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the simulation is the DMRG algorithm, in which the run-time is dominated by the iterative diagonalization of the Hamilton operator. Since the most time-consuming step of the diagonalization can be expressed as a list of dense matrix operations, the DMRG is an appealing candidate to fully utilize the computing power residing in novel kilo-processor architectures. In our pilot project a smart hybrid CPU-GPU implementation was presented, which exploited the power of both CPU (Central Processing Unit) and GPU (Graphical Processing Unit) and tolerates problems exceeding the GPU memory size. Furthermore, a new NVIDIA CUDA kernel has been designed for asymmetric matrix-vector multiplication to accelerate the rest of the diagonalization. Besides the evaluation of the GPU implementation, the practical limits of a field-programmable gate array (FPGA) implementation were also discussed.
Entanglement in uranium-based complexes. — The accurate description of the complexation of the CUO molecule by Ne and Ar noble gas matrices represents a challenging task for present-day quantum chemistry. Especially, the accurate prediction of the spin ground state of different CUO-noble-gas complexes remains elusive. In our work, the interaction of the CUO unit with the surrounding noble gas matrices has been investigated in terms of complexation energies and dissected into its molecular orbital quantum entanglement patterns. Our analysis elucidated the anticipated singlet-triplet ground-state reversal of the CUO molecule diluted in different noble gas matrices and demonstrated that the strongest uranium-noble gas interaction is found for CUOAr4 in its triplet configuration.
Entanglement and chemical bonding. — The chemical bond is an important local concept to understand chemical compounds and processes. Unfortunately, like most local concepts, the chemical bond and the bond order do not correspond to any physical observable and thus cannot be determined as an expectation value of a quantum chemical operator. We have recently demonstrated that one- and two-orbital-based entanglement measures can be utilized to interpret electronic wave functions in terms of orbital correlation. Orbital entanglement emerged to be a powerful tool to provide a qualitative understanding of bond-forming and bond-breaking processes, and allowed for an estimation of bond orders of simple diatomic molecules beyond the classical bonding models. In our work we demonstrated that the orbital entanglement analysis can be extended to polyatomic molecules to understand chemical bonding.
Metal-Insulator-like Transition (MIT) and entanglement. — We have studied the Metal-Insulator-like Transition (MIT) in lithium and beryllium ring-shaped clusters through ab-initio Density Matrix Renormalization Group (DMRG) method. Performing accurate calculations for different interatomic distances and using Quantum Information Theory (QIT), we investigated the changes occurring in the wave function between a metallic-like state and an insulating state built from free atoms. We also discussed entanglement and relevant excitations among the molecular orbitals in the Li and Be rings and showed that the transition bond length can be detected using orbital entropy functions. Also, the effect of different orbital bases on the effectiveness of the DMRG procedure has been analyzed by comparing the convergence behavior.
Nickel-ethene bond-formation. — We presented a conceptually new approach to dissect bond-formation processes in metal-driven catalysis by using concepts from quantum
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information theory. Our method used the entanglement and correlation among molecular orbitals to analyze changes in electronic structure that accompany chemical processes. As a proof-of-principle example, the evolution of nickel-ethene bond-formation was dissected which allowed us to monitor the interplay of back-bonding and π-donation along the reaction coordinate. Furthermore, the reaction pathway of nickel-ethene complexation was analyzed by using quantum chemistry methods, revealing the presence of a transition state.
Our study supported the crucial role of metal-to-ligand back-donation in the bond-forming process of nickel-ethene.
Entanglement and correlations in quantum many-body systems. — We defined a generalized, entanglement-based correlation function related to the mutual information of two localized, typically single-site, subsystems of a larger many-body system. The two-site mutual information is defined in terms of the von Neumann entropy of the single-site and two-site density matrices, which, in turn, can be written in terms of expectation values of transition operators between localized states. It can be used to map out entanglement patterns between the subsystems (that is, sites) of the system. By defining generalized correlation functions as two-point correlation functions of transition operators, we found that the long-distance decay of the mutual information follows the square of that of the most slowly decaying generalized correlation function. We showed how the generalized correlation functions are related to conventional correlation functions for spin and fermion lattice models. We explored the behavior of the mutual information, the generalized correlation functions, and their relation for the general spin-1/2 Heisenberg model and for SU(n) Hubbard models with n=2,3,4, and 5, and demonstrated the principles on known phases of the spin and SU(2) Hubbard models and obtained results characterizing the dimerized, trimerized, and quadrimerized phases in the SU(3), SU(4), and SU(5) Hubbard models, respectively. In addition, we extended the picture of the two-site mutual information and the corresponding generalized correlation functions to the n-site case.
