Construction Of Wave Functions Research Articles

Anyon condensation: coherent states, symmetry enriched topological phases, Goldstone theorem, and dynamical rearrangement of symmetry

Although the mathematics of anyon condensation in topological phases has been studied intensively in recent years, a proof of its physical existence is tantamount to constructing an effective Hamiltonian theory. In this paper, we concretely establish the physical foundation of anyon condensation by building the effective Hamiltonian and the Hilbert space, in which we explicitly construct the vacuum of the condensed phase as the coherent states that are the eigenstates of the creation operators creating the condensate anyons. Along with this construction, which is analogous to Laughlin’s construction of wavefunctions of fractional quantum hall states, we generalize the Goldstone theorem in the usual spontaneous symmetry breaking paradigm to the case of anyon condensation. We then prove that the condensed phase is a symmetry enriched (protected) topological phase by directly constructing the corresponding symmetry transformations, which can be considered as a generalization of the Bogoliubov transformation.

Quantitative Resonance Theory Based on the Clar Sextet Model.

Despite its great explanatory power in understanding the chemistry of polycyclic aromatic hydrocarbons (PAHs) and related systems, the Clar sextet rule still remains an intuitive and qualitative model with notable exceptions in some cases. Here we develop a quantitative theory of chemical resonance based on semilocalized Clar-type resonance structures (named the Clar resonators) consisting of variable numbers of Clar sextets and C═C bonds. The constructed wave functions of the Clar resonators are used to expand the actual wave function of the π-conjugated system obtained from a DFT or Hartree-Fock calculation. The resultant weights and one-electron energies of the Clar resonators can serve as a quantitative measure of the importance of these resonators. Implementing the theory in our open-source python code EzReson and applying it to over a thousand PAH molecules of different sizes and shapes, we show that the weight of the Clar resonators increases exponentially with increasing number of sextets and that their energy decreases linearly with the latter, thus confirming the general validity of the Clar rule. On the basis of such a large-scale resonance analysis, we propose three extended Clar rules, along with a unified quantitative model, that are able to evaluate the importance of all Clar resonators and the ring aromaticity for PAHs. Using the present theories, we uncover the essential role that the minor Clar resonators may play in correctly understanding the resonance stabilization and local aromaticity of rings, which was totally overlooked in the original Clar model.

A quantum moat barrier, realized with a finite square well

The notion of a double well potential typically involves two regions of space separated by a repulsive potential barrier. The ground state is a wave function that is suppressed in the barrier region and localized in the two surrounding regions. We illustrate that an attractive potential well (a quantum moat) with a finite nonzero width also acts as a barrier, using a simple square well model. We also show how the pseudopotential method both explains the role of the well as a barrier, and greatly improves the efficiency of constructing wave functions for this system using matrix diagonalization. With this simplified model we provide an introduction to the ideas typically used to simplify calculations in solids, where in place of the double well potential, multiple potentials occur in a periodic array.

Elastic waves with correlated directional and orbital angular momentum degrees of freedom

We introduce a formalism that enables the calculation of elastic wave functions supported by parallel arrays of coupled one-dimensional elastic waveguides. These wave functions are expressed as tensor products of a spinor part associated with directional degrees of freedom and an orbital angular momentum (OAM) part associated with the phase of the coupled waveguides. We demonstrate that one can construct wave functions as a superposition of these elastic waves, which cannot be written as a tensor product of a spinor part and an OAM part. These elastic wave functions are not separable in the tensor product Hilbert space of directional and OAM subspaces. We show that we can construct maximally nonseparable states that are similar to Bell states.

Size-Extensive Wave Functions for Quantum Monte Carlo: A Linear Scaling Generalized Valence Bond Approach.

We propose a new class of multideterminantal Jastrow-Slater wave functions constructed with localized orbitals and designed to describe complex potential energy surfaces of molecular systems for use in quantum Monte Carlo (QMC). Inspired by the generalized valence bond formalism, we elaborate a coupling scheme between electron pairs which progressively includes new classes of excitations in the determinantal component of the wave function. In this scheme, we exploit the local nature of the orbitals to construct wave functions which have increasing complexity but scale linearly. The resulting wave functions are compact, can correlate all valence electrons, and are size extensive. We assess the performance of our wave functions in QMC calculations of the homolytic fragmentation of N-N, N-O, C-O, and C-N bonds, very common in molecules of biological interest. We find excellent agreement with experiments, and, even with the simplest forms of our wave functions, we satisfy chemical accuracy and obtain dissociation energies of equivalent quality to the CCSD(T) results computed with the large cc-pV5Z basis set.

