Density Matrix Renormalization Group Research Articles

Chelate Complexes of 3d Transition Metal Ions─A Challenge for Electronic-Structure Methods?

Different electronic-structure methods were assessed for their ability to predict two important properties of the industrially relevant chelating agent nitrilotriacetic acid (NTA): its selectivity with respect to six different first-row transition metal ions and the spin-state energetics of its complex with Fe(III). The investigated methods encompassed density functional theory (DFT), the random phase approximation (RPA), coupled cluster (CC) theory, and the auxiliary-field quantum Monte Carlo (AFQMC) method, as well as the complete active space self-consistent field (CASSCF) method and the respective on-top methods: second-order N-electron valence state perturbation theory (NEVPT2) and multiconfiguration pair-density functional theory (MC-PDFT). Different strategies for selecting active spaces were explored, and the density matrix renormalization group (DMRG) approach was used to solve the largest active spaces. Despite somewhat ambiguous multi-reference diagnostics, most methods gave relatively good agreement with experimental data for the chemical reactions connected to the selectivity, which only involved transition-metal complexes in their high-spin state. CC methods yielded the highest accuracy followed by range-separated DFT and AFQMC. We discussed in detail that even higher accuracies can be obtained with NEVPT2, under the prerequisite that consistent active spaces along the entire chemical reaction can be selected, which was not the case for reactions involving Fe(III). A bigger challenge for electronic-structure methods was the prediction of the spin-state energetics, which additionally involved lower spin states that exhibited larger multi-reference diagnostics. Conceptually different, typically accurate methods ranging from CC theory via DMRG-NEVPT2 in combination with large active spaces to AFQMC agreed well that the high-spin state is energetically significantly favored over the other spin states. This was in contrast to most DFT functionals and RPA which yielded a smaller stabilization and some common DFT functionals and MC-PDFT even predicting the low-spin state to be energetically most favorable.

Detecting hidden order in fractional Chern insulators

Topological phase transitions go beyond Ginzburg and Landau's paradigm of spontaneous symmetry breaking and occur without an associated local order parameter. Instead, such transitions can be characterized by the emergence of nonlocal order parameters, which require measurements on extensively many particles simultaneously—an impossible venture in real materials. On the other hand, quantum simulators have demonstrated such measurements, making them prime candidates for experimental confirmation of nonlocal topological order. Here, building upon the recent advances in preparing few-particle fractional Chern insulators using ultracold atoms and photons, we propose a realistic scheme for detecting the hidden off-diagonal long-range order (HODLRO) characterizing Laughlin states. Furthermore, we demonstrate the existence of this hidden order in fractional Chern insulators, specifically for the ν=1/2-Laughlin state in the isotropic Hofstadter-Bose-Hubbard model. This is achieved by large-scale numerical density matrix renormalization group (DMRG) simulations based on matrix product states, for which we formulate an efficient sampling procedure providing direct access to HODLRO in close analogy to the proposed experimental scheme. We confirm the characteristic power-law scaling of HODLRO, with an exponent 1/ν=2, and show that its detection requires only a few thousand snapshots. This makes our scheme realistically achievable with current technology and paves the way for further analysis of nonlocal topological orders, e.g., in topological states with non-Abelian anyonic excitations. Published by the American Physical Society 2024

Two-dimensional quantum lattice models via mode optimized hybrid CPU-GPU density matrix renormalization group method

Two-dimensional quantum lattice models via mode optimized hybrid CPU-GPU density matrix renormalization group method

A numerical study of bounds in the correlations of fractional quantum Hall states

We numerically compute the guiding center static structure factor \bar S(k)S‾(k) of various fractional quantum Hall (FQH) states to \mathcal{O}(k\ell)^6𝒪(kℓ)6 where kk is the wavenumber and \ellℓ is the magnetic length. Employing density matrix renormalization group on an infinite cylinder of circumference L_yLy, we study the two-dimensional limit using L_y/\xi \gg 1Ly/ξ≫1, where \xiξ is the correlation length. The main findings of our work are: 1) the ground states that deviate away from the ideal conformal block wavefunctions, do not saturate the Haldane bound, and 2) the coefficient of O(k\ell)^6O(kℓ)6 term appears to be bounded above by a value predicted by field theories proposed in the literature. The first finding implies that the graviton mode is not maximally chiral for experimentally relevant FQH states.

