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Journal articles on the topic 'Density Matrix Renormalization Group (DMRG) algorithm'

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Published: 22 February 2023

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1

Schollwöck, Ulrich. "The density-matrix renormalization group: a short introduction." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, no. 1946 (July 13, 2011): 2643–61. http://dx.doi.org/10.1098/rsta.2010.0382.

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The density-matrix renormalization group (DMRG) method has established itself over the last decade as the leading method for the simulation of the statics and dynamics of one-dimensional strongly correlated quantum lattice systems. The DMRG is a method that shares features of a renormalization group procedure (which here generates a flow in the space of reduced density operators) and of a variational method that operates on a highly interesting class of quantum states, so-called matrix product states (MPSs). The DMRG method is presented here entirely in the MPS language. While the DMRG generally fails in larger two-dimensional systems, the MPS picture suggests a straightforward generalization to higher dimensions in the framework of tensor network states. The resulting algorithms, however, suffer from difficulties absent in one dimension, apart from a much more unfavourable efficiency, such that their ultimate success remains far from clear at the moment.

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SCHOLLWÖCK, ULRICH. "RECENT PROGRESS IN THE DENSITY-MATRIX RENORMALIZATION GROUP." International Journal of Modern Physics B 21, no. 13n14 (May 30, 2007): 2564–75. http://dx.doi.org/10.1142/s0217979207043890.

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Over the last decade, the density-matrix renormalization group (DMRG) has emerged as the most powerful method for the simulation of strongly correlated one-dimensional (1D) quantum systems. Input from quantum information has allowed to trace the method's performance to the entanglement properties of quantum states, revealing why it works so well in 1D and not so well in 2D; it has allowed to devise algorithms for time-dependent quantum systems and, by clarifying the link between DMRG and Wilson's numerical renormalization group (NRG), for quantum impurity systems.

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3

Devakul, Trithep, Vedika Khemani, Frank Pollmann, David A. Huse, and S. L. Sondhi. "Obtaining highly excited eigenstates of the localized XX chain via DMRG-X." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 375, no. 2108 (October 30, 2017): 20160431. http://dx.doi.org/10.1098/rsta.2016.0431.

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We benchmark a variant of the recently introduced density matrix renormalization group (DMRG)-X algorithm against exact results for the localized random field XX chain. We find that the eigenstates obtained via DMRG-X exhibit a highly accurate l-bit description for system sizes much bigger than the direct, many-body, exact diagonalization in the spin variables is able to access. We take advantage of the underlying free fermion description of the XX model to accurately test the strengths and limitations of this algorithm for large system sizes. We discuss the theoretical constraints on the performance of the algorithm from the entanglement properties of the eigenstates, and its actual performance at different values of disorder. A small but significant improvement to the algorithm is also presented, which helps significantly with convergence. We find that, at high entanglement, DMRG-X shows a bias towards eigenstates with low entanglement, but can be improved with increased bond dimension. This result suggests that one must be careful when applying the algorithm for interacting many-body localized spin models near a transition. This article is part of the themed issue ‘Breakdown of ergodicity in quantum systems: from solids to synthetic matter’.

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4

MARTÍN-DELGADO, MIGUEL A., and GERMÁN SIERRA. "ANALYTIC FORMULATIONS OF THE DENSITY MATRIX RENORMALIZATION GROUP." International Journal of Modern Physics A 11, no. 17 (July 10, 1996): 3145–74. http://dx.doi.org/10.1142/s0217751x96001516.

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We present two new analytic formulations of the density matrix renormalization group (DMRG) method. In these formulations we combine the block renormalization group (BRG) procedure with the variational and Fokker-Planck methods. The BRG method is used to reduce the lattice size while the latter are used to construct approximate target states to compute the block density matrix. We apply our DMRG methods to the Ising model in a transverse field (ITF model) and compute several of its critical properties, which are then compared with the old BRG results.

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MA, HAIBO, CHUNGEN LIU, and YUANSHENG JIANG. "BLOCK DENSITY MATRIX RENORMALIZATION GROUP WITH EFFECTIVE INTERACTIONS." Journal of Theoretical and Computational Chemistry 08, no. 05 (October 2009): 837–48. http://dx.doi.org/10.1142/s0219633609005064.

