A fast impurity solver for the dynamical mean field theory(DMFT) named Two Mode Approxi- mation (TMA) is proposed based on the Gutzwiller variational approach, which captures the main features of both the coherent and incoherent motion of the electrons. The new solver works with real frequency at zero temperature and it provides directly the spectral function of the electrons. It can be easily generalized to multi-orbital impurity problems with general on-site interactions, which makes it very useful in LDA+DMFT. Benchmarks on one and two band Hubbard models are presented, and the results agree well with those of Exact Diagonalization (ED).
We study Gutzwiller-correlated wave functions as variational ground states for the two-impurity Anderson model (TIAM) at particle-hole symmetry as a function of the impurity separation ${\bf R}$. Our variational state is obtained by applying the Gutzwiller many-particle correlator to a single-particle product state. We determine the optimal single-particle product state fully variationally from an effective non-interacting TIAM that contains a direct electron transfer between the impurities as variational degree of freedom. For a large Hubbard interaction $U$ between the electrons on the impur…
We analyze the ground-state energy, magnetization, magnetic susceptibility, and Kondo screening cloud of the symmetric single-impurity Anderson model (SIAM) that is characterized by the band width $W$, the impurity interaction strength $U$, and the local hybridization $V$. We compare Gutzwiller variational and magnetic Hartree-Fock results in the thermodynamic limit with numerically exact data from the Density-Matrix Renormalization Group (DMRG) method on large rings. To improve the DMRG performance, we use a canonical transformation to map the SIAM onto a chain with half the system size and o…
Renormalization of non-magnetic impurity potential by strong electron correlation is investigated in detail. We adopt the t-t'-t"-J model and consider mainly a delta-function impurity potential. The variational Monte Carlo method shows that impurity potential scattering matrix elements between Gutzwiller-projected quasi-particle excited states are as strongly renormalized as the hopping terms. Such renormalization is also seen by the Bogoliubov-de Gennes equation with an impurity, where the strong correlation is treated by a Gutzwiller mean-field theory with local Gutzwiller factors and local …
Recent STM measurements have observed many inhomogeneous patterns of the local density of states on the surface of high-T_c cuprates. As a first step to study such disordered strong correlated systems, we use the BdG equation for the t-t'-t"-J model with an impurity. The impurity is taken into account by a local potential or local variation of the hopping/exchange terms. Strong correlation is treated by a Gutzwiller mean-field theory with local Gutzwiller factors and local chemical potentials. It turned out that the potential impurity scattering is greatly suppressed, while the local variation…
We propose a fast impurity solver for the general quantum impurity model based on the perturbation theory around the atomic limit, which can be used in combination with the local density approximation (LDA) and the dynamical mean field theory (DMFT). We benchmark the solver in the two band Hubbard model within DMFT against quantum Monte Carlo (QMC) and numerical renormalization group (NRG) results. We find that the solver works very well in the paramagnetic Mott insulator phase. We also apply this impurity solver to the DMFT study of the anti-ferromagnetic phase transition in the unfrustrated …
We have developed a new efficient and accurate impurity solver for the single impurity Anderson model (SIAM), which is based on a non-perturbative recursion technique in a space of operators and involves expanding the self-energy as a continued fraction. The method has no special occupation number or temperature restrictions; the only approximation is the number of levels of the continued fraction retained in the expansion. We also show how this approach can be used as a new approach to Dynamical Mean Field Theory (DMTF) and illustrate this with the Hubbard model. The three lowest orders of re…
The energy-dependent scattering of fermions from a localized orbital at an energy-dependent rate $Γ(ε)\propto |ε|^r$ gives rise to quantum critical points (QCPs) in the pseudogap single-impurity Anderson model separating a local moment phase with an unscreened spin moment from a strong-coupling phase which slightly deviates from the screened phase of standard Kondo problem. Using the time-dependent numerical renormalization group (TD-NRG) approach we show that local dynamic properties always equilibrate towards a steady-state value even for quenches across the QCP but with systematic deviation…
We develop a variational scheme called "Gutzwiller renormalization group" (GRG), which enables us to calculate the ground state of Anderson impurity models (AIM) with arbitrary numerical precision. Our method can exploit the low-entanglement property of the ground state in combination with the framework of the Gutzwiller wavefunction, and suggests that the ground state of the AIM has a very simple structure, which can be represented very accurately in terms of a surprisingly small number of variational parameters. We perform benchmark calculations of the single-band AIM that validate our theor…
