**2018**

**Tensor factorization in high dimensional problems and applications to strongly correlated systems in condensed matter physics and quantum chemistry.**

In this year we have continued our research on various strongly correlated systems using the Density Matrix Renormalization Group (DMRG), Matrix Product State (MPS) and Tree Tensor Network State (TTNS) methods. We have also given close to twenty talks on different conferences and seminars, and we have presented some ten posters. In addition, we have further developed our scientific softwares (**Budapest QC-DMRG program package**), which have been used with great success in numerous research institutes and universities around the world, for, e.g., simulating material properties of solid state systems or molecules, or for the quantum simulation of the information technology itself. Further algorithmic developments have also been carried out concerning the quantum chemistry DMRG and Coupled-Cluster (CC) algorithms. In addition, in collaboration with Prof. Karol Kowalski, PNNL, Richland, Washington State, USA we have worked on the migration of the DMRG algorithm into the NWChem (commercial) program package, which ensures the possibility of massive parallelization. In collaboration with guest researchers from the groups of Uni Ghent and Uni Marburg, we have been working on new algorithmic solutions on the tree-TNS algorithm. As will be presented below, among many others, we have examined strongly correlated electrons in magnetic materials in several quantum phases, exotic quantum phases in ultracold atomic systems, and we have determined multi-orbital correlation and entanglement patterns in molecules, playing important role in chemical compounds.

**Numerical and Theoretical Aspects of the DMRG-TCC Method Exemplified by the Nitrogen Dimer** – We have investigated the numerical and theoretical aspects of the coupled-cluster method tailored by matrix-product states. We have investigated chemical properties of the used method, such as energy size extensivity and the equivalence of linked and unlinked formulation. The existing mathematical analysis was elaborated in a quantum chemical framework. In particular, we highlighted the use of a so-called CAS-ext gap describing the basis splitting between the complete active space and the external part. Moreover, the behavior of the energy error as a function of the optimal basis splitting were discussed. We have shown numerical investigations on the robustness with respect to the bond dimensions of the single orbital entropy and the mutual information, which are quantities that are used to choose the complete active space. Furthermore, we have extended the mathematical analysis with a numerical study on the complete active space dependence of the error.

**Ground-state properties of the symmetric single-impurity Anderson model on a ring from Density-Matrix Renormalization Group, Hartree-Fock, and Gutzwiller theory** - We have analyzed the ground-state energy, magnetization, magnetic susceptibility, and Kondo screening cloud of the symmetric single-impurity Anderson model (SIAM) that is characterized by the band width, the impurity interaction strength, and the local hybridization. We have compared Gutzwiller variational and magnetic Hartree-Fock results in the thermodynamic limit with numerically exact data from the DMRG method on large rings. To improve the DMRG performance, we have used a canonical transformation to map the SIAM onto a chain with half the system size and open boundary conditions. We have compared to Bethe-Ansatz results for the ground-state energy, magnetization, and spin susceptibility that become exact in the wide-band limit. Our detailed comparison have shown that the field-theoretical description is applicable to the SIAM on a ring for a broad parameter range. Hartree-Fock theory gives an excellent ground-state energy and local moment for intermediate and strong interactions. However, it lacks spin fluctuations and thus cannot screen the impurity spin. The Gutzwiller variational energy bound becomes very poor for large interactions because it does not describe properly the charge fluctuations. Nevertheless, the Gutzwiller approach provides a qualitatively correct description of the zero-field susceptibility and the Kondo screening cloud. The DMRG provides excellent data for the ground-state energy and the magnetization for finite external fields. At strong interactions, finite-size effects make it extremely difficult to recover the exponentially large zero-field susceptibility and the mesoscopically large Kondo screening cloud.