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Based on the contractor renormalization group (CORE) method and the density matrix renormalization group (DMRG) method, a new computational scheme, which is called the block density matrix renormalization group with effective interactions (BDMRG-EI), is proposed to deal with the numerical computation of quantum correlated systems. Different from the conventional CORE method in calculating the blocks and the fragments, where the DMRG method instead of the exact diagonalization is employed in BDMRG-EI, BDMRG-EI makes the calculations of larger blocks and fragments applicable. Integrating DMRG's advantage of high accuracy and CORE's advantage of low computational costs, BDMRG-EI can be widely used for the theoretical calculations of the ground state and low-lying excited states of large systems with simple or complicated connectivity. Test calculations on a 240 site one-dimensional chain and a double-layer polyacene oligomer containing 48 hexagons with the spin-1/2 Heisenberg Hamiltonian demonstrate the efficiency and potentiality of the method.

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6

Sharma, Prachi, Varinia Bernales, Stefan Knecht, Donald G. Truhlar, and Laura Gagliardi. "Density matrix renormalization group pair-density functional theory (DMRG-PDFT): singlet–triplet gaps in polyacenes and polyacetylenes." Chemical Science 10, no. 6 (2019): 1716–23. http://dx.doi.org/10.1039/c8sc03569e.

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7

Friesecke, Gero, and Benedikt R. Graswald. "Two-electron wavefunctions are matrix product states with bond dimension three." Journal of Mathematical Physics 63, no. 9 (September 1, 2022): 091901. http://dx.doi.org/10.1063/5.0072261.

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We prove the statement in the title for a suitable (wavefunction-dependent) choice of the underlying orbitals and show that 3 is optimal. Thus, for two-electron systems, the quantum chemistry density matrix renormalization group (QC-DMRG) method with bond dimension 3 combined with fermionic mode optimization exactly recovers the full configuration-interaction (FCI) energy.

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8

Akutsu, Noriko. "Density-matrix renormalization-group study for a two-dimensional lattice-gas on the Ih-ice prism surface." Journal of Physics: Conference Series 2207, no. 1 (March 1, 2022): 012036. http://dx.doi.org/10.1088/1742-6596/2207/1/012036.

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Abstract The phase transition temperature of a two-dimensional (2D) lattice-gas model for the Ih-ice prism surface ((1010) surface on a wurtzite crystal structure) is calculated using the density-matrix renormalization-group (DMRG) method. Since the unit cell on the prism surface contains four atoms without inversion symmetry, the asymmetric version of the DMRG is required to obtain a precise value of the phase transition temperature. The effective bond energy on the surface is obtained by comparing the phase transition temperature with the faceting transition temperature observed for a real ice crystal. Using the effective bond energy, the temperature dependence of the coverage on sub-lattice sites is calculated. The surface tensions on the (0001) surface (the basal plane) and the prism plane are also calculated. The surface tension of the basal plane for the ice/water interface agrees well with the recent calculated values obtained by large scale molecular dynamics.

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9

LI, SHUHUA, JING MA, and YUANSHENG JIANG. "HEISENBERG MODEL AND ITS APPLICATIONS TO π-CONJUGATED SYSTEMS." Journal of Theoretical and Computational Chemistry 01, no. 02 (October 2002): 351–71. http://dx.doi.org/10.1142/s0219633602000270.

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This paper briefly reviewed the Heisenberg model and its improvement by including higher order corrections, and their applications to bond lengths, stability, and reactivity of non-benzenoids and the low-lying excitation spectra of conjugated systems. Two efficient computational methods, the Lanczos method and the density-matrix renormalization group (DMRG), for exactly and approximately solving various Heisenberg models, respectively, were briefly introduced.

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10

McCulloch, I. P., and M. Gulácsi. "The non-Abelian density matrix renormalization group algorithm." Europhysics Letters (EPL) 57, no. 6 (March 2002): 852–58. http://dx.doi.org/10.1209/epl/i2002-00393-0.

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11

Liu, Fengyi, Yuki Kurashige, Takeshi Yanai, and Keiji Morokuma. "Multireference Ab Initio Density Matrix Renormalization Group (DMRG)-CASSCF and DMRG-CASPT2 Study on the Photochromic Ring Opening of Spiropyran." Journal of Chemical Theory and Computation 9, no. 10 (September 10, 2013): 4462–69. http://dx.doi.org/10.1021/ct400707k.

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12

Henelius, Patrik. "Two-dimensional infinite-system density-matrix renormalization-group algorithm." Physical Review B 60, no. 13 (October 1, 1999): 9561–65. http://dx.doi.org/10.1103/physrevb.60.9561.

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13

Mcculloch, Ian P., and Miklós Gulácsi. "Total spin in the density matrix renormalization group algorithm." Philosophical Magazine Letters 81, no. 6 (June 2001): 447–53. http://dx.doi.org/10.1080/09500830110040009.

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14

Marti, Konrad Heinrich, and Markus Reiher. "The Density Matrix Renormalization Group Algorithm in Quantum Chemistry." Zeitschrift für Physikalische Chemie 224, no. 3-4 (April 2010): 583–99. http://dx.doi.org/10.1524/zpch.2010.6125.

