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arXiv:0808.2949 · Journal of Mathematical Physics

On the use of group theoretical and graphical techniques toward the solution of the general N-body problem

W. Blake Laing, Martin Dunn, D. K. Watson

Published Jun 1, 2009• 13 citations
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Abstract

Group theoretic and graphical techniques are used to derive the N-body wave function for a system of identical bosons with general interactions through first-order in a perturbation approach. This method is based on the maximal symmetry present at lowest order in a perturbation series in inverse spatial dimensions. The symmetric structure at lowest order has a point group isomorphic with the SN group, the symmetric group of N particles, and the resulting perturbation expansion of the Hamiltonian is order-by-order invariant under the permutations of the SN group. This invariance under SN imposes severe symmetry requirements on the tensor blocks needed at each order in the perturbation series. We show here that these blocks can be decomposed into a basis of binary tensors invariant under SN. This basis is small (25 terms at first order in the wave function), independent of N, and is derived using graphical techniques. This checks the N6 scaling of these terms at first order by effectively separating the N scaling problem away from the rest of the physics. The transformation of each binary tensor to the final normal coordinate basis requires the derivation of Clebsch–Gordon coefficients of SN for arbitrary N. This has been accomplished using the group theory of the symmetric group. This achievement results in an analytic solution for the wave function, exact through first order, that scales as N0, effectively circumventing intensive numerical work. This solution can be systematically improved with further analytic work by going to yet higher orders in the perturbation series.

