Impulzustérbeli sűrűségmátrix-renormálásicsoport-algoritmus fejlesztése és alkalmazása fermion modellekre. = Development and application of momentum space density matrix renormalization group method for fermion models

Az OTKA pályázat kutatási tervének megfelelően a négyéves időszak alatt a sűrűségmátrixos renormálásicsoport-algoritmus (DMRG) számos fejlesztését végeztük el, kiváltképpen a kvantuminformáció-elmélet (QIT) összefüggéseinek alkalmazása mellett. E két terület kapcsolatát legjobban leíró ún. összefonódottságot, mely egyben a kvantumos rendszerek alapvető tulajdonságait is megszabó mennyiség, hatékonyan alkalmaztuk spin és fermionrendszerek vizsgálatára rövid és hosszú távú kölcsönhatások figyelembevétele mellett. Kutatásaink mindezek tükrében interdiszciplináris jellegűek voltak, melyet jól tükröznek az általunk vizsgált szilárdtestfizikai, kvantumkémiai és statisztikus fizikai problémák. A numerikus vizsgálatok mellett számos esetben elvégeztük a modellek analitikus módszerekkel való közelítését is, így végső megállapításainkat az analitikus predikciók és numerikus számítások együtteseként alakítottuk ki. Fontos megjegyezni, hogy világszinten - legjobb tudomásom szerint - mindösszesen hat olyan DMRG kód létezik, mely hosszú távú kölcsönhatások kezelését is lehetővé teszi. | In the past for years we have developed the most general version of the densitymatrix-renormalization group (DMRG) method treating non local interactions and implemented various concepts of quantum information theory (QIT) in order to optimize the RG procedure. Quantum systems with short and long ranged interactions have been studied with great success. Models of such fields include mixed-stack compounds, quantum chemical systems, various spin systems in one dimensions. We have also analyzed quantum fluctuations between subsystems and defined an effective temperature to describe such effects. In addition, we have developed new methods based on the entanglement between subsystems to detect and locate quantum phase transition

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. In this article, we analyze the nature and the type of electron correlation effects that can be captured by wavefunctions restricted to electron-pair states. We focus on the Antisymmetric Product of 1-reference orbital Geminal (AP1roG) method combined with an orbital optimization protocol presented in [Phys. Rev. B, 89, 201106(R), 2014] whose performance is assessed against electronic structures obtained form DMRG reference data. Our numerical analysis covers 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 is 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 indicates that contributions from singly occupied states become important in the strong correlation regime which highlights the limitations of the AP1roG method. Furthermore, we examine the effect of orbital rotations within the AP1roG model on correlations between orbital pairs.Comment: 15 pages, 8 figure

Second-order Peierls transition in the spin-orbital Kumar-Heisenberg model

We add a Heisenberg interaction term $\propto\lambda$ in the one-dimensional SU(2)$\otimes$XY spin-orbital model introduced by B. Kumar. At $\lambda=0$ the spin and orbital degrees of freedom can be separated by a unitary transformation leading to an exact solution of the model. We show that a finite $\lambda>0$ leads to spontaneous dimerization of the system which in the thermodynamic limit becomes a smooth phase transition at $\lambda\to 0$, whereas it remains discontinuous within the first order perturbation approach. We present the behavior of the entanglement entropy, energy gap and dimerization order parameter in the limit of $\lambda\to 0$ confirming the critical behavior. Finally, we show the evidence of another phase transition in the Heisenberg limit, $\lambda\to\infty$, and give a qualitative analytical explanation of the observed dimerized states both in the limit of small and large $\lambda$.Comment: 11 pages, 12 figure

On the calculation of complete dissociation curves of closed-shell pseudo-onedimensional systems through the multireference method of increments

The Method of Increments (MoI) has been employed using a multireference approach to calculate the dissociation curve of beryllium ring-shaped clusters Be$_n$ of different sizes. Benchmarks obtained through different single and multireference methods including the ab initio Density Matrix Renormalization Group (DMRG) were used to verify the validity of the MoI truncation which showed a reliable behavior for the whole dissociation curve. Moreover we investigated the size dependence of the correlation energy at different distances in order to extrapolate the values for the periodic chain and to discuss the transition from a metal-like to a insulating-like behavior of the wave function through quantum chemical considerations

Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks

We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the density matrix renormalization group (DMRG) method have long been applied to such systems, the spatial topology of DMRG-based methods allows efficient optimizations to be carried out with respect to one spatial dimension only. Extending the matrix-product-state picture, we formulate a more general approach by allowing the local sites to be coupled to more than two neighboring auxiliary subspaces. Following Shi. et. al. [Phys. Rev. A, 74, 022320 (2006)], we treat a tree-like network ansatz with arbitrary coordination number z, where the z=2 case corresponds to the one-dimensional scheme. For this ansatz, the long-range correlation deviates from the mean-field value polynomially with distance, in contrast to the matrix-product ansatz, which deviates exponentially. The computational cost of the tree-tensor-network method is significantly smaller than that of previous DMRG-based attempts, which renormalize several blocks into a single block. In addition, we investigate the effect of unitary transformations on the local basis states and present a method for optimizing such transformations. For the 1-d interacting spinless fermion model, the optimized transformation interpolates smoothly between real space and momentum space. Calculations carried out on small quantum chemical systems support our approach

Predicting the FCI energy of large systems to chemical accuracy from restricted active space density matrix renormalization group calculations

We theoretically derive and validate with large scale simulations a remarkably accurate power law scaling of errors for the restricted active space density matrix renormalization group (DMRG-RAS) method [arXiv:2111.06665] 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 like the FeMoco with significantly lower computational demands than 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.Comment: 16 pages, 10 figure

Generic Mott-Hubbard phase diagram for extended Hubbard models without Umklapp scattering

We determine 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 cattering, 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 conclude that generic short-range interactions do not change the nature of the Mott transition qualitatively.Comment: 17 pages, 15 figure

H\"uckel--Hubbard-Ohno modeling of π\boldsymbol{\pi}-bonds in ethene and ethyne with application to trans-polyacetylene

Quantum chemistry calculations provide the potential energy between two carbon atoms in ethane (H$_3$C$-$CH$_3$), ethene (H$_2$C$=$CH$_2$), and ethyne (HC$\equiv$CH) as a function of the atomic distance. Based on the energy function for the $\sigma$-bond in ethane, $V_{\sigma}(r)$, we use the H\"uckel model with Hubbard--Ohno interaction for the $\pi$~electrons to describe the energies $V_{\sigma\pi}(r)$ and $V_{\sigma\pi\pi}(r)$ for the $\sigma\pi$ double bond in ethene and the $\sigma\pi\pi$ triple bond in ethyne, respectively. The fit of the force functions shows that the Peierls coupling can be estimated with some precision whereas the Hubbard-Ohno parameters are insignificant at the distances under consideration. We apply the H\"uckel-Hubbard-Ohno model to describe the bond lengths and the energies of elementary electronic excitations of trans-polyacetylene, (CH)$_n$, and adjust the $\sigma$-bond potential for conjugated polymers.Comment: 10 pages, 7 figures, 3 table