**Elucidating cation--cation interactions in neptunyl dications using multireference ab initio theory** - Understanding the binding mechanism in neptunyl clusters formed due to cation-cation interactions is of crucial importance in nuclear waste reprocessing and related areas of research. Since experimental manipulations with such species are often rather limited, we have to rely on quantum-chemical predictions of their electronic structures and spectroscopic parameters. We have presented a state-of-the-art quantum chemical study of the T-shaped and diamond-shaped neptunyl(V) and neptunyl(VI) dimers. Specifically, we have scrutinized their molecular structures, solvation effects, the interplay of static and dynamical correlation, and the influence of spin-orbit coupling on the ground state and lowest-lying excited states for different total spin states and total charges of the neptunyl dications. Furthermore, we have used the picture of interacting orbitals (quantum entanglement and correlation analysis) to identify strongly correlated orbitals in the cation-cation complexes that should be included in complete active space calculations. Most importantly, we have highlighted the complex interplay of correlation effects and relativistic corrections in the description of the ground and lowest-lying excited states of neptunyl dications.

**Imaging the Wigner Crystal of Electrons in One Dimension** - The quantum crystal of electrons, predicted more than eighty years ago by Eugene Wigner, is still one of the most elusive states of matter. Recently it became possible to design experiments that observe the one-dimensional Wigner crystal directly, by imaging its charge density in real-space. The obtained images, of few electrons confined in one-dimension, match those of strongly interacting crystals, with electrons ordered like pearls on a necklace. In order to further support the existence of such state, we have performed large scale DMRG calculations on the given sysetm. Comparison to theoretical modeling demonstrates the dominance of Coulomb interactions over kinetic energy and the weakness of exchange interactions. Our experiments together with numerical simulations provide direct evidence for this long-sought electronic state, and open the way for studying other fragile interacting states by imaging their many-body density in real-space.

**Analysis of The Coupled-Cluster Method Tailored by Tensor-Network States in Quantum Chemistry **- We have analyzed the tailored coupled-cluster (TCC) method, which is a multi-reference formalism that combines the single-reference coupled-cluster (CC) approach with a full configuration interaction (FCI) solution covering the static correlation. This covers in particular the high efficiency coupled-cluster method tailored by tensor-network states (TNS-TCC). For statically correlated systems, we have introduced the conceptually new CAS-ext-gap assumption for multi-reference problems which replaces the unreasonable HOMO-LUMO gap. We have characterized the TCC function and have shown local strong monotonicity and Lipschitz continuity such that Zarantonello's Theorem yields locally unique solutions fulfilling a quasi-optimal error bound for the TCC method. We have performed an energy error analysis revealing the mathematical complexity of the TCC-method. Due to the basis-splitting nature of the TCC formalism, the error decomposes into several parts. Using the Aubin-Nitsche-duality method we have derived a quadratic (Newton type) error bound valid for the linear-tensor-network TCC scheme DMRG-TCC and other TNS-TCC methods.

**Three-Legged Tree Tensor Network States** - We have presented a new variational tree tensor network state (TTNS) ansatz, the three-legged tree tensor network state (T3NS). Physical tensors have been interspersed with branching tensors. Physical tensors have one physical index and at most two virtual indices, as in the matrix product state (MPS) ansatz of the DMRG ansatz. Branching tensors have no physical index, but up to three virtual indices. In this way, advantages of DMRG, in particular a low computational cost and a simple implementation of symmetries, have been combined with advantages of TTNS, namely incorporating more entanglement. Our code has been capable of simulating quantum chemical Hamiltonians, and we have presented several proof-of-principle calculations on LiF, N_{2}, and the bis(μ-oxo) and μ–η^{2}:η^{2} peroxo isomers of [Cu_{2}O2]^{2+}.

**Full Configuration Interaction Quantum Monte Carlo Benchmark and Multireference Coupled Cluster Studies for Tetramethyleneethane** - We have performed a full configuration interaction (FCI) quality benchmark calculation for the tetramethyleneethane molecule in the cc-pVTZ basis set employing a subset of complete active space second order perturbation theory, CASPT2(6,6), natural orbitals for the FCI quantum Monte Carlo calculation. The results have been in an excellent agreement with the previous large scale diffusion Monte Carlo calculations by Pozun et al. and available experimental results. Our computations have verified that there is a maximum on the potential energy surface (PES) of the ground singlet state (^{1}A) 45° torsional angle, and the corresponding vertical singlet–triplet energy gap is 0.01 eV. We have employed this benchmark for the assessment of the accuracy of Mukherjee’s coupled clusters with up to triple excitations (MkCCSDT) and CCSD tailored by the DMRG method. Multireference MkCCSDT with CAS(2,2) model space, though giving good values for the singlet–triplet energy gap, has not been able to properly describe the shape of the multireference singlet PES. Similarly, DMRG(24,25) has not been able to correctly capture the shape of the singlet surface, due to the missing dynamic correlation. On the other hand, the DMRG-tailored CCSD method has described the shape of the ground singlet state with excellent accuracy but for the correct ordering requires computation of the zero-spin-projection component of the triplet state (^{3}B_{1}).