We present an implementation of the hybridization expansion impurity solver which employs sparse matrix exact-diagonalization techniques to compute the time evolution of the local Hamiltonian. This method avoids computationally expensive matrix-matrix multiplications and becomes advantageous over the conventional implementation for models with 5 or more orbitals. In particular, this method will allow the systematic investigation of 7-orbital systems (lanthanide and actinide compounds) within single-site dynamical mean field theory. We illustrate the power and usefulness of our approach with dy…
We use the time dependent variational matrix product state (tVMPS) approach to investigate the dynamical properties of the single impurity Anderson model (SIAM). Under the Jordan-Wigner transformation, the SIAM is reformulated into two spin-1/2 XY chains with local magnetic fields along the z-axis. The chains are connected by the longitudinal Ising coupling at the end points. The ground state of the model is searched variationally within the space spanned by the matrix product state (MPS). The temporal Green's functions are calculated both by the imaginary and real time evolutions, from which …
We apply the Green's function coupled cluster singles and doubles (GFCCSD) impurity solver to realistic impurity problems arising for strongly correlated solids within the self-energy embedding theory (SEET) framework. We describe the details of our GFCC solver implementation, investigate its performance, and highlight potential advantages and problems on examples of impurities created during the self-consistent SEET for antiferromagnetic MnO and paramagnetic SrMnO$_{3}$. GFCCSD provides satisfactory descriptions for weakly and moderately correlated impurities with sizes that are intractable b…
In this paper a fast impurity solver is proposed for dynamical mean field theory (DMFT) based on a decoupling of the equations of motion for the impurity Greens function. The resulting integral equations are solved efficiently with a method based on genetic algorithms. The Hubbard and periodic Anderson models are studied with this impurity solver. The method describes the Mott metal insulator transition and works for a large range of parameters at finite temperature on the real frequency axis. This makes it useful for the exploration of real materials in the framework of LDA+DMFT.
We present an efficient impurity solver for the dynamical mean-field theory (DMFT). It is based on the separation of bath degrees of freedom into the low energy and the high energy parts. The former is solved exactly using exact diagonalization and the latter is treated approximately using Green's function equation of motion decoupling approximation. The two parts are combined coherently under the standard basis operator formalism. The impurity solver is applied to the Anderson impurity model and, combined with DMFT, to the one-band Hubbard model. Qualitative agreement is found with other well…
Tree tensor network impurity solver based on Cayley-tree mapping
Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
An impurity solver for the dynamical mean field (DMFT) study of the Mott insulators is proposed, which is based on the second order perturbation of the hybridization function. After carefully benchmarking it with Quantum Monte Carlo results on the anti-ferromagnetic phase of the Hubbard model, we conclude that this impurity solver can capture the main physical features in the strong coupling regime and can be a very useful tool for the LDA+DMFT studies of the Mott insulators with long range order.
Impurity solvers play an essential role in the numerical investigation of strongly correlated electrons systems within the "dynamical mean field" approximation. Recently, a new class of continuous-time solvers has been developed, based on a diagrammatic expansion of the partition function in either the interactions or the impurity-bath hybridization. We investigate the performance of these two complementary approaches and compare them to the well-established Hirsch-Fye method. The results show that the continuous-time methods, and in particular the version which expands in the hybridization, p…
We propose a fast multi-orbital impurity solver for the dynamical mean field theory (DMFT). Our DMFT solver is based on the equations of motion (EOM) for local Green's functions and constructed by generalizing from the single-orbital case to the multi-orbital case with inclusion of the inter-orbital hybridizations and applying a mean field approximation to the inter-orbital Coulomb interactions. The two-orbital Hubbard model is studied using this impurity solver within a large range of parameters. The Mott metal-insulator transition and the quasiparticle peak are well described. A comparison o…
We describe an open-source implementation of the continuous-time hybridization-expansion quantum Monte Carlo method for impurity models with general instantaneous two-body interactions and complex hybridization functions. The code is built on the ALPS libraries (Applications and Libraries for Physics Simulations).
