithms, i.e., algorithms that gather all gatherable configurations. The first algorithm works under the assumption that a common upper bound $$N$$N on the size of the network is known to all agents. In this case our algorithm guarantees gathering with detection, i.e., the existence of a round for any gatherable configuration, such that all agents are at the same node and all declare that gathering is accomplished. If no upper bound on the size of the network is known, we show that a universal algorithm for gathering with detection does not exist. Hence, for this harder scenario, we construct a second universal gathering algorithm, which guarantees that, for any gatherable configuration, all agents eventually get to one node and stop, although they cannot tell if gathering is over. The time of the first algorithm is polynomial in the upper bound $$N$$N on the size of the network, and the time of the second algorithm is polynomial in the (unknown) size itself. Our results have an important consequence for the leader election problem for anonymous agents in arbitrary graphs. Leader election is a fundamental symmetry breaking problem in distributed computing. Its goal is to assign, in some common round, value 1 (leader) to one of the entities and value 0 (non-leader) to all others. For anonymous agents in graphs, leader election turns out to be equivalent to gathering with detection. Hence, as a by-product, we obtain a complete solution of the leader election problem for anonymous agents in arbitrary graphs.

Phase transitions in small-world systems: application to functional brain networks

In the present paper the problem of symmetry breaking in the systems with a small- world property is considered. The obtained results are applied to the description of the functional brain networks. Origin of the entropy of fractal and multifractal small-world systems is discussed. Applying the maximum entropy principle the topology of these networks has been determined. The symmetry of the regular subgroup of a small-world system is described by a discrete subgroup of the Galilean group. The algorithm of determination of this group and transformation properties of the order parameter have been proposed. The integer basis of the irreducible representation is constructed and a free energy functional is introduced. It has been shown that accounting the presence of random connections leads to an integro- differential equation for the order parameter. For q-exponential distributions an equation of motion for the order parameter takes the form of a fractional differential equation. We consider the system that is described by a two-component order parameter and discuss the features of the spatial distribution of solutions.

Qualitative breakdown of the unrestricted Hartree-Fock energy.

The stretching of closed-shell molecules is a qualitative problem for restricted Hartree-Fock that is usually circumvented by the use of unrestricted Hartree-Fock (UHF). UHF is well known to break the spin symmetry at the Coulson-Fischer point, leading to a discontinuous derivative in the potential energy surface and incorrect spin density. However, this is generally not considered as a major drawback. In this work, we present a set of two electron molecules which magnify the problem of symmetry breaking and lead to drastically incorrect potential energy surfaces with UHF. These molecules also fail with unrestricted density-functional calculations where a functional such as B3LYP gives both symmetry breaking and an unphysically low energy due to the delocalization error. The implications for density functional theory are also discussed.

Large-radius Holstein polaron and the problem of spontaneous symmetry breaking

A translation-invariant solution is found for a large-radius Holstein polaron whose energy in the strong coupling limit is lower than that obtained by Holstein. The wave function corresponding to this solution is delocalized. A conclusion is made about the absence of a spontaneous symmetry breaking in the quantum system discussed.

Symmetry breaking and cross-streamline migration of three-dimensional vesicles in an axial Poiseuille flow.

We analyze numerically the problem of spontaneous symmetry breaking and migration of a three-dimensional vesicle [a model for red blood cells (RBCs)] in axisymmetric Poiseuille flow. We explore the three relevant dimensionless parameters: (i) capillary number, Ca, measuring the ratio between the flow strength over the membrane bending mode, (ii) the ratio of viscosities of internal and external liquids, λ, and (iii) the reduced volume, ν=[V/(4/3)π]/(A/4π)3/2 (A and V are the area and volume of the vesicle). The overall picture turns out to be quite complex. We find that the parachute shape undergoes spontaneous symmetry-breaking bifurcations into a croissant shape and then into slipper shape. Regarding migration, we find complex scenarios depending on parameters: The vesicles either migrate towards the center, or migrate indefinitely away from it, or stop at some intermediate position. We also find coexisting solutions, in which the migration is inwards or outwards depending on the initial position. The revealed complexity can be exploited in the problem of cell sorting out and can help understanding the evolution of RBCs' in vivo circulation.