**Analysis of electron-correlation effects in strongly correlated systems (N _{2} and N_{2}^{+}) by applying the DMRG method and quantum information theory** - The dissociation of N

_{2}and N

^{2+}has been studied by using the ab initio density-matrix renormalization-group (DMRG) method. Accurate potential energy surfaces (PESs) have been obtained for the electronic ground states of N

_{2}(X

_{1}Σ

_{g}

^{+}) and N

_{2}

^{+}(X

_{2}Σ

_{g}

^{+}) as well as for the N

_{2}

^{+}excited state B

_{2}Σ

_{u}

^{+}. Inherent to the DMRG approach, the eigenvalues of the reduced density matrix and their correlation functions have been at hand. Thus we could apply quantum information theory directly and we have investigated how the wave function changes along the PES and depicted differences between the different states. Moreover, by characterizing quantum entanglement between different pairs of orbitals and analyzing the reduced density matrix, we have achieved a better understanding of the multireference character featured by these systems.

**Towards a multiconfigurational method of increments**- The method of increments (MoI) allows one to successfully calculate cohesive energies of bulk materials with high accuracy, but it encounters difficulties when calculating dissociation curves. The reason is that its standard formalism is based on a single Hartree–Fock (HF) configuration whose orbitals are localized and used for the many-body expansion. In situations where HF does not allow a size-consistent description of the dissociation, the MoI cannot be guaranteed to yield proper results either. We have addressed the problem by employing a size-consistent multiconfigurational reference for the MoI formalism. This has led to a matrix equation where a coupling derived by the reference itself is employed. In principle, such an approach allows one to evaluate approximate values for the ground as well as excited states energies. While the latter are accurate close to the avoided crossing only, the ground state results are very promising for the whole dissociation curve, as has been shown by the comparison with DMRG benchmarks. We have tested this two-state constant-coupling MoI on beryllium rings of different sizes and studied the error introduced by the constant coupling.

**The classification of multipartite quantum correlation**- In multipartite entanglement theory, the partial separability properties have an elegant, yet complicated structure, which becomes simpler in the case when multipartite correlations are considered. We have elaborate this, by giving necessary and sufficient conditions for the existence and uniqueness of the class of a given class-label, by the use of which we have worked out the structure of the classification for some important particular cases, namely, for the finest classification, for the classification based on k-partitionability and k-producibility, and for the classification based on the atoms of the correlation properties.

**An entropy production based method for determining the position diffusion’s coefficient of a quantum Brownian motion**- Quantum Brownian motion of a harmonic oscillator in the Markovian approximation has been described by the respective Caldeira–Leggett master equation. This master equation can be brought into Lindblad form by adding a position diffusion term to it. The coefficient of this term is either customarily taken to be the lower bound dictated by the Dekker inequality or determined by more detailed derivations on the linearly damped quantum harmonic oscillator. We have explored the theoretical possibilities of determining the position diffusion term’s coefficient by analyzing the entropy production of the master equation.

**An Isolated Molecule of Iron(II) Phthalocyanin Exhibits Quintet Ground‐State: A Nexus between Theory and Experiment**- Iron(II) phthalocyanine (FePc) is an important member of the phthalocyanines family with potential applications in the fields of electrocatalysis, magnetic switching, electrochemical sensing, and phototheranostics. Despite the importance of electronic properties of FePc in these applications, a reliable determination of its ground‐state is still challenging. We have presented combined state of the art computational methods and experimental approaches, that is, Mössbauer spectroscopy and Superconducting Quantum Interference Device magnetic measurements to identify the ground state of FePc. While the nature of the ground state obtained with density functional theory depends on the functional, giving mostly the triplet state, multi‐reference complete active space second‐order perturbation theory and DMRG methods assign quintet as the FePc ground‐state in gas‐phase. This has been confirmed by the hyperfine parameters obtained from

^{57}Fe Mössbauer spectroscopy performed in frozen monochlorobenzene. The use of monochlorobenzene guarantees an isolated nature of the FePc as indicated by a zero Weiss temperature. The results open doors for exploring the ground state of other metal porphyrin molecules and their controlled spin transitions via external stimuli.

