Arbitrary Permutational Symmetry Research Articles

One-electron densities of freely rotating Wigner molecules

A formalism enabling computation of the one-particle density of a freely rotating assembly of identical particles that vibrate about their equilibrium positions with amplitudes much smaller than their average distances is presented. It produces densities as finite sums of products of angular and radial functions, the length of the expansion being determined by the interplay between the point-group and permutational symmetries of the system in question. Obtaining from a convolution of the rotational and bosonic components of the parent wavefunction, the angular functions are state-dependent. On the other hand, the radial functions are Gaussians with maxima located at the equilibrium lengths of the position vectors of individual particles and exponents depending on the scalar products of these vectors and the eigenvectors of the corresponding Hessian as well as the respective eigenvalues. Although the new formalism is particularly useful for studies of the Wigner molecules formed by electrons subject to weak confining potentials, it is readily adaptable to species (such as ʻballiums’ and Coulomb crystals) composed of identical particles with arbitrary spin statistics and permutational symmetry. Several examples of applications of the present approach to the harmonium atoms within the strong-correlation regime are given.

REDUCED DENSITY MATRIX AND ENTANGLEMENT ENTROPY OF PERMUTATIONALLY INVARIANT QUANTUM MANY-BODY SYSTEMS

In this paper we discuss the properties of the reduced density matrix of quantum many body systems with permutational symmetry and present basic quantification of the entanglement in terms of the von Neumann (VNE), Renyi and Tsallis entropies. In particular, we show, on the specific example of the spin 1/2 Heisenberg model, how the RDM acquires a block diagonal form with respect to the quantum number k fixing the polarization in the subsystem conservation of Sz and with respect to the irreducible representations of the Sn group. Analytical expression for the RDM elements and for the RDM spectrum are derived for states of arbitrary permutational symmetry and for arbitrary polarizations. The temperature dependence and scaling of the VNE across a finite temperature phase transition is discussed and the RDM moments and the Rényi and Tsallis entropies calculated both for symmetric ground states of the Heisenberg chain and for maximally mixed states.

Hyperspherical functions with arbitrary permutational symmetry: Reverse construction

An algorithm is formulated for the construction of hyperspherical functions with an arbitrary permutational symmetry. As proposed by Novoselsky and Katriel [Phys. Rev. A $49,$ 833 (1994)], we use a recursive procedure, introducing hyperspherical coefficients of fractional parentage (hscfps). These coefficients are the eigenvectors of the transposition class sum of the symmetric group in an appropriate basis. Utilizing a reversed-order set of the Jacobi coordinates we obtain a set of symmetrized basis functions well suited for few-body calculations as the evaluation of matrix elements of two-body and three-body forces involves the hscfps but no further rotation of the Jacobi coordinates. The results are applicable to the study of atomic, molecular and nuclear few-body problems. Numerical results for nuclear soft-core model potential are presented for 3, 4, and 5 body systems.

Hyperspherical functions with arbitrary permutational symmetry.

An algorithm is formulated for the construction of many-particle permutational symmetry adapted functions in hyperspherical coordinates. A recursive procedure is proposed, introducing hyperspherical coefficients of fractional parentage (hscfps). These coefficients are the eigenvectors of the transposition class sum of the symmetric group in an appropriate basis. Only the matrix element of the transposition of the last two particles has to be calculated in each step. This matrix element is obtained by using the hscfps calculated in the preceding step as well as the Raynal-Revai and the T coefficients. The results are applicable to the study of the atomic, molecular, and nuclear few-body problem.

Harmonic Oscillator SU3 States with Arbitrary Permutational Symmetry

Many-particle harmonic oscillator states that belong to a given SU 3 irreducible representation and are at the same time characterized by a Yamanouchi symbol specifying their permutational symmetry, are constructed recursively, using the SU3 coefficients of fractional parentage. These coefficients are the eigenvectors of the two-cycle class operators of the symmetric group in the appropriate basis. In the recursion procedure only the matrix element of the transposition of the last two particles has to be calculated in each step. This matrix element is obtained by using the SU 3 Racah coefficients. The significance of this procedure for nuclear physics as well as for quark calculations is briefly pointed out.

