2023
In this year we have continued our research on various strongly correlated systems using the Density Matrix Renormalization Group (DMRG) and Matrix Product State (MPS). We have wide range international collaborations with more than twenty institutes around the world, resulting in twelve research articles in high rank international journals, and five preprints. We have given some twenty talks on different conferences and seminars, and we have presented six posters. We have applied our scientific software (Budapest QC-DMRG program package) to various spin and electron systems, 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. As will be presented below, among many others, we have examined strongly correlated electrons in magnetic materials in several quantum phases, identified exotic quantum phases of matter, investigated time dependent phenomena, studied entanglement in nucleon systems, and identified classical and quantum correlations in molecules, playing important role in chemical compounds. We have also presented detailed mathematical analysis of various concepts in quantum entanglement theory of qubits and fermions, and dynamics of open quantum systems. Here we summarize only selected results due to length restrictions.
Condensed matter and statistical physics:
We determined the ground-state phase diagram for the 1/r-Hubbard model with repulsive nearest-neighbor interaction at half band-filling using the density-matrix renormalization group (DMRG) method. Due to the absence of Umklapp scattering, the phase diagram displays finite regions for the three generic phases, namely, a Luttinger liquid metal for weak interactions, a Mott-Hubbard insulator for dominant Hubbard interactions, and a charge-density-wave insulator for dominant nearest-neighbor interactions. Up to moderate interactions strengths, the quantum phase transitions between the metallic and insulating phases are continuous, i.e., the gap opens continuously as a function of the interaction strength. We concluded that generic short-range interactions do not change the nature of the Mott transition qualitatively. [1]
We showed that dynamical hadron formation can be spectroscopically detected in an ultracold atomic setting within the most paradigmatic and simplest model of condensed matter physics, the repulsive SU(N) Hubbard model. We found that by starting from an appropriately engineered high-energy initial state of the strongly interacting SU(3) Hubbard model, doublons (mesons) and trions (barions) naturally emerged during time evolution and thermalized to a negative temperature quantum gas, as demonstrated by extensive one-dimensional simulations and exact diagonalization calculations. For strong interactions, trions become heavy and attract each other strongly. Their residual interaction with doublons generates doublon diffusion, that we captured by the evolution of the equal time density correlation function. Although our numerical calculations are performed on one-dimensional chains, many of our conclusions extend to a large variety of initial conditions and hold for other spatial dimensions and all SU(N > 2) Hubbard models. [2]
The collective tunneling of a Wigner necklace – a crystalline state of a small number of strongly interacting electrons confined to a suspended nanotube and subject to a double well potential – was theoretically analyzed and compared with experiments in [Shapir et al., Science 364, 870 (2019)]. Density Matrix Renormalization Group computations, exact diagonalization, and instanton theoryprovided a consistent description of this very strongly interacting system, and showed good agreement with experiments. Experimentally extracted and theoretically computed tunneling amplitudes exhibited a scaling collapse. We found that collective quantum fluctuations renormalized the tunneling, and substantially enhanced it as the number of electrons increased. [3]
We investigated the quantum quench dynamics of the interacting Hatano-Nelson model with open boundary conditions using both Abelian bosonization and numerical methods. Specifically, we followed the evolution of the particle density and current profile in real space over time by turning the imaginary vector potential on or off in the presence of weak interactions. Our results revealed spatiotemporal Friedel oscillations in the system with light cones propagating ballistically from the open ends, accompanied by local currents of equal magnitude for both switch-off and -on protocols. Remarkably, the bosonization method accurately accounted for the density and current patterns with a single overall fitting parameter. The continuity equation was satisfied by the long-wavelength part of the density and current, despite the nonunitary time evolution when the Hatano-Nelson term was switched on. [4]