**Interaction quench and thermalization in a one-dimensional topological Kondo insulator**– We have studied the nonequilibrium dynamics of a one-dimensional topological Kondo insulator, modelled by a p-wave Anderson lattice model, following a quantum quench of the on-site interaction strength. Our goal was to examine how the quench influences the topological properties of the system, therefore our main focus was the time evolution of the string order parameter, entanglement spectrum and the topologically-protected edge states. We have pointed out that postquench local observables can be well captured by a thermal ensemble up to a certain interaction strength. Our results have demonstrated that the topological properties after the interaction quench are preserved at finite times; however, the absolute value of the string order parameter decays in time. These predictions could be directly tested in state-of-the-art cold-atom experiments.

**2017**

**Tensor factorization in high dimensional problems and applications to strongly correlated systems in condensed matter physics and quantum chemistry.** — In this year we have continued our research on various strongly correlated systems using the Density Matrix Renormalization Group (DMRG), Matrix Product State (MPS) and Tree Tensor Network State (TTNS) methods. In addition, we have further developed our scientific softwares (**Budapest QC-DMRG program package**), which have been used with great success in numerous research institutes and universities around the world for, e.g., simulating material properties of solid state systems or molecules, or for the quantum simulation of the information technology itself. Further algorithmic developments have also been carried out concerning the quantum chemistry DMRG and Coupled-Cluster (CC) algorithms, and we have carried out the most large-scale calculations available in the literature for tetramethyleneethane molecule. We have also implemented parallelization in several parts of the code, and worked on further MPI, open-MP and GPU based developments. In addition, during a two-week visit at PNNL (Prof. Karol Kowalski, PNNL, Richland, Washington State, USA) we started the migration of the DMRG algorithm into the NWChem (commercial) program package, which ensures the possibility of massive parallelization. In collaboration with guest researchers from the groups of Uni Ghent and Uni Marburg, we have been working on new algorithmic solutions on the tree-TNS algorithm. As will be presented below, among many others, we have examined strongly correlated electrons in magnetic materials in several quantum phases, exotic quantum phases in ultracold atomic systems, and we have determined multi-orbital correlation and entanglement patterns in molecules, playing important role in chemical compounds.

**Interaction effects in a chaotic graphene quantum billiard.** — We have investigated the local electronic structure of a Sinai-like, quadrilateral graphene quantum billiard with zigzag and armchair edges using scanning tunneling microscopy (STM) at room temperature in collaboration with the Research Institute for Materials Science, Centre for Energy Research, HAS. We have revealed that besides the (√3×√3)R30∘ superstructure, which is caused by the intervalley scattering, its overtones also appear in the STM measurements, which are attributed to the Umklapp processes. We have pointed out that these results can be well understood by taking into account the Coulomb interaction in the quantum billiard, accounting for both the measured density of state values and the experimentally observed topography patterns. The analysis of the level-spacing distribution substantiates the experimental findings as well. We have also revealed the magnetic properties of our system which should be relevant in future graphene based electronic and spintronic applications.

**A magnetic phase-transition graphene transistor with tunable spin polarization. **— Graphene nanoribbons have been proposed as potential building blocks for field effect transistor (FET) devices due to their quantum confinement bandgap. We have proposed a novel graphene nanoribbon device concept, enabling the control of both charge and spin signals, integrated within the simplest three-terminal device configuration. In a conventional FET device, a gate electrode is employed to tune the Fermi level of the system in and out of a static bandgap. By contrast, in the switching mechanism we proposed, the applied gate voltage can dynamically open and close an interaction gap, with only a minor shift of the Fermi level. Furthermore, the strong interplay of the band structure and edge spin configuration in zigzag ribbons enables such transistors to carry spin polarized current without employing an external magnetic field or ferromagnetic contacts. Using an experimentally validated theoretical model, we have shown that such transistors can switch at low voltages and high speed, and the spin polarization of the current can be tuned from 0% to 50% by using the same back gate electrode. Furthermore, such devices are expected to be robust against edge irregularities and can operate at room temperature. Controlling both charge and spin signal within the simplest FET device configuration could open up new routes in data processing with graphene based devices.

