The dimension of the Hilbert space needed for the description of an interacting electron system increases exponentially with electron number $N$. As pointed out by W. Kohn this exponential wall problem (EWP) limits the concept of many-electron wavefunctions based on solutions of Schr\"odinger's equation to $N \leq N_0$, where $N_0 \approx 10^3$ \cite{Kohn1999}. This limitation does not hold when the electronic interactions are neglected or treated in a mean-field approximation. The EWP has directed electronic structure calculations for solids to methods like density-functional theory which avoid dealing with many-electron wavefunctions. We show that the highly unsatisfactory limitation of many-electron wavefunctions to $N \leq N_0$ can be overcome by generalizing their definition. We show for the ground state of a large system that it is the logarithm of the solution of Schr\"odinger's equation which should be used to characterize this system. In this case the wavefunctions of independent subsystems add up rather than multiply. This feature is realized by the action function. It provides a simple physical picture for the resolution of the EWP.

The Hilbert space for an interacting electron system increases exponentially with electron number $N$. This limits the concept of wavefunctions $\psi$ based on solutions of the Schr\"odinger equation to $N \leq N_0$ with $N_0 \simeq 10^3$ \cite{Kohn1999}. It is argued that this exponential wall problem (EWP) is connected with an increasing redundance of information contained, e.g., in the ground-state of the system and it's wavefunction. The EWP as well as redundance of information are avoided when the characterization of the ground state is based on the action function $R$ rather than on the solutions $\psi$ of the Sch\"odinger equation. Both are related through a logarithm, i.e., $R = -i \hbar \ ln \psi$. Working with the logarithm is made possible by the use of cumulants. It is pointed out the way electronic structure calculations for periodic solids may use this concept.

Wavefunctions for large electron numbers N are plagued by the Exponential Wall Problem (EWP), i.e., an exponential increase in the dimensions of Hilbert space with N. Therefore, they lose their meaning for macroscopic systems, a point stressed, in particular, by Kohn. The EWP has to be resolved in order to provide a solid basis for wavefunction based electronic structure calculations of macroscopic systems, e.g., solids. The origin of the EWP is the multiplicative property of wavefunctions when independent subsystems are considered. Therefore, it can only be avoided when wavefunctions are formulated so that they are additive instead, in particular, when matrix elements involving them are calculated. We describe how this is done for the ground state of a macroscopic electron system. Going over from a multiplicative to an additive quantity requires taking a logarithm. Here it implies going over from Hilbert space to the operator- or Liouville space with a metric based on cumulants. The operators which define the ground-state wavefunction generate fluctuations from a mean-field state. The latter does not suffer from an EWP and therefore may serve as a vacuum state. The fluctuations have to be connected like the ones caused by pair interactions in a classical gas when the free energy is calculated (Meyer's cluster expansion). This fixes the metric in Liouville space. The scheme presented here provides a solid basis for electronic structure calculations for the ground state of solids. In fact, its applicability has already been proven. We discuss also matrix product states, which have been applied to one-dimensional systems with results of high precision. Although these states are formulated in Hilbert space, they are processed by using operators in Liouville space. We show that they fit into the general formalism described above.

Wavefunctions for large electron numbers suffer from an exponential growth of the Hilbert space which is required for their description. In fact, as pointed out by W. Kohn, for electron numbers N > N_0 where N_0 ≈ 10^3 they become meaningless (exponential wall problem). Nevertheless, despite of the enormous successes of density functional theory, one would also like to develop electronic structure calculations for large systems based on wavefunctions. This is possible if one defines the latter in Liouville space with a cumulant metric rather than in Hilbert space. The cluster expansion of the free energy of a classical monoatomic gas makes it plausible that cumulants are a proper tool for electronic structure calculations.