We present a numerically exact Inchworm Monte Carlo method for equilibrium multiorbital quantum impurity problems with general interactions and hybridizations. We show that the method, originally developed to overcome the dynamical sign problem in certain real-time propagation problems, can also overcome the sign problem as a function of temperature for equilibrium quantum impurity models. This is shown in several cases where the current method of choice, the continuous-time hybridization expansion, fails due to the sign problem. Our method therefore enables simulations of impurity problems as…
We present an impurity solver based on adaptively truncated Hilbert spaces. The solver is particularly suitable for dynamical mean-field theory in circumstances where quantum Monte Carlo approaches are ineffective. It exploits the sparsity structure of quantum impurity models, in which the interactions couple only a small subset of the degrees of freedom. We further introduce an adaptive truncation of the particle or hole excited spaces, which enables computations of Green functions with an accuracy needed to avoid unphysical (sign change of imaginary part) self-energies. The method is benchma…
We present an analytic universal impurity solver for strongly correlated electrons. We extend the many-body perturbation expansion via suitable two-particle renormalizations from the Fermi-liquid regime to the critical region of the metal-insulator transition. The reliability of the approximation in the strong-coupling limit is demonstrated by reproducing the Kondo scale in the single-impurity Anderson model. We disclose the origin of the Kondo resonance in terms of Feynman diagrams and find criteria for the existence of the proper Kondo asymptotic behavior in approximate theories.
A recently developed continuous time solver based on an expansion in hybridization about an exactly solved local limit is reformulated in a manner appropriate for general classes of quantum impurity models including spin exchange and pair hopping terms. The utility of the approach is demonstrated via applications to the dynamical mean field theory of the Kondo lattice and two-orbital models. The algorithm can handle low temperatures and strong couplings without encountering a sign problem.
We present a new continuous time solver for quantum impurity models such as those relevant to dynamical mean field theory. It is based on a stochastic sampling of a perturbation expansion in the impurity-bath hybridization parameter. Comparisons to quantum Monte Carlo and exact diagonalization calculations confirm the accuracy of the new approach, which allows very efficient simulations even at low temperatures and for strong interactions. As examples of the power of the method we present results for the temperature dependence of the kinetic energy and the free energy, enabling an accurate loc…
In the present paper, we present an efficient continuous-time quantum Monte Carlo impurity solver with high acceptance rate at low temperature for multi-orbital quantum impurity models with general interaction. In this hybridization expansion impurity solver, the imaginary time evolution operator for the high energy multiplets, which decays very rapidly with the imaginary time, is approximated by a probability normalized $δ$-function. As the result, the virtual charge fluctuations of $f^{n}\rightarrow f^{n\pm1}$ are well included on the same footing without applying Schrieffer-Wolff transforma…
We present a continuous-time Monte Carlo impurity solver for multiorbital impurity models which combines a strong-coupling hybridization expansion and a weak-coupling expansion in the Hund's coupling parameter J. This double-expansion approach allows to treat the dominant density-density interactions U within the efficient segment representation. We test the approach for a two-orbital model with static interactions, and then explain how the double-expansion allows to simulate models with frequency dependent U(ω) and J(ω). The method is used to investigate spin state transitions in a toy model …
Multi-orbital quantum impurity models with general interaction and hybridization terms appear in a wide range of applications including embedding, quantum transport, and nanoscience. However, most quantum impurity solvers are restricted to a few impurity orbitals, discretized baths, diagonal hybridizations, or density-density interactions. Here, we generalize the inchworm quantum Monte Carlo method to the interaction expansion and explore its application to typical single- and multi-orbital problems encountered in investigations of impurity and lattice models. Our implementation generically ou…
