Numerical Renormalization Group Research Articles

Smoothly vanishing density in the contact process by an interplay of disorder and long-distance dispersal

Realistic modeling of ecological population dynamics requires spatially explicit descriptions that can take into account spatial heterogeneity as well as long-distance dispersal. Here, we present Monte Carlo simulations and numerical renormalization group results for the paradigmatic model, the contact process, in the combined presence of these factors in both one and two-dimensional systems. Our results confirm our analytic arguments stating that the density vanishes smoothly at the extinction threshold, in a way characteristic of infinite-order transitions. This extremely smooth vanishing of the global density entails an enhanced exposure of the population to extinction events. At the same time, a reverse order parameter, the local persistence displays a discontinuity characteristic of mixed-order transitions, as it approaches a non-universal critical value algebraically with an exponent \beta_p'<1βp′<1.

Probing magnetic and triplet correlations in spin-split superconductors with magnetic impurities

A superconductor (SC) in proximity to a ferromagnetic insulator (FMI) is predicted to exhibit mixed singlet and triplet correlations. The magnetic proximity effect of FMI spin splits the energy of Bogoliubov excitations and leads to a spin polarization at the surface for superconducting films that are thinner than the superconducting coherence length. In this work, we study manifestations of these phenomena in the properties of a magnetic impurity coupled via Kondo coupling to this FMI/SC system. Using the numerical renormalization group (NRG) method, we compute the properties of the ground state and low-lying excited states of a model that incorporates the Kondo interaction and a Ruderman-Kittel-Kasuya-Yosida (RKKY)-like interaction with the surface spin polarization. Our main finding is an energy splitting of the lowest even fermion-parity states caused by the proximity to the FMI. As the Kondo coupling increases, the splitting grows and saturates to a universal value equal to twice the exchange field of the FMI. We introduce a two-site model that can be solved analytically and provides a qualitative understanding of this and other NRG results. In addition, using perturbation theory, we demonstrate that the mechanism behind the splitting involves the RKKY field and the triplet correlations of the spin-split superconductor. A scaling analysis combined with NRG shows that the splitting can be written as a single-parameter scaling function of the ratio of the Kondo temperature and the superconducting gap, which is also numerically obtained. Published by the American Physical Society 2024

Proximate Tomonaga-Luttinger liquid in an anisotropic Kitaev-Gamma model

The characterization of quantum spin liquid phases in Kitaev materials has been a subject of intensive studies over the recent years, both theoretically and experimentally. Most theoretical studies have focused on an isotropically interacting model with its coupling strength being equivalent on each bond in an attempt to simplify the problem. In this Letter, we study an extended spin-1/2 Kitaev-Γ model on a honeycomb lattice with an additional tuning parameter that controls the coupling strength on one of the bonds: We connect the limit of isolated Kitaev-Γ chains, which is known to exhibit an emergent SU(2)1 Tomonaga-Luttinger liquid phase [Yang , ], to the two-dimensional model. We report on an instance, in which properties of the Tomonaga-Luttinger liquid persist for finite interchain coupling forming a phase we call the proximate Tomonaga-Luttinger liquid (pTLL). This pTLL is strongly anisotropic in character analogous to , and features spinon-like excitations similar to those of the antiferromagnetic Heisenberg chain. We use numerical exact diagonalization and density matrix renormalization group on various cluster geometries in a complementary way to overcome finite-size limitations. Published by the American Physical Society 2024

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

Atomic-scale magnetic doping of monolayer stanene by revealing Kondo effect from self-assembled Fe spin entities

Atomic-scale spin entity in a two-dimensional topological insulator lays the foundation to manufacture magnetic topological materials with single atomic thickness. Here, we have successfully fabricated Fe monomer, dimer and trimer doped in the monolayer stanene/Cu(111) through a low-temperature growth and systematically investigated Kondo effect by combining scanning tunneling microscopy/spectroscopy (STM/STS) with density functional theory (DFT) and numerical renormalization group (NRG) method. Given high spatial and energy resolution, tunneling conductance (dI/dU) spectra have resolved zero-bias Kondo resonance and resultant magnetic-field-dependent Zeeman splitting, yielding an effective spin Seff = 3/2 with an easy-plane magnetic anisotropy on the self-assembled Fe atomic dopants. Reduced Kondo temperature along with attenuated Kondo intensity from Fe monomer to trimer have been further identified as a manifestation of Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between Sn-separated Fe atoms. Such magnetic Fe atom assembly in turn constitutes important cornerstones for tailoring topological band structures and developing magnetic phase transition in the single-atom-layer stanene.