P. Fulde∗ Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany (Dated: August 12, 2018) Abstract Wavefunctions for large electron numbers N are plagued by the Exponential Wall Problem (EWP), i.e., an exponential increase in the dimensions of Hilbert space with N . Therefore they loose their meaning for macroscopic systems, a point stressed in particular by W. Kohn. The EWP has to be resolved in order to be able to perform electronic structure calculations, e.g., for solids. The origin of the EWP is the multiplicative property of wavefunctions when independent subsystems are considered. Therefore it can only be avoided when wavefunctions are formulated so that they are additive instead, in particular when matrix elements involving them are calculated. We describe how this is done for the ground state of a macroscopic electron system. Going over from a multiplicative to an additive quantity requires taking a logarithm. Here it implies going over from Hilbert space to the operatoror Liouville space with a metric based on cumulants. The operators which define the ground-state wavefunction generate fluctuations from a mean-field state. The latter does not suffer from an EWP and therefore may serve as a vacuum state. The fluctuations have to be connected like the ones caused by pair interactions in a classical gas when the free energy is calculated (Meyer’s cluster expansion). This fixes the metric in Liouville space. The scheme presented here provides a solid basis for electronic structure calculations for the ground state of solids. In fact, its applicability has already been proven. We discuss also matrix product states, which have been applied to one-dimensional systems with results of high precision. Although these states are formulated in Hilbert space they are processed by using operators in Liouville space. We show that they fit into the general formalism described above.

An alternative to the density functional theory is the use of wavefunction based electronic structure calculations for solids. In order to perform them, the Exponential Wall (EW) problem has to be resolved. It is caused by an exponential increase of the number of configurations with increasing electron number N. There are different routes one may follow. One is to characterize a many-electron wavefunction by a vector in Liouville space with a cumulant metric rather than in Hilbert space. This removes the EW problem. Another is to model the solid by an impurity or fragment embedded in a bath which is treated at a much lower level than the former. This is the case in the Density Matrix Embedding Theory (DMET) or the Density Embedding Theory (DET). The latter two are closely related to a Schmidt decomposition of a system and to the determination of the associated entanglement. We show here the connection between the two approaches. It turns out that the DMET (or DET) has an identical active space as a previously used Local Ansatz, based on a projection and partitioning approach. Yet, the EW problem is resolved differently in the two cases. By studying a H10 ring, these differences are analyzed with the help of the method of increments.

We derive an analytic expression for the temperature dependent critical magnetic field parallel to ultra-thin superconducting films with Rashba spin-orbit interaction. Thereby we cover the range from small to large spin-orbit interactions. compared with the gap parameter Delta(0). We find that at a critical spin-orbit energy lambda(c) a first-order phase transition takes place at which the pairing momentum of the Cooper pairs changes discontinuously. We speculate that this might give raise to new phenomena. With increasing lambda/Delta(0) the pair formation changes from inter-band to intra-band pairing. For lambda > lambda(c) a dimensional cross-over of the critical field from two to one dimension is taking place.

Many-electron wavefunctions face the exponential-wall problem at large electron numbers. Formulating wavefunctions with the help of cumulants effectively avoids this problem and provides a valuable starting point for electronic-structure calculations for solids.

For 4 d 1 and 5 d 1 spin–orbit-coupled electron configurations, the notion of nonmagnetic j =3/2 quartet ground state discussed in classical textbooks is at odds with the observed variety of magnetic properties. Here we throw fresh light on the electronic structure of 4 d 1 and 5 d 1 ions in molybdenum- and osmium-based double-perovskite systems and reveal different kinds of on-site many-body physics in the two families of compounds: although the sizable magnetic moments and g -factors measured experimentally are due to both metal d –ligand p hybridisation and dynamic Jahn–Teller interactions for 4 d electrons, it is essentially d − p covalency for the 5 d 1 configuration. These results highlight the subtle interplay of spin–orbit interactions, covalency and electron–lattice couplings as the major factor in deciding the nature of the magnetic ground states of 4 d and 5 d quantum materials. Cation charge imbalance in the double-perovskite structure is further shown to allow a fine tuning of the gap between the t 2 g and e g levels, an effect of much potential in the context of orbital engineering in oxide electronics.

Electronic structure calculations for solids based on many-electron wavefunctions have been hampered by the argument that for large electron numbers wavefunctions are not a legitimate scientific concept, because they face an exponential wall problem. We show that this problem can be avoided by using cumulant techniques in formulating wavefunctions. Therefore calculations for solids based on many-electron functions are possible and useful. This includes also systems with strongly correlated electrons.

