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For a convex function we can always draw two straight lines with a non-vertical slope on any region R in the interior of its domain around x, so a convex function on a one dimensional space is locally Lipschitz continuous. Note that at the boundary of its domain, the convex function can jump, so a convex function is not necessarily continuous at the boundary of its domain.
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In this review we provide a rigorous and self-contained presentation of one-body reduced density-matrix (1RDM) functional theory. We do so for the case of a finite basis set, where density-functional theory (DFT) implicitly becomes a 1RDM functional theory. To avoid non-uniqueness issues we consider the case of fermionic and bosonic systems at elev...
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... can readily convince oneself of the correctness of this theorem by sketching a convex function and consid- ering all straight lines one can draw between a some point x on the graph and all points in the neighborhood. Since these lines never become vertical, there is some maximum slope for these lines. This is illustrated in Fig. 2. A more rigorous proof can be found in Appendix ...
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... Note that it already contains information about quantum correlations via the correlation entropy S[γ] = −Tr[γ ln γ] [33,[73][74][75]. Surprisingly, 1-RDMFT is mainly focused on the goal of computing the ground-state energy and some associated observables [57,[76][77][78][79][80][81], but this powerful formalism has not been used to scrutinize multipartite entanglement or nonlocality. Indeed, the map γ, u → |ψ[γ, u] can be used for the calculation of functionals of QFIM; by using Eq. ...
Quantum Fisher information (QFI) is a central concept in quantum sciences used to quantify the ultimate precision limit of parameter estimation, detect quantum phase transitions, witness genuine multipartite entangle-ment, or probe nonlocality. Despite this widespread range of applications, computing the QFI value of quantum many-body systems is, in general, a very demanding task. Here we combine ideas from functional theories and quantum information to develop a functional framework for the QFI of fermionic and bosonic ground states. By relying upon the constrained-search approach, we demonstrate that the QFI matrix terms can universally be determined by the one-body reduced density matrix (1-RDM), thus avoiding the use of exponentially large wave functions. Furthermore, we show that QFI functionals can be determined from the universal 1-RDM functional by calculating its derivatives with respect to the coupling strengths, thus becoming the generating functional of the QFI. We showcase our approach with the Bose-Hubbard model and present exact analytical and numerical QFI functionals. Our results provide the first connection between the one-body reduced density matrix functional theory and the quantum Fisher information.
... Noteworthy, it already contains information about quantum correlations, via the correlation entropy S[γ] = −Tr[γ ln γ] [31,[68][69][70]. Surprisingly, 1-RDMFT is mainly focused on the goal of computing the ground-state energy and some associated observables [54,[71][72][73][74][75][76], but this powerful formalism has not been used to scrutiny multipartite entanglement or nonlocality. Indeed, the map γ → |ψ[γ]⟩ can be used for the calculation of functionals of QFIM: by using Eq. ...
Quantum Fisher information (QFI) is a central concept in quantum sciences used to quantify the ultimate precision limit of parameter estimation, detect quantum phase transitions, witness genuine multipartite entanglement, or probe nonlocality. Despite this widespread range of applications, computing the QFI value of quantum many-body systems is, in general, a very demanding task. Here we combine ideas from functional theories and quantum information to develop a novel functional framework for the QFI of fermionic and bosonic ground states. By relying upon the constrained-search approach, we demonstrate that the QFI matricial values can universally be determined by the one-body reduced density matrix (1-RDM), avoiding thus the use of exponentially large wave functions. Furthermore, we show that QFI functionals can be determined from the universal 1-RDM functional by calculating its derivatives with respect to the coupling strengths, becoming thus the generating functional of the QFI. We showcase our approach with the Bose-Hubbard model and present exact analytical and numerical QFI functionals. Our results provide the first connection between the one-body reduced density matrix functional theory and the quantum Fisher information.