Recent experiments demonstrated that single-particle quantum walks can reveal the topological properties of single-particle states. Here, we generalized this picture to the many-body realm by focusing on multiparticle quantum walks of strongly interacting fermions. After injecting N particles with multiple flavors in the interacting SU(N) Su-Schrieffer-Heeger chain, their multiparticle continuous-time quantum walk was monitored by a variety of methods. We found that the many-body Berry phase in the N-body part of the spectrum signaled a topological transition upon varying the dimerization, similarly to the single-particle case. This topological transition was captured by the single- and many-body mean chiral displacement during the quantum walk and remains present for strong interaction as well as for moderate disorder. Our predictions were well within experimental reach for cold atomic gases and can be used to detect the topological properties of many-body excitations through dynamical probes. [5]
We explored the Kardar-Parisi-Zhang (KPZ) scaling in the one-dimensional Hubbard model, which exhibits global SUc(2)xSUs(2) symmetry at half filling, for the pseudocharge and the total spin. We analyzed dynamical scaling properties of high-temperature charge and spin correlations and transport. At half filling, we observed a clear KPZ scaling in both charge and spin sectors. Away from half filling, the SUc(2) charge symmetry is reduced to Uc(1), while the SUs(2) symmetry for the total spin is retained. Consequently, transport in the charge sector becomes ballistic, while KPZ scaling is preserved in the spin sector. These findings confirmed the link between non-Abelian symmetries and KPZ scaling in the presence of integrability. We studied two settings of the model: one involving a quench from a bipartitioned state asymptotically close to the T to infinity equilibrium state of the system, and another where the system is coupled to two Markovian reservoirs at the two edges of the chain. [6]
Parafermions are anyons with the potential for realizing non-local qubits that are resilient to local perturbations. Compared to Majorana zero modes, braiding of parafermions implements an extended set of topologically protected quantum gates. This, however, comes at the price that parafermionic zero modes can not be realized in the absence of strong interactions whose theoretical description is challenging. We constructed a simple lattice model for interacting spinful electrons with parafermionic zero energy modes. The explicit microscopic nature of the considered model highlights new realization avenues for these exotic excitations in recently fabricated quantum dot arrays. By density matrix renormalization group calculations, we identified a broad range of parameters, with well-localized zero modes, whose parafermionic nature is substantiated by their unique 8𝜋 periodic Josephson spectrum. [7]
Quantum chemistry:
We theoretically derived and validated with large scale simulations a remarkably accurate power law scaling of errors for the restricted active space density matrix renormalization group (DMRG-RAS) method [J. Phys. Chem. A 126, 9709] in electronic structure calculations. This yields a new extrapolation method, DMRG-RAS-X, which reaches chemical accuracy for strongly correlated systems such as the chromium dimer, dicarbon up to a large cc-pVQZ basis and even a large chemical complex such as the FeMoco with significantly lower computational demands than those of previous methods. The method is free of empirical parameters, performed robustly and reliably in all examples we tested, and has the potential to become a vital alternative method for electronic structure calculations in quantum chemistry and more generally for the computation of strong correlations in nuclear and condensed matter physics. [8]
Tailored-CC (TCC) approach works well in many situations, however, in exactly degenerate cases (with two or more determinants of equal weight), it exhibits a bias towards the reference determinant representing the Fermi vacuum. In order to overcome the single-reference bias of the TCC method, we developed a Hilbert-space multireference version of tailored CC, which can treat several determinants on an equal footing. We employed a multireference analysis of the DMRG wave function in the matrix product state form to get the active amplitudes for each reference determinant and their constant contribution to the effective Hamiltonian. We have implemented and compared the performance of three Hilbert-space MRCC variants - the state universal one, and the Brillouin-Wigner and Mukherjee's state specific ones. We have assessed these approaches on the cyclobutadiene and tetramethylenethane (TME) molecules, which are both diradicals with exactly degenerate determinants at a certain geometry. [9]
