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The Pauli exclusion principle is fundamental to understanding electronic quantum systems. It namely constrains the expected occupancies $n_i$ of orbitals $\varphi_i$ according to $0 \leq n_i \leq 2$. In this work, we first refine the underlying one-body $N$-representability problem by taking into account simultaneously spin symmetries and a potential degree of mixedness $\boldsymbol w$ of the $N$-electron quantum state. We then derive a comprehensive solution to this problem by using basic tools from representation theory, convex analysis and discrete geometry. Specifically, we show that the set of admissible orbital one-body reduced density matrices is fully characterized by linear spectral constraints on the natural orbital occupation numbers, defining a convex polytope $\Sigma_{N,S}(\boldsymbol w) \subset [0,2]^d$. These constraints are independent of $M$ and the number $d$ of orbitals, while their dependence on $N, S$ is linear, and we can thus calculate them for arbitrary system sizes and spin quantum numbers. Our results provide a crucial missing cornerstone for ensemble density (matrix) functional theory.

It is the ultimate goal of this work to foster synergy between quantum chemistry and the flourishing field of quantum information theory. For this, we first translate quantum information concepts such as entanglement and correlation into the context of quantum chemical systems. In particular, we establish two conceptually distinct perspectives on `electron correlation' leading to a notion of orbital and particle correlation. We then demonstrate that particle correlation equals total orbital correlation minimized over all orbital bases. Accordingly, particle correlation resembles the minimal, thus intrinsic, complexity of many-electron wave functions while orbital correlation quantifies their complexity relative to a basis. We illustrate these concepts of intrinsic and extrinsic correlation complexity in molecular systems, which also manifests the crucial link between the two correlation pictures. Our results provide theoretical justification for the long-favored natural orbitals for simplifying electronic structures, and open new pathways for developing more efficient approaches towards the electron correlation problem.

Predictive modeling of CO2 storage sites requires a detailed understanding of physico-chemical processes and potential challenges for scale-up. Dramatic injectivity decline may occur due to salt precipitation pore clogging in high-salinity reservoirs, even over a short time frame. This study aims to elucidate the adverse impact of CO2-induced salt crystallization in porous media on the geomechanical properties of near-wellbore reservoir sandstones. As the impact of salt precipitation cannot be isolated from the precursor effects of interaction with CO2 and carbonic acid, we initiated our study by a comprehensive review of CO2 chemo-mechanical interactions with sandstones. We conducted laboratory geochemical CO2-brine-rock interactions at elevated pressures and temperatures on two sets of porous sandstone with contrasting petrophysical qualities. Two paths were followed: treatment with (a) CO2-acidified brine at 10 MPa fluid pressure and 60C for 7 days, and a second subset continuation with (b) supercritical injection until complete dry-out and salt precipitation. Afterward, the core samples were tested in a triaxial apparatus at varying stresses and temperatures. The elastic moduli of intact, CO2-reacted, and salt-damaged sandstones were juxtaposed to elucidate the extent of crystallization damages. The salt-affected specimens showed a maximum of 50 percent reduction in Young's and shear moduli and twice an increase in Poisson's ratio compared to intact condition. The deterioration was notably higher for the tighter rocks with higher initial stiffness.

Traditional design cycles for new materials and assemblies have two fundamental drawbacks. The underlying physical relationships are often too complex to be precisely calculated and described. Aside from that, many unknown uncertainties, such as exact manufacturing parameters or materials composition, dominate the real assembly behavior. Machine learning (ML) methods overcome these fundamental limitations through data-driven learning. In addition, modern approaches can specifically increase system knowledge. Representation Learning allows the physical, and if necessary, even symbolic interpretation of the learned solution. In this way, the most complex physical relationships can be considered and quickly described. Furthermore, generative ML approaches can synthesize possible morphologies of the materials based on defined conditions to visualize the effects of uncertainties. This modern approach accelerates the design process for new materials and enables the prediction and interpretation of realistic materials behavior., Comment: This paper has been accepted for publication at the AAAI 2024 Workshop on AI to Accelerate Science and Engineering

To advance the foundation of one-particle reduced density matrix functional theory (1RDMFT) we refine and relate some of its fundamental features and underlying concepts. We define by concise means the scope of a 1RDMFT, identify its possible natural variables and explain how symmetries could be exploited. In particular, for systems with time-reversal symmetry, we explain why there exist six equivalent universal functionals, prove concise relations among them and conclude that the important notion of $v$-representability is relative to the scope and choice of variable. All these fundamental concepts are then comprehensively discussed and illustrated for the Hubbard dimer and its generalization to arbitrary pair interactions $W$. For this, we derive by analytical means the pure and ensemble functionals with respect to both the real- and complex-valued Hilbert space. The comparison of various functionals allows us to solve the underlying $v$-representability problems analytically and the dependence of its solution on the pair interaction is demonstrated. Intriguingly, the gradient of each universal functional is found to always diverge repulsively on the boundary of the domain. In that sense, this key finding emphasizes the universal character of the fermionic exchange force, recently discovered and proven in the context of translationally-invariant one-band lattice models.

