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arXiv:1210.5748 · Journal of Physics A Mathematical and Theoretical

The Weyl expansion for systems of independent identical particles

Quirin Hummel, Juan Diego Urbina, Klaus Richter

Published Dec 11, 2013• 8 citations
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Abstract

We present a novel analytical approach for the calculation of the mean density of states in many-body systems consisting of confined indistinguishable and independent particles. Our method makes explicit the intrinsic geometry inherent in the symmetrization postulate. In the spirit of the usual Weyl expansion for the smooth part of the density of states in confined single-particle systems, our results take the form of a sum over clusters of particles moving freely around manifolds in configuration space invariant under permutations. In our approach the emergence of the fermionic ground state is a consequence of a delicate cancellation effect of cluster contributions. As an asymptotic expansion, our approximation gives increasingly better results for large excitation energies, and we show that it coincides with the Bethe estimate in the appropriate region. Moreover, our construction gives the correct high-energy asymptotics expected from general considerations. Our expansion in cluster zones is naturally incorporated for systems of interacting particles, opening an alternative road to address the interplay between symmetry, confinement and interactions in many-body systems of identical bosonic or fermionic particles.

DOI
10.1088/1751-8113/47/1/015101

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We present a novel analytical approach for the calculation of the mean density of states in many-body systems consisting of confined indistinguishable and independent particles. Our method makes explicit the intrinsic geometry inherent in the symmetrization postulate. In the spirit of the usual Weyl expansion for the smooth part of the density of states in confined single-particle systems, our results take the form of a sum over clusters of particles moving freely around manifolds in configuration space invariant under permutations. In our approach the emergence of the fermionic ground state is a consequence of a delicate cancellation effect of cluster contributions. As an asymptotic expansion, our approximation gives increasingly better results for large excitation energies, and we show that it coincides with the Bethe estimate in the appropriate region. Moreover, our construction gives the correct high-energy asymptotics expected from general considerations. Our expansion in cluster zones is naturally incorporated for systems of interacting particles, opening an alternative road to address the interplay between symmetry, confinement and interactions in many-body systems of identical bosonic or fermionic particles.
We address the fundamental interplay between indistinguishability and interactions when discreteness effects are neglected in systems with strictly fixed number of particles. For this end we supplement cluster expansions (many-body canonical techniques where quantum statistics is treated exactly) with short-time/large volume dynamical information where interparticle forces are described non-perturbatively. This approach, specially suitable for the few-body case where it overcomes the inappropriate use of virial expansions, can be consistently combined with scaling considerations, minimal ground-state information and strong coupling expansions in such a way that a single interaction event provides most of the thermodynamic and spectral properties of 1D systems with short range interactions. Our analytical results, in excellent agreement with numerical simulations, show a form of universal integrability of interaction effects for arbitrary confinements.
Abstract The modeling of out-of-equilibrium many-body quantum systems requires to go beyond low-energy physics and single or few bodies densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including highly excited ones, contribute to dynamics and therefore affect the system’s properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical particles due to the difficulty to enumerate accessible states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the general case of non-interacting systems of identical quantum particles. The starting point of our method is the principal component analysis of a filling matrix F describing how N particles can be distributed into L single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of singular vectors of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions : (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.
We develop an approach of calculating the many-body path integral based on the linked cluster expansion method. First, we derive a linked cluster expansion and we give the diagrammatic rules for calculating the free energy and the pair distribution function g(r)g(r)g(r) as a systematic power-series expansion in the particle density. We also present a structured Pad\'e approximation scheme in the momentum space to determine g(r)g(r)g(r). The calculated g(r)g(r)g(r) for distinguishable particles interacting with Lennard-Jones and hard-sphere potential in various attempted schemes of approximation of the diagrammatic series compares very well with the results of path-integral Monte Carlo simulation. Our method is applicable to a wide range of problems of current general interest and may be extended to the case of identical particles and, in particular, to the case of the many-fermion problem.
For a system of identical particles (we take fermions as an example), we derive the analogue of the Weyl formula for the average level density. This is done by introducing the projection operator onto states in Hilbert space which have the symmetry imposed by particle identity. The semi-classical result is obtained by using the Wigner representation for this operator and by letting ℏ → 0. Numerical examples for few degrees of freedom show in general good agreement with exact diagonalization even near the ground state.
A coupled-cluster approach for systems of NNN bosons in external traps is developed. In the coupled-cluster approach the exact many-body wave function is obtained by applying an exponential operator expT\mathrm{exp}{T}expT to the ground configuration $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$. The natural ground configuration for bosons is, of course, when all reside in a single orbital. Because of this simple structure of $\ensuremath{\mid}{\ensuremath{\phi}}_{0}⟩$, the appearance of excitation operators $T={\ensuremath{\sum}}_{n=1}^{N}{T}_{n}$ for bosons is much simpler than for fermions. We can treat very large numbers of bosons with coupled-cluster expansions. In a substantial part of this work, we address the issue of size consistency for bosons and enquire whether truncated coupled-cluster expansions are size consistent. We show that, in contrast to the familiar situation for fermions for which coupled-cluster expansions are size consistent, for bosons the answer to this question depends on the choice of ground configuration. Utilizing the natural ground configuration, working equations for the truncated coupled-cluster with T=T1+T2T={T}_{1}+{T}_{2}T=T1​+T2​, i.e., coupled-cluster singles doubles are explicitly derived. Finally, an illustrative numerical example for a condensate with up to N=10000N=10\phantom{\rule{0.2em}{0ex}}000N=10000 bosons in an harmonic trap is provided and analyzed. The results are highly promising.
We derive the analogue of the Weyl formula for the average level density, in the case of a system of identical particles (taking fermions as an example). This is accomplished by introducing the projection operator onto states in Hilbert space which have the symmetry imposed by particle identity. The semiclassical result is obtained by using the Wigner representation for this operator and by letting h {r_arrow} 0. Numerical examples for few degrees of freedom show in general good agreement with exact diagonalization even near the ground state of the system.