Competitive autocatalytic reactions in chaotic flows with diffusion: Prediction using finite-time Lyapunov exponents

We investigate chaotic advection and diffusion in autocatalytic reactions for time-periodic sine flow computationally using a mapping method with operator splitting. We specifically consider three different autocatalytic reaction schemes: a single autocatalytic reaction, competitive autocatalytic reactions, which can provide insight into problems of chiral symmetry breaking and homochirality, and competitive autocatalytic reactions with recycling. In competitive autocatalytic reactions, species B and C both undergo an autocatalytic reaction with species A such that [Formula: see text] and [Formula: see text]. Small amounts of initially spatially localized B and C and a large amount of spatially homogeneous A are advected by the velocity field, diffuse, and react until A is completely consumed and only B and C remain. We find that local finite-time Lyapunov exponents (FTLEs) can accurately predict the final average concentrations of B and C after the reaction completes. The species that starts in the region with the larger FTLE has, with high probability, the larger average concentration at the end of the reaction. If B and C start in regions with similar FTLEs, their average concentrations at the end of the reaction will also be similar. When a recycling reaction is added, the system evolves towards a single species state, with the FTLE often being useful in predicting which species fills the entire domain and which is depleted. The FTLE approach is also demonstrated for competitive autocatalytic reactions in journal bearing flow, an experimentally realizable flow that generates chaotic dynamics.

The extended Koopmans' theorem for orbital-optimized methods: Accurate computation of ionization potentials

The extended Koopmans' theorem (EKT) provides a straightforward way to compute ionization potentials (IPs) from any level of theory, in principle. However, for non-variational methods, such as Møller-Plesset perturbation and coupled-cluster theories, the EKT computations can only be performed as by-products of analytic gradients as the relaxed generalized Fock matrix (GFM) and one- and two-particle density matrices (OPDM and TPDM, respectively) are required [J. Cioslowski, P. Piskorz, and G. Liu, J. Chem. Phys. 107, 6804 (1997)]. However, for the orbital-optimized methods both the GFM and OPDM are readily available and symmetric, as opposed to the standard post Hartree-Fock (HF) methods. Further, the orbital optimized methods solve the N-representability problem, which may arise when the relaxed particle density matrices are employed for the standard methods, by disregarding the orbital Z-vector contributions for the OPDM. Moreover, for challenging chemical systems, where spin or spatial symmetry-breaking problems are observed, the abnormal orbital response contributions arising from the numerical instabilities in the HF molecular orbital Hessian can be avoided by the orbital-optimization. Hence, it appears that the orbital-optimized methods are the most natural choice for the study of the EKT. In this research, the EKT for the orbital-optimized methods, such as orbital-optimized second- and third-order Møller-Plesset perturbation [U. Bozkaya, J. Chem. Phys. 135, 224103 (2011)] and coupled-electron pair theories [OCEPA(0)] [U. Bozkaya and C. D. Sherrill, J. Chem. Phys. 139, 054104 (2013)], are presented. The presented methods are applied to IPs of the second- and third-row atoms, and closed- and open-shell molecules. Performances of the orbital-optimized methods are compared with those of the counterpart standard methods. Especially, results of the OCEPA(0) method (with the aug-cc-pVTZ basis set) for the lowest IPs of the considered atoms and closed-shell molecules are substantially accurate, the corresponding mean absolute errors are 0.11 and 0.15 eV, respectively.

Analytic energy gradients for the orbital-optimized third-order Møller–Plesset perturbation theory

Analytic energy gradients for the orbital-optimized third-order Møller-Plesset perturbation theory (OMP3) [U. Bozkaya, J. Chem. Phys. 135, 224103 (2011)] are presented. The OMP3 method is applied to problematic chemical systems with challenging electronic structures. The performance of the OMP3 method is compared with those of canonical second-order Møller-Plesset perturbation theory (MP2), third-order Møller-Plesset perturbation theory (MP3), coupled-cluster singles and doubles (CCSD), and coupled-cluster singles and doubles with perturbative triples [CCSD(T)] for investigating equilibrium geometries, vibrational frequencies, and open-shell reaction energies. For bond lengths, the performance of OMP3 is in between those of MP3 and CCSD. For harmonic vibrational frequencies, the OMP3 method significantly eliminates the singularities arising from the abnormal response contributions observed for MP3 in case of symmetry-breaking problems, and provides noticeably improved vibrational frequencies for open-shell molecules. For open-shell reaction energies, OMP3 exhibits a better performance than MP3 and CCSD as in case of barrier heights and radical stabilization energies. As discussed in previous studies, the OMP3 method is several times faster than CCSD in energy computations. Further, in analytic gradient computations for the CCSD method one needs to solve λ-amplitude equations, however for OMP3 one does not since λ(ab)(ij(1))=t(ij)(ab(1)) and λ(ab)(ij(2))=t(ij)(ab(2)). Additionally, one needs to solve orbital Z-vector equations for CCSD, but for OMP3 orbital response contributions are zero owing to the stationary property of OMP3. Overall, for analytic gradient computations the OMP3 method is several times less expensive than CCSD (roughly ~4-6 times). Considering the balance of computational cost and accuracy we conclude that the OMP3 method emerges as a very useful tool for the study of electronically challenging chemical systems.

