Global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on bipartite lattices

In this paper the global symmetry of the Hubbard model on a bipartite lattice is found to be larger than SO(4). The model is one of the most studied many-particle quantum problems, yet except in one dimension it has no exact solution, so that there remain many open questions about its properties. Symmetry plays an important role in physics and often can be used to extract useful information on unsolved non-perturbative quantum problems. Specifically, here it is found that for on-site interaction U ≠ 0 the local SU(2) × SU(2) × U(1) gauge symmetry of the Hubbard model on a bipartite lattice with N_a^D sites and vanishing transfer integral t = 0 can be lifted to a global [SU(2) × SU(2) × U(1)]/Z₂² = SO(3) × SO(3) × U(1) symmetry in the presence of the kinetic-energy hopping term of the Hamiltonian with t > 0. (Examples of a bipartite lattice are the D-dimensional cubic lattices of lattice constant a and edge length L = N_aa for which D = 1, 2, 3,... in the number N_a^D of sites.) The generator of the new found hidden independent charge global U(1) symmetry, which is not related to the ordinary U(1) gauge subgroup of electromagnetism, is one half the rotated-electron number of singly occupied sites operator. Although addition of chemical-potential and magnetic-field operator terms to the model Hamiltonian lowers its symmetry, such terms commute with it. Therefore, its 4^N_aD energy eigenstates refer to representations of the new found global [SU(2) × SU(2) × U(1)]/Z₂² = SO(3) × SO(3) × U(1) symmetry. Consistently, we find that for the Hubbard model on a bipartite lattice the number of independent representations of the group SO(3) × SO(3) × U(1) equals the Hilbert-space dimension 4^N_aD. It is confirmed elsewhere that the new found symmetry has important physical consequences.     

动力学对称群方法对三原子分子高激发振动态的理论研究

利用动力学对称群方法研究了弯曲三原子分子的振动高激发态能谱-该方法显示：三原子分子的动力学对称性为U₁(4)U₂(4),则三原子分子的Hamiltonian量可写成代数的各元素之和,通过李代数处理而求得分子代数Hamiltonian量的本征值,进而得到分子的振动能谱-并具体计算了O₃分子- 关键词：     

Study of the stretching modes of the arsine molecule

To study local mode XY₃ molecules, we use properties of the group chain U(4)⊃U(3)⊃K(3)⊃S(3)≈C_3v. For the Hamiltonian, we deduce diagonal terms and coupling terms between bonds. We analyze the stretching modes of the arsine molecule. An algebraic transition operator is built and applied to the same molecular system.     

A perturbation theory study of H2 on LiF(0 0 1)

In response to recent helium atom scattering (HAS) results, Monte Carlo (MC) simulations and perturbation theory have been performed for H₂ on LiF(0 0 1). MC simulations predict that H₂ molecules form a series of structures, p(2 × 2), p(8 × 2), p(4 × 2) with coverages Θ = 0.5, 0.625 and 0.75, respectively, that are stable up to 8 K. This is in partial agreement with the HAS results that report c(2 × 2) and c(8 × 2) structures; they agree in terms of coverage and stability, but disagree in terms of symmetry. To reconcile the results of the simulations and experiments, the orientation of the adsorbed H₂ molecules was studied using perturbation theory. These calculations show that the adsorbed H₂ molecules are azimuthally delocalized and that the structures are c-type rather then p-type. The calculations also indicate that p-H₂ and helicoptering o-H₂ prefer cationic sites, while cartwheeling o-H₂ prefers anionic sites.     

On the study of the vibrational energy levels of Arsine molecule

We compare two formalisms applied to the vibrational modes of the molecule of AsH₃ of C_3v molecular symmetry group. Indeed, the close stretching modes of this molecule may be considered as those of a three-dimensional oscillator whereas the bending modes may be considered either as a one-dimensional oscillator of symmetry A₁ and a two-dimensional oscillator of symmetry E or as an approximate three-dimensional oscillator. So, we have applied the U(p + 1) formalism to the both stretching and bending modes and introduced coupling terms acting on an appropriate coupled vibrational basis through a local mode formalism. We have then compared the result of our fitting with those obtained with the coupling of a local mode formalism adapted to the stretching vibrations with a normal mode formalism for the bending ones. Finally we compare our results with other methods recently proposed in the literature.     

Algebraic treatments of the problems of the spin-1/2 particles in the one- and two-dimensional geometry: A systematic study

We consider solutions of the 2 × 2 matrix Hamiltonians of the physical systems within the context of the su (2) and su (1, 1) Lie algebras. Our technique is relatively simple when compared with those of others and treats those Hamiltonians which can be treated in a unified framework of the Sp (4, R) algebra. The systematic study presented here reproduces a number of earlier results in a natural way as well as leads to a novel finding. Possible generalizations of the method are also suggested.     

