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arXiv:cond-mat/0511734

Cluster states of Fermions in the single l-shell model

András Csordás, Szoke, Eva, P. Szépfalusy

Published Nov 30, 2005• 0 citations
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Abstract

The paper concerns the ground state structure of the partly filled l-shell of a fermionic gas of atoms of spin s in a spherically symmetric spin independent trap potential. At particle numbers N=n(2s+1), n=1,2,...,2l+1 the basic building blocks are clusters consisting of (2s+1) atoms, whose wave functions are completely symmetric and antisymmetric in space and spin variables, respectively. The creation operator of a cluster is constructed whose repeated application to the vacuum leads to the multi-cluster state. Ground state energy expressions are derived for the n-cluster states at different l,s values and interpreted in simple terms.

DOI
10.48550/arxiv.cond-mat/0511734

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การคำนวณคาเบองตนของพลงงานสถานะพนของสสารประเภทเฟอรมออนในสองมตทประกอบดวยไฮโดรเจนหนงและสองอะตอม (Basic Calculations of Ground-State Energy of Two Dimensional Fermionic Matter Consisting of One and Two Hydrogen Atoms)

สิริ สิรินิลกุล

การคำนวณเบื้องต้นของพลังงานสถานะพื้นของสสารประเภทเฟอร์มิออนในสองมิติที่ประกอบด้วยอะตอมไฮโดรเจน ได้จากการพิจารณาฟังก์ชันดีเทอร์มิเนน ที่สอดคล้องกับฟังก์ชันคลื่นนอร์มัลไลซ์และฟังก์ชันสปินนอร์มัลไลซ์ พลังงานสถานะพื้นสองค่าที่คำนวณได้ ได้แก่พลังงานสถานพื้นแบบแม่นตรงของอะตอมไฮโดรเจนจำนวนหนึ่งอะตอมเท่ากับ ซึ่งพิจารณาภายใต้อันตรกิริยาแบบคูลอมป์ และค่าขอบเขตบนของพลังงานสถานะพื้นของสสารประเภทเฟอร์มิออน ที่ประกอบด้วยอะตอมไฮโดรเจนจำนวนสองอะตอมเท่ากับ The basic calculations of the ground-state energy of two dimensional fermionic matter consisting of hydrogen atom have been shown by considering the determinantal function with normalized wavefunction and normalized spin functions. Two ground-state energies are derived. The exact ground-state energy, based on considering the one hydrogen atom wavefunction in the ground-state under Coulomb interaction in two dimensions, is . The upper bound of the ground-state energy, based on considering 2 hydrogen atoms with infinitely separated 2 clusters, each in its ground-state, with nuclear charges each having one electron, is .

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On the spin asymmetry of ground states in trapped two-component Fermi gases with repulsive interactions

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We examine the spin asymmetry of ground states for two-dimensional, harmonically trapped two-component gases of fermionic atoms at zero temperature with weakly repulsive short range interactions. Our main result is that, in contrast to the three-dimensional case, in two dimensions a non-trivial spin-asymmetric phase can only be caused by shell structure. A simple, qualitative description is given in terms of an approximate single particle model, comparing well to the standard results of Hartree-Fock or direct diagonalization methods.

