Symmetry Breaking Solution Research Articles

Hybrid DFT study of electronic structure on quasi-one-dimensional halogen-bridged binuclear metal complexes (MMX)

The electronic structure of quasi-one-dimensional halogen-bridged binuclear metal complex Ni2(dta)4I (dta=CH3CS2−) was investigated by hybrid density functional theory. UB3LYP was successfully applied to reproduce averaged-valence spin density wave state. The magnetic interactions between Ni dimers were estimated by calculating effective exchange integrals (Jab) using Ni2(dta)4I dimer and tetramer models. Calculated J values were consistent with that of experimental results. The natural orbital analysis of the broken-symmetry UB3LYP solution were performed to elucidate symmetry-adapted molecular orbitals and their occupation numbers. Several chemical indices such as polyradical character and information entropy were introduced on the basis of the occupation numbers to discuss the bonding character of MMX chain. All these indices supports that Ni2(dta)4I was in the strongly correlating electron system.

Extensions to three-dimensional flow in a porous channel

We consider the flow of a viscous, incompressible fluid contained between two parallel, porous walls. The flow is driven by a spatially uniform injection/suction of fluid through the bounding walls. We extend the solution structure of previous investigations to a more general three-dimensional stagnation-point form which can capture a whole range of phenomena in a single class of states. In particular, we show that this form of solution contains states previously discussed under more restrictive assumptions on the flow field. We show that a range of two- and three-dimensional states exist, together with symmetry-broken solutions and periodic states. We discuss the stability of these states and relate the previous results of Drazin, Banks, Zaturska and co-workers to those of Goldshtik and Javorsky on the “bifurcation to swirl” and of Hewitt and Duck on non-axisymmetric von Kármán flows.

Inhomogeneous Gutzwiller approximation with random phase fluctuations for the Hubbard model

We present a detailed study of the time-dependent Gutzwiller approximation for the Hubbard model. The formalism, labelled GA+RPA, allows us to compute random-phase approximation-like (RPA) fluctuations on top of the Gutzwiller approximation (GA). No restrictions are imposed on the charge and spin configurations which makes the method suitable for the calculation of linear excitations around symmetry-broken solutions. Well-behaved sum rules are obeyed as in the Hartree-Fock (HF) plus RPA approach. Analytical results for a two-site model and numerical results for charge-charge and current-current dynamical correlation functions in one and two dimensions are compared with exact and HF+RPA results, supporting the much better performance of GA+RPA with respect to conventional HF+RPA theory.

SO(5) as a critical dynamical symmetry in the SU(4) model of high-temperature superconductivity

An SU(4) model of high-temperature superconductivity and antiferromagnetism has recently been proposed. The SO(5) group employed by Zhang is embedded in this SU(4) as a subgroup, suggesting a connection between our SU(4) model and the Zhang SO(5) model. In order to understand the relationship between the the two models, we have used generalized coherent states to analyze the nature of the SO(5) subgroup. By constructing coherent-state energy surfaces, we demonstrate explicitly that the SU(4) supset SO(5) symmetry can be interpreted as a critical dynamical symmetry interpolating between superconducting and antiferromagnetic phases, and that this critical dynamical symmetry has many similarities to critical dynamical symmetries identified previously in other fields of physics. More generally, we demonstrate with this example that the mathematical techniques associated with generalized coherent states may have powerful applications in condensed matter physics because they provide a clear connection between microscopic many-body theories and their broken-symmetry approximate solutions. In addition, these methods may be interpreted as defining the most general Bogoliubov transformation subject to a Lie group symmetry constraint, thus providing a mathematical connection between algebraic formulations and the language of quasiparticle theory. Finally, we suggest that the identification of the SO(5) symmetry as a critical dynamical symmetry implies deep algebraic connections between high-temperature superconductors and seemingly unrelated phenomena in other field of physics.

Bose-Einstein condensates in a one-dimensional double square well: Analytical solutions of the nonlinear Schrödinger equation

We present a representative set of analytic stationary state solutions of the Nonlinear Schr\"odinger equation for a symmetric double square well potential for both attractive and repulsive nonlinearity. In addition to the usual symmetry preserving even and odd states, nonlinearity introduces quite exotic symmetry breaking solutions - among them are trains of solitons with different number and sizes of density lumps in the two wells. We use the symmetry breaking localized solutions to form macroscopic quantum superpositions states and explore a simple model for the exponentially small tunneling splitting.