Distributed Graph Coloring: Fundamentals and Recent Developments

The focus of this monograph is on symmetry breaking problems in the message-passing model of distributed computing. In this model a communication network is represented by a n-vertex graph G = (V,E), whose vertices host autonomous processors. The processors communicate over the edges of G in discrete rounds. The goal is to devise algorithms that use as few rounds as possible. A typical symmetry-breaking problem is the problem of graph coloring. Denote by [delta] the maximum degree of G. While coloring G with [delta]+ 1 colors is trivial in the centralized setting, the problem becomes much more challenging in the distributed one. One can also compromise on the number of colors, if this allows for more efficient algorithms. Other typical symmetry-breaking problems are the problems of computing a maximal independent set (MIS) and a maximal matching (MM). The study of these problems dates back to the very early days of distributed computing. The founding fathers of distributed computing laid firm foundations for the area of distributed symmetry breaking already in the eighties. In particular, they showed that all these problems can be solved in randomized logarithmic time. Also, Linial showed that an O([delta]2)-coloring can be solved very efficiently deterministically. However, fundamental questions were left open for decades. In particular, it is not known if the MIS or the ([delta] + 1)-coloring can be solved in deterministic polylogarithmic time. Moreover, until recently it was not known if in deterministic polylogarithmic time one can color a graph with significantly fewer than [delta]2 colors. Additionally, it was open (and still open to some extent) if one can have sublogarithmic randomized algorithms for the symmetry breaking problems. Recently, significant progress was achieved in the study of these questions. More efficient deterministic and randomized ([delta] + 1)-coloring algorithms were achieved. Deterministic [delta]1 + o(1)-coloring algorithms with polylogarithmic running time were devised. Improved (and often sublogarithmic-time) randomized algorithms were devised. Drastically improved lower bounds were given. Wide families of graphs in which these problems are solvable much faster than on general graphs were identified. The objective of our monograph is to cover most of these developments, and as a result to provide a treatise on theoretical foundations of distributed symmetry breaking in the message-passing model. We hope that our monograph will stimulate further progress in this exciting area. Table of Contents: Acknowledgments / Introduction / Basics of Graph Theory / Basic Distributed Graph Coloring Algorithns / Lower Bounds / Forest-Decomposition Algorithms and Applications / Defective Coloring / Arbdefective Coloring / Edge-Coloring and Maximal Matching / Network Decompositions / Introduction to Distributed Randomized Algorithms / Conclusion and Open Questions / Bibliography / Authors' Biographies

Symmetry Breaking and Hole Localization in Multiple Core Electron Ionization

Motivated by recent opportunitites to study hollow molecules with multiple core holes offered by X-ray free electron lasers, we revisit the core-hole localization and symmetry breaking problem, now studying ionization of more than one core electron. It is shown, using a N2 molecule with one, two, three, and four core holes, for example, that in a multiconfigurational determination of the core ionization potentials employing a molecular point group with broken inversion symmetry, one particular configuration is sufficient to account for the symmetry breaking relaxation energy in an independent particle approximation in the case of one or three holes, whereas the choice of point group symmetry is unessential for two and four holes. The relaxation energy follows a quadratic dependence on the number of holes in both representations.

Orbital optimized double-hybrid density functionals

This paper advocates development of a new class of double-hybrid (DH) density functionals where the energy is fully orbital optimized (OO) in presence of all correlation, rather than using a final non-iterative second order perturbative correction. The resulting OO-DH functionals resolve a number of artifacts associated with conventional DH functionals, such as first derivative discontinuities. To illustrate the possibilities, two non-empirical OO-DH functionals are obtained from existing DH functionals based on PBE: OO-PBE0-DH and OO-PBE0-2. Both functionals share the same functional form, with parameters determined on the basis of different physical considerations. The new functionals are tested on a variety of bonded, non-bonded and symmetry-breaking problems.