Periodic Anderson model. — We investigated the effect of the Coulomb interaction, Ucf, between conduction and f-electrons in the periodic Anderson model using the Density Matrix Renormalization Group algorithm. We calculated the excitation spectrum of the half-filled symmetric model with an emphasis on the spin and charge excitations. In the one-dimensional version of the model, it was found that the spin gap was smaller than the charge gap below a certain value of Ucf and the reversed inequality was valid for stronger Ucf. This behavior was also verified by the behavior of the spin and density correlation functions. We also performed a quantum information analysis of the model and determined the entanglement map of f- and conduction electrons. It was revealed that for a certain Ucf the ground state is dominated by the configuration in which the conduction and f electrons are strongly entangled, and the ground state is almost a product state. For larger Ucf, the sites are occupied alternately dominantly by two f-electrons or by two conduction electrons.
Ultracold Atoms. — In the work “Phase Separation of Super Fluids in the Chain of Four-Component Ultracold Atoms” we have investigated the competition of various exotic superfluid states in a chain of spin-polarized ultracold fermionic atoms with hyper spin F=3/2 and s-wave contact interactions. We showed that the ground state is an exotic inhomogeneous mixture in which two distinct superfluid phases – spin-carrying pairs and singlet quartets – form alternating domains in an extended region of the parameter space.
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Graphene nanoribbons. — The possibility that non-magnetic materials such as carbon could exhibit a novel type of s-p electron magnetism has attracted much attention over the years, not least because such magnetic order is predicted to be stable at high temperatures. We found that the magnetic order on graphene edges of controlled zigzag orientation can be stable even at room temperature, raising hopes of graphene-based spintronic devices operating under ambient conditions.
Grants
OTKA K100908 Simulating strongly correlated systems with fermionic alkaline earth atom isotopes in optical lattices and related quantum chemistry of transition metal complexes (Ö.
Legeza, 2012–2016)
European Research Area Chemistry(ERA-Chemistry) “Generalized tensor methods in quantum chemistry” under OTKA NN110360, DFG SCHN 530/9-1 project under Grant No.
10041620 and FWF-E1243-N19
“Momentum” Program of the H.A.S.: Tensor factorization in high-dimensional spaces and applications to ultracold atomic systems and transition metal complexes (Ö. Legeza 2012-2017).
Ányos Jedlik Predoctoral Scholarship (I. Hagymási, 2013.12-2014.11)
International cooperation
ETH Zürich, (Zürich, Switzerland), Development of the relativistic DMRG algorithm (S.