Fractional quantum Hall effect studied by means of a nonspurious basis set.

A method for constructing a nonspurious set of N-particle harmonic wave functions with arbitrary permutational symmetry is employed to study the behavior of two-dimensional few-electron clusters on an infinite flat plane in a strong magnetic field. The Coulomb interaction is exactly diagonalized within the first Landau level and energy cusps are identified at the appropriate filling factors. The exact energies and pair distribution functions are compared with the corresponding results obtained using Laughlin's trial wave functions. Very good agreement is found with Laughlin's m=3 wave function, but some deviation is indicated for m=5.

Multi-cluster wavefunctions with arbitrary permutational symmetry

We consider a system of identical particles distributed in several clusters. Each cluster is characterized by a Yamanouchi symbol specifying its permutational symmetry. A procedure for coupling the various clusters into an overall well-defined angular momentum and permutational symmetry, is presented. It is based on diagonalizing an appropriate set of single-cycle class operators of the symmetric group. The techniques for the evaluation of the necessary matrix-elements are developed. They provide interesting generalizations of the corresponding techniques for a system consiting of a single-cluster of particles. The results are immediately applicable to the study of nuclear cluster models, nuclei as quark clusters, as well as the interaction among gas-phase atomic clusters.

Multi-cluster wavefunctions with arbitrary permutational symmetry: Jacob Katriel. Department of Chemistry, Technion—Israel Institute of Technology, Haifa 32000, Israel; and Akiva Novoselsky. Department of Nuclear Physics, The Weizmann Institute of Science, Rehovot 76100, Israel

Multi-cluster wavefunctions with arbitrary permutational symmetry: Jacob Katriel. Department of Chemistry, Technion—Israel Institute of Technology, Haifa 32000, Israel; and Akiva Novoselsky. Department of Nuclear Physics, The Weizmann Institute of Science, Rehovot 76100, Israel

Generalized creation and annihilation operators: Physical interpretation and reordering properties.

Matrix elements for fermion and boson operators describing the creation or annihilation of particles with multiple angular-momentum quantum numbers can be factored into products of matrix elements of generalized operators, one for each quantum number. The generalized operators have nonzero matrix elements between states of arbitrary permutational symmetry. This factorization greatly increases the efficiency of shell-model calculations. We provide a physical interpretation for these generalized operators and introduce a simple operator-reordering procedure.

Nonspurious harmonic oscillator states in single particle coordinates

A procedure for the construction of nonspurious harmonic oscillator wave functions with arbitrary permutational symmetry has recently been proposed. The resulting wave functions are expressed in terms of normalized Jacobi coordinates, and involve a new type of harmonic oscillator coefficients of fractional parentage. A simple algorithm to transform these states from the Jacobi coordinates to the single particle coordinates is presented. This is a generalization to an arbitrary number of particles of the harmonic oscillator transformation from center-of-mass and relative coordinates to single particle coordinates.

Non-spurious harmonic oscillator states with arbitrary symmetry

An algorithm for the construction of non-spurious harmonic oscillator wave functions with arbitrary permutational symmetry is formulated by adapting a procedure recently developed for building general single shell wave functions. The harmonic oscillator functions, expressed in Jacobi coordinates, are calculated recursively using a new type of harmonic oscillator coefficients of fractional parentage. These coefficients are the eigenvectors of the two-cycle class operator of the permutation group in the appropriate basis. The matrix elements of the class operators are evaluated by using a specific version of the harmonic oscillator brackets. The significance of this procedure to atomic, molecular and nuclear physics is pointed out. The presently proposed method is expected to be computationally very efficient.

Matrix elements of shell model Hamiltonians in multiple angular momentum coupling schemes.

