Theoretical research of the Strongly Correlated Sysytems Momentum Research Group focuses on the development of new mathematic algorithms based on the fundamental laws of quantum mechanics, that make the simulated examination of more complex systems possible than previously; also forecasting their behaviour, or even designing their properties. Computer programs developed by the group are successfully used in numerous research institutes and universities worldwide, for example in the simulation of material properties in solid bodies, in molecular quantumchemistry, and even the quantum simulation of information technology itself. By their calculations they can effectively simulate such quantum systems (called ultra cold atoms) that can be built experimentally, from which the evolvement of quantum computers or high-temperature superconductors is hoped.

Örs Legeza

Peter Balla

Gergely Barcza

Mihály Csirik

Imre Hagymási

Kazumasa Itai

Tamás Mosoni

Jenõ Sólyom

Szilárd Szalay

Edina Szirmai

Ferenc Woynarovich

(* ***2016**)

**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, Energy Research Center, 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}.** —

**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.

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) algorithms. 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. Major algorithmic developments have also been carried out concerning the *quantum chemistry DMRG* and *Coupled-Cluster* (CC) algorithms. 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 the correlation and entanglement patterns in molecules, playing important role in chemical reactions.

**Entanglement, excitations and correlation effects in narrow zigzag graphene nanoribbons.** — We have investigated the low-lying excitation spectrum and ground-state properties of narrow graphene nanoribbons with zigzag edge configurations. Such nanoribbons have been synthesized very recently, and their descriptions require more sophisticated methods since in this regime conventional methods like mean-field or density-functional theory with local density approximation fail to capture the enhanced quantum fluctuations. Using the unbiased DMRG algorithm, we have calculated the charge gaps with high accuracy for different widths and interaction strengths and compared them with mean-field results. It turned out that the gaps are much smaller in the former case due to the proper treatment of quantum fluctuations. Applying the elements of quantum information theory, we also revealed the entanglement structure inside a ribbon and examined the spectrum of subsystem density matrices to understand the origin of entanglement. We examined the possibility of magnetic ordering and the effect of magnetic field. Our findings are relevant for understanding the gap values in different recent experiments and the deviations between them.

**Characterization of a correlated topological Kondo insulator in one dimension.** — We have investigated the ground-state of a p-wave Kondo-Heisenberg model introduced by Alexandrov and Coleman with an Ising-type anisotropy in the Kondo interaction and with correlated conduction electrons. Our aim was to understand how they affect the stability of the Haldane state obtained in the SU(2) symmetric case without the Hubbard interaction. By applying the DMRG algorithm and calculating the entanglement entropy, we have shown that in the anisotropic case a phase transition occurs and a Néel state emerges above a critical value of the Coulomb interaction. These findings were also corroborated by the examination of the entanglement spectrum and the spin profile of the system which clarify the structure of each phase.

**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 calculate the average bond-length, dimerization, and optical phonon frequencies for finite straight and zigzag chains using the DMRG method. We have checked 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.

**Coupled-cluster method with single and double excitations tailored by MPS wave functions. **— In the last decade, the quantum chemical version of the DMRG method has established itself as the method of choice for calculations of strongly correlated molecular systems. Despite its favourable scaling, it is not suitable for computations of dynamic correlation in practice. We have presented a novel method for accurate "post-DMRG" treatment of dynamic correlation based on the tailored CC theory, in which the DMRG method is responsible for the proper description of non-dynamic correlation, whereas dynamic correlation is incorporated through the framework of the CC theory. We have illustrated the potential of this method on prominent multireference systems, in particular N_{2}, Cr_{2} molecules and also oxo-Mn(Salen) for which we have performed the first "post-DMRG" computations in order to shed light on the energy ordering of the lowest spin states.

**The correlation theory of the chemical bond.** — The quantum mechanical description of the chemical bond is 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 an algorithm for the multiorbital correlation clustering. These make it possible to characterize quantitatively how well a bonding picture describes the chemical system. As 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.