Prediction of impurity spectrum function by deep learning algorithm

By using the numerical renormalization group (NRG) method, we construct a large dataset with about one million spectral functions of the Anderson quantum impurity model. The dataset contains the density of states (DOS) of the host material, the strength of Coulomb interaction between on-site electrons (U), and the hybridization between the host material and the impurity site (Γ). The continued DOS and spectral functions are stored with Chebyshev coefficients and wavelet functions, respectively. From this dataset, we build seven different machine learning networks to predict the spectral function from the input data, DOS, U, and Γ. Three different evaluation indexes, mean absolute error (MAE), relative error (RE) and root mean square error (RMSE), are used to analyze the prediction abilities of different network models. Detailed analysis shows that, for the two kinds of widely used recurrent neural networks (RNNs), gate recurrent unit (GRU) has better performance than the long short term memory (LSTM) network. A combination of bidirectional GRU (BiGRU) and GRU has the best performance among GRU, BiGRU, LSTM, and BiLSTM. The MAE peak of BiGRU+GRU reaches 0.00037. We have also tested a one-dimensional convolutional neural network (1DCNN) with 20 hidden layers and a residual neural network (ResNet), we find that the 1DCNN has almost the same performance of the BiGRU+GRU network for the original dataset, while the robustness testing seems to be a little weak than BiGRU+GRU when we test all these models on two other independent datasets. The ResNet has the worst performance among all the seven network models. The datasets presented in this paper, including the large data set of the spectral function of Anderson quantum impurity model, are openly available at https://doi.org/10.57760/sciencedb.j00113.00192.

Z3 and (×Z3)3 symmetry protected topological paramagnets

We identify two-dimensional three-state Potts paramagnets with gapless edge modes on a triangular lattice protected by (×Z3)3 ≡ Z3 × Z3 × Z3 symmetry and smaller Z3 symmetry. We derive microscopic models for the gapless edge, uncover their symmetries and analyze the conformal properties. We study the properties of the gapless edge by employing the numerical density-matrix renormalization group (DMRG) simulation and exact diagonalization. We discuss the corresponding conformal field theory, its central charge, and the scaling dimension of the corresponding primary field. We argue, that the low energy limit of our edge modes defined by the SUk(3)/SUk(2) coset conformal field theory with the level k = 2. The discussed two-dimensional models realize a variety of symmetry-protected topological phases, opening a window for studies of the unconventional quantum criticalities between them.

The PointGroupNRG code for numerical renormalization group calculations with discrete point-group symmetries

The numerical renormalization group (NRG) has been widely used as a magnetic impurity solver since the pioneering works by Wilson. Over the past decades, a significant attention has been focused on the application of symmetries in order to reduce the computational cost of the calculations and to improve their accuracy. In particular, a notable progress has been made in implementing continuous symmetries such as SO(3), useful for studying impurities in an isotropic medium, or SU(N), which is applicable to a wide range of systems. In this work, we focus on the application of discrete point group symmetries, which are particularly relevant for impurity systems in metals where crystal field effects are important. With this aim, we have developed an original NRG code written in the Julia language, PointGroupNRG, where we have implemented crystal point-group symmetries for the Anderson impurity model, as well as the continuous spin and charge symmetries. Among other results, we demonstrate the advantage of our procedure by applying the code to a two-impurity model with RKKY interaction and an impurity system with two orbitals of Eg symmetry and two channels. We also provide benchmarks to show the performance improvements obtained by exploiting the orbital symmetries.

Modulated Kondo screening along magnetic mirror twin boundaries in monolayer MoS2

When a single electron is confined to an impurity state in a metal, a many-body resonance emerges at the Fermi energy if the electron bath screens the impurity’s magnetic moment. This is the Kondo effect, originally introduced to explain the abnormal resistivity behaviour in bulk magnetic alloys, and it has been realized in many quantum systems over the past decades, ranging from heavy-fermion lattices down to adsorbed single atoms. Here we describe a Kondo system that allows us to experimentally resolve the spectral function consisting of impurity levels and a Kondo resonance in a large Kondo temperature range, as well as their spatial modulation. Our approach is based on a discrete half-filled quantum confined state within a MoS2 grain boundary, which—in conjunction with numerical renormalization group calculations—enables us to test the predictive power of the Anderson model that is the basis of the microscopic understanding of Kondo physics.