Symmetry-protected fractional Chern insulators and fractional topological insulators

In this paper we construct fully symmetric wave functions for the spin-polarized fractional Chern insulators (FCIs) and time-reversal-invariant fractional topological insulators (FTIs) in two dimensions using the parton approach. We show that the lattice symmetry gives rise to many different FCI and FTI phases even with the same filling fraction $\ensuremath{\nu}$ (and the same quantized Hall conductance ${\ensuremath{\sigma}}_{xy}$ in the FCI case). They have different symmetry-protected topological orders, which are characterized by different projective symmetry groups. We mainly focus on FCI phases which are realized in a partially filled band with Chern number 1. The low-energy gauge groups of a generic ${\ensuremath{\sigma}}_{xy}=1/m\ifmmode\cdot\else\textperiodcentered\fi{}{e}^{2}/h$ FCI wave function can be either $\text{SU}(m)$ or the discrete group ${Z}_{m}$, and in the latter case the associated low-energy physics are described by Chern-Simons-Higgs theories. We use our construction to compute the ground-state degeneracy. Examples of FCI/FTI wave functions on honeycomb lattice and checkerboard lattice are explicitly given. Possible non-Abelian FCI phases which may be realized in a partially filled band with Chern number 2 are discussed. Generic FTI wave functions in the absence of spin conservation are also presented whose low-energy gauge groups can be either $\text{SU}(m)\ifmmode\times\else\texttimes\fi{}\text{SU}(m)$ or ${Z}_{m}\ifmmode\times\else\texttimes\fi{}{Z}_{m}$. The constructed wave functions also set up the framework for future variational Monte Carlo simulations.

ELECTRONIC STATES AND LOCAL DENSITY OF STATES NEAR GRAPHENE CORNER EDGE

We study that stability of edge localized states in semi-infinite graphene with a corner edge of the angles 60°, 90°, 120° and 150°. We adopt a nearest-neighbor tight-binding model to calculate the local density of states (LDOS) near each corner edge using Haydock's recursion method. The results of the LDOS indicate that the edge localized states stably exist near the 60°, 90°, and 150° corner, but locally disappear near the 120° corner. By constructing wave functions for a graphene ribbon with three 120° corners, we show that the local disappearance of the LDOS is caused by destructive interference of edge states and evanescent waves.

Independent pair-potential correlated wave functions

An approximation for calculating the correlation energy is proposed which lies intermediate between the independent electron pair approximation and the total pair-excited variational wave function. In this approximation the correlation contribution from each electron is calculated separately by constructing wave functions which include all the pair excitations of an electron occupying a particular reference orbital and the remaining electrons in the system. These fragments are then summed in order to obtain the total correlation energy. Some illustrative calculations on boron hydride are discussed.

Little type IIB matrix model

We study the zero-dimensional reduced model of D=6 pure super Yang-Mills theory and argue that the large N limit describes the (2,0) Little String Theory. The one-loop effective action shows that the force exerted between two diagonal blocks of matrices behaves as 1/r{sup 4}, implying a six-dimensional spacetime. We also observe that it is due to nongravitational interactions. We construct wave functions and vertex operators which realize the D=6 (2,0) tensor representation. We also comment on other little analogues of the IIB matrix model and Matrix Theory with less supercharges.

Fundamental importance of the Coulomb hole sum rule to the understanding of the Colle-Salvetti wave function functional

In this paper we consider the general form of the correlated-determinantal wave function functional of Colle and Salvetti (CS) for the He atom. The specific form employed by CS is the basis for the widely used CS correlation energy formula and the Lee-Yang-Parr correlation energy density functional of Kohn-Sham density functional theory. We show the following: (i) The key assumption of CS for the determination of this wave function functional, viz., that the resulting single-particle density matrix and the Hartree-Fock theory Dirac density matrix are the same, is equivalent to the satisfaction of the Coulomb hole sum rule for each electron position. The specific wave function functional derived by CS does not satisfy this sum rule for any electron position. (ii) Application of the theorem on the one-to-one correspondence between the Coulomb hole sum rule for each electron position and the constraint of normalization for approximate wave functions then proves that the wave function derived by CS violates charge conservation. (iii) Finally, employing the general form of the CS wave function functional, the exact satisfaction of the Coulomb hole sum rule at each electron position then leads to a wave function that is normalized. The structure of the resulting approximate Coulomb holes is reasonably accurate, reproducing both the short- and the long-range behavior of the hole for this atom. Thus, the satisfaction of the Coulomb hole sum rule by an approximate wave function is a necessary condition for constructing wave functions in which electron-electron repulsion is represented reasonably accurately.