Analytic energy gradients for the orbital-optimized second-order Møller–Plesset perturbation theory

Analytic energy gradients for the orbital-optimized second-order Møller-Plesset perturbation theory (OMP2) are presented. The OMP2 method is applied to difficult chemical systems, including those where spatial or spin symmetry-breaking instabilities are observed. The performance of the OMP2 method is compared with that of second-order Møller-Plesset perturbation theory (MP2) for investigating geometries and vibrational frequencies of the cis-HOOH(+), trans-HOOH(+), LiO2, C3(+), and NO2 molecules. For harmonic vibrational frequencies, the OMP2 method eliminates the singularities arising from the abnormal response contributions observed for MP2 in case of symmetry-breaking problems, and provides significantly improved vibrational frequencies for the above molecules. We also consider the hydrogen transfer reactions between several free radicals, for which MP2 provides poor reaction energies. The OMP2 method again exhibits a considerably better performance than MP2, providing a mean absolute error of 2.3 kcal mol(-1), which is more than 5 times lower than that of MP2 (13.2 kcal mol(-1)). Overall, the OMP2 method seems quite helpful for electronically challenging chemical systems such as symmetry-breaking molecules, hydrogen transfer reactions, or other cases where standard MP2 proves unreliable. For such systems, we recommend using OMP2 instead of MP2 as a more robust method with the same computational scaling.

How may confinement affect technicolour?

Confinement has been introduced into the quark gap equation, as proposed by Cornwall, as a possible solution to the problem of chiral symmetry breaking in quantum chromodynamics with dynamically massive gluons. We argue that the same mechanism can be applied for technicolour with dynamically massive technigluons. Within this approach both theories develop a hard self-energy dynamic, resulting from an effective four-fermion interaction, which does not lead to the known technicolour phenomenological problems. We outline a quite general type of technicolour model within this proposal that may naturally explain the masses of different fermion generations.

Orbital-optimized third-order Møller-Plesset perturbation theory and its spin-component and spin-opposite scaled variants: Application to symmetry breaking problems

In this research, orbital-optimized third-order Møller-Plesset perturbation theory (OMP3) and its spin-component and spin-opposite scaled variants (SCS-OMP3 and SOS-OMP3) are introduced. Using a Lagrangian-based approach, an efficient, quadratically convergent algorithm for variational optimization of the molecular orbitals (MOs) for third-order Møller-Plesset perturbation theory (MP3) is presented. Explicit equations for response density matrices, the MO gradient, and Hessian are reported in spin-orbital form. The OMP3, SCS-OMP3, and SOS-OMP3 approaches are compared with the second-order Møller-Plesset perturbation theory (MP2), MP3, coupled-cluster doubles (CCD), optimized-doubles (OD), and coupled-cluster singles and doubles (CCSD) methods. All these methods are applied to the O(4)(+), O(3), and seven diatomic molecules. Results demonstrate that the OMP3 and its variants provide significantly better vibrational frequencies than MP3, CCSD, and OD for the molecules where the symmetry-breaking problems are observed. For O(4)(+), the OMP3 prediction, 1343 cm(-1), for ω(6) (b(3u)) mode, where symmetry-breaking appears, is even better than presumably more reliable methods such as Brueckner doubles (BD), 1194 cm(-1), and OD, 1193 cm(-1), methods (the experimental value is 1320 cm(-1)). For O(3), the predictions of SCS-OMP3 (1143 cm(-1)) and SOS-OMP3 (1165 cm(-1)) are remarkably better than the more robust OD method (1282 cm(-1)); the experimental value is 1089 cm(-1). For the seven diatomics, again the SCS-OMP3 and SOS-OMP3 methods provide the lowest average errors, ∣Δω(e)∣ = 44 and ∣Δω(e)∣ = 35 cm(-1), respectively, while for OD, ∣Δω(e)∣ = 161 cm(-1)and CCSD ∣Δω(e)∣ = 106 cm(-1). Hence, the OMP3 and especially its spin-scaled variants perform much better than the MP3, CCSD, and more robust OD approaches for considered test cases. Therefore, considering both the computational cost and the reliability, SCS-OMP3 and SOS-OMP3 appear to be the best methods for the symmetry-breaking cases, based on present application results. The OMP3 method offers certain advantages: it provides reliable vibrational frequencies in case of symmetry-breaking problems, especially with spin-scaling tricks, its analytic gradients are easier to compute since there is no need to solve the coupled-perturbed equations for the orbital response, and the computation of one-electron properties are easier because there is no response contribution to the particle density matrices. The OMP3 has further advantages over standard MP3, making it promising for excited state properties via linear response theory.