Extracting correlation length in Mott insulators by strong-field driving

The breakdown of a Mott-insulator when subjected to intense laser fields is characterized by the formation of doublon-hole pairs. This breakdown is furthermore evidenced by the production of high harmonics that can be experimentally measured. Here, we present an approach for extracting the doublon-hole correlation length of a Mott insulator. The method is based on a dynamical calculation of the Mott insulator’s rate of charge production in response to an applied strong-field laser pulse. We find that coupling the Mott insulator to a metal drastically increases the correlation length, in support of our recent hypothesis [Phys. Rev. B 108, 144434 (2023)2469-995010.1103/PhysRevB.108.144434] that coupling to a metal enhances the charge fluctuations in the insulator. We confirm our conclusions using density matrix renormalization group (DMRG) calculations. The proposed method can be applied to experimentally measured observables, such as differential reflectivity or the high harmonic generation (HHG) spectrum to extract doublon-hole correlation length.

Theoretical investigation of distal charge separation in a perylenediimide trimer

An exciton–phonon (ex–ph) model based on our recently developed block interaction product basis framework is introduced to simulate the distal charge separation (CS) process in aggregated perylenediimide (PDI) trimer incorporating the quantum dynamic method, i.e., the time-dependent density matrix renormalization group. The electronic Hamiltonian in the ex–ph model is represented by nine constructed diabatic states, which include three local excited (LE) states and six charge transfer (CT) states from both the neighboring and distal chromophores. These diabatic states are automatically generated from the direct products of the leading localized neutral or ionic states of each chromophore’s reduced density matrix, which are obtained from ab initio quantum chemical calculation of the subsystem consisting of the targeted chromophore and its nearest neighbors, thus considering the interaction of the adjacent environment. In order to quantum-dynamically simulate the distal CS process with massive coupled vibrational modes in molecular aggregates, we used our recently proposed hierarchical mapping approach to renormalize these modes and truncate those vibrational modes that are not effectively coupled with electronic states accordingly. The simulation result demonstrates that the formation of the distal CS process undergoes an intermediate state of adjacent CT, i.e., starts from the LE states, passes through an adjacent CT state to generate the intermediates (∼200 fs), and then formalizes the targeted distal CS via further charge transference (∼1 ps). This finding agrees well with the results observed in the experiment, indicating that our scheme is capable of quantitatively investigating the CS process in a realistic aggregated PDI trimer and can also be potentially applied to exploring CS and other photoinduced processes in larger systems.

Magnetic transition of ytterbium atoms confined in optical superlattice with local ferromagnetic interaction

We used the density matrix renormalization group to study the ground state of ytterbium atoms (171Yb) for the Hund lattice model, where the delocalized atoms are confined in a onc-dimensional optical superlattice and his number is one third of the lattice sites. We found a paramagnetic-ferromagnetic quantum phase transition for any value of the potential strength. The local critical ferromagnetic coupling decreases as the superlattice potential increases.

Using Kondo entanglement to induce spin correlations between disconnected quantum dots

We investigate the entanglement between the spins of two quantum dots that are not simultaneously connected to the same system. Quantum entanglement among localized spins is a crucial property for the advancement of quantum computing and quantum information. Generating and controlling an entangled state between quantum dots have garnered significant attention in recent years for this reason. In this study, we demonstrate that information about the spin orientation of a quantum dot can be preserved, utilizing Kondo entanglement, within a reservoir of electrons. Subsequently, this information can be transmitted to another dot after the initial dot has been decoupled from the reservoirs. We employ a double quantum dot system in a parallel geometry to establish the initial state, where each dot interacts with reservoirs of different symmetries. A specific phase in the couplings is chosen to induce antiferromagnetic spin correlation between the dots. The time evolution of the initial state is analyzed using the time-dependent density matrix renormalization group method. Our findings reveal that a partially entangled state between the dots can be achieved, even when they are not simultaneously connected. This entangled state arises transiently and dissipates in the stationary state. The stability of the state observed during the transient phase is demonstrated. To comprehend the details of these phenomena, we employ a canonical transformation of real space.