We present a very efficient solver for the general Anderson impurity problem. It is based on the perturbation around a solution obtained from exact diagonalization using a small number of bath sites. We formulate a perturbation theory which is valid for both weak and strong coupling and interpolates between these limits. Good agreement with numerically exact quantum Monte-Carlo results is found for a single bath site over a wide range of parameters. In particular, the Kondo resonance in the intermediate coupling regime is well reproduced for a single bath site and the lowest order correction. …
We introduce a new implementation of hybridization expansion continuous time quantum impurity solver which is relevant to dynamical mean-field theory. It employs Newton interpolation at a sequence of real Leja points to compute the time evolution of the local Hamiltonian efficiently. Since the new algorithm avoids not only computationally expansive matrix-matrix multiplications in conventional implementations but also huge memory consumptions required by Lanczos/Arnoldi iterations in recently developed Krylov subspace approach, it becomes advantageous over the previous algorithms for quantum i…
A nonperturbative quantum impurity solver is proposed based on a formally exact hierarchical equations of motion (HEOM) formalism for open quantum systems. It leads to quantitatively accurate evaluation of physical properties of strongly correlated electronic systems, in the framework of dynamical mean-field theory (DMFT). The HEOM method is also numerically convenient to achieve the same level of accuracy as that using the state-of-the-art numerical renormalization group impurity solver at finite temperatures. The practicality of the novel HEOM+DMFT method is demonstrated by its applications …
Using an imaginary-time matrix-product state (MPS) based quantum impurity solver we perform a realistic dynamical mean-field theory (DMFT) calculation combined with density functional theory (DFT) for Sr$_2$RuO$_4$. We take the full Hubbard-Kanamori interactions and spin-orbit coupling (SOC) into account. The MPS impurity solver works at essentially zero temperature in the presence of SOC, a regime of parameters currently inaccessible to continuous-time quantum Monte Carlo (CTQMC) methods, due to a severe sign problem. We show that earlier results obtained at high temperature, namely that the …
Quantum impurity solvers have a broad range of applications in theoretical studies of strongly correlated electron systems. Especially, they play a key role in dynamical mean-field theory calculations of correlated lattice models and realistic materials. Therefore, the development and implementation of efficient quantum impurity solvers is an important task. In this paper, we present an open source interacting quantum impurity solver toolkit (dubbed $i$QIST). This package contains several highly optimized quantum impurity solvers which are based on the hybridization expansion continuous-time q…
We propose an improved fast multi-orbital impurity solver for the dynamical mean field theory (DMFT) based on equations of motion (EOM) of Green's functions and decoupling scheme. In this scheme the inter-orbital Coulomb interactions are treated fully self-consistently, involving the inter-orbital fluctuations. As an example of the derived multi-orbital impurity solver, the two-orbital Hubbard model is studied for various cases. Comparisons are made between numerical results obtained with our EOM scheme and those obtained with quantum Monte Carlo and numerical renormalization group methods. Th…
Strongly correlated quantum impurity problems appear in a wide variety of contexts ranging from nanoscience and surface physics to material science and the theory of strongly correlated lattice models, where they appear as auxiliary systems within dynamical mean-field theory. Accurate and unbiased solutions must usually be obtained numerically, and continuous-time quantum Monte Carlo algorithms, a family of algorithms based on the stochastic sampling of partition function expansions, perform well for such systems. With the present paper we provide an efficient and generic implementation of the…
Recently solvers for the Anderson impurity model (AIM) working directly on the real-frequency axis have gained much interest. A simple and yet frequently used impurity solver is exact diagonalization (ED), which is based on a discretization of the AIM bath degrees of freedom. Usually, the bath parameters cannot be obtained directly on the real-frequency axis, but have to be determined by a fit procedure on the Matsubara axis. In this work we present an approach where the bath degrees of freedom are first discretized directly on the real-frequency axis using a large number of bath sites ($\appr…