Effective Bi-Layer Model Hamiltonian and Density-Matrix Renormalization Group Study for the High-T c Superconductivity in La3Ni2O7 under High Pressure

High-T c superconductivity with possible T c ≈ 80 K has been reported in the single crystal of La3Ni2O7 under high pressure. Based on the electronic structure given by the density functional theory calculations, we propose an effective bi-layer model Hamiltonian including both 3d z 2 and 3d x 2–y 2 orbital electrons of the nickel cations. The main feature of the model is that the 3d z 2 electrons form inter-layer σ-bonding and anti-bonding bands via the apical oxygen anions between the two layers, while the 3d x 2–y 2 electrons hybridize with the 3d z 2 electrons within each NiO2 plane. The chemical potential difference of these two orbital electrons ensures that the 3d z 2 orbitals are close to half-filling and the 3d x 2–y 2 orbitals are near quarter-filling. The strong on-site Hubbard repulsion of the 3d z 2 orbital electrons gives rise to an effective inter-layer antiferromagnetic spin super-exchange J. Applying pressure can self dope holes on the 3d z 2 orbitals with the same amount of electrons doped on the 3d x 2–y 2 orbitals. By performing numerical density-matrix renormalization group calculations on a minimum setup and focusing on the limit of large J and small doping of 3d z 2 orbitals, we find the superconducting instability on both the 3d z 2 and 3d x 2–y 2 orbitals by calculating the equal-time spin singlet pair–pair correlation function. Our numerical results may provide useful insights in the high-T c superconductivity in single crystal La3Ni2O7 under high pressure.

Quantum correlations in the frustrated XY model on the honeycomb lattice

We have investigated the spin-1/2 XY frustrated antiferromagnetic Heisenberg honeycomb model, which features an intermediate region in its ground state phase diagram that is not well understood. The two dominant phases in the diagram are the quantum spin-liquid (QSL) and the antiferromagnetic Ising order. Quantum correlations suggest that the QSL phase is likely to exhibit entanglement. To explore this possibility, we utilized numerical Lanczos and density matrix renormalization group (DMRG) methods to calculate concurrence, quantum discord (QD), and entanglement entropy. The results of our study indicate the existence of quantum entanglement within the intermediate region, implying a greater probability for the dominance of the quantum spin-liquid (QSL) phase over the antiferromagnetic Ising order. This discovery underscores the importance of considering quantum correlations in comprehending the model’s behavior and provides insight into the complex nature of quantum systems.

Impurity Knight shift in quantum dot Josephson junctions

Spectroscopy of a Josephson junction device with an embedded quantum dot reveals the presence of a contribution to level splitting in external magnetic field that is proportional to \cos\phicosϕ, where \phiϕ is the gauge-invariant phase difference across the junction. To elucidate the origin of this unanticipated effect, we systematically study the Zeeman splitting of spinful subgap states in the superconducting Anderson impurity model. The magnitude of the splitting is renormalized by the exchange interaction between the local moment and the continuum of Bogoliubov quasiparticles in a variant of the Knight shift phenomenon. The leading term in the shift is linear in the hybridisation strength \GammaΓ (quadratic in electron hopping), while the subleading term is quadratic in \GammaΓ (quartic in electron hopping) and depends on \phiϕ due to spin-polarization-dependent corrections to the Josephson energy of the device. The amplitude of the \phiϕ-dependent part is largest for experimentally relevant parameters beyond the perturbative regime where it is investigated using numerical renormalization group calculations. Such magnetic-field-tunable coupling between the quantum dot spin and the Josephson current could find wide use in superconducting spintronics.

