arXiv subjects

Christian Schilling

Publications and source records attributed to Christian Schilling.

85 records · Page 5Linked to original sources

arXiv2017

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.

quant-ph↗

arXiv2016

Instrumenting an SMT Solver to Solve Hybrid Network Reachability Problems

PDDL+ planning has its semantics rooted in hybrid automata (HA) and recent work has shown that it can be modeled as a network of HAs. Addressing the complexity of nonlinear PDDL+ planning as HAs requires both space and time efficient reasoning. Unfortunately, existing solvers either do not address nonlinear dynamics or do not natively support networks of automata. We present a new algorithm, called HNSolve, which guides the variable selection of the dReal Satisfiability Modulo Theories (SMT) solver while reasoning about network encodings of nonlinear PDDL+ planning as HAs. HNSolve tightly integrates with dReal by solving a discrete abstraction of the HA network. HNSolve finds composite runs on the HA network that ignore continuous variables, but respect mode jumps and synchronization labels. HNSolve admissibly detects dead-ends in the discrete abstraction, and posts conflict clauses that prune the SMT solver's search. We evaluate the benefits of our HNSolve algorithm on PDDL+ benchmark problems and demonstrate its performance with respect to prior work.

cs.AI↗

arXiv2016

Pinning of fermionic occupation numbers: Higher spatial dimensions and spin

The role of the generalized Pauli constraints (GPCs) in higher spatial dimensions and by incorporating spin degrees of freedom is systematically explored for a system of interacting fermions confined by a harmonic trap. Physical relevance of the GPCs is confirmed by analytical means for the ground state in the regime of weak couplings by finding its vector of natural occupation numbers close to the boundary of the allowed region. Such quasipinning is found to become weaker in the intermediate and strong coupling regime. The study of crossovers between different spatial dimensions by detuning the harmonic trap frequencies suggests that quasipinning is essentially an effect for systems with reduced spatial dimensionality. In addition, we find that quasipinning becomes stronger by increasing the degree of spin-polarization. Consequently, the number of states available around the Fermi level plays a key role for the occurrence of quasipinning. This suggests that quasipinning emerges from the conflict between energy minimization and fermionic exchange symmetry.

quant-ph↗

arXiv2016

Invariant Clusters for Hybrid Systems

In this paper, we propose an approach to automatically compute invariant clusters for semialgebraic hybrid systems. An invariant cluster for an ordinary differential equation (ODE) is a multivariate polynomial invariant g(u,x)=0, parametric in u, which can yield an infinite number of concrete invariants by assigning different values to u so that every trajectory of the system can be overapproximated precisely by a union of concrete invariants. For semialgebraic systems, which involve ODEs with multivariate polynomial vector flow, invariant clusters can be obtained by first computing the remainder of the Lie derivative of a template multivariate polynomial w.r.t. its Groebner basis and then solving the system of polynomial equations obtained from the coefficients of the remainder. Based on invariant clusters and sum-of-squares (SOS) programming, we present a new method for the safety verification of hybrid systems. Experiments on nonlinear benchmark systems from biology and control theory show that our approach is effective and efficient.

math.OC↗

arXiv2016

Pinning of Fermionic Occupation Numbers: General Concepts and One Dimension

Analytical evidence for the physical relevance of generalized Pauli constraints (GPCs) has recently been provided in [PRL 110, 040404]: Natural occupation numbers $\vecλ\equiv (λ_i)$ of the ground state of a model system in the regime of weak couplings $κ$ of three spinless fermions in one spatial dimension were found extremely close, in a distance $D_{min}\sim κ^8$ to the boundary of the allowed region. We provide a self-contained and complete study of this quasipinning phenomenon. In particular, we develop tools for its systematic exploration and quantification. We confirm that quasipinning in one dimension occurs also for larger particle numbers and extends to intermediate coupling strengths, but vanishes for very strong couplings. We further explore the non-triviality of our findings by comparing quasipinning by GPCs to potential quasipinning by the less restrictive Pauli exclusion principle constraints. This allows us to eventually confirm the significance of GPCs beyond Pauli's exclusion principle.