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15

Maiellaro, Alfonso, Fabrizio Illuminati, and Roberta Citro. "Topological Phases of an Interacting Majorana Benalcazar–Bernevig–Hughes Model." Condensed Matter 7, no. 1 (March 4, 2022): 26. http://dx.doi.org/10.3390/condmat7010026.

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We study the effects of Coulomb repulsive interactions on a Majorana Benalcazar–Bernevig–Huges (MBBH) model. The MBBH model belongs to the class of second-order topological superconductors (HOTSC2), featuring robust Majorana corner modes. We consider an interacting strip of four chains of length L and perform a density matrix renormalization group (DMRG) numerical simulation based on a tensor-network approach. Study of the non-local fermionic correlations and the degenerate entanglement spectrum indicates that the topological phases are robust in the presence of interactions, even in the strongly interacting regime.

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16

Ueda, Hiroshi. "Infinite-Size Density Matrix Renormalization Group with Parallel Hida’s Algorithm." Journal of the Physical Society of Japan 87, no. 7 (July 15, 2018): 074005. http://dx.doi.org/10.7566/jpsj.87.074005.

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17

Chan, Garnet Kin-Lic. "An algorithm for large scale density matrix renormalization group calculations." Journal of Chemical Physics 120, no. 7 (February 15, 2004): 3172–78. http://dx.doi.org/10.1063/1.1638734.

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18

Ma, Yingjin, Jing Wen, and Haibo Ma. "Density-matrix renormalization group algorithm with multi-level active space." Journal of Chemical Physics 143, no. 3 (July 21, 2015): 034105. http://dx.doi.org/10.1063/1.4926833.

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19

Nakatani, Naoki, and Sheng Guo. "Density matrix renormalization group (DMRG) method as a common tool for large active-space CASSCF/CASPT2 calculations." Journal of Chemical Physics 146, no. 9 (March 7, 2017): 094102. http://dx.doi.org/10.1063/1.4976644.

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20

CHOUDHURY, P., and A. N. DAS. "SMALL-TO-LARGE POLARON CROSSOVER IN ONE DIMENSION USING VARIATIONAL PHONON-AVERAGING TECHNIQUES." International Journal of Modern Physics B 15, no. 13 (May 30, 2001): 1923–37. http://dx.doi.org/10.1142/s0217979201004939.

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The ground-state properties of polarons in a one-dimensional chain is studied analytically within the modified Lang–Firsov (MLF) transformation using various phonon-averaging techniques. The object of the work is to examine how the analytical approaches may be improved to give rise to the real picture of polaronic properties as predicted by extensive numerical studies. The results are compared with those obtained from numerical analyses using the density matrix renormalization group (DMRG) and other variational techniques. It is observed that our results agree well with the numerical results particularly in the low and intermediate range of phonon coupling.

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21

KUMAR, MANORANJAN, SUJIT SARKAR, and S. RAMASESHA. "SUPERSOLID PHASE IN ONE-DIMENSIONAL BOSE–HUBBARD MODEL WITH EXTENDED RANGE INTERACTIONS: DMRG AND FIELD THEORETIC STUDY AT DIFFERENT DENSITIES." International Journal of Modern Physics B 25, no. 01 (January 10, 2011): 159–69. http://dx.doi.org/10.1142/s0217979211057700.

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We use the Density Matrix Renormalization Group and the Abelian bosonization method to study the effect of density on quantum phases of one-dimensional extended Bose–Hubbard model. We predict the existence of supersolid phase and also other quantum phases for this system. We have analyzed the role of extended range interaction parameters on solitonic phase near half-filling. We discuss the effects of dimerization in nearest neighbor hopping and interaction as well as next nearest neighbor interaction on the plateau phase at half-filling.

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22

Guo, Sheng, Zhendong Li, and Garnet Kin-Lic Chan. "A Perturbative Density Matrix Renormalization Group Algorithm for Large Active Spaces." Journal of Chemical Theory and Computation 14, no. 8 (June 21, 2018): 4063–71. http://dx.doi.org/10.1021/acs.jctc.8b00273.

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23

Allerdt, Andrew, Hasnain Hafiz, Bernardo Barbiellini, Arun Bansil, and Adrian E. Feiguin. "Many-Body Effects in FeN4 Center Embedded in Graphene." Applied Sciences 10, no. 7 (April 7, 2020): 2542. http://dx.doi.org/10.3390/app10072542.