Keldysh crossover in one-dimensional Mott insulators

Recent advancements in pulse laser technology have facilitated the exploration of nonequilibrium spectroscopy of electronic states in the presence of strong electric fields across a broad range of photon energies. The Keldysh crossover serves as an indicator that distinguishes between excitations resulting from photon absorption triggered by near-infrared multicycle pulses and those arising from quantum tunneling induced by terahertz pulses. Using a time-dependent density-matrix renormalization group, we investigate the emergence of the Keldysh crossover in a one-dimensional (1D) Mott insulator. We find that the Drude weight is proportional to photo-doped doublon density when a pump pulse induces photon absorption. In contrast, the Drude weight is suppressed when a terahertz pulse introduces doublons and holons via quantum tunneling. The suppressed Drude weight accompanies glassy dynamics with suppressed diffusion, which is a consequence of strong correlations and exhibits finite polarization decaying slowly after pulse irradiation. In the quantum tunneling region, entanglement entropy slowly grows logarithmically. These contrasting behaviors between the photon-absorption and quantum tunneling regions are a manifestation of the Keldysh crossover in 1D Mott insulators and provide a novel methodology for designing the localization and symmetry of electronic states called subcycle-pulse engineering.

Terminable Transitions in a Topological Fermionic Ladder.

Interacting fermionic ladders are versatile platforms to study quantum phases of matter, such as different types of Mott insulators. In particular, there are D-Mott and S-Mott states that hold preformed fermion pairs and become paired-fermion liquids upon doping (d wave and s wave, respectively). We show that the D-Mott and S-Mott phases are in fact two facets of the same topological phase and that the transition between them is terminable. These results provide a quantum analog of the well-known terminable liquid-to-gas transition. However, the phenomenology we uncover is even richer, as the order of the transition may alternate between continuous and first order, depending on the interaction details. Most importantly, the terminable transition is robust in the sense that it is guaranteed to appear for weak, but arbitrary couplings. We discuss a minimal model where some analytical insights can be obtained, a generic model where the effect persists; and a model-independent field-theoretical study demonstrating the general phenomenon. The role of symmetry and the edge states is briefly discussed. The numerical results are obtained using the variational uniform matrix-product state (VUMPS) formalism for infinite systems, as well as the density-matrix renormalization group (DMRG) algorithm for finite systems.

Topological Phase Diagram of an Interacting Kitaev Chain: Mean Field versus DMRG Study

In this work, we study the topological phase transitions of a Kitaev chain generalized by the addition of nearest-neighbor Coulomb interaction. We show the presence of a robust topological phase as a function of the interaction strength and of the on-site energy with associated non-zero energy Majorana states localized at the chain edges. We provide an effective mean-field model that allows for the self-consistent computation of the mean value of the local particle number operator, and we also perform Density Matrix Renormalization Group numerical simulations based on a tensor network approach. We find that the two methods show a good agreement in reporting the phase transition between trivial and topological superconductivity. Temperature robustness within a physically relevant threshold has also been demonstrated. These findings shed light on an entire class of topological interacting one-dimensional systems in which the effects of residual Coulomb interactions play a relevant role.

Zero-Field Splitting Tensor of the Triplet Excited States of Aromatic Molecules: A Valence Full-π Complete Active Space Self-Consistent Field Study.