We initiate the recently proposed $\boldsymbol{w}$-ensemble one-particle reduced density matrix functional theory ($\boldsymbol{w}$-RDMFT) by deriving the first functional approximations and illustrate how excitation energies can be calculated in practice. For this endeavour, we first study the symmetric Hubbard dimer, constituting the building block of the Hubbard model, for which we execute the Levy-Lieb constrained search. Second, due to the particular suitability of $\boldsymbol{w}$-RDMFT for describing Bose-Einstein condensates, we demonstrate three conceptually different approaches for deriving the universal functional in a homogeneous Bose gas for arbitrary pair interaction in the Bogoliubov regime. Remarkably, in both systems the gradient of the functional is found to diverge repulsively at the boundary of the functional's domain, extending the recently discovered Bose-Einstein condensation force to excited states. Our findings highlight the physical relevance of the generalized exclusion principle for fermionic and bosonic mixed states and the curse of universality in functional theories., Comment: Application of w-RDMFT which has been proposed in arXiv:2204.12715; close to published version

Density functional theory constitutes the workhorse of modern electronic structure calculations due to its favourable computational cost despite the fact that it usually fails to describe strongly correlated systems. A particularly promising approach to overcome those difficulties is reduced density matrix functional theory (RDMFT): It abandons the complexity of the $N$-particle wave function and at the same time explicitly allows for fractional occupation numbers. It is the goal of this thesis to initiate and establish a bosonic RDMFT for both ground state and excited state energy calculations. Motivated by the Onsager and Penrose criterion which identifies RDMFT as a particularly suitable approach to describe Bose-Einstein condensates (BECs), we derive the universal functional for a homogeneous BEC in the Bogoliubov regime. Remarkably, the gradient of the universal functional is found to diverge repulsively in the regime of complete condensation. This introduces the new concept of a BEC force, which provides a universal explanation for quantum depletion since it is merely based on the geometry of quantum states. In the second part of the thesis, we propose and work out an ensemble RDMFT targeting excitations in bosonic quantum systems. This endeavour further highlights the potential of convex analysis for the development of functional theories in the future. Indeed by resorting to several concepts from convex analysis, we succeeded to provide a comprehensive foundation of $\boldsymbol{w}$-ensemble RDMFT for bosons which is further based on a generalization of the Ritz variational principle and a constrained search formalism. In particular, we solve the emerging $N$-representability problem leading to a generalization of Pauli's famous exclusion principle to bosonic mixed states., Comment: Master thesis, LMU Munich, 103 pages. Thesis based on arXiv:2010.06634, arXiv:2106.03918, arXiv:2105.06459, arXiv:2204.12715

Motivated by the Penrose-Onsager criterion for Bose-Einstein condensation we propose a functional theory for targeting low-lying excitation energies of bosonic quantum systems through the one-particle picture. For this, we employ an extension of the Rayleigh-Ritz variational principle to ensemble states with spectrum $\boldsymbol{w}$ and prove a corresponding generalization of the Hohenberg-Kohn theorem: The underlying one-particle reduced density matrix determines all properties of systems of $N$ identical particles in their $\boldsymbol{w}$-ensemble states. Then, to circumvent the $v$-representability problem common to functional theories, and to deal with energetic degeneracies, we resort to the Levy-Lieb constrained search formalism in combination with an exact convex relaxation. The corresponding bosonic one-body $\boldsymbol{w}$-ensemble $N$-representability problem is solved comprehensively. Remarkably, this reveals a complete hierarchy of bosonic exclusion principle constraints in conceptual analogy to Pauli's exclusion principle for fermions and recently discovered generalizations thereof., Comment: In particular, bosonic counterpart of fermionic w-RDMFT (arXiv:2106.02560, arXiv:2106.03918) is presented; v2: title has been changed, new section proving a generalization of the Hohenberg-Kohn theorem for w-RDMFT has been added

The emergence and impact of tipping points have garnered significant interest in both the social and natural sciences. Despite widespread recognition of the importance of feedbacks between human and natural systems, it is often assumed that the observed nonlinear dynamics in these coupled systems rests within either underlying human or natural processes, rather than the rates at which they interact. Using adoption of agricultural diversification practices as a case study, we show how two stable management paradigms (one dominated by conventional, homogeneous practices, the other by diversified practices) can emerge purely from temporal feedbacks between human decisions and ecological responses. We explore how this temporal mechanism of tipping points provides insight into designing more effective interventions that promote farmers transitions towards sustainable agriculture. Moreover, our flexible modeling framework could be applied to other cases to provide insight into numerous questions in social-ecological systems research and environmental policy., Comment: One Earth (2022)

In [Phys. Rev. Lett. 128, 013001 (2022)] a novel ground state method was proposed. It has been suggested that this $i$-DMFT would be a method within one-particle reduced density matrix functional theory (DMFT), capable of describing accurately molecules at various geometries with an information-theoretical nature. We reassess this work and its suggestions from a conceptual and practical point of view, leading to the following conclusions: i) A method which assigns to each molecule $\mathcal{M}$ its own functional $\mathcal{F}_{\!\mathcal{M}}$ is not a functional theory (striking violation of "universality") ii) even for the simplest systems $i$-DMFT yields incorrect one-particle reduced density matrices and iii) the use of an information-theoretical concept to describe molecular dissociation limits was not essential. The latter insight may help to fix the deficiency of $i$-DMFT to not reproduce correctly the smaller occupation numbers and thus to not recover the important dynamic correlations., Comment: Comments are very welcome!