Quadratically convergent algorithm for orbital optimization in the orbital-optimized coupled-cluster doubles method and in orbital-optimized second-order Møller-Plesset perturbation theory

Using a Lagrangian-based approach, we present a more elegant derivation of the equations necessary for the variational optimization of the molecular orbitals (MOs) for the coupled-cluster doubles (CCD) method and second-order Møller-Plesset perturbation theory (MP2). These orbital-optimized theories are referred to as OO-CCD and OO-MP2 (or simply "OD" and "OMP2" for short), respectively. We also present an improved algorithm for orbital optimization in these methods. Explicit equations for response density matrices, the MO gradient, and the MO Hessian are reported both in spin-orbital and closed-shell spin-adapted forms. The Newton-Raphson algorithm is used for the optimization procedure using the MO gradient and Hessian. Further, orbital stability analyses are also carried out at correlated levels. The OD and OMP2 approaches are compared with the standard MP2, CCD, CCSD, and CCSD(T) methods. All these methods are applied to H(2)O, three diatomics, and the O(4)(+) molecule. Results demonstrate that the CCSD and OD methods give nearly identical results for H(2)O and diatomics; however, in symmetry-breaking problems as exemplified by O(4)(+), the OD method provides better results for vibrational frequencies. The OD method has further advantages over CCSD: its analytic gradients are easier to compute since there is no need to solve the coupled-perturbed equations for the orbital response, the computation of one-electron properties are easier because there is no response contribution to the particle density matrices, the variational optimized orbitals can be readily extended to allow inactive orbitals, it avoids spurious second-order poles in its response function, and its transition dipole moments are gauge invariant. The OMP2 has these same advantages over canonical MP2, making it promising for excited state properties via linear response theory. The quadratically convergent orbital-optimization procedure converges quickly for OMP2, and provides molecular properties that are somewhat different than those of MP2 for most of the test cases considered (although they are similar for H(2)O). Bond lengths are somewhat longer, and vibrational frequencies somewhat smaller, for OMP2 compared to MP2. In the difficult case of O(4)(+), results for several vibrational frequencies are significantly improved in going from MP2 to OMP2.

About the chiral symmetry breaking in QED3

The problem of the chiral symmetry breaking in QED 3 is considered by solving the Schwinger–Dyson equation for the fermion propagator in the ladder approximation using the Landau gauge for the photon propagator. Within the framework of the indicated approximation, different simplifications that allow expressions for the fermion mass function to be retrieved in an explicit form are analyzed. The results obtained are compared with the data of numerical analysis. It appears that the neglect of higher Gegenbauer harmonics in the kernel of the initial integral equation for the fermion mass function influences the dynamic mass value and the asymptotics of the mass function only weakly. On the other hand, it is established that the conclusion about a complicated structure of the fermion vacuum of the massive phase is an artifact of linearization of the Schwinger–Dyson equation kernel: consideration of the kernel nonlinearity yields a simple massive phase structure of the fermion vacuum.

Symmetry breaking of solitons in two-component Gross-Pitaevskii equations

We revisit the problem of the spontaneous symmetry breaking (SSB) of solitons in two-component linearly coupled nonlinear systems, adding the nonlinear interaction between the components. With this feature, the system may be realized in new physical settings, in terms of optics and the Bose-Einstein condensate (BEC). SSB bifurcation points are found analytically, for both symmetric and antisymmetric solitons (the symmetry between the two components is meant here). Asymmetric solitons, generated by the bifurcations, are described by means of the variational approximation (VA) and numerical methods, demonstrating good accuracy of the variational results. In the space of the self-phase-modulation (SPM) parameter and soliton's norm, a border separating stable symmetric and asymmetric solitons is identified. The nonlinear coupling may change the character of the SSB bifurcation, from subcritical to supercritical. Collisions between moving asymmetric and symmetric solitons are investigated too. Antisymmetric solitons are destabilized by a supercritical bifurcation, which gives rise to self-confined modes featuring Josephson oscillations, instead of stationary states with broken antisymmetry. An additional instability against delocalized perturbations is also found for the antisymmetric solitons.