DOI
10.1063/1.3118883

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Group theoretic and graphical techniques are used to derive the N-body wave function for a system of identical bosons with general interactions through first-order in a perturbation approach. This method is based on the maximal symmetry present at lowest order in a perturbation series in inverse spatial dimensions. The symmetric structure at lowest order has a point group isomorphic with the SN group, the symmetric group of N particles, and the resulting perturbation expansion of the Hamiltonian is order-by-order invariant under the permutations of the SN group. This invariance under SN imposes severe symmetry requirements on the tensor blocks needed at each order in the perturbation series. We show here that these blocks can be decomposed into a basis of binary tensors invariant under SN. This basis is small (25 terms at first order in the wave function), independent of N, and is derived using graphical techniques. This checks the N6 scaling of these terms at first order by effectively separating the N scaling problem away from the rest of the physics. The transformation of each binary tensor to the final normal coordinate basis requires the derivation of Clebsch–Gordon coefficients of SN for arbitrary N. This has been accomplished using the group theory of the symmetric group. This achievement results in an analytic solution for the wave function, exact through first order, that scales as N0, effectively circumventing intensive numerical work. This solution can be systematically improved with further analytic work by going to yet higher orders in the perturbation series.
We discuss a basis set developed to calculate perturbation coefficients in an expansion of the general N-body problem. This basis has two advantages. First, the basis is complete order-by-order for the perturbation series. Second, the number of independent basis tensors spanning the space for a given order does not scale with N, the number of particles, despite the generality of the problem. At first order, the number of basis tensors is 23 for all N although the problem at first order scales as N^6. The perturbation series is expanded in inverse powers of the spatial dimension. This results in a maximally symmetric configuration at lowest order which has a point group isomorphic with the symmetric group, S_N. The resulting perturbation series is order-by-order invariant under the N! operations of the S_N point group which is responsible for the slower than exponential growth of the basis. In this paper, we perform the first test of this formalism including the completeness of the basis through first order by comparing to an exactly solvable fully-interacting problem of N particles with a two-body harmonic interaction potential.
We discuss a basis set developed to calculate perturbation coefficients in an expansion of the general N -body problem. This basis has two advantages. First, the basis is complete order-by-order for the perturbation series. Second, the number of independent basis tensors spanning the space for a given order does not scale with N , the number of particles, despite the generality of the problem. At first order, the number of basis tensors is 25 for all N , i.e. the problem scales as N 0 , although one would initially expect an N 6 scaling at first order. The perturbation series is expanded in inverse powers of the spatial dimension. This results in a maximally symmetric configuration at lowest order which has a point group isomorphic with the symmetric group, S N . The resulting perturbation series is order-by-order invariant under the N ! operations of the S N point group which is responsible for the slower than exponential growth of the basis. In this paper, we demonstrate the completeness of the basis and perform the first test of this formalism through first order by comparing to an exactly solvable fully interacting problem of N particles with a two-body harmonic interaction potential.
The resources required to solve the general interacting quantum NNN-body problem scale exponentially with NNN, making the solution of this problem very difficult when NNN is large. In a previous series of papers we develop an approach for a fully interacting wave function for a confined system of identical bosons with a general two-body interaction. This method tames the NNN scaling by developing a perturbation series that is order-by-order invariant under a point group isomorphic with SN{S}_{N}SN​. Group theory and graphical techniques are then used to solve for the wave function exactly and analytically at each order, yielding a solution for the general NNN-body problem which scales as N0{N}^{0}N0 at any given order. Recently this formalism has been used to obtain the first-order fully interacting wave function for a system of harmonically confined bosons interacting harmonically. In this paper, we report the application of this NNN-body wave function to a system of NNN fully interacting bosons in three dimensions. We derive an expression for the density profile for a confined system of harmonically interacting bosons. Choosing this simple interaction is not necessary or even advantageous for our method, however this choice allows a direct comparison of our exact results through first order with exact results obtained in an independent solution. Our density profile for the wave function through first order in three dimensions is indistinguishable from the first-order exact result obtained independently and shows strong convergence to the exact result to all orders.
The resources required to solve the general interacting quantum N-body problem scale exponentially with N, making the solution of this problem very difficult when N is large. In a previous series of papers we develop an approach for a fully-interacting wave function with a general two-body interaction which tames the N-scaling by developing a perturbation series that is order-by-order invariant under a point group isomorphic with S_N . Group theory and graphical techniques are then used to solve for the wave function exactly and analytically at each order. Recently this formalism has been used to obtain the first-order, fully-interacting wave function for a system of harmonically-confined bosons interacting harmonically. In this paper, we report the first application of this N-body wave function to a system of N fully-interacting bosons in three dimensions. We determine the density profile for a confined system of harmonically-interacting bosons. Choosing this simple interaction is not necessary or even advantageous for our method, however this choice allows a direct comparison of our exact results through first order with exact results obtained in an independent solution. Our density profile through first-order in three dimensions is indistinguishable from the first-order exact result obtained independently and shows strong convergence to the exact result to all orders.
In this paper, the second in a series of two, we complete the derivation of the lowest-order wave function of a dimensional perturbation theory (DPT) treatment for the N-body quantum-confined system. Taking advantage of the symmetry of the zeroth-order configuration, we use group theoretic techniques and the FG matrix method from quantum chemistry to obtain analytic results for frequencies and normal modes. This method directly accounts for each two-body interaction, rather than an average interaction so that even lowest-order results include beyond-mean-field effects. It is thus appropriate for the study of both weakly and strongly interacting systems and the transition between them. While previous work has focused on energies, lowest-order wave functions yield important information such as the nature of excitations and expectation values of physical observables at low orders including density profiles. Higher orders in DPT also require as input the zeroth-order wave functions. In the earlier paper we presented a program for calculating the analytic normal-mode coordinates of the large-D system and illustrated the procedure by deriving the two simplest normal modes. In this paper we complete this analysis by deriving the remaining, and more complex, normal coordinates of the system.
We generalize the harmonic-order N-body DPT method for an isotropically confined quantum system of identical interacting bosons to first-anharmonic order and, in principle, higher orders. We introduce a graphical decomposition of the perturbative expansion of the N-body Hamiltonian. We calculate the Clebsch-Gordon coefficient tensors that couple together 3 irreducible representations of SN analytically, and analytically transformed the graphical basis to collective coordinates. We calculate the N-body wavefunction and density profile in general and have demonstrated agreement with an analytic model. We apply this formalism to the exactly-solvable example of a trapped gas of atoms interacting with a (fully-interacting) harmonic-oscillator "Hooke's law" interaction and compare with the dimensional expansion of the exact ground-state wavefunction and density profile. We report progress on the example of a cold gas BEC (with zero angular momentum).