We provide new insights regarding the headline result that Medicaid increased emergency department (ED) use from the Oregon experiment. We find meaningful heterogeneous impacts of Medicaid on ED use using causal machine learning methods. The individualized treatment effect distribution includes a wide range of negative and positive values, suggesting the average effect masks substantial heterogeneity. A small group-about 14% of participants-in the right tail of the distribution drives the overall effect. We identify priority groups with economically significant increases in ED usage based on demographics and previous utilization. Intensive margin effects are an important driver of increases in ED utilization.

This paper studies reputation in the online market for illegal drugs in which no legal institutions exist to alleviate uncertainty. Trade takes place on platforms that offer rating systems for sellers, thereby providing an observable measure of reputation. The analysis exploits the fact that one of the two dominant platforms unexpectedly disappeared. Re-entering sellers reset their rating. The results show that on average prices decreased by up to 9% and that a 1% increase in rating causes a price increase of 1%. Ratings and prices recover after about three months. We calculate that identified good types earn 1,650 USD more per week.

In [Phys. Rev. Lett. 127, 023001 (2021)] a reduced density matrix functional theory (RDMFT) has been proposed for calculating energies of selected eigenstates of interacting many-fermion systems. Here, we develop a solid foundation for this so-called $\boldsymbol{w}$-RDMFT and present the details of various derivations. First, we explain how a generalization of the Ritz variational principle to ensemble states with fixed weights $\boldsymbol{w}$ in combination with the constrained search would lead to a universal functional of the one-particle reduced density matrix. To turn this into a viable functional theory, however, we also need to implement an exact convex relaxation. This general procedure includes Valone's pioneering work on ground state RDMFT as the special case $\boldsymbol{w}=(1,0,\ldots)$. Then, we work out in a comprehensive manner a methodology for deriving a compact description of the functional's domain. This leads to a hierarchy of generalized exclusion principle constraints which we illustrate in great detail. By anticipating their future pivotal role in functional theories and to keep our work self-contained, several required concepts from convex analysis are introduced and discussed., Comment: published version; in particular various derivations of the letter arXiv:2106.02560 are provided

From a geometric point of view, Pauli's exclusion principle defines a hypersimplex. This convex polytope describes the compatibility of $1$-fermion and $N$-fermion density matrices, therefore it coincides with the convex hull of the pure $N$-representable $1$-fermion density matrices. Consequently, the description of ground state physics through $1$-fermion density matrices may not necessitate the intricate pure state generalized Pauli constraints. In this article, we study the generalization of the $1$-body $N$-representability problem to ensemble states with fixed spectrum $\mathbf{w}$, in order to describe finite-temperature states and distinctive mixtures of excited states. By employing ideas from convex analysis and combinatorics, we present a comprehensive solution to the corresponding convex relaxation, thus circumventing the complexity of generalized Pauli constraints. In particular, we adapt and further develop tools such as symmetric polytopes, sweep polytopes, and Gale order. For both fermions and bosons, generalized exclusion principles are discovered, which we determine for any number of particles and dimension of the $1$-particle Hilbert space. These exclusion principles are expressed as linear inequalities satisfying hierarchies determined by the non-zero entries of $\mathbf{w}$. The two families of polytopes resulting from these inequalities are part of the new class of so-called lineup polytopes., Comment: 59 pages, 17 figures, 3 appendices Version 2: several minor changes and fixes, now the source files include two rst files containing the obtained generalized exclusion principles as linear inequalities

We present a novel ultrastable superconducting radio-frequency (RF) ion trap realized as a combination of an RF cavity and a linear Paul trap. Its RF quadrupole mode at 34.52 MHz reaches a quality factor of $Q\approx2.3\times 10^5$ at a temperature of 4.1 K and is used to radially confine ions in an ultralow-noise pseudopotential. This concept is expected to strongly suppress motional heating rates and related frequency shifts which limit the ultimate accuracy achieved in advanced ion traps for frequency metrology. Running with its low-vibration cryogenic cooling system, electron beam ion trap and deceleration beamline supplying highly charged ions (HCI), the superconducting trap offers ideal conditions for optical frequency metrology with ionic species. We report its proof-of-principle operation as a quadrupole mass filter with HCI, and trapping of Doppler-cooled ${}^9\text{Be}^+$ Coulomb crystals., Comment: 16 pages, 19 figures