Universal scaling of tunable Yu-Shiba-Rusinov states across the quantum phase transition

Quantum magnetic impurities give rise to a wealth of phenomena attracting tremendous research interest in recent years. On a normal metal, magnetic impurities generate the correlation-driven Kondo effect. On a superconductor, bound states emerge inside the superconducting gap called the Yu-Shiba-Rusinov (YSR) states. Theoretically, quantum impurity problems have been successfully tackled by numerical renormalization group (NRG) theory, where the Kondo and YSR physics are shown to be unified and the normalized YSR energy scales universally with the Kondo temperature divided by the superconducting gap. However, experimentally the Kondo temperature is usually extracted from phenomenological approaches, which gives rise to significant uncertainties and cannot account for magnetic fields properly. Using scanning tunneling microscopy at 10 mK, we apply a magnetic field to several YSR impurities on a vanadium tip to reveal the Kondo effect and employ the microscopic single impurity Anderson model with NRG to fit the Kondo spectra in magnetic fields accurately and extract the corresponding Kondo temperature unambiguously. Some YSR states move across the quantum phase transition (QPT) due to the changes in atomic forces during tip approach, yielding a continuous universal scaling with quantitative precision for quantum spin-frac{1}{2} impurities.

Yu-Shiba-Rusinov states assisted by asymmetric Coulomb repulsion in a bipartite molecular device

Interfacing magnetism with superconducting condensates are promising candidates holding Majorana bound states with which fault-tolerant quantum computation could be implemented. Within this field, understanding the detailed dynamics is important both for fundamental reasons and for the development of innovative quantum technologies. Herein, motivated by a molecular magnet Tb2Pc3 interacting with a superconducting Pb(111) substrate, which results in spin-orbital Yu-Shiba-Rusinov (YSR) states, as is affirmed by a theoretical simulation with the aid of the numerical renormalization group technique (see Xia et al 2022 Nat. Commun. 13 6388), we study the YSR states and quantum phase transitions (QPTs) in a bipartite molecular device adsorbed on an wave superconducting substrate. We highlight the effect of asymmetric Coulomb repulsion by computing the spectrum function and spin correlation function in various parameter regimes. We demonstrate that if one impurity is non-interacting, there are no YSR states in both impurities with any repulsion value in the other impurity. Whereas if the repulsion in one impurity is strong, the YSR states are observed in both impurities, and a QPT arises as the repulsion in the other impurity sweeps, assisted by the competition between the superconducting singlet (Cooper pair) and the Kondo singlet. The evolution of YSR states distinguishes from the single impurity case and can be well interpreted by the energy scales of the isotropic superconducting gap parameter, as well as the two Kondo temperatures. Our findings provide theoretical insights into the phase diagram of two magnetic impurities on a superconducting host and shine light on the effects induced by asymmetric Coulomb repulsion on many-body interactions.

Manipulating the optimal transport orbital and virtual states in a supramolecular transistor

Supramolecular electronics bridges the gap between single-molecule electronics and bulk, plastic electronics, and is embodied as one of the biggest promises of nanometer electronics made by supramolecular functional materials. Within this field, the way in which electrons propagate through the supramolecule plays a profound role both in designing supramolecular devices and in understanding the structure–property relationships at the molecular level. Herein, we consider a supramolecular transistor, which contains bipartite molecular monomers, with one molecular orbital is non-interacting, whereas the other one is strongly correlated. Modeling this device using the two-impurity Anderson model, we unravel the orbital selections and the virtual states in the electronic transport process with the aid of the celebrated numerical renormalization group method. We demonstrate that if both orbitals are in their particle-hole symmetric regimes, electrons favor to transmit through the non-interacting molecular orbital via the resonant tunneling process, rather than through the strongly correlated molecular orbital via the Kondo effect, despite the energy of the strongly correlated orbital is lower. This phenomenon indicates that if the Kondo effect occurs, it experiences a temporarily doubly occupied virtual state. Interestingly, the situation reverses if the inter-molecule hopping integral is applied, where the strongly correlated orbital becomes optimal and the virtual state turns to be an empty state. To understand these behaviors, both the static and the dynamical properties, as well as those thermodynamic quantities are studied in detail, and necessary physical arguments are given. Our findings afford deep insights into the manipulations of the optimal transport orbital and the Kondo behaviors within quantum impurity devices.

Field-induced local Fermi-liquid states in the underscreened Kondo model for Eu compounds

We have studied an underscreened Kondo model for the impurity spin S=7/2S=7/2 capturing the 4f4f characteristics of a nearly divalent Eu ion. By using the numerical renormalization group method, it is shown that an unusual heavy-fermion state is induced from the underscreened spin state when a weak crystal field is introduced. It is also found that the application of the magnetic field enhances the effective mass of the heavy fermions, reflecting the field-induced level crossing of the spin states. We will discuss the mechanism of these mass enhancements in terms of a realization of the two-stage Kondo effect.