Knecht, M. Reiher)
Philipps Universität Marburg, (Marburg, Germany), Optical properties of polydiacetylenes (F. Gebhard); Entanglement scaling in momentum space DMRG (G. Ehlers, R.M. Noack) Freie Universität, (Berlin, Germany), Basis optimization using matrix product state (MPS) based approach (C. Krumnow, R. Schneider, J. Eisert); Ab initio description of metal Insulator transitions (E. Fertitta, B. Paulus)
Universität Wien, (Vienna, Austria), Development of tree tensor network state (TTNS) algorithm (V. Murg, F. Verstraete)
Technische Universität Berlin, (Berlin, Germany), Tensor factorizations in high dimensional problems (M. Pfeffer, R. Schneider)
McMaster University, (McMaster, Canada), Bond braking and formation through entanglement (K. Boguslawski, P. Tecmer, P. Ayers)
Ustav Fyzikalni Chemie J. Heyrovskeho AV CR, (Prague, Czech Republic), Development of the quantum chemistry version of the DMRG method (L. Veis, J. Pittner)
Long-term visitor
Libor Veis, Ustav Fyzikalni Chemie J. Heyrovskeho AV CR, Prague, Czech Republic (Oct. 2014- March 2015; host: Ö. Legeza)
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Publications
Articles
1. Fertitta E, Paulus B, Barcza G, Legeza Ö: Investigation of metal-insulator-like transition through the ab initio density matrix renormalization group approach. PHYS REV B, 90:(24) Paper 245129. (2014)
2. Hagymasi I, Solyom J, Legeza O: Interorbital interaction in the one-dimensional periodic Anderson model: A density-matrix renormalization-group study. PHYS REV B, 90: Paper 125137. 10 p. (2014)
3. Knecht S, Legeza Ö, Reiher M: Communication: Four-component density matrix renormalization group. J CHEM PHYS, 140:(4) Paper 041101. 5 p. (2014)
4. Magda GZs, Jin XZ, Hagymási I, Vancsó P, Osváth Z, Nemes-Incze P, Hwang CY, Biro LP, Tapasztó L: Room-temperature magnetic order on zigzag edges of narrow graphene nanoribbons. NATURE, 514: pp. 608-611. (2014)
5. Mottet M, Tecmer P, Boguslawski K, Legeza Ö, Reiher M: Quantum entanglement in carbon-carbon, carbon-phosphorus and silicon-silicon bonds. PHYS CHEM CHEM PHYS, 16:(19) pp. 8872-8880. (2014)
6. Nemes Cs, Barcza G, Nagy Z, Legeza Ö, Szolgay P: The density matrix renormalization group algorithm on kilo-processor architectures: Implementation and trade-offs.
COMPUT PHYS COMMUN, 185:(6) pp. 1570-1581. (2014)
7. Sólyom J: Wigner crystals: New realizations of an old idea. EPJ WEB OF CONFERENCES, 78: Paper 01009. 8 p. (2014)
8. Tecmer P, Boguslawski K, Legeza O, Reiher M: Unravelling the quantum-entanglement effect of noble gas coordination on the spin ground state of CUO. PHYS CHEM CHEM PHYS, 16:(2) pp. 719-727. (2014)
Book chapter
9. Legeza Ö, Rohwedder Th, Schneider R, Szalay Sz: Tensor product approximation (DMRG) and coupled cluster method in quantum chemistry. In: Many-Electron Approaches in Physics, Chemistry and Mathematics, Eds.: Bach V, Delle Site L, Heidelberg: Springer International Publishing, 2014. pp. 53-76.
Conference proceeding
10. Legeza Ő: Generalized tensor methods and entanglement measurements for strongly correlated systems. In: Proc. II. International Summer School on Exact and Numerical Methods for Low-Dimensional Quantum Structures, Izmir, Turkey, 23.08.2014-31.08.2014, Paper e-only. 149 p.
Others
11. Legeza Ő: Generalized tensor methods and entanglement optimizations in quantum
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chemistry. New wavefunction methods and entanglement optimizations in quantum chemistry, Workshop on Theoretical Chemistry in Mariapfarr 18.2.2014-21.2.2014, 105 slides (2014)
12. Legeza Ő (ed.): New wavefunction methods and entanglement optimizations in quantum chemistry. Workshop on Theoretical Chemistry in Mariapfarr 18.2.2014-21.2.2014 (2014)
13. Szalay Sz: Entanglement and correlations: an introduction. Entanglement Day(s) 04.09.2014, Wigner RCP, Budapest, 74 p.
14. Szalay Sz, Pfeffer M, Murg V, Barcza G, Verstraete F, Schneider R, Legeza Ö: Tensor product methods and entanglement optimization for ab initio quantum chemistry.
Tutorial-review paper, 107 pages, 44 figures, 277 references (2014)
15. Woynarovich F: Gondolatok a "modell" fogalom használatáról (Thoughts about the use of the “model” conception, in Hungarian). FIZIKAI SZEMLE 64:(3) pp. 103-106. (2014)
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