Structural glass on a lattice in the limit of infinite dimensions

We construct a mean-field theory for the lattice model of a structural glass and solve it using the replica method and one-step replica symmetry breaking ansatz; this theory becomes exact in the limit of infinite dimensions. Analyzing the stability of this solution we conclude that the metastable states remain uncorrelated in a finite temperature range below the transition, but become correlated at a certain low temperature; below this temperature metastable states bifurcate and the one step replica symmetry breaking solution fails. We find dynamic and thermodynamic transition temperatures as functions of the density, and construct a full thermodynamic description of a typical physical process in which the system is trapped in one metastable state when cooled below the vitrification temperature. We find that for such physical process the entropy and pressure at the glass transition are continuous across the transition, while their temperature derivatives have jumps.

Dynamical symmetry breaking ofλφ4theory in the two loop effective potential

The two loop effective potential of massless $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory is presented in several regularization and renormalization prescriptions and the dynamical symmetry breaking solution is obtained in the strong-coupling situation in several prescriptions except the Coleman-Weinberg prescription. The beta function in the broken phase becomes negative and the UV fixed point turns out to be a strong-coupling one, and its numeric value varies with the renormalization prescriptions, a detail which is different from the asymptotic-free solution in the one loop case. The symmetry-breaking phase is shown to be an entirely strong-coupling phase. The reason for the relevance of the renormalization prescriptions is shown to be due to the nonperturbative nature of the effective potential. We also reanalyze the two loop effective potential by adopting a differential equation approach based on the understanding that all the quantum field theories are ill-defined formulations of the ``low-energy'' effective theories of a complete underlying theory. The relevance of the prescriptions of fixing the local ambiguities to physical properties such as symmetry breaking is further emphasized. We also tentatively propose a rescaling insensitivity argument for fixing the quadratic ambiguities. Some detailed properties of the strongly coupled broken phase and related issues are discussed.

Dynamical Generation of Four-Dimensional Space-Time in the IIB Matrix Model

We study the spontaneous breakdown of SO(10) symmetry in the IIB matrix model, a conjectured nonperturbative definition of type IIB superstring theory in ten dimensions. Our analysis is based on a Gaussian expansion technique, which was originally proposed by Kabat-Lifschytz and applied successfully to the strong coupling dynamics of the Matrix Theory. We propose a prescription for including higher order corrections, which yields a rapid convergence in a simple example. This prescription is then applied to the IIB matrix model up to the third order. We find that the `self-consistency equations' allow various symmetry breaking solutions. Among them, the solution preserving SO(4) symmetry is found to have the smallest free energy. The value of the free energy comes closer to the analytic formula obtained by Krauth-Nicolai-Staudacher as we increase the order. The extent of the space-time in the 4 directions is larger than the remaining 6 directions, and the ratio increases with the order. These results provide the first analytical evidence that four-dimensional space-time is generated dynamically in the IIB matrix model.

Analysis of the∞-replica symmetry breaking solution of the Sherrington-Kirkpatrick model

In this work we analyze the Parisi infinity-replica symmetry breaking solution of the Sherrington-Kirkpatrick model without external field using high order perturbative expansions. The predictions are compared with those obtained from the numerical solution of the infinity-replica symmetry breaking equations, which are solved using a pseudospectral code that allows for very accurate results. With these methods we are able to get more insight into the analytical properties of the solutions. We are also able to determine numerically the end point x(max) of the plateau of q(x) and find that lim(T-->0)x(max)(T)>0.5.

Electronic properties of model quantum-dot structures in zero and finite magnetic fields

We have computed electronic structures and total energies of circularly confined two-dimensional quantum dots and their lateral dimers in zero and finite uniform external magnetic fields using different theoretical schemes: the spin-density-functional theory (SDFT), the current-and-spin-density-functional theory (CSDFT), and the variational quantum Monte Carlo (VMC) method. The SDFT and CSDFT calculations employ a recently-developed, symmetry-unrestricted real-space algorithm allowing solutions which break the spin symmetry. Results obtained for a six-electron dot in the weak confinement limit and in zero magnetic field as well as in a moderate confinement and in finite magnetic fields enable us to draw conclusions about the reliability of the more approximative SDFT and CSDFT schemes in comparison with the VMC method. The same is true for results obtained for the two-electron quantum dot dimer as a function of inter-dot distance. The structure and role of the symmetry-breaking solutions appearing in the SDFT and CSDFT calculations for the above systems are discussed.