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We introduce a computational approach to study porphyrin-like transition metal complexes, bridging density functional theory and exact many-body techniques, such as the density matrix renormalization group (DMRG). We first derive a multi-orbital Anderson impurity Hamiltonian starting from first principles considerations that qualitatively reproduce generalized gradient approximation (GGA)+U results when ignoring inter-orbital Coulomb repulsion U ′ and Hund exchange J. An exact canonical transformation is used to reduce the dimensionality of the problem and make it amenable to DMRG calculations, including all many-body terms (both intra- and inter-orbital), which are treated in a numerically exact way. We apply this technique to FeN 4 centers in graphene and show that the inclusion of these terms has dramatic effects: as the iron orbitals become single occupied due to the Coulomb repulsion, the inter-orbital interaction further reduces the occupation, yielding a non-monotonic behavior of the magnetic moment as a function of the interactions, with maximum polarization only in a small window at intermediate values of the parameters. Furthermore, U ′ changes the relative position of the peaks in the density of states, particularly on the iron d z 2 orbital, which is expected to affect the binding of ligands greatly.

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24

Parker, Shane M., and Toru Shiozaki. "Communication: Active space decomposition with multiple sites: Density matrix renormalization group algorithm." Journal of Chemical Physics 141, no. 21 (December 3, 2014): 211102. http://dx.doi.org/10.1063/1.4902991.

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25

Mcculloch, I. P., A. R. Bishop, and M. Gulacsi. "Density matrix renormalization group algorithm and the two-dimensional t-J model." Philosophical Magazine B 81, no. 10 (October 2001): 1603–13. http://dx.doi.org/10.1080/13642810110051782.

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26

Weyrauch, M., and M. V. Rakov. "Efficient MPS Algorithm for Periodic Boundary Conditions and Applications Section: Solid matter Original Author's Text: English Abstract: We present the implementation of an efficient algorithm for the calculation of the spectrum of one-dimensional quantum systems with periodic boundary conditions. This algorithm is based on a matrix product representation for quantum states (MPS) and a similar representation for Hamiltonians and other operators (MPO). It is significantly more efficient for systems of about 100 sites and more than for small quantum systems. We apply the formalism to calculate the ground state and the first excited state of a spin-1 Heisenberg ring and deduce the size of a Haldane gap. The results are compared to previous high-precision DMRG calculations. Furthermore, we study the spin-1 systems with a biquadratic nearest-neighbor interaction and show the first results of an application to a mesoscopic Hubbard ring of spinless fermions, which carries a persistent current. Key words: matrix product representation for quantum states (MPS) and Hamiltonians (MPO), spin-1 Heisenberg ring, density matrix renormalization group (DMRG). List Issues Configure." Ukrainian Journal of Physics 58, no. 7 (July 2013): 657–65. http://dx.doi.org/10.15407/ujpe58.07.0657.

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27

Moritz, Gerrit, Bernd Artur Hess, and Markus Reiher. "Convergence behavior of the density-matrix renormalization group algorithm for optimized orbital orderings." Journal of Chemical Physics 122, no. 2 (January 8, 2005): 024107. http://dx.doi.org/10.1063/1.1824891.

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28

Tamascelli, Dario. "Excitation Dynamics in Chain-Mapped Environments." Entropy 22, no. 11 (November 19, 2020): 1320. http://dx.doi.org/10.3390/e22111320.

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The chain mapping of structured environments is a most powerful tool for the simulation of open quantum system dynamics. Once the environmental bosonic or fermionic degrees of freedom are unitarily rearranged into a one dimensional structure, the full power of Density Matrix Renormalization Group (DMRG) can be exploited. Beside resulting in efficient and numerically exact simulations of open quantum systems dynamics, chain mapping provides an unique perspective on the environment: the interaction between the system and the environment creates perturbations that travel along the one dimensional environment at a finite speed, thus providing a natural notion of light-, or causal-, cone. In this work we investigate the transport of excitations in a chain-mapped bosonic environment. In particular, we explore the relation between the environmental spectral density shape, parameters and temperature, and the dynamics of excitations along the corresponding linear chains of quantum harmonic oscillators. Our analysis unveils fundamental features of the environment evolution, such as localization, percolation and the onset of stationary currents.

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29

Deng, Dan, Bingbing Suo, and Wenli Zou. "New Light on an Old Story: Breaking Kasha’s Rule in Phosphorescence Mechanism of Organic Boron Compounds and Molecule Design." International Journal of Molecular Sciences 23, no. 2 (January 14, 2022): 876. http://dx.doi.org/10.3390/ijms23020876.