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The paper concerns the ground state structure of the partly filled l-shell of a fermionic gas of atoms of spin s in a spherically symmetric spin independent trap potential. At particle numbers N=n(2s+1), n=1,2,...,2l+1 the basic building blocks are clusters consisting of (2s+1) atoms, whose wave functions are completely symmetric and antisymmetric in space and spin variables, respectively. The creation operator of a cluster is constructed whose repeated application to the vacuum leads to the multi-cluster state. Ground state energy expressions are derived for the n-cluster states at different l,s values and interpreted in simple terms.
It was recently shown in self-consistent Hartree-Fock calculations that a harmonically trapped dilute gas of fermionic atoms with a repulsive two-body interaction exhibits a pronounced {\it super-shell} structure: the shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of the beating mode depends on the strength of the interaction. We give a qualitative interpretation of the beat structure in terms of a semiclassical trace formula that uniformly describes the symmetry breaking U(3) →\to→ SO(3) in a 3D harmonic oscillator potential perturbed by an anharmonic term ∝r4\propto r^4∝r4 with arbitrary strength. We show that at low Fermi energies (or particle numbers), the beating gross-shell structure of this system is dominated solely by the two-fold degenerate circular and (diametrically) pendulating orbits.
We show that a dilute harmonically trapped two-component gas of fermionic atoms with a weak repulsive interaction has a pronounced super-shell structure: The shell fillings due to the spherical harmonic trapping potential are modulated by a beat mode. This changes the ``magic numbers'' occurring between the beat nodes by half a period. The length and amplitude of this beating mode depend on the strength of the interaction. We give a simple interpretation of the beat structure in terms of a semiclassical trace formula for the symmetry breaking $\mathrm{U}(3)\ensuremath{\rightarrow}\mathrm{SO}(3)$.
We introduce a mean-field and a perturbative approach, based on clusters, to describe the ground state of fermionic strongly correlated systems. In the cluster mean-field approach, the ground-state wave function is written as a simple tensor product over optimized cluster states. The optimization of the single-particle basis where the cluster mean field is expressed is crucial in order to obtain high-quality results. The mean-field nature of the Ansatz allows us to formulate a perturbative approach to account for intercluster correlations; other traditional many-body strategies can be easily devised in terms of the cluster states. We present benchmark calculations on the half-filled 1D and (square) 2D Hubbard model, as well as the lightly doped regime in 2D, using cluster mean-field and second-order perturbation theory. Our results indicate that, with sufficiently large clusters or to second-order in perturbation theory, a cluster-based approach can provide an accurate description of the Hubbard model in the considered regimes. Several avenues to improve upon the results presented in this work are discussed.
การคำนวณเบื้องต้นของพลังงานสถานะพื้นของสสารประเภทเฟอร์มิออนในสองมิติที่ประกอบด้วยอะตอมไฮโดรเจน ได้จากการพิจารณาฟังก์ชันดีเทอร์มิเนน ที่สอดคล้องกับฟังก์ชันคลื่นนอร์มัลไลซ์และฟังก์ชันสปินนอร์มัลไลซ์ พลังงานสถานะพื้นสองค่าที่คำนวณได้ ได้แก่พลังงานสถานพื้นแบบแม่นตรงของอะตอมไฮโดรเจนจำนวนหนึ่งอะตอมเท่ากับ ซึ่งพิจารณาภายใต้อันตรกิริยาแบบคูลอมป์ และค่าขอบเขตบนของพลังงานสถานะพื้นของสสารประเภทเฟอร์มิออน ที่ประกอบด้วยอะตอมไฮโดรเจนจำนวนสองอะตอมเท่ากับ The basic calculations of the ground-state energy of two dimensional fermionic matter consisting of hydrogen atom have been shown by considering the determinantal function with normalized wavefunction and normalized spin functions. Two ground-state energies are derived. The exact ground-state energy, based on considering the one hydrogen atom wavefunction in the ground-state under Coulomb interaction in two dimensions, is . The upper bound of the ground-state energy, based on considering 2 hydrogen atoms with infinitely separated 2 clusters, each in its ground-state, with nuclear charges each having one electron, is .
We examine the spin asymmetry of ground states for two-dimensional, harmonically trapped two-component gases of fermionic atoms at zero temperature with weakly repulsive short range interactions. Our main result is that, in contrast to the three-dimensional case, in two dimensions a non-trivial spin-asymmetric phase can only be caused by shell structure. A simple, qualitative description is given in terms of an approximate single particle model, comparing well to the standard results of Hartree-Fock or direct diagonalization methods.