Magnetic properties of quantum dots and rings

Exact many-body methods as well as current-spin-density functional theory are used to study the magnetism and electron localization in two-dimensional quantum dots and quasi-one-dimensional quantum rings. Predictions of broken-symmetry solutions within the density functional model are confirmed by exact configuration interaction (CI) calculations: In a quantum ring the electrons localize to form an antiferromagnetic chain which can be described with a simple model Hamiltonian. In a quantum dot the magnetic field localizes the electrons as predicted with the density functional approach.

A Stefan problem for a protocell model with symmetry-breaking bifurcations of analytic solutions

A simple model of a living cell which undergoes processes of growth and dissolution is described as a free boundary problem for a system of two reaction-diffusion equations; the condition on the free boundary is of the Stefan type. The special case of radially symmetric cells was studied in earlier work. This paper is concerned with the existence of symmetry-breaking stationary solutions, i.e. with solutions which are not radially symmetric. It is proved, in the two-dimensional case, that there exist branches of non-radial stationary solutions bifurcating from radially symmetric solutions; indeed, for any mode l, l \[ge] 2, there exists a unique bifurcation branch whose free boundary has the form r = Rl + \[epsilon] cos l\[theta] + \[Sigma]n\[ge]2 \[epsilon]n\[lambda]n (\[theta]), | \[epsilon] | small, with \[lambda]n (\[theta]) orthogonal to cos l\[theta].

Multiscale reference function analysis of the 𝒫𝒯 symmetry breaking solutions for theP2+ iX3+ iαXHamiltonian

The recent work of Delabaere and Trinh (Delabaere E and Trinh D T 2000 J. Phys. A: Math. Gen. 33 8771) discovered the existence of symmetry breaking, complex energy, L2 solutions for the one-dimensional Hamiltonian, P2 + iX 3 + iαX, in the asymptotic limit α→-∞. Their asymptotic analysis produced questionable results for moderate values of α. We can easily confirm the existence of symmetry breaking solutions by explicitly computing the low-lying states for |α|<O(10). Our analysis makes use of the multiscale reference function (MRF) approach, developed by Tymczak et al (Tymczak C J, Japaridze G S, Handy C R and Wang Xiao-Qian 1998a Phys. Rev. Lett. 80 3678; 1998b Phys. Rev. A 58 2708). The MRF results can be validated by comparing them with the converging eigenenergy bounds generated through the eigenvalue moment method, as recently argued by Handy (2001a, b). Given the reliability of the MRF analysis, its fast numerical implementation, high accuracy and theoretical simplicity, the present formalism defines an effective and efficient procedure for analysing many related problems that have appeared in the recent literature.

A condition on the chiral symmetry breaking solution of the Dyson–Schwinger equation in three-dimensional QED

In three-dimensional QED, which is analyzed in the 1/$N$ expansion, we obtain a sufficient and necessary condition for a nontrivial solution of the Dyson-Schwinger equation to be chiral symmetry breaking solution. In the derivation, a normalization condition of the Goldstone bound state is used. It is showed that the existent analytical solutions satisfy this condition.

On the three-sublattice random antiferromagnet

Some properties of a model for random antiferromagnets with a three-sublattice structure (Ising model) which is a generalization of the well-known Sherrington–Kirkpatrick model are studied. Using standard techniques, we obtain the replica symmetric and permutation broken replica symmetry (RSB) solutions (numerically) and the phase diagram in the field versus temperature plane. The solution indicates that metastability is greatly enhanced in many sublattice systems as compared with simple systems. It is found that there may appear two kinds of RSB solutions for some values of the temperature and external field one of them with a Parisi's order-parameter function displaying non-monotonic behavior.

Macroscopically frustrated Ising model.