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In this work, the phosphorescence mechanism of (E)-3-(((4-nitrophenyl)imino)methyl)-2H-thiochroman-4-olate-BF2 compound (S-BF2) is investigated theoretically. The phosphorescence of S-BF2 has been reassigned to the second triplet state (T2) by the density matrix renormalization group (DMRG) method combined with the multi-configurational pair density functional theory (MCPDFT) to approach the limit of theoretical accuracy. The calculated radiative and non-radiative rate constants support the breakdown of Kasha’s rule further. Our conclusion contradicts previous reports that phosphorescence comes from the first triplet state (T1). Based on the revised phosphorescence mechanism, we have purposefully designed some novel compounds in theory to enhance the phosphorescence efficiency from T2 by replacing substitute groups in S-BF2. Overall, both S-BF2 and newly designed high-efficiency molecules exhibit anti-Kasha T2 phosphorescence instead of the conventional T1 emission. This work provides a useful guidance for future design of high-efficiency green-emitting phosphors.

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30

Roy, Dipayan, R. Torsten Clay, and Sumit Mazumdar. "Absence of Superconductivity in the Hubbard Dimer Model for κ-(BEDT-TTF)2X." Crystals 11, no. 6 (May 22, 2021): 580. http://dx.doi.org/10.3390/cryst11060580.

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In the most studied family of organic superconductors κ-(BEDT-TTF)2X, the BEDT-TTF molecules that make up the conducting planes are coupled as dimers. For some anions X, an antiferromagnetic insulator is found at low temperatures adjacent to superconductivity. With an average of one hole carrier per dimer, the BEDT-TTF band is effectively 12-filled. Numerous theories have suggested that fluctuations of the magnetic order can drive superconducting pairing in these models, even as direct calculations of superconducting pairing in monomer 12-filled band models find no superconductivity. Here, we present accurate zero-temperature Density Matrix Renormalization Group (DMRG) calculations of a dimerized lattice with one hole per dimer. While we do find an antiferromagnetic state in our results, we find no evidence for superconducting pairing. This further demonstrates that magnetic fluctuations in the effective 12-filled band approach do not drive superconductivity in these and related materials.

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31

Patel, Niravkumar D., and Nandini Trivedi. "Magnetic field-induced intermediate quantum spin liquid with a spinon Fermi surface." Proceedings of the National Academy of Sciences 116, no. 25 (May 31, 2019): 12199–203. http://dx.doi.org/10.1073/pnas.1821406116.

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The Kitaev model with an applied magnetic field in the H∥[111] direction shows two transitions: from a nonabelian gapped quantum spin liquid (QSL) to a gapless QSL at Hc1≃0.2K and a second transition at a higher field Hc2≃0.35K to a gapped partially polarized phase, where K is the strength of the Kitaev exchange interaction. We identify the intermediate phase to be a gapless U(1) QSL and determine the spin structure function S(k) and the Fermi surface ϵFS(k) of the gapless spinons using the density matrix renormalization group (DMRG) method for large honeycomb clusters. Further calculations of static spin-spin correlations, magnetization, spin susceptibility, and finite temperature-specific heat and entropy corroborate the gapped and gapless nature of the different field-dependent phases. In the intermediate phase, the spin-spin correlations decay as a power law with distance, indicative of a gapless phase.

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32

Nemes, Csaba, Gergely Barcza, Zoltán Nagy, Örs Legeza, and Péter Szolgay. "The density matrix renormalization group algorithm on kilo-processor architectures: Implementation and trade-offs." Computer Physics Communications 185, no. 6 (June 2014): 1570–81. http://dx.doi.org/10.1016/j.cpc.2014.02.021.

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33

Stokes, James, and John Terilla. "Probabilistic Modeling with Matrix Product States." Entropy 21, no. 12 (December 17, 2019): 1236. http://dx.doi.org/10.3390/e21121236.

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Inspired by the possibility that generative models based on quantum circuits can provide a useful inductive bias for sequence modeling tasks, we propose an efficient training algorithm for a subset of classically simulable quantum circuit models. The gradient-free algorithm, presented as a sequence of exactly solvable effective models, is a modification of the density matrix renormalization group procedure adapted for learning a probability distribution. The conclusion that circuit-based models offer a useful inductive bias for classical datasets is supported by experimental results on the parity learning problem.

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34

BOGNER, THORSTEN. "General variational model reduction applied to incompressible viscous flows." Journal of Fluid Mechanics 617 (December 25, 2008): 31–50. http://dx.doi.org/10.1017/s002211200800387x.