quant-ph↗

arXiv2016

Number-parity effect for confined fermions in one dimension

For $N$ spin-polarized fermions with harmonic pair interactions in a $1$-dimensional trap an odd-even effect is found. The spectrum of the $1$-particle reduced density matrix of the system's ground state differs qualitatively for $N$ odd and $N$ 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 $1$-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 $N$-dependency than density functional theory.

quant-ph↗

arXiv2015

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 $\vecλ=(λ_i)$. For few-site Hubbard models we explore the occurrence of this pinning effect. By varying the on-site interaction $U$ for the fermions we find sharp transitions from pinning of $\vecλ$ 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 $λ_{i}(U), λ_{i+1}(U)$ or a crossing of $N$-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.

cond-mat.quant-gas↗

arXiv2015

Quantum Marginal Problem and its Physical Relevance

The Pauli exclusion principle as constraint on fermionic occupation numbers is a consequence of the much deeper fermionic exchange statistics. Just recently, it was shown by Klyachko that this antisymmetry of fermionic wave functions leads to further restrictions on natural occupation numbers. These so-called generalized Pauli constraints (GPC) significantly strengthen Pauli's exclusion principle. Our first goal is to develop an understanding of the mathematical concepts behind Klyachko's work, in the context of quantum marginal problems. Afterwards, we explore the physical relevance of GPC and study concrete physical systems from that new viewpoint. In the first part of this thesis we review Klyachko's solution of the univariate quantum marginal problem. In particular we break his abstract derivation based on algebraic topology down to a more elementary level and reveal the geometrical picture behind it. The second part explores the possible physical relevance of GPC. We review the effect of pinning, i.e. the saturation of some GPC by given natural occupation numbers and explain its consequences. Although this effect would be quite spectacular we argue that pinning is unnatural. Instead, we conjecture the effect of quasipinning, defined by occupation numbers close to (but not exactly on) the boundary of the allowed region. In the third part we study concrete fermionic quantum systems from the new viewpoint of GPC. In particular, we compute the natural occupation numbers for the ground state of a family of interacting fermions in a harmonic potential. Intriguingly, we find that the occupation numbers are strongly quasipinned, even up to medium interaction strengths. We identify this as an effect of the lowest few energy eigenstates, which provides first insights into the mechanism behind quasipinning.

quant-ph↗

arXiv2014

Quasipinning and its relevance for $N$-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 $N$-fermion quantum state $|Ψ_N\rangle$ 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 $|Ψ_N\rangle$, 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.

quant-ph↗

arXiv2014

Duality of reduced density matrices and their eigenvalues

For states of quantum systems of $N$ particles with harmonic interactions we prove that each reduced density matrix $ρ$ obeys a duality condition. This condition implies duality relations for the eigenvalues $λ_k$ of $ρ$ and relates a harmonic model with length scales $l_1,l_2, \ldots, l_N $ with another one with inverse lengths $1/l_1, 1/l_2,\ldots, 1/l_N$. Entanglement entropies and correlation functions inherit duality from $ρ$. Self-duality can only occur for noninteracting particles in an isotropic harmonic trap.

quant-ph↗

arXiv2014

The Quantum Marginal Problem

The question of whether given density operators for subsystems of a multipartite quantum system are compatible to one common total density operator is known as the quantum marginal problem. We briefly review the solution of a subclass of such problems found just recently. In particular, this provides the solution of the $1$-body $N$-representability problem. Its solution, the so-called generalized Pauli constraints, restrict the set of mathematically possible fermionic occupation numbers significantly, and strengthens Pauli's exclusion principle. Moreover, we review the study of a concrete physical model of interacting fermions confined to a harmonic trap. There, we found occupation numbers close, but not exactly on the boundary of the allowed region. This new effect of quasipinning is physically relevant since it corresponds to a simplified structure of the corresponding $N$-fermion quantum state.

quant-ph↗

arXiv2013

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 $ρ_1$ analytically. In case of bosons $ρ_1$ 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.

quant-ph↗

arXiv2012

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.

quant-ph↗