A disordered spin-glass model in which both static and dynamical properties depend on macroscopic magnetizations is presented. These magnetizations interact via random couplings and, therefore, the typical quenched realization of the system exhibits a macroscopic frustration. The model is solved by using a revisited replica approach, and the broken symmetry solution turns out to coincide with the symmetric solution. Some dynamical aspects of the model are also discussed, showing how it could be a useful tool for describing some properties of real systems such as, for example, natural ecosystems or human social systems.

Charge Density Analysis of Triplet and Broken Symmetry States Relevant to Magnetic Coupling in Systems with Localized Spin Moments

Comparison of electron charge density for triplet and broken symmetry solutions obtained from different computational methods together with the atoms-in-molecules (AIM) and electron localization function (ELF) topological analyses support the description of these systems through the Heisenberg Hamiltonian. The reduction of the low-energy spectrum to a purely spin Hamiltonian holds for all studied methods, although local density approximation (LDA) exhibits some noticeable deviation. The analysis of charge difference density plots clearly shows that the failure of LDA to describe magnetic coupling is due to the too-strong delocalization that leads to a qualitatively incorrect electron density in the region near the nuclei. Gradient-corrected and hybrid functionals correct this defect but in an exaggerated way. The role of mixing Fock exchange is also discussed.

Symmetry-breaking bifurcation of analytic solutions to free boundary problems: An application to a model of tumor growth

In this paper we develop a general technique for establishing analyticity of solutions of partial differential equations which depend on a parameter e. The technique is worked out primarily for a free boundary problem describing a model of a stationary tumor. We prove the existence of infinitely many branches of symmetry-breaking solutions which bifurcate from any given radially symmetric steady state; these asymmetric solutions are analytic jointly in the spatial variables and in e. 1. THE MODEL AND MAIN RESULT In this paper we present a general technique for establishing analyticity of solutions of systems of partial differential equations which depend analytically on a parameter e. The method works not only for boundary value problems but also for free boundary problems. In this latter context it can be used to establish long time existence of transient solutions, and also to study the existence of spatially asymmetric steady solutions. Since free boundary problems are typically more challenging than their boundary-value counterparts, we shall concentrate here on a free boundary problem from developmental biology, namely, a model of tumor growth. To further exemplify the generality of our approach an instance of a boundary value problem (in a fixed domain) is presented in the last section of the paper. A variety of other problems are amenable to the same analysis, including, in particular, the Hele-Shaw model of fluid flow [11]. Within the last several decades a number of mathematical models have been developed that aimed at describing the evolution of carcinomas (see. e.g., [1, 5, 6, 8, 12, 13] and the references cited there). The main objective of these models has been to qualitatively describe, under various simplifying assumptions, the growth and stability of tumor tissue. Analysis and simulations of such models are helping to assess the relative importance of various mechanisms affecting tumor growth as well as the efficacy of certain cancer treatments. On the other hand, the description of the stationary (dormant) configurations that arise from the models has only been addressed in the case of spherical tumors, but otherwise it remains largely unexplored. In this paper we develop a method for establishing analyticity of Received by the editors August 17, 1999. 1991 Mathematics Subject Classification. Primary 35B32, 35R35; Secondary 35B30, 35B60, 35C10, 35J85, 35Q80, 92C15, 95C15.

Information dynamics of neural networks with the aid of supersymmetry fields as the microscopic thermal flow

Information dynamics is discussed from the point of view of microscopic thermal flow. In the fields of learning, association, storage and so on, the concepts of neural networks (NNWs) have been widely used. Their neurodynamics have been investigated as a stochastic process of an infinite neuron system, using the replica method. This approach includes unsettled points; regimes where replica symmetry (RS) solutions and replica symmetry breaking (RSB) solutions are valid, low-dimensional behaviour regimes, and so forth.

Collective oscillations in quantum rings: A broken symmetry case

We present calculations within density functional theory of the ground state and collective electronic oscillations in small two-dimensional quantum rings. No spatial symmetries are imposed to the solutions and, as in a recent contribution, a transition to a broken symmetry solution in the intrinsic reference frame for an increasingly narrow ring is found. The oscillations are addressed by using real-time simulation. Conspicuous effects of the broken symmetry solution on the spectra are pointed out.