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In this paper, a method is introduced that allows calculation of an approximate proper orthogonal decomposition (POD) without the need to perform a simulation of the full dynamical system. Our approach is based on an application of the density matrix renormalization group (DMRG) to nonlinear dynamical systems, but has no explicit restriction on the spatial dimension of the model system. The method is not restricted to fluid dynamics. The applicability is exemplified on the incompressible Navier–Stokes equation in two spatial dimensions. Merging of two equal-signed vortices with periodic boundary conditions is considered for low Reynolds numbers Re≤800 using a spectral method. We compare the accuracy of a reduced model, obtained by our method, with that of a reduced model obtained by standard POD. To this end, error functionals for the reductions are evaluated. It is observed that the proposed method is able to find a reduced system that yields comparable or even superior accuracy with respect to standard POD method results.

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Ma, Yong-Jun, Jia-Xiang Wang, Xin-Ye Xu, Qi Wei, and Kais Sabre. "Error Analysis of the Density-Matrix Renormalization Group Algorithm for a Chain of Harmonic Oscillators." Chinese Physics Letters 31, no. 6 (June 2014): 060501. http://dx.doi.org/10.1088/0256-307x/31/6/060501.

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36

Sharma, Sandeep. "A general non-Abelian density matrix renormalization group algorithm with application to the C2 dimer." Journal of Chemical Physics 142, no. 2 (January 14, 2015): 024107. http://dx.doi.org/10.1063/1.4905237.

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37

ten Brink, M., S. Gräber, M. Hopjan, D. Jansen, J. Stolpp, F. Heidrich-Meisner, and P. E. Blöchl. "Real-time non-adiabatic dynamics in the one-dimensional Holstein model: Trajectory-based vs exact methods." Journal of Chemical Physics 156, no. 23 (June 21, 2022): 234109. http://dx.doi.org/10.1063/5.0092063.

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We benchmark a set of quantum-chemistry methods, including multitrajectory Ehrenfest, fewest-switches surface-hopping, and multiconfigurational-Ehrenfest dynamics, against exact quantum-many-body techniques by studying real-time dynamics in the Holstein model. This is a paradigmatic model in condensed matter theory incorporating a local coupling of electrons to Einstein phonons. For the two-site and three-site Holstein model, we discuss the exact and quantum-chemistry methods in terms of the Born–Huang formalism, covering different initial states, which either start on a single Born–Oppenheimer surface, or with the electron localized to a single site. For extended systems with up to 51 sites, we address both the physics of single Holstein polarons and the dynamics of charge-density waves at finite electron densities. For these extended systems, we compare the quantum-chemistry methods to exact dynamics obtained from time-dependent density matrix renormalization group calculations with local basis optimization (DMRG-LBO). We observe that the multitrajectory Ehrenfest method, in general, only captures the ultrashort time dynamics accurately. In contrast, the surface-hopping method with suitable corrections provides a much better description of the long-time behavior but struggles with the short-time description of coherences between different Born–Oppenheimer states. We show that the multiconfigurational Ehrenfest method yields a significant improvement over the multitrajectory Ehrenfest method and can be converged to the exact results in small systems with moderate computational efforts. We further observe that for extended systems, this convergence is slower with respect to the number of configurations. Our benchmark study demonstrates that DMRG-LBO is a useful tool for assessing the quality of the quantum-chemistry methods.

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Chan, Garnet Kin-Lic, and Martin Head-Gordon. "Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group." Journal of Chemical Physics 116, no. 11 (March 15, 2002): 4462–76. http://dx.doi.org/10.1063/1.1449459.

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39

Guo, Sheng, Zhendong Li, and Garnet Kin-Lic Chan. "Communication: An efficient stochastic algorithm for the perturbative density matrix renormalization group in large active spaces." Journal of Chemical Physics 148, no. 22 (June 14, 2018): 221104. http://dx.doi.org/10.1063/1.5031140.

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40

Huang, Chen-How, Masaki Tezuka, and Miguel A. Cazalilla. "Topological Lifshitz transitions, orbital currents, and interactions in low-dimensional Fermi gases in synthetic gauge fields." New Journal of Physics 24, no. 3 (March 1, 2022): 033043. http://dx.doi.org/10.1088/1367-2630/ac5a87.

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Abstract Low-dimensional systems of interacting fermions in a synthetic gauge field have been experimentally realized using two-component ultra-cold Fermi gases in optical lattices. Using a two-leg ladder model that is relevant to these experiments, we have studied the signatures of topological Lifshitz transitions and the effects of the inter-species interaction U on the gauge-invariant orbital current in the regime of large intra-leg hopping Ω. Focusing on non-insulating regimes, we have carried out numerically exact density-matrix renormalization-group (DMRG) calculations to compute the orbital current at fixed particle number as a function of the interaction strength and the synthetic gauge flux per plaquette. Signatures of topological Lifshitz transitions where the number Fermi points changes are found to persist even in the presence of very strong repulsive interactions. This numerical observation suggests that the orbital current can be computed from an appropriately renormalized mean-field band structure, which is also described here. Quantitative agreement between the mean-field and the DMRG results in the intermediate interaction regime where U ≲ Ω is demonstrated. We also have observed that interactions can change the sign of the current susceptibility at zero field and induce Lifshitz transitions between two metallic phases, which is also captured by the mean-field theory. Correlation effects beyond mean-field theory in the oscillations of the local inter-leg current are also reported. We argue that the observed robustness against interactions makes the orbital current a good indicator of the topological Lifshitz transitions.