Generation of Clebsch–Gordan coefficients for the point and double groups

The theory of the point and double groups has been widely used in quantum physics to understand the structure and dynamical properties of molecules and solids. In order to construct wave functions for such systems, one often needs the Clebsch–Gordan coefficients for the symmetry groups. Here, we present an extension of the Bethe program to support the calculation of the Clebsch–Gordan coefficients as applied, for instance, in crystal field theory. Apart from the generation of the Clebsch–Gordan coefficients, the program also provides a simple access to the group theoretical data for all frequently applied point and double groups. Program summary Title of program: Bethe Catalogue number:ADUH_v3_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADUH_v3_0 Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Reference in CPC to previous versions: Comput. Phys. Comm. 162 (2004) 124–142; Comput. Phys. Comm. 171 (2005) 119–132 Catalog identifiers of previous versions: ADUH, ADVU Does the new version supersede the old version?: Yes Licensing provisions: None Computers for which the program is designed: All computers with a license of the computer algebra package Maple [Maple is a registered trademark of Waterloo Maple Inc.] Installations: University of Kassel (Germany) Operating systems under which the program has been tested: Linux 8.1+ and Windows 2000 Programming language used: Maple 7 and 8 Memory required to execute with typical data: 10–30 MB No. of lines in distributed program, including test data, etc.: 11 024 No. of bytes in distributed program, including test data, etc.: 210 244 Distribution format:tar.gz Nature of the physical problem: The energy levels of atoms, placed into a crystal environment, can be classified by using group theory. In order to represent, for instance, the wave functions, which are associated with these atomic levels, one often requires the Clebsch–Gordan coefficients of the underlying symmetry group of the overall system. These coefficients arise in the coupling of the electronic wave functions (subsystems) and therefore help investigate the interaction between the many-electron atom and the external field of the crystal. Method of solution: In the framework of the Bethe program [K. Rykhlinskaya, S. Fritzsche, Comput. Phys Comm. 162 (2004) 124–142; K. Rykhlinskaya, S. Fritzsche, Comput. Phys Comm. (2005), in press], we previously defined data structures to deal with a large number of group parameters of the point and double groups. Among other parameters, here we also implemented the irreducible (matrix) representations of these groups which are utilized in the present extension of the program in order to generate the Clebsch–Gordan coefficients for the point and double groups. In practice, of course, these coefficients are obtained by means of a proper summation over the matrix elements of the irreducible representations of the group. Reasons for the new version: Extension of the program. Summary of revision: A number additional procedure have been created to generate the Clebsch–Gordan coefficients for the symmetry groups (Bethe_CG_matrix(), Bethe_CG_coefficient(), Bethe_group_direct_product(), etc.) Restrictions onto the complexity of the problem: The generation of the Clebsch–Gordan coefficients is supported for the cyclic and related groups C i , C s , C n , C n h , C n v , the dihedral groups D n , D n h , D n d , the improper cyclic groups S 2 n ( n ⩽ 10 ) , the cubic groups O, T, O h , T h , T d as well as the icosahedral groups I and I h . Both the point and the double groups are supported. Unusual features of the program: All commands of the Bethe program are available for interactive work. Apart from the generation of the Clebsch–Gordan coefficients, the program also provides a simple access to the group theoretical data for all the groups specified above. The notation of the symmetry operations and of the irreducible representations follows the compilation by Altmann and Herzig [S. Altmann, P. Herzig, Point-Group Theory Tables, Clarendon Press, Oxford, 1994]. For a quick reference to the program, a description of all user-relevant commands is given in the (user) manual Bethe-commands.pdf which is distributed together with the code. Typical running time: Although the program replies ‘promptly’ on most requests, the running time depends strongly on the particular task.

Supergravity Backgrounds in IIB Matrix Model

We construct wave functions and vertex operators in the type IIB matrix model by using a supersymmetric Wilson line operator. It is shown that wave functions can be intepreted as overlaps of the D-instanton boundary states with the closed string massless states. We then calculate the one-loop effective action for N D-instantons under the supergravity backgrounds by integrating out one D-instanton with an appropriate wave function.

Fermionic backgrounds and condensation of supergravity fields in the type IIB matrix model

In a previous paper [1] we constructed wave functions of a D-instanton and vertex operators in type IIB matrix model by expanding supersymmetric Wilson line operators. They describe couplings of a D-instanton and type IIB matrix model to the massless closed string fields, respectively, and form a multiplet of $D=10$ $\mathcal{N}=2$ supersymmetries. In this paper we consider fermionic backgrounds and condensation of supergravity fields in IIB matrix model by using these wave functions. We start from the type IIB matrix model in a flat background whose matrix size is $(N+1)\ifmmode\times\else\texttimes\fi{}(N+1)$, or equivalently the effective action for $(N+1)$ D-instantons. We then calculate an effective action for $N$ D-instantons by integrating out 1 D-instanton (which we call a mean-field D-instanton) with an appropriate wave function and show that various terms can be induced corresponding to the choice of the wave functions. In particular, a Chern-Simons-like term is induced when the mean-field D-instanton has a wave function of the antisymmetric tensor field. A fuzzy sphere becomes a classical solution to the equation of motion for the effective action. We also give an interpretation of the above wave functions in the superstring theory side as overlaps of the D-instanton boundary state with the closed string massless states in the Green-Schwarz formalism.