Coefficients of fractional parentage (CFP's) in an L-S (or J-T) coupling scheme have recently been computed by using a new algorithm extending two procedures introduced many years ago: a factorization procedure introduced by Jahn and a combination of numerical and analytic methods introduced by Bayman and Lande. As a consequence of the factorization, coefficients of fractional parentage between states of arbitrary permutational symmetry must be computed within each angular momentum subspace separately. The new algorithm is extended here to calculate Hamiltonian matrix elements in the second quantized formalism. The matrix elements of the creation and annihilation operators, within each individual subspace, between states of arbitrary symmetry, are defined to be proportional to the corresponding coefficients of fractional parentage. The matrix elements for the one- and two-body operators in the Hamiltonian are constructed by combining the matrix elements in the individual subspaces using a new sum-over-path-overlaps technique. This procedure is extremely efficient for quark shell model calculations, where the quarks carry multiple quantum numbers of angular momentum, or more general, type (L, S, color, and flavor).

Coefficients of fractional parentage in the L–S coupling scheme

An efficient procedure for the evaluation of the coefficients of fractional parentage (cfp’s) for L–S coupled wave functions is presented. The cfp’s are calculated separately for N particles, each with angular momentum l (s), coupled into a total angular momentum L (S). The N-particle states formed can belong to any permutational symmetry. The procedure for the evaluation of the L and the S cfp’s for arbitrary permutational symmetry is a generalization of the procedure proposed by Bayman and Lande [Nucl. Phys. 77, 1 (1966)] for symmetric and antisymmetric states. It involves the construction and diagonalization of the matrices representing the quadratic Casimir operators for the appropriate special unitary and symplectic (or orthogonal) groups. The cfp’s of the antisymmetric L–S coupled states are obtained in terms of products of cfp’s for L and S corresponding to conjugate representations of the symmetric group. This method is demonstrated to provide cfp’s for L–S states for systems with a considerably larger number of particles than is feasible using the procedures heretofore available.

A group theoretical approach to geminal product matrix elements

Expressions enabling systematic compilation of Hamiltonian and overlap matrix elements for an antisymmetrized multiterm geminal product trial function are derived, using double coset (DC) decompositions and subgroup adapted irreducible representations of the symmetric group, SN. The trial function may describe an even electron atomic or molecular system in any total spin eigenstate, and the geminals may be nonorthogonal, have arbitrary permutational symmetry, and be explicit functions of interelectronic distance. A DC decomposition is used to factor out permutations not exchanging particle labels between geminals (elements of the interior pair group, Sn2, n≡N/2). This reduces the sum over N! permutations to a sum over DC generators. If the irreducible representation λ (S) of SN is adapted to Sn2 each geminal is projected into its singlet or triplet component. The DC generators are chosen such that each has the form QP, where Q permutes odd particle labels only and P is a permutation of geminals (element of the exterior pair group, S?). With the aid of matrices called DC symbols an algorithm for these generators is derived, and used to find explicit sets for N=2, 4, 6, and 8. The N-electron Hamiltonian and overlap integrals arising with a particular DC generator QP are factored into products of smaller integrals, called cluster integrals, according to the cycle structure of Q. The cluster integrals are of only three main types—overlap, one-cluster energy, and two-cluster energy (analogs of orbital overlap, 1-electron, and 2-electron integrals) —and are further classified by order (number of geminals), geminal permutational symmetry, and in some cases pattern of connection. Matrix element compilation is systematic in that all N-electron integrals are products of a relatively small number of different types of cluster integral, and that N-electron integrals with similar factored forms are collected together in the summation. From counting the different cluster integrals required, it is concluded that a geminal product calculation not using orbital expansion is feasible only for systems with eight or less electrons. In some cases semiempirical calculations with correlated geminals might be considered, for the more complex cluster integrals (those of high order) are quite small for a system approximating a collection of localized electron pairs. The matrix element expressions are specialized for three cases—all geminals being singlets, strongly orthogonal geminals, and identical geminals. Comparison is made with a recently developed diagrammatical method.