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41

Fazzini, Serena, and Arianna Montorsi. "Hidden Charge Orders in Low-Dimensional Mott Insulators." Applied Sciences 9, no. 4 (February 22, 2019): 784. http://dx.doi.org/10.3390/app9040784.

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The opening of a charge gap driven by interaction is a fingerprint of the transition to a Mott insulating phase. In strongly correlated low-dimensional quantum systems, it can be associated to the ordering of hidden non-local operators. For Fermionic 1D models, in the presence of spin–charge separation and short-ranged interaction, a bosonization analysis proves that such operators are the parity and/or string charge operators. In fact, a finite fractional non-local parity charge order is also capable of characterizing some two-dimensional Mott insulators, in both the Fermionic and the bosonic cases. When string charge order takes place in 1D, degenerate edge modes with fractional charge appear, peculiar of a topological insulator. In this article, we review the above framework, and we test it to investigate through density-matrix-renormalization-group (DMRG) numerical analysis the robustness of both hidden orders at half-filling in the 1D Fermionic Hubbard model extended with long range density-density interaction. The preliminary results obtained at finite size including several neighbors in the case of dipolar, screened and unscreened repulsive Coulomb interactions, confirm the phase diagram of the standard extended Hubbard model. Besides the trivial Mott phase, the bond ordered and charge density wave insulating phases are also not destroyed by longer ranged interaction, and still manifest hidden non-local orders.

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42

Bursill, R. J. "Density-matrix renormalization-group algorithm for quantum lattice systems with a large number of states per site." Physical Review B 60, no. 3 (July 15, 1999): 1643–49. http://dx.doi.org/10.1103/physrevb.60.1643.

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43

Nakatani, Naoki, and Garnet Kin-Lic Chan. "Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix renormalization group algorithm." Journal of Chemical Physics 138, no. 13 (April 7, 2013): 134113. http://dx.doi.org/10.1063/1.4798639.

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44

Mainali, Samrit, Fabien Gatti, Dmitri Iouchtchenko, Pierre-Nicholas Roy, and Hans-Dieter Meyer. "Comparison of the multi-layer multi-configuration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalization group (DMRG) for ground state properties of linear rotor chains." Journal of Chemical Physics 154, no. 17 (May 7, 2021): 174106. http://dx.doi.org/10.1063/5.0047090.

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45

Yamada, Susumu, Masahiko Okumura, and Masahiko Machida. "Direct Extension of Density-Matrix Renormalization Group to Two-Dimensional Quantum Lattice Systems: Studies of Parallel Algorithm, Accuracy, and Performance." Journal of the Physical Society of Japan 78, no. 9 (September 15, 2009): 094004. http://dx.doi.org/10.1143/jpsj.78.094004.

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46

Li, Zhendong. "Time-reversal symmetry adaptation in relativistic density matrix renormalization group algorithm." Journal of Chemical Physics, January 11, 2023. http://dx.doi.org/10.1063/5.0127621.

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In the nonrelativistic Schr\"{o}dinger equation, the total spin $S$ and spin projection $M$ are good quantum numbers. In contrast, spin symmetry is lost in the presence of spin-dependent interactions such as spin-orbit couplings in relativistic Hamiltonians. Therefore, relativistic density matrix renormalization group algorithm (R-DMRG) only employing particle number symmetry is much more expensive than nonrelativistic DMRG. Besides, artificial breaking of Kramers degeneracy can happen in the treatment of systems with odd number of electrons. To overcome these issues, we propose time-reversal symmetry adaptation for R-DMRG. Since the time-reversal operator is antiunitary, this cannot be simply achieved in the usual way. We introduce a time-reversal symmetry-adapted renormalized basis and present strategies to maintain the structure of basis functions during the sweep optimization. With time-reversal symmetry adaptation, only half of the renormalized operators are needed and the computational costs of Hamiltonian-wavefunction multiplication and renormalization are reduced by half. The present construction of time-reversal symmetry-adapted basis also directly applies to other tensor network states without loops.