Wavefunctions for the time-dependent linear oscillator and Lie point symmetries

We investigate the solution of the quantal time-dependent linear oscillator from the viewpoint of the Lie point symmetries of its time-dependent Schrodinger equation. Four of the five nongeneric symmetries can be used for the construction of wavefunctions, two as creation symmetries and two as annihilation symmetries. The fifth nongeneric symmetry provides an eigenvalue for the Lie Bracket which maps a solution symmetry to itself. The general treatment indicates that care must be taken with the selection of solutions of the classical equation for the time-dependent linear oscillator and its third-order self-adjoint counterpart to obtain results which are consistent with the standard results for the autonomous harmonic oscillator.

Wilson loops and vertex operators in a matrix model

We systematically construct wave functions and vertex operators in the type IIB (IKKT) matrix model by expanding a supersymmetric Wilson loop operator. They form a massless multiplet of the $\mathcal{N}=2$ type IIB supergravity and automatically satisfy conservation laws.

Continuum discretization using orthogonal polynomials

A method for discretizing the continuum by using a transformed harmonic oscillator basis has recently been presented [Phys. Rev. A 63, 052111 (2001)]. In the present paper, we propose a generalization of that formalism which does not rely on the harmonic oscillator for the inclusion of the continuum in the study of weakly bound systems. In particular, we construct wave functions that represent the continuum by making use of families of orthogonal polynomials whose weight function is the square of the ground state wave function, expressed in terms of a suitably scaled variable. As an illustration, the formalism is applied to one-dimensional Morse, P\"oschl-Teller, and square well potentials. We show how the method can deal with potentials having several bound states, and for the square well case we present a comparison of the discretized and exact continuum wave functions.

Valence bond studies of theD2h isomer of O4: An interim report

Single-zeta and π-electron double-zeta basis sets are used to examine some theories of the origin of the stability of the D2h isomer of O4, using ab initio valence-bond procedures. With these basis sets, resonance between covalent-type (i.e., O2 ·· O2) valence-bond structures does not lead to a stabilization of the dimer relative to the separated monomers. When basis sets of the same size are used to construct wave functions for covalent and ionic structures, covalent-ionic resonance (i.e., O2 ·· O2 ↔ O2+ ·· O2− ↔ O2− ·· O2+) is also unable to stabilize the dimer. Without consideration of the basis-set superposition error, stability is obtained when the size of the AO basis is increased for the dimer relative to the monomer, either via the basis for the ionic structures or by the inclusion of midbond functions. Brief consideration is given to an increased-valence structure for the dimer. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 63: 547–555, 1997

Calculation of energy subband structures of corrugated lateral-surface superlattices based on variational principles

A theoretical method is developed for calculating energy subbands of carriers in lateral-surface superlattices with corrugated interfaces (CLSSLs). Based on the variational principle and a coordinate transformation, the method overcomes difficulties in constructing wave functions which must satisfy complicated boundary conditions on corrugated interfaces. The method is tested numerically via calculations of electron subbands, probability distributions and intersubband optical absorptions of CLSSLs with periodic variations of well thicknesses. Sensitive dependencies of electron subbands, probability distributions and intersubband optical absorptions on structural parameters of CLSSLs are predicted.

Exact solutions of generalized Calogero-Sutherland models: BCN and CN cases

Using a collective field method, we obtain explicit solutions of the generalized Calogero-Sutherland models that are characterized by the roots of the classical groups B N and C N . Starting from the explicit wave functions for the A N−1 type expressed in terms of the singular vectors of the W N algebra, we give a systematic method to construct wave functions and derive energy eigenvalues for other types of theories.

Generalization of a result of Matsuo and Cherednik to the Calogero-Sutherland-Moser integrable models with exchange terms

A few years ago, Matsuo and Cherednik proved that from some solutions of the Knizhnik-Zamolodchikov (KZ) equations, which first appeared in conformal field theory, one can obtain wave functions for the Calogero integrable system. In the present communication, it is shown that from some solutions of generalized KZ equations, one can construct wave functions, characterized by any given permutational symmetry, for some Calogero-Sutherland-Moser integrable models with exchange terms. Such models include the spin generalizations of the original Calogero and Sutherland ones, as well as that with $\delta$-function interaction.