Kondo effect in twisted bilayer graphene

The emergence of flat bands in twisted bilayer graphene at the magic angle can be understood in terms of a vanishing Fermi velocity of the Dirac cone. This is associated with van Hove singularities approaching the Fermi energy and becoming higher-order. In the density of states, this is reflected by flanking logarithmic van Hove divergences pinching off the central Dirac cone in energy space. The low-energy pseudogap of the Dirac cone away from the magic angle is replaced by a power-law divergence due to the higher-order van Hove singularity at the magic angle. This plays an important role in the exotic phenomena observed in this material, such as superconductivity and magnetism, by amplifying electronic correlation effects. Here we investigate one such correlation effect---the Kondo effect due to a magnetic impurity embedded in twisted bilayer graphene. We use the Bistritzer-MacDonald model to extract the low-energy density of states of the material as a function of twist angle and study the resulting quantum impurity physics using perturbative and numerical renormalization group methods. Although at zero temperature the impurity is only Kondo screened precisely at the magic angle, we find highly nontrivial behavior at finite temperatures relevant to experiments, due to the complex interplay between Dirac, van Hove, and Kondo physics.

Z2 non-Hermitian skin effect in equilibrium heavy-fermion systems

We demonstrate that a correlated equilibrium $f$-electron system with time-reversal symmetry can exhibit a ${\mathbb{Z}}_{2}$ non-Hermitian skin effect of quasiparticles. In particular, we analyze a two-dimensional periodic Anderson model with spin-orbit coupling by combining the dynamical mean-field theory (DMFT) and the numerical renormalization group. We prove the existence of the ${\mathbb{Z}}_{2}$ skin effect by explicitly calculating the topological invariant and show that spin-orbit interaction is essential to this effect. Our DMFT analysis demonstrates that the ${\mathbb{Z}}_{2}$ skin effect of quasiparticles is reflected on the pseudospectrum. Furthermore, we analyze temperature effects on this skin effect using the generalized Brillouin zone technique, which clarifies that the skin modes are strongly localized above the Kondo temperature.

Chiral numerical renormalization group

The interplay between the Kondo screening of quantum impurities (by the electronic channels to which they couple) and the interimpurity RKKY interactions (mediated by the same channels) has been extensively studied. However, the effect of unidirectional channels (e.g., chiral or helical edge modes of 2D topological materials) which greatly restrict the mediated interimpurity interactions, has only more recently come under scrutiny, and it can drastically alter the physics. Here we take Wilson's numerical renormalization group (NRG), the most established numerical method for treating quantum impurity models, and extend it to systems consisting of two impurities coupled at different locations to unidirectional channel(s). This is challenging due to the incompatibility of unidirectionality with one of the main ingredients in NRG -- the mapping of the channel(s) to a Wilson chain -- a tight-binding chain with the impurity at one end and hopping amplitudes which decay exponentially with the distance. We bridge this gap by introducing a "Wilson ladder" consisting of two coupled Wilson chains, and demonstrate that this construction successfully captures the unidirectionality of the channel(s), as well as the distance between the two impurities. We use this mapping in order to study two Kondo impurities coupled to a single chiral channel, showing that all local properties and thermodynamic quantities are indifferent to the interimpurity distance, and correspond to two separate single-impurity models. Extensions to more impurities and/or helical channels are possible.

Effective low-energy models for superconducting impurity systems

We present two complementary methods to calculate the Andreev bound state energies of a single-level quantum dot connected to superconducting leads described by the superconducting impurity Anderson model. The first method, which is based on a mapping to a low-energy model, can be utilized to extract the Andreev bound state energies from finite-temperature, imaginary-time quantum Monte Carlo data without the necessity of any analytic continuation technique. The second method maps the full model on an exactly solvable superconducting atomic limit with renormalized parameters. As such, it represents a fast and reliable method for a quick scan of the parameter space. We demonstrate that after adding a simple band correction this method can provide predictions for measurable quantities, including the Josephson current, that are in a solid quantitative agreement with precise results obtained by the numerical renormalization group and quantum Monte Carlo.