1,135 research outputs found
Number-parity effect for confined fermions in one dimension
For spin-polarized fermions with harmonic pair interactions in a
-dimensional trap an odd-even effect is found. The spectrum of the
-particle reduced density matrix of the system's ground state differs
qualitatively for odd and even. This effect does only occur for strong
attractive and repulsive interactions. Since it does not exists for bosons, it
must originate from the repulsive nature implied by the fermionic exchange
statistics. In contrast to the spectrum, the -particle density and
correlation function for strong attractive interactions do not show any
sensitivity on the number parity. This also suggests that
reduced-density-matrix-functional theory has a more subtle -dependency than
density functional theory.Comment: published versio
Duality of reduced density matrices and their eigenvalues
For states of quantum systems of particles with harmonic interactions we
prove that each reduced density matrix obeys a duality condition. This
condition implies duality relations for the eigenvalues of
and relates a harmonic model with length scales with
another one with inverse lengths . Entanglement
entropies and correlation functions inherit duality from . Self-duality
can only occur for noninteracting particles in an isotropic harmonic trap
Hubbard model: Pinning of occupation numbers and role of symmetries
Fermionic natural occupation numbers do not only obey Pauli's exclusion
principle, but are even further restricted by so-called generalized Pauli
constraints. Such restrictions are particularly relevant whenever they are
saturated by given natural occupation numbers . For
few-site Hubbard models we explore the occurrence of this pinning effect. By
varying the on-site interaction for the fermions we find sharp transitions
from pinning of to the boundary of the allowed region to
nonpinning. We analyze the origin of this phenomenon which turns out be either
a crossing of natural occupation numbers or
a crossing of -particle energies. Furthermore, we emphasize the relevance of
symmetries for the occurrence of pinning. Based on recent progress in the field
of ultracold atoms our findings suggest an experimental set-up for the
realization of the pinning effect.Comment: published versio
Natural Orbitals and Occupation Numbers for Harmonium: Fermions vs. Bosons
For a quantum system of N identical, harmonically interacting particles in a
one-dimensional harmonic trap we calculate for the bosonic and fermionic ground
state the corresponding 1-particle reduced density operator
analytically. In case of bosons is a Gibbs state for an effective
harmonic oscillator. Hence the natural orbitals are Hermite functions and their
occupation numbers obey a Boltzmann distribution. Intriguingly, for fermions
with not too large couplings the natural orbitals coincide up to just a very
small error with the bosonic ones. In case of strong coupling this still holds
qualitatively. Moreover, the decay of the decreasingly ordered fermionic
natural occupation numbers is given by the bosonic one, but modified by an
algebraic prefactor. Significant differences to bosons occur only for the
largest occupation numbers. After all the "discontinuity" at the "Fermi level"
decreases with increasing coupling strength but remains well pronounced even
for strong interaction
Quasipinning and its relevance for -Fermion quantum states
Fermionic natural occupation numbers (NON) do not only obey Pauli's famous
exclusion principle but are even further restricted to a polytope by the
generalized Pauli constraints, conditions which follow from the fermionic
exchange statistics. Whenever given NON are pinned to the polytope's boundary
the corresponding -fermion quantum state simplifies due to
a selection rule. We show analytically and numerically for the most relevant
settings that this rule is stable for NON close to the boundary, if the NON are
non-degenerate. In case of degeneracy a modified selection rule is conjectured
and its validity is supported. As a consequence the recently found effect of
quasipinning is physically relevant in the sense that its occurrence allows to
approximately reconstruct , its entanglement properties and
correlations from 1-particle information. Our finding also provides the basis
for a generalized Hartree-Fock method by a variational ansatz determined by the
selection rule
Influence of the Fermionic Exchange Symmetry beyond Pauli's Exclusion Principle
Pauli's exclusion principle has a strong impact on the properties of most
fermionic quantum systems. Remarkably, the fermionic exchange symmetry implies
further constraints on the one-particle picture. By exploiting those
generalized Pauli constraints we derive a measure which quantifies the
influence of the exchange symmetry beyond Pauli's exclusion principle. It is
based on a geometric hierarchy induced by the exclusion principle constraints.
We provide a proof of principle by applying our measure to a simple model. In
that way, we conclusively confirm the physical relevance of the generalized
Pauli constraints and show that the fermionic exchange symmetry can have an
influence on the one-particle picture beyond Pauli's exclusion principle. Our
findings provide a new perspective on fermionic multipartite correlation since
our measure allows one to distinguish between static and dynamic correlations.Comment: title has been changed; very close to published versio
Universal upper bounds on the Bose-Einstein condensate and the Hubbard star
For hard-core bosons on an arbitrary lattice with sites and
independent of additional interaction terms we prove that the hard-core
constraint itself already enforces a universal upper bound on the Bose-Einstein
condensate given by . This bound can only be attained for
one-particle states with equal amplitudes with respect to the
hard-core basis (sites) and when the corresponding -particle state
is maximally delocalized. This result is generalized to the
maximum condensate possible within a given sublattice. We observe that such
maximal local condensation is only possible if the mode entanglement between
the sublattice and its complement is minimal. We also show that the maximizing
state is related to the ground state of a bosonic `Hubbard star'
showing Bose-Einstein condensation.Comment: to appear in Phys. Rev.
Pinning of Fermionic Occupation Numbers
The Pauli exclusion principle is a constraint on the natural occupation
numbers of fermionic states. It has been suspected since at least the 1970's,
and only proved very recently, that there is a multitude of further constraints
on these numbers, generalizing the Pauli principle. Here, we provide the first
analytic analysis of the physical relevance of these constraints. We compute
the natural occupation numbers for the ground states of a family of interacting
fermions in a harmonic potential. Intriguingly, we find that the occupation
numbers are almost, but not exactly, pinned to the boundary of the allowed
region (quasi-pinned). The result suggests that the physics behind the
phenomenon is richer than previously appreciated. In particular, it shows that
for some models, the generalized Pauli constraints play a role for the ground
state, even though they do not limit the ground-state energy. Our findings
suggest a generalization of the Hartree-Fock approximation
Diverging exchange force and form of the exact density matrix functional
For translationally invariant one-band lattice models, we exploit the ab
initio knowledge of the natural orbitals to simplify reduced density matrix
functional theory (RDMFT). Striking underlying features are discovered: First,
within each symmetry sector, the interaction functional depends
only on the natural occupation numbers . The respective sets
and of pure and ensemble -representable
one-matrices coincide. Second, and most importantly, the exact functional is
strongly shaped by the geometry of the polytope , described by linear constraints . For
smaller systems, it follows as . This generalizes
to systems of arbitrary size by replacing each by a linear
combination of and adding a non-analytical term involving
the interaction . Third, the gradient
is shown to diverge on the boundary
, suggesting that the fermionic exchange symmetry
manifests itself within RDMFT in the form of an "exchange force". All findings
hold for systems with non-fixed particle number as well and can be
any -particle interaction. As an illustration, we derive the exact
functional for the Hubbard square.Comment: published versio
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