**Method of Increments (MoI).** — We have further developed the method of increments (MoI) that allows one to successfully calculate cohesive energies of bulk materials with high accuracy, but it encounters difficulties when calculating whole 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. Therefore, in those situations where HF does not allow a size-consistent description of the dissociation, the MoI cannot yield proper results either. We have addressed the problem by employing a size-consistent multiconfigurational reference for the MoI formalism. This led to a matrix equation where a coupling derived by the reference itself is employed. In principle, such 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 we have shown by the comparison with DMRG benchmarks. We tested this two-state constant-coupling (TSCC)-MoI on beryllium rings of different sizes and studied the error introduced by the constant coupling.

**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 introduce a protocol to determine optimal active spaces based on the use of the DMRG algorithm and concepts of quantum information theory. We applied the protocol to elucidate the electronic structure and bonding mechanism of volatile plutonium oxides (PuO

**Analysis of two-orbital correlations in wavefunctions restricted to electron-pair states.** — Wavefunctions constructed from electron-pair states can accurately model strong electron correlation effects and are promising approaches especially for larger many-body systems. We have analyzed the nature and the type of electron correlation effects that can be captured by wavefunctions restricted to electron-pair states. We focused on the Antisymmetric Product of 1-reference orbital Geminal (AP1roG) method combined with an orbital optimization protocol whose performance was assessed against electronic structures obtained from DMRG reference data. Our numerical analysis covered model systems for strong correlation: the one-dimensional Hubbard model with periodic boundary condition as well as metallic and molecular hydrogen rings. Specifically, the accuracy of AP1roG was benchmarked using the single-orbital entropy, the orbital-pair mutual information as well as the eigenvalue spectrum of the one-orbital and two-orbital reduced density matrices. Our study indicated that contributions from singly occupied states become important in the strong correlation regime which highlights the limitations of the AP1roG method. Furthermore, we have examined the effect of orbital rotations within the AP1roG model on correlations between orbital pairs.

**Fermionic orbital optimization in tensor network states.** — We have further developed and implemented the novel fermionic mode transformation algorithm in the Budapest QC-DMRG code and performed additional large scale calculations by keeping more than 8000 block states. For the investigated chemical systems we have obtained significantly more optimal basis sets compared to those generated by conventional approaches.

**Nuclear structure theory.** — We have further improved the nuclear shell variant of the DMRG algorithm that includes an optimal ordering of the proton and neutron orbitals and an efficient expansion of the active space, utilizing various concepts of quantum information theory. We have generalized the implementation of non-Abelian symmetries.

**Post-DMRG methods.** —We have developed algorithms that rely on the MPS representation of the wave function obtained by the DMRG procedure. These so-called post-DMRG algorithms allow one to calculate expectation values of arbitrary operators between wave function generated by independent DMRG calculations. These methods were applied to various spin and fermionic models.

**Multipartite correlations in fermionic systems.** — We have started to investigate the theoretical foundations of the notions of correlations in second-quantized fermionic systems, mostly in the Jordan-Wigner representation. The results of our investigations make possible to use the main parts of the correlation theory of distinguishable systems for fermionic systems, providing firm theoretical grounds for the projects dealing with fermionic systems, e.g., in molecular physics or fermionic lattice models.

**N-heterocyclic carbenes.** — The N-heterocyclic carbenes are one of the most important "experimental tools" of the modern main group chemistry, which can be applied to stabilize compounds with unidentified bonds. Nowadays, the understanding of the stabilization (whether the process is realized through dative or covalent bond) is in the focus of theoretical literature to shed light on the unidentified bonds. This year we began to study various carbenes, using our previously developed quantum information theory based analysis which is capable to distinguish covalent and dative structures.

**Ultracold atomic systems.** — The interest in the gas of optically trapped ultracold atoms rapidly increases due to the high controllability and tunability of the experimental setups and the versatility of the realized quantum phases. We have studied the interacting system of atoms with spin 3/2 trapped in a square lattice using different numerical approaches. In the strongly repulsive limit, the system can be described by a generalized bilinear-biquadratic Heisenberg model. We have investigated this model in terms of mean field approaches, exact diagonalization, DMRG and cluster mean field approximation. Our primary goal is to understand the conflicting results published so far for SU(4) symmetric couplings, and to extend the analysis to general cases.

**Graphene nanoribbons.** — We have investigated the effect of long-range interaction on graphene nanoribbons and their low-lying excitation spectrum. We have also studied triangular-shaped graphene nanoflakes to reveal the correlation effects and magnetic properties induced by the electron-electron interaction.