**Entanglement and magnetism in high-spin graphene nanodisks. **— We have investigated the ground-state properties of triangular graphene nanoflakes with zigzag edge configurations. The description of zero-dimensional nanostructures requires accurate many-body techniques since the widely used density-functional theory with local density approximation or Hartree-Fock methods cannot handle the strong quantum fluctuations. Applying the unbiased density-matrix renormalization group algorithm, we have calculated the magnetization and entanglement patterns with high accuracy for different interaction strengths and compared them to the mean-field results. With the help of quantum information analysis and subsystem density matrices, we have revealed that the edges are strongly entangled with each other. We have also addressed the effect of electron and hole doping and demonstrated that the magnetic properties of triangular nanoflakes can be controlled by electric field, which reveals features of flat-band ferromagnetism. This may open up new avenues in graphene based spintronics.

**Optical phonons for Peierls chains with long-range Coulomb interactions. **— We have considered a chain of atoms that are bound together by a harmonic force. Spin-1/2 electrons that move between neighboring chain sites (Hückel model) induce a lattice dimerization at half band filling (Peierls effect). We have supplemented the Hückel model with a local Hubbard interaction and a long-range Ohno potential, and calculated the average bond-length, dimerization, and optical phonon frequencies for finite straight and zigzag chains using the DMRG method. We have tested our numerical approach against analytic results for the Hückel model. The Hubbard interaction mildly affects the average bond length but substantially enhances the dimerization and increases the optical phonon frequencies whereas, for moderate Coulomb parameters, the long-range Ohno interaction plays no role.

**Interplay between exotic superfluidity and magnetism in a chain of four-component ultracold atoms.** — We have investigated the spin-polarized chain of ultracold alkaline-earth-metal atoms with spin-3/2 described by the fermionic Hubbard model with SU(4) symmetric attractive interaction. The competition of bound pairs, trions, quartets, and unbound atoms has been studied analytically and by DMRG. We have found several distinct states where bound particles coexist with the ferromagnetic state of unpaired fermions. In particular, an exotic inhomogeneous Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-type superfluid of quartets in a magnetic background of uncorrelated atoms has been found for weaker interactions. We have shown that the system can be driven from this quartet-FFLO state to a molecular state of localized quartets where spatial segregation between molecular crystals and ferromagnetic liquids emerges, and this transition is reflected in the static structure factor.

**Role of the pair potential for the saturation of generalized Pauli constraints. **— The dependence of the (quasi-)saturation of the generalized Pauli constraints on the pair potential is studied for ground states of few-fermion systems. For this, we have considered spinless fermions in one dimension which are harmonically confined and interact by pair potentials of the form |x_{i}−x_{j}|s with −1≤s≤5. Using the DMRG approach and large orbital basis sets ensures the convergence on more than ten digits of both the variational energy and the natural occupation numbers. Our results confirm that the conflict between energy minimization and fermionic exchange symmetry results in a quasi-saturation of the generalized Pauli constraints (quasipinning), implying structural simplifications of the fermionic ground state. However, a self-consistent perturbation theory reveals that most of that relevance has to be assigned to Pauli's original exclusion principle, except for the harmonic case, i.e., s=2. This emphasizes the unique nature of the strong, non-trivial quasipinning found recently for the Harmonium model.

**An entropy production based method for determining the position diffusion's coefficient of a quantum Brownian motion. **— Quantum Brownian motion of a harmonic oscillator in the Markovian approximation is described by the respective Caldeira-Leggett master equation. This master equation can be brought into Lindblad form by adding a position diffusion term to it. The coefficient of this term is either customarily taken to be the lower bound dictated by the Dekker inequality or determined by more detailed derivations on the linearly damped quantum harmonic oscillator. We have explored the theoretical possibilities of determining the position diffusion term's coefficient by analyzing the entropy production of the master equation. We have shown that the obtained value has a linear dependence on the temperature, which is in marked contrast to previous studies.