... This and other recent progress in the field of ground state RDMFT for bosons [3][4][5][6][7] and w -ensemble RDMFT for excited states [8][9][10] urges us to systematically derive in this paper the first functionals for w -ensemble RDMFT. To initiate the development of w -ensemble functionals in different fields of physics we derive analytically the universal functional for both the symmetric Hubbard dimer with on-site interaction and the homogeneous Bose gas in the Bogoliubov regime. ...
We initiate the recently proposed w w -ensemble one-particle reduced density matrix functional theory ( w 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 w 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.
... Historically, the search for alternatives to KS-DFT produced successful approaches like density matrix functional theory [8][9][10][11][12][13], which aims at chemical accuracy and transferability across electronic structure problems-and whose technical aspects are very much related to our agenda in the present article-as well as modern orbital-free DFT in all its variety, e.g., density-potential functional theory (DPFT) [14][15][16][17][18], which aims at scalability toward large particle numbers and transferability across interactions and dimensionality. Still, these alternative methods are currently implemented in position space. ...
... We have thus explicitly formulated the constrained search in Eq. (11) over ρ mp as a constrained search over density matrices in the 1pEx-basis, where derives from ρ mp through Eqs. (25), (18), and (13). In any particular application, we need (i) to find the values of the H ab,cd coefficients; (ii) to write down one (0) matrix for ρ with (0) aa = n a and (0) 2 = 2 (0) ; and (iii) to minimize the right-hand-side of Eq. (33) with ...
We introduce 'single-particle-exact density functional theory' (1pEx-DFT), a novel density functional approach that represents all single-particle contributions to the energy with exact functionals. Here, we parameterize interaction energy functionals by utilizing two new schemes for constructing density matrices from 'participation numbers' of the single-particle states of quantum many-body systems. These participation numbers play the role of the variational variables akin to the particle densities in standard orbital-free density functional theory. We minimize the total energies with the help of evolutionary algorithms and obtain ground-state energies that are typically accurate at the one-percent level for our proof-of-principle simulations that comprise interacting Fermi gases as well as the electronic structure of atoms and ions, with and without relativistic corrections. We thereby illustrate the ingredients and practical features of 1pEx-DFT and reveal its potential of becoming an accurate, scalable, and transferable technology for simulating mesoscopic quantum many-body systems.
... He noted explicitly that Lieb's proofs are in L 3/2 (R 3 ) + L ∞ (R 3 ) which is not in K 3 but then he says "K 3 nearly contains the Lieb class in some sense." For the thermal case, the only rigorous proofs of the one-to-one HK theorem seem to be in a finite-basis [20] or on a lattice [21]. For explicitly finite systems, the results for graphs are perhaps more perplexing from the conventional viewpoint, since the first HK theorem demonstrably does not hold [22] for those systems. ...
Though calculations based on density functional theory (DFT) are used remarkably widely in chemistry, physics, materials science, and biomolecular research and though the modern form of DFT has been studied for almost 60 years, some mathematical problems remain. From a physical science perspective, it is far from clear whether those problems are of major import. For context, we provide an outline of the basic structure of DFT as it is presented and used conventionally in physical sciences, note some unresolved mathematical difficulties with those conventional demonstrations, then pose several questions regarding both the time-independent and time-dependent forms of DFT that could benefit from attention in applied mathematics. Progress on any of these would aid in development of better approximate functionals and in interpretation of DFT.
... In the case of bosons, the natural occupation numbers are only restricted through 0 ⩽ λ i ⩽ N and there are no additional constraints as the generalized Pauli constraints for fermions. Therefore, the pure state N-representability constraints for bosons are known and, in particular, E 1 N = P 1 N [15,24]. Nevertheless, relaxing the minimization in equation (15) to a convex one is also advantageous for bosons, at least from a numerical point of view. ...
... Hence, Σ(w) takes indeed the form of a polytope as anticipated above. In figure 5, as a further illustration of the spectral polytopes, we demonstrate the important inclusion relation equation (24) for the setting (N, d) = (2, 3). Indeed the six spectral polytopes Σ(w) (and Σ ↓ (w)) are contained in each other since the six respective weight vectors w are related by majorization. ...
