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arXiv:1501.06182 · Physical Review A

Sum rules for spin- 1/2 quantum gases in states with well-defined spins: Spin-independent interactions and spin-dependent external fields

Vladimir Yurovsky

Published May 1, 2015• 7 citations
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Abstract

Analytical expressions are derived for sums of matrix elements and their squared moduli over many-body states with given total spin---the states built from spin and spatial wave functions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. For spin-dependent one-body interactions with external fields and spin-independent two-body ones between the particles, the sum dependence on the many-body states is given by universal factors, which are independent of the interaction details and Hamiltonians of noninteracting particles. The sum rules are applied to perturbative analysis of energy spectra.

DOI
10.1103/physreva.91.053601

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Analytical expressions are derived for sums of matrix elements and their squared moduli over many-body states with given total spin---the states built from spin and spatial wave functions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. For spin-dependent one-body interactions with external fields and spin-independent two-body ones between the particles, the sum dependence on the many-body states is given by universal factors, which are independent of the interaction details and Hamiltonians of noninteracting particles. The sum rules are applied to perturbative analysis of energy spectra.
Sum rules for spin-1/21/21/2 quantum gases in well-defined-spin states: spin-independent interactions and spin-dependent external fields
Many-body eigenstates of spin-1/21/21/2 particles with defined total spins contain spin and spatial wavefunctions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. Matrix elements in the basis of such eigenstates are analyzed for spin-dependent interactions with external fields and spin-independent ones between the particles. Analytical expressions are obtained for sums of the matrix elements and sums of their squared modules. The sum rules are applied to perturbative analysis of energy spectra.
Sums of matrix elements of spin-dependent two-body momentum-independent interactions and sums of their products are calculated analytically in the basis of many-body states with given total spin---the states built from spin and spatial wave functions belonging to multidimensional irreducible representations of the symmetric group, unless the total spin has the maximal allowed value. As in the first part of the series [V. A. Yurovsky, Phys. Rev. A 91, 053601 (2015)], the sum dependence on the many-body states is given by universal factors, which are independent of the Hamiltonians of noninteracting particles. The sum rules are applied to perturbative analysis of energy spectra and to calculation of two-body spin-dependent local correlations.
Many-body systems of identical arbitrary-spin particles, with separable spin and spatial degrees of freedom, are considered. Their eigenstates can be classified by Young diagrams, corresponding to nontrivial permutation symmetries (beyond the conventional paradigm of symmetric-antisymmetric states). The present work obtains the following. (a) Selection rules for additional nonseparable (dependent on spins and coordinates) k-body interactions: the Young diagrams, associated with the initial and final states of a transition, can differ by relocation of no more than k boxes between their rows. (b) Correlation rules in which eigenstate-averaged local correlations of k particles vanish if k exceeds the number of columns (for bosons) or rows (for fermions) in the associated Young diagram. It also elucidates the physical meaning of the quantities conserved due to permutation symmetry-in 1929, Dirac identified those with characters of the symmetric group-relating them to experimentally observable correlations of several particles. The results provide a way to control the formation of entangled states belonging to multidimensional non-Abelian representations of the symmetric group. These states can find applications in quantum computation and metrology.
The sum rules for the energy eigenvalues of the ring Heisenberg Hamiltonian are discussed according to the classification of eigenstates through the total spin, the z-component of the total spin and the wave number. It is shown that the number of the eigenstates belonging to the energy sum obtained by our method substantially corresponds to one-n-th of that obtained by Van Vleck. With respect to the average energy defined as the quotient of the energy sum divided by the number of states, it is found that there exists a significant difference in its values between ours and Heisenberg's in the case of small n. The algebraic properties of the energy levels for atoms as many as eight are made clear by using the moments of the energy eigenvalues.
The wave functions of boson and fermion gases are known even when the particles have harmonic interactions. Here we generalize these results by solving exactly the N–body Schrödinger equation for potentials V that can be any function of the sum of the squares of the distances of the particles from one another in three dimensions. For the harmonic case that function is linear in r2. Explicit N–body solutions are given when U(r) = –2Mh–2V(r) = 𝛇r–1 – 𝛇2r–2. Here M is the sum of the masses and r2 = 1/2 M–2ΣmImJ(xI – xJ)2. For general U(r) the solution is given in terms of the one– or two–body problem with potential U(r) in three dimensions. The degeneracies of the levels are derived for distinguishable particles, for bosons of spin zero and for spin 1/2 fermions. The latter involve significant combinatorial analysis which may have application to the shell model of atomic nuclei. For large N the fermionic ground state gives the binding energy of a degenerate white dwarf star treated as a giant atom with an N–body wave function. The N–body forces involved in these extraordinary N–body problems are not the usual sums of two–body interactions, but nor are forces between quarks or molecules. Bose–Einstein condensation of particles in three dimensions interacting via these strange potentials can be treated by this method.
A sum rule is developed that allows one to write a closed form expression for d2EndV2\frac{{d}^{2}{E}_{n}}{d{V}^{2}}dV2d2En​​; where En(V){E}_{n}(V)En​(V) is the nthn\mathrm{th}nth eigenvalue and VVV the volume of an enclosed many-body system with arbitrary interaction potential. This results in a simple differential equation for En(V){E}_{n}(V)En​(V). Unfortunately, the solution, En(V){E}_{n}(V)En​(V), does not appear to be physically meaningful. The reason for this is the appearance of unacceptable functions in the formal development of the sum rule. Because these functions do not belong to the linear manifold with respect to which the Hamiltonian is Hermitian, considerable care must be exercised in applying reduction formulas. Several examples are considered in detail.