**The correlation theory of the chemical bond. **— The quantum mechanical description of the chemical bond is generally given in terms of delocalized bonding orbitals, or, alternatively, in terms of correlations of occupations of localized orbitals. However, in the latter case, multiorbital correlations were treated only in terms of two-orbital correlations, although the structure of multiorbital correlations is far richer; and, in the case of bonds established by more than two electrons, multiorbital correlations represent a more natural point of view. For the first time, we have introduced the true multiorbital correlation theory, consisting of a framework for handling the structure of multiorbital correlations, a toolbox of true multiorbital correlation measures, and the formulation of the multiorbital correlation clustering, together with an algorithm for obtaining that. These make it possible to characterize quantitatively how well a bonding picture describes the chemical system. As a proof of concept, we have applied the theory for the investigation of the bond structures of several molecules. We have shown that the non-existence of well-defined multiorbital correlation clustering provides a reason for debated bonding picture.

**Correlation analysis of electron-deficit bonds. **— We have extended the use of quantum information theory to classify chemical bonds based on multiorbital correlations within the molecule to electron deficient bonds, employing the adequate multireference method of DMRG. First, we have analyzed the bonding structure of the standard diborane(6) molecule and the newly discovered prototype, diborane(4), and we have confirmed that this theory is capable of correct description of bonding in electron deficient bonds, like diborans. Subsequently, we have studied neutral zerovalent s-block beryllium complex, a substance first synthesized in 2016, whose surprising stability was attributed to a strong three-center two-electron π-bond stretching over the C-Be-C core. Our calculations have provided quantitatively adequate images of correlations within the beryllium complex and confirmed the theoretical suggestions on the molecule’s stability.

**On the multi-reference nature of plutonium oxides PuO _{2}^{2+}, PuO_{2}, PuO_{3} and PuO_{2}(OH)_{2}. **— Actinide-containing complexes present formidable challenges for electronic structure methods due to the large number of degenerate or quasi-degenerate electronic states arising from partially occupied 5f and 6d shells. Conventional multi-reference methods can treat active spaces that are often at the upper limit of what is required for a proper treatment of species with complex electronic structures, leaving no room for verifying their suitability. We have addressed the issue of properly defining the active spaces in such calculations, and introduced a protocol to determine optimal active spaces based on the use of the DMRG algorithm and concepts of quantum information theory. We have applied the protocol to elucidate the electronic structure and bonding mechanism of volatile plutonium oxides (PuO

_{3 }and PuO

_{2}(OH)

_{2}), species associated with nuclear safety issues for which little is known about the electronic structure and energetics. We have shown how, within a scalar relativistic framework, orbital-pair correlations can be used to guide the definition of optimal active spaces which provides an accurate description of static/non-dynamic electron correlation, as well as to analyze the chemical bonding beyond a simple orbital model. From this bonding analysis, we have been able to show that the addition of oxo- or hydroxo-groups to the plutonium dioxide species considerably changes the π-bonding mechanism with respect to the bare triatomics, resulting in bent structures with a considerable multi-reference character.

**Full configuration interaction quantum Monte Carlo benchmark and multireference coupled cluster studies of tetramethyleneethane diradical. **— We have performed an FCI-quality benchmark calculation for the tetramethyleneethane molecule in cc-pVTZ basis set employing a subset of CASPT2(6,6) natural orbitals for the FCIQMC calculation. The results are in an excellent agreement with the previous large scale diffusion Monte Carlo calculations by Pozun et al. and available experimental results. Our computations have verified that there is a maximum on potential energy surface of the ground singlet state (^{1}A) 45∘ torsional angle and the corresponding vertical singlet-triplet energy gap is 0.01 eV. We have employed this benchmark for the assessment of the accuracy of MkCCSDT and DMRG-tailored CCSD (TCCSD) methods. Multireference MkCCSDT with CAS(2,2) model space, though giving good values for the singlet-triplet energy gap, is not able to properly describe the shape of the multireference singlet potential energy surface. Similarly, DMRG(24,25) is not able to correctly capture the shape of the singlet surface, due to the missing dynamic correlation. On the other hand, the DMRG-tailored CCSD method describes the shape of the ground singlet state with an excellent accuracy, but for the correct ordering, computation of the zero-spin-projection component of the triplet state (^{3}B_{1}) is required.