A method to predict the D tensor in the molecular frame with multiconfigurational wave functions in large active space was proposed, and the spin properties of the lowest triplets of aromatic molecules were examined with full-π active space; such calculations were challenging because the size of active space grows exponentially with the number of π electrons. In this method, the exponential growth of complexity is resolved by the density matrix renormalization group (DMRG) algorithm. From the D tensor, we can directly determine the direction of the magnetic axes and the ZFS parameters, D- and E-values, of the phenomenological spin Hamiltonian with their signs, which are not usually obtained in ESR experiments. The method using the DMRG-CASSCF wave function can give correct results even when the sign of D- and E-values is sensitive to the accuracy of the prediction of the D tensor and existing methods fail to predict the correct magnetic axes.

QuTree: A tree tensor network package.

We present QuTree, a C++ library for tree tensor network approaches. QuTree provides class structures for tensors, tensor trees, and related linear algebra functions that facilitate the fast development of tree tensor network approaches such as the multilayer multiconfigurational time-dependent Hartree approach or the density matrix renormalization group approach and its various extensions. We investigate the efficiency of relevant tensor and tensor network operations and show that the overhead for managing the network structure is negligible, even in cases with a million leaves and small tensors. QuTree focuses on providing simple, high-level routines while retaining easy access to the backend to facilitate novel developments. We demonstrate the capabilities of the package by computing the eigenstates of coupled harmonic oscillator Hamiltonians and performing random circuit simulations on a virtual quantum computer.

Quantum criticality in chains of planar rotors with dipolar interactions.

In this work, we perform a density matrix renormalization group study of chains of planar rotors interacting via dipolar interactions. By exploring the ground state from weakly to strongly interacting rotors, we find the occurrence of a quantum phase transition between a disordered and a dipole-ordered quantum state. We show that the nature of the ordered state changes from ferroelectric to antiferroelectric when the relative orientation of the rotor planes varies and that this change requires no modification of the overall symmetry. The observed quantum phase transitions are characterized by critical exponents and central charges, which reveal different universality classes ranging from that of the (1 + 1)D Ising model to the 2D classical XY model.

Topological phases in the dynamics of the simple exclusion process.

We study the dynamical large deviations of the classical stochastic symmetric simple exclusion process (SSEP) by means of numerical matrix product states. We show that for half filling, long-time trajectories with a large enough imbalance between the number hops in even and odd bonds of the lattice belong to distinct symmetry-protected topological (SPT) phases. Using tensor network techniques, we obtain the large deviation (LD) phase diagram in terms of counting fields conjugate to the dynamical activity and the total hop imbalance. We show the existence of high activity trivial and nontrivial SPT phases (classified according to string order parameters) separated by either a critical phase or a critical point. Using the leading eigenstate of the tilted generator, obtained from infinite-system density-matrix renormalization group simulations, we construct a near-optimal dynamics for sampling the LDs, and show that the SPT phases manifest at the level of rare stochastic trajectories. We also show how to extend these results to other filling fractions, and discuss generalizations to asymmetric SEPs.

Matrix‐Product State Approach to the Generalized Nuclear Pairing Hamiltonian

AbstractIt is shown that from the point of view of the generalized pairing Hamiltonian, the atomic nucleus is a system with small entanglement and can thus be described efficiently using a 1D tensor network (matrix‐product state) despite the presence of long‐range interactions. The ground state can be obtained using the density‐matrix renormalization group (DMRG) algorithm, which is accurate up to machine precision even for large nuclei, is numerically as cheap as the widely used Bardeen‐Cooper‐Schrieffer (BCS) approach, and does not suffer from any mean‐field artifacts. This framework is applied to compute the even‐odd mass differences of all known lead isotopes from to in a very large configuration space of 13 shells between the neutron magic numbers 82 and 184 (i.e., two major shells) and find good agreement with the experiment. Pairing with non‐zero angular momentum is also considered and the lowest excited states in the full configuration space of one major shell is determined, which is demonstrated for the , isotones. To demonstrate the capabilities of the method beyond low‐lying excitations, the first 100 excited states of with singlet pairing and the two‐neutron removal spectral function of are calculated, which relate to a two‐neutron pickup experiment.