The electronic structure of the low-dimensional 4d(5) oxides Sr2RhO4 and Ca3CoRhO6 is herein investigated by embedded-cluster quantum chemistry calculations. A negative tetragonal-like t2g splitting is computed in Sr2RhO4 and a negative trigonal-like splitting is predicted for Ca3CoRhO6, in spite of having positive tetragonal distortions in the former material and cubic oxygen octahedra in the latter. Our findings bring to the foreground the role of longer-range crystalline anisotropy in generating noncubic potentials that compete with local distortions of the ligand cage, an issue not addressed in standard textbooks on crystal-field theory. We also show that sizable t2g(5)-t2g(4)eg(1) couplings via spin-orbit interactions produce in Sr2RhO4 ⟨Z⟩ = ⟨Σ(i)l(i)·s(i)⟩ ground-state expectation values significantly larger than 1, quite similar to theoretical and experimental data for 5d(5) spin-orbit-driven oxides such as Sr2IrO4. On the other hand, in Ca3CoRhO6, the ⟨Z⟩ values are lower because of larger t2g-eg splittings. Future X-ray magnetic circular dichroism experiments on these 4d oxides will constitute a direct test for the ⟨Z⟩ values that we predict here, the importance of many-body t2g-eg couplings mediated by spin-orbit interactions, and the role of low-symmetry fields associated with the extended surroundings.

The quasiparticle bands of diamond, a prototype covalent insulator, are herein studied by means of a wavefunction electronic-structure theory, with emphasis on the nature of the correlation hole around a bare particle. Short-range correlations are in such a system conveniently described by using a real-space representation and many-body techniques from ab initio quantum chemistry. To account for long-range polarization effects, on the other hand, we adopt the approximation of a dielectric continuum. Having as "uncorrelated" reference the Hartree-Fock band structure, the post-Hartree-Fock treatment is carried out in terms of localized Wannier functions derived from the Hartree-Fock solution. The computed correlation-induced corrections to the relevant real-space matrix elements are important and give rise to a strong reduction, in the range of 50%, of the initial Hartree-Fock gap. While our final results for the indirect and direct gaps, 5.4 and 6.9 eV, respectively, compare very well with the experimental data, the width of the valence band comes out by 10% to 15% too large as compared to experiment. This overestimation of the valence-band width appears to be related to size-consistency effects in the configuration-interaction correlation treatment.

In the search for topological phases in correlated electron systems, materials with 5d transition-metal ions, in particular the iridium-based pyrochlores A2Ir2O7, provide fertile grounds. Several topological states have been predicted but the actual realization of such states is believed to critically depend on the strength of local potentials arising from distortions of the IrO6 cages. We test this hypothesis by measuring with resonant inelastic x-ray scattering the electronic level splittings in the A = Y, Eu systems, which we show to agree very well with ab initio quantum chemistry electronic-structure calculations for the series of materials with A = Sm, Eu, Lu, and Y. We find, however, that the primary source for quenching the spin-orbit interaction is not a distortion of the IrO6 octahedra but longer-range lattice anisotropies which inevitably break the local cubic symmetry.

The electronic structure of the low-dimensional 4d 5 oxides Sr2RhO4 and Ca3CoRhO6 is herein investigated by embedded-cluster quantum chemistry calculations. A negative tetragonal-like t2g splitting is computed in Sr2RhO4 and a negative trigonal-like splitting is predicted for Ca3CoRhO6, in spite of having positive tetragonal distortions in the former material and cubic oxygen octahedra in the latter. Our findings bring to the foreground the role of longer- range crystalline anisotropy in generating noncubic potentials that compete with local distortions of the ligand cage, an issue not addressed in standard textbooks on crystal-field theory. We also show that sizable t2g 5 −t2g 4 eg 1 couplings via spin− orbit interactions produce in Sr2RhO4 ⟨ ⟩=⟨ ∑ · ⟩