... Their existence might be surprising at first sight. This is due to the fact that the solution of the pure state N-representability problem for bosons is trivial, i.e. there are no constraints on occupation numbers beyond normalization and positivity [15,24]. This also relates to the fact that the complete hierarchy of Pauli exclusion principles [41] has its origin in both the exchange symmetry and the enforced mixedness of the N-particle state while the bosonic constraints emerge primarily from the latter. ...
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 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 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 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.
... Here and throughout, we assume that the partition sum (22) and hence the free energy (20) exists, see Giesbertz and Ruggenthaler's [39] account of the divergences that occur in even simple unbounded systems. In our case, we assume (as we do classically [31]) that the system is bounded via the influence of appropriate container walls, as modelled by a corresponding form of the external potential V ext (r). ...
We address the consequences of invariance properties of the free energy of spatially inhomogeneous quantum many-body systems. We consider a specific position-dependent transformation of the system that consists of a spatial deformation and a corresponding locally resolved change of momenta. This operator transformation is canonical and hence equivalent to a unitary transformation on the underlying Hilbert space of the system. As a consequence, the free energy is an invariant under the transformation. Noether’s theorem for invariant variations then allows to derive an exact sum rule, which we show to be the locally resolved equilibrium one-body force balance. For the special case of homogeneous shifting, the sum rule states that the average global external force vanishes in thermal equilibrium.
... This and other recent progress in the field of ground state RDMFT for bosons [3][4][5][6][7] and w-ensemble RDMFT for excited states [8][9][10] urges us to systematically derive in this paper the first functionals for w-ensemble RDMFT. To initiate the development of w-ensemble functionals in different fields of physics we derive analytically the universal functional for both the symmetric Hubbard dimer with on-site interaction and the homogeneous Bose gas in the Bogoliubov regime. ...
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.
... One advantage over DFT is that we have also an explicit expression for the kinetic energy while still having the total energy as a functional of the 1RDM [6]. However, in the zero temperature setting, mapping back from 1RDMs to (non-local) potentials is problematic as already noted by Gilbert [6] and others [7][8][9][10]. This is most clear in the case of non-interacting * s.m.sutter@vu.nl ...
... In Ref. [10] 1RDM functional theory (1RDMFT) was presented for the grand canonical ensemble within a finite basis set. However, the use of a grand canonical ensemble is inappropriate if the number of particles is relatively low as in ultra cold atom experiments [18], but also in the low temperature limit the grand canonical ensemble can lead to unphysical results [19,20]. ...
We establish one-body reduced density-matrix functional theory for the canonical ensemble in a finite basis set at elevated temperature. Including temperature guarantees differentiability of the universal functional by occupying all states and additionally not fully occupying the states in a fermionic system. We use convexity of the universal functional and invertibility of the potential-to-1RDM map to show that the subgradient contains only one element which is equivalent to differentiability. This allows us to show that all 1RDMs with a purely fractional occupation number spectrum ($0 < n_i < 1 \; \forall_i$) are uniquely $v$-representable up to a constant.
... dr N · and the asterisk denotes complex conjugation. Here and throughout, we assume that the partition sum (22) and hence the free energy (20) exists, see Giesbertz and Ruggenthaler's [39] account of the divergences that occur in even simple unbounded systems. In our case, we assume (as we do classically [29]) that the system is bounded via the influence of appropriate container walls, as modelled by a corresponding form of the external potential V ext (r). ...
We address the consequences of invariance properties of the free energy of spatially inhomogeneous quantum many-body systems. We consider a specific position-dependent transformation of the system that consists of a spatial deformation and a corresponding locally resolved change of momenta. This operator transformation is canonical and hence equivalent to a unitary transformation on the underlying Hilbert space of the system. As a consequence, the free energy is an invariant under the transformation. Noether's theorem for invariant variations then allows to derive an exact sum rule, which we show to be the locally resolved equilibrium one-body force balance. For the special case of homogeneous shifting, the sum rule states that the average global external force vanishes in thermal equilibrium.