Microwave structural studies of organoselenium compounds 1. Microwave spectra, molecular structure, and methyl barrier to internal rotation of dimethyl diselenide

The rotational spectra of nine isotopomers of dimethyl diselenide, CH₃SeSeCH₃, have been measured with a molecular-beam Fourier transform microwave spectrometer. The spectra were complex due to the presence of many isotopomers in natural abundance and the splitting caused by the interactions with two methyl internal rotors. The spectra were assigned and fit to experimental precision to an effective rotational Hamiltonian for molecules with two periodic internal motions. The spectra of the symmetric isotopomers are consistent with a C₂ equilibrium structure. The rotational constants were used to determine the r_s structure of the C-Se-Se-C frame with the results r(SeSe)=2.306(3) Å, r(SeC)=1.954(6) Å, ?(CSeSe)=99.8(2)°, ?(CSeSeC)=85.2(1)°. A barrier to internal rotation of the methyl groups of 395 ± 2 cm⁻¹ was derived from the internal rotation splittings.     

Dynamical Lie algebra method for highly excited vibrational state of asymmetric linear tetratomic molecules

The highly excited vibrational states of asymmetric linear tetratomic molecules are studied in the framework of Lie algebra. By using symmetric groupU ₁(4)U ₂(4)⊗U ₃(4), we construct the Hamiltonian that includes not only Casimir operators but also Majorana operators M₁₂, M₁₃ and M₂₃, which are useful for getting potential energy surface and force constants in Lie algebra method. By Lie algebra treatment, we obtain the eigenvalues of the Hamiltonian, and make the concrete calculation for molecule C₂HF.     

Supersymmetries of the spin-1/2 particle in the field of magnetic vortex, and anyons

The quantum non-relativistic spin-1/2 planar systems in the presence of a perpendicular magnetic field are known to possess the N = 2 supersymmetry. We consider such a system in the field of a magnetic vortex, and find that there are just two self-adjoint extensions of the Hamiltonian that are compatible with the standard N = 2 supersymmetry. We show that only in these two cases one of the subsystems coincides with the original spinless Aharonov-Bohm model and comes accompanied by the super-partner Hamiltonian which allows a singular behavior of the wave functions. We find a family of additional, nonlocal integrals of motion and treat them together with local supercharges in the unifying framework of the tri-supersymmetry. The inclusion of the dynamical conformal symmetries leads to an infinitely generated superalgebra, that contains several representations of the superconformal osp(2∣2) symmetry. We present the application of the results in the framework of the two-body model of identical anyons. The nontrivial contact interaction and the emerging N = 2 linear and nonlinear supersymmetries of the anyons are discussed.     

A Monte Carlo simulation study of H2 layers on NaCl(0 0 1)

Monte Carlo simulations show that, at one monolayer coverage, H₂ molecules adsorbed on a NaCl(0 0 1) surface occupy all Na⁺ sites and form a commensurate c(2 × 2) structure. If the Cl⁻ sites are occupied as well, a bi-layer p(2 × 1) structure forms. An examination of the H₂ molecules’ rotational motion shows the molecular axes are azimuthally delocalized and so both of the structures acquire (1 × 1) symmetry in accord with experimental observations. These calculations also show that helicoptering o-H₂ (J = 1, m = ±1) prefer to sit on top of Na⁺ sites, while cartwheeling o-H₂ (J = 1, m = 0) prefers to locate over Cl⁻ sites, in agreement with other work.     

SUSY QM, symmetries and spectrum generating algebras for two-dimensional systems

We show in a systematic and clear way how factorization methods can be used to construct the generators for hidden and dynamical symmetries. This is shown by studying the 2D problems of hydrogen atom, the isotropic harmonic oscillator and the radial potential Aρ^2ζ−2 − Bρ^ζ−2. We show that in these cases the non-compact (compact) algebra corresponds to so(2, 1) (su(2)).     

The su q(2) algebra in the off-diagonal basis and applications to quantum optics

We consider a new exactly solvable nonlinear quantum model as a Hamiltonian defined in terms of the generators of the su _q(2) algebra. The corresponding matrix elements of finite rotations (the q-deformed Wigner d functions) are introduced. It is shown that the quantum optical model of the three-wave interaction has an approximate su _q(2) dynamical symmetry given by this Hamiltonian. Such q symmetry allows us to investigate the spectral and dynamical properties of the three wave model through new perturbation techniques.     

Toroidal p-branes, anharmonic oscillators and (hyper)elliptic solutions

Exact solvability of brane equations is studied, and a new U(1)×U(1)×?×U(1) invariant anzats for the solution of p-brane equations in D=(2p+1)-dimensional Minkowski space is proposed. The reduction of the p-brane Hamiltonian to the Hamiltonian of p-dimensional relativistic anharmonic oscillator with the monomial potential of the degree equal to 2p is revealed. For the case of degenerate p-torus with equal radii it is shown that the p-brane equations are integrable and their solutions are expressed in terms of elliptic (p=2) or hyperelliptic (p>2) functions. The solution describes contracting p-brane with the contraction time depending on p and the brane energy density. The toroidal brane elasticity is found to break down linear Hooke law as it takes place for the anharmonic elasticity of smectic liquid crystals.