We also presented a brief overview of the fermionic mode optimization within the framework of tensor network state methods (Krumnow et al. in Phys Rev Lett 117:210402, 2016), and demonstrated that it has the potential to compress the multireference character of the wave functions after finding optimal molecular orbitals (modes), based on entanglement minimization. Numerical simulations were performed for the nitrogen dimer in the cc-pVDZ basis for the equilibrium and for stretched geometries. [10]
We have presented a tutorial paper to bring novel concepts on orbital optimization protocols employed in the DMRG procedure closer to the quantum chemistry community. While a more rigorous mathematical approach has been presented previously, here we focused on technical aspects to obtain natural orbital (NO) and Rényi entropy based orbital (REMO) optimization in practice via low-cost DMRG calculations. We also showed how large scale DMRG calculations can be brought closer to the FCI limit at the negligible additional computational cost of NO- or REMO-generation. [11]
Material science:
We studied the symmetric carbon tetramer clusters in hexagonal boron nitride and proposed them as spin qubits for sensing. We utilized periodic-DFT and quantum chemistry approaches to reliably and accurately predict the electronic, optical, and spin properties of the studied defect. We showed that the nitrogen-centered symmetric carbon tetramer gives rise to spin state-dependent optical signals with strain-sensitive intersystem crossing rates. Furthermore, the weak hyperfine coupling of the defect to their spin environments results in a reduced electron spin resonance linewidth that can enhance sensitivity. [12]
We proposed the negatively charged nitrogen split interstitial defect in hBN as a plausible microscopic model for the blue emitter. We carefully analyzed the accuracy of first principles methods and showed that the commonly used HSE hybrid exchange-correlation functional fails to describe the electronic structure of this defect. Using the generalized Koopman’s theorem, we fine tuned the functional and obtained a zero-phonon photoluminescence (ZPL) energy in the blue spectral range. We showed that the defect exhibits high emission rate in the ZPL line and features a characteristic phonon side band that resembles the blue emitter’s spectrum. We also studied the electric field dependence of the ZPL and numerically showed that the defect exhibits a quadratic Stark shift for perpendicular to plane electric fields, making the emitter insensitive to electric field fluctuations in first order. [13]
Nuclear physics:
We proposed a novel many-body framework combining the density matrix renormalization group (DMRG) with the valence-space (VS) formulation of the in-medium similarity renormalization group. This hybrid scheme admitted for favorable computational scaling in large-space calculations compared to direct diagonalization. The capacity of the VS-DMRG approach was highlighted in ab initio calculations of neutron-rich nickel isotopes based on chiral two- and three-nucleon interactions, and allowed us to perform converged ab initio computations of ground and excited state energies. We also studied orbital entanglement in the VS-DMRG, and investigated nuclear correlation effects in oxygen, neon, and magnesium isotopes. The explored entanglement measures revealed nuclear shell closures as well as pairing correlations. [14]
Algorithmic aspects:
The interplay of quantum and classical simulation and the delicate divide between them is in the focus of massively parallelized tensor network state (TNS) algorithms designed for high performance computing (HPC). We presented novel algorithmic solutions together with implementation details to extend current limits of TNS algorithms on HPC infrastructure building on state-of-the-art hardware and software technologies. Benchmark results obtained via large-scale density matrix renormalization group (DMRG) simulations are presented for selected strongly correlated molecular systems addressing problems on Hilbert space dimensions up to 2.88×1036. [15]
We also presented novel algorithmic solutions together with implementation details utilizing non-Abelian symmetries via hybrid CPU-multiGPU solution where scheduling is decentralized, threads are autonomous and inter-thread communications are solely limited to interactions with globally visible lock-free constructs. Benchmark tests demonstrated the utilization of NVIDIA's highly specialized tensor cores, leading to performance around 110 TFLOPS on a single node supplied with eight NVIDIA A100 devices. In comparison to U(1) implementations with matching accuracy, our solution had an estimated effective performance of 250-500 TFLOPS. [16]