A tensor network view of multilayer multiconfiguration time-dependent Hartree methods

The multilayer multiconfiguration time-dependent Hartree (ML-MCTDH) method and the density matrix renormalisation group (DMRG) are powerful workhorses applied mostly in different scientific fields. Although both methods are based on tensor network states, very different mathematical languages are used for describing them. This severely limits knowledge transfer and sometimes leads to re-inventions of ideas well known in the other field. Here, we review ML-MCTDH and DMRG theory using both MCTDH expressions and tensor network diagrams. We derive the ML-MCTDH equations of motions using diagrams and compare them with time-dependent and time-independent DMRG algorithms. We further review two selected recent advancements. The first advancement is related to optimising unoccupied single-particle functions in MCTDH, which corresponds to subspace enrichment in the DMRG. The second advancement is related to finding optimal tree structures and on highlighting similarities and differences of tensor networks used in MCTDH and DMRG theories. We hope that this contribution will foster more fruitful cross-fertilisation of ideas between ML-MCTDH and DMRG.

Emergent Superconductivity and Competing Charge Orders in Hole-Doped Square-Lattice t-J Model.

The square-lattice Hubbard and closely related t-J models are considered as basic paradigms for understanding strong correlation effects and unconventional superconductivity (SC). Recent large-scale density matrix renormalization group simulations on the extended t-J model have identified d-wave SC on the electron-doped side (with the next-nearest-neighbor hopping t_{2}>0) but a dominant charge density wave (CDW) order on the hole-doped side (t_{2}<0), which is inconsistent with the SC of hole-doped cuprate compounds. We re-examine the ground-state phase diagram of the extended t-J model by employing the state-of-the-art density matrix renormalization group calculations with much enhanced bond dimensions, allowing more accurate determination of the ground state. On six-leg cylinders, while different CDW phases are identified on the hole-doped side for the doping range δ=1/16-1/8, a SC phase emerges at a lower doping regime, with algebraically decaying pairing correlations and d-wave symmetry. On the wider eight-leg systems, the d-wave SC also emerges on the hole-doped side at the optimal 1/8 doping, demonstrating the winning of SC over CDW by increasing the system width. Our results not only suggest a new path to SC in general t-J model through weakening the competing charge orders, but also provide a unified understanding on the SC of both hole- and electron-doped cuprate superconductors.

On the magnetization of the 120° order of the spin-1/2 triangular lattice Heisenberg model: a DMRG revisited.

We revisit the issue about the magnetization of the 120° order in the spin-1/2 triangular lattice Heisenberg model with density matrix renormalization group (DMRG). The accurate determination of the magnetization of this model is challenging for numerical methods and its value exhibits substantial disparities across various methods. We perform a large-scale DMRG calculation of this model by employing bond dimension as large asD=24000and by studying the system with width as large asLy=12. With careful extrapolation with truncation error and suitable finite size scaling, we give a conservative estimation of the magnetization asM0=0.208(8). The ground state energy per site we obtain isEg=-0.5503(8). Our results provide valuable benchmark values for the development of new methods in the future.

Possible chiral spin liquid state in the S = 1/2 kagome Heisenberg model

The nature of the ground state for the S = 1/2 Kagome Heisenberg antiferromagnet (KHAF) has been elusive. We revisit this challenging problem and provide numerical evidence that its ground state is a chiral spin liquid. Combining the density matrix renormalization group method with analytical analyses, we demonstrate that the previously observed chiral spin liquid phase in the KHAF with longer-range couplings is stable in a broader region of the phase diagram. We characterize the nature of the ground state by computing energy derivatives, revealing the ground-state degeneracy arising from the spontaneous breaking of the time-reversal symmetry, and targeting the semion sector. We further investigate the phase diagram in the vicinity of the KHAF and observe a 3×3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sqrt{3}\ imes \\sqrt{3}$$\\end{document} magnetically ordered phase and two valence-bond crystal phases.