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47

Tian, Yingqi, and Haibo Ma. "High-Performance Computing for Density Matrix Renormalization Group." Current Chinese Science 03 (November 25, 2022). http://dx.doi.org/10.2174/2210298103666221125162959.

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Abstract: In the last decades, many algorithms have been developed to use high-performance computing (HPC) techniques to accelerate the density matrix renormalization group (DMRG) method, an effective method for solving large active space strong correlation problems. In this article, the previous DMRG parallelization algorithms at different levels of the parallelism are introduced. The heterogeneous computing acceleration methods and the mixed-precision implementation are also presented and discussed. This mini-review concludes with some summary and prospects for future works.

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48

Li, Weitang, Jiajun Ren, Hengrui Yang, and Zhigang Shuai. "On the Fly Swapping Algorithm for Ordering of Degrees of Freedom in Density Matrix Renormalization Group." Journal of Physics: Condensed Matter, April 4, 2022. http://dx.doi.org/10.1088/1361-648x/ac640e.

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Abstract Abstract: Density matrix renormalization group (DMRG) and its time-dependent variants have found widespread applications in quantum chemistry, including ab initio electronic structure of complex bio-molecules, spectroscopy for molecular aggregates, and charge transport in bulk organic semiconductors. The underlying wavefunction ansatz for DMRG, matrix product state (MPS), requires mapping degrees of freedom (DOF) into a one-dimensional topology. DOF ordering becomes a crucial factor for DMRG accuracy. In this work, we propose swapping neighbouring DOFs during the DMRG sweeps for DOF ordering, which we term ``on the fly swapping'' (OFS) algorithm. We show that OFS is universal for both static and time-dependent DMRG with minimum computational overhead. Examples are given for one dimensional antiferromagnetic Heisenberg model, ab initio electronic structure of N2 molecule, and the S1/S2 internal conversion dynamics of pyrazine molecule. It is found that OFS can indeed improve accuracy by finding better DOF ordering in all cases.

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49

Araz, Jack Y., and Michael Spannowsky. "Quantum-inspired event reconstruction with Tensor Networks: Matrix Product States." Journal of High Energy Physics 2021, no. 8 (August 2021). http://dx.doi.org/10.1007/jhep08(2021)112.

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Abstract Tensor Networks are non-trivial representations of high-dimensional tensors, originally designed to describe quantum many-body systems. We show that Tensor Networks are ideal vehicles to connect quantum mechanical concepts to machine learning techniques, thereby facilitating an improved interpretability of neural networks. This study presents the discrimination of top quark signal over QCD background processes using a Matrix Product State classifier. We show that entanglement entropy can be used to interpret what a network learns, which can be used to reduce the complexity of the network and feature space without loss of generality or performance. For the optimisation of the network, we compare the Density Matrix Renormalization Group (DMRG) algorithm to stochastic gradient descent (SGD) and propose a joined training algorithm to harness the explainability of DMRG with the efficiency of SGD.

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50

Iino, Tsubasa, Toru Shiozaki, and Takeshi Yanai. "Algorithm for Analytic Nuclear Energy Gradients of State Averaged DMRG-CASSCF Theory with Newly Derived Coupled-Perturbed Equations." Journal of Chemical Physics, December 16, 2022. http://dx.doi.org/10.1063/5.0130636.

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We present an algorithm for evaluating analytic nuclear energy gradients of the state-averaged density matrix renormalization group complete-active-space self-consistent field (SA-DMRG-CASSCF) theory, based on the newly derived coupled-perturbed (CP) DMRG-CASSCF equations. The Lagrangian for the conventional SA-CASSCF analytic gradient theory is extended to the SA-DMRG-CASSCF variant that can fully consider a whole set of constraints on the parameters of multi-root canonical matrix product states (MPSs) formed at all the DMRG block configurations. An efficient algorithm to solve the CP-DMRG-CASSCF equations for determining the multipliers was developed. The complexity of the resultant analytic gradient algorithm is overall the same as that of the unperturbed SA-DMRG-CASSCF algorithm. In addition, a reduced-scaling approach was developed to directly compute the SA reduced density matrices (SA-RDMs) and their perturbed ones without calculating separate state-specific RDMs. As part of our implementation scheme, we neglect the term associated with the constraint on the active orbitals in terms of the active-active rotation in the Lagrangian. Thus, errors from the true analytic gradients may be caused in this scheme. The proposed gradient algorithm was tested with the spin-adapted implementation by checking how accurately the computed analytic energy gradients reproduce numerical gradients of the SA-DMRG-CASSCF energies using a common number of renormalized basis. The illustrative applications show that the errors are sufficiently small when using a typical number of the renormalized basis, which is required to attain adequate accuracy in DMRG's total energies.

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