We presented a hybrid numerical approach to simulate quantum many body problems on two spatial dimensional quantum lattice models via the non-Abelian ab initio version of the density matrix renormalization group method on state-of-the-art high performance computing infrastructures. We demonstrated for the two dimensional spinless fermion model and for the Hubbard model on torus geometry that altogether several orders of magnitude in computational time can be saved by performing calculations on an optimized basis and by utilizing hybrid CPU-multiGPU parallelization. At least an order of magnitude reduction in computational complexity resulted from mode optimization, while a further order of reduction in wall time is achieved by massive parallelization. Our results were measured directly in FLOP and seconds. A detailed scaling analysis of the obtained performance as a function of matrix ranks and as a function of system size up to 12x12 lattice topology was discussed. [17]
References:
[1] Florian Gebhard, Kevin Bauerbach, Örs Legeza, Generic Mott-Hubbard phase diagram for extended Hubbard models without Umklapp scattering, Phys. Rev. B 108, 205130
[2] Miklós Antal Werner, Cătălin Paşcu Moca, Márton Kormos, Örs Legeza, Balázs Dóra, Gergely Zaránd, Spectroscopic evidence for engineered hadron formation in repulsive fermionic SU(N) Hubbard Models, Phys. Rev. Research 5, 043020 (2023)
[3] Dominik Szombathy, Miklós Antal Werner, Cătălin Paşcu Moca, Örs Legeza, Assaf Hamo, Shahal Ilani, Gergely Zaránd, Collective Wigner crystal tunneling in carbon nanotubes, arXiv:2306.15985 [cond-mat.mes-hall]
[4] Balázs Dóra, Miklós Antal Werner, and Cătălin Paşcu Moca, Quantum quench dynamics in the Luttinger liquid phase of the Hatano-Nelson model, Phys. Rev. B 108, 035104
[5] Bogdan Ostahie, Doru Sticlet, Cătălin Paşcu Moca, Balázs Dóra, Miklós Antal Werner, János K. Asbóth, and Gergely Zaránd, Multiparticle quantum walk: A dynamical probe of topological many-body excitations, Phys. Rev. B 108, 035126
[6] Cătălin Paşcu Moca, Miklós Antal Werner, Angelo Valli, Tomaž Prosen, and Gergely Zaránd, Kardar-Parisi-Zhang scaling in the Hubbard model, Phys. Rev. B 108, 235139
[7] Botond Osváth, Gergely Barcza, Örs Legeza, Balázs Dóra, László Oroszlány, A simple electronic ladder model harboring ℤ4 parafermions, arXiv:2311.07359 [cond-mat.str-el]
[8] Gero Friesecke, Gergely Barcza, Örs Legeza, Predicting the FCI energy of large systems to chemical accuracy from restricted active space density matrix renormalization group calculations, J. Chem. Theory Comput. 2024, 20, 1, 87–102
[9] Ondrej Demel, Jan Brandejs, Jakub Lang, Jiri Brabec, Libor Veis, Ors Legeza, Jiri Pittner, Hilbert space multireference coupled cluster tailored by matrix product states, J. Chem. Phys. 159, 224115 (2023)
[10] Mihály Máté, Klára Petrov, Szilárd Szalay, Örs Legeza, Compressing multireference character of wave functions via fermionic mode optimization, J. Math. Chem. 61, 362–375, (2023)
[11] Klára Petrov, Zsolt Benedek, Ádám Ganyecz, Gergely Barcza, András Olasz, and Örs Legeza, Low-cost generation of optimal molecular orbitals for multireference CI expansion:natural orbitals versus Rényi entropy minimized orbitals provided by the Density Matrix Renormalization Group, Progress in Theoretical Chemistry and Physics, accepted
[12] Zsolt Benedek, Rohit Babar, Ádám Ganyecz, Tibor Szilvási, Örs Legeza, Gergely Barcza, Viktor Ivády, Symmetric carbon tetramers forming spin qubits in hexagonal boron nitride, npj Comput Mater 9, 187 (2023)
[13] Ádám Ganyecz, Rohit Babar, Zsolt Benedek, Igor Aharonovich, Gergely Barcza, Viktor Ivády, First principles theory of the nitrogen interstitial in hBN: a plausible model for the blue emitter, arXiv:2308.01687 [cond-mat.mtrl-sci]
[14] A. Tichai, S. Knecht, A.T. Kruppa, Ö. Legeza, C.P. Moca, A. Schwenk, M.A. Werner, G. Zarand, Combining the in-medium similarity renormalization group with the density matrix renormalization group: Shell structure and information entropy, Phy. Lett. B, 845, 138139 (2023)
[15] Andor Menczer, Örs Legeza, Massively Parallel Tensor Network State Algorithms on Hybrid CPU-GPU Based Architectures, arXiv:2305.05581 [quant-ph]
[16] Andor Menczer, Örs Legeza, Boosting the effective performance of massively parallel tensor network state algorithms on hybrid CPU-GPU based architectures via non-Abelian symmetries, arXiv:2309.16724 [physics.comp-ph]
[17] Andor Menczer, Kornél Kapás, Miklós Antal Werner, Örs Legeza, Two dimensional quantum lattice models via mode optimized hybrid CPU-GPU density matrix renormalization group