We present a new impurity solver for dynamical mean-field theory based on imaginary-time evolution of matrix product states. This converges the self-consistency loop on the imaginary-frequency axis and obtains real-frequency information in a final real-time evolution. Relative to computations on the real-frequency axis, required bath sizes are much smaller and less entanglement is generated, so much larger systems can be studied. The power of the method is demonstrated by solutions of a three band model in the single and two-site dynamical mean-field approximation. Technical issues are discuss…
In the dynamics of driven impurity models, there is a fundamental asymmetry between the processes of emission and absorption of environment excitations: most of the emitted excitations are rapidly and irreversibly scattered away, and only a small amount of them is reabsorbed back. We propose to use a stochastic simulation of the irreversible quantum emission processes in real-time dynamics, while taking into account the reabsorbed virtual excitations by the bath discretization. The resulting method delivers a fast convergence with respect to the number of bath sites, on a wide time interval, w…
We present an algorithm for solving the self-consistency equations of the dynamical mean-field theory (DMFT) with high precision and efficiency at low temperatures. In each DMFT iteration, the impurity problem is mapped to an auxiliary Hamiltonian, for which the Green function is computed by combining determinantal quantum Monte Carlo (BSS-QMC) calculations with a multigrid extrapolation procedure. The method is numerically exact, i.e., yields results which are free of significant Trotter errors, but retains the BSS advantage, compared to direct QMC impurity solvers, of linear (instead of cubi…
Since the inception of the dynamical mean-field theory, numerous numerical studies have relied on the Hirsch-Fye quantum Monte Carlo (HF-QMC) method for solving the associated impurity problem. Recently developed continuous-time algorithms (CT-QMC) avoid the Trotter discretization error and allow for faster configuration updates, which makes them candidates for replacing HF-QMC. We demonstrate, however, that a state-of-the-art implementation of HF-QMC (with extrapolation of discretization delta_tau -> 0) is competitive with CT-QMC. A quantitative analysis of Trotter errors in HF-QMC estimates …
Neural network impurity solver for real-frequency dynamical mean-field theory
Suzhou Institute for Advanced Research, University of Science and Technology of China（中国科学技术大学苏州高等研究院）；School of Artificial Intelligence and Data Science, University of Science and Technology of China（中国科学技术大学人工智能与数据科学学院）；National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University（南京大学固体微结构物理实验室和物理系）；Collaborative Innovation Center of Advanced Microstructures, Nanjing University（南京大学先进微结构协同创新中心）
Although several impurity solvers in the dynamical mean field theory (DMFT) have been proposed, especially in multi-band systems, there are practical difficulties arising from a trade-off between numerical costs and reliability. In this study, we re-interpret the iterative perturbation theory (IPT) as an approximation which captures the strong correlation effects by mimicking the particular frequency structures of the exact full vertex, and extend it such that it can have efficiency and reliability simultaneously by modifying IPT vertex using the parquet equations. We apply this method to seve…
We present TRIQS/CTHYB, a state-of-the art open-source implementation of the continuous-time hybridisation expansion quantum impurity solver of the TRIQS package. This code is mainly designed to be used with the TRIQS library in order to solve the self-consistent quantum impurity problem in a multi-orbital dynamical mean field theory approach to strongly-correlated electrons, in particular in the context of realistic calculations. It is implemented in C++ for efficiency and is provided with a high-level Python interface. The code is ships with a new partitioning algorithm that divides the loca…
We describe the use of coupled-cluster theory as an impurity solver in dynamical mean-field theory (DMFT) and its cluster extensions. We present numerical results at the level of coupled-cluster theory with single and double excitations (CCSD) for the density of states and self-energies of cluster impurity problems in the one- and two-dimensional Hubbard models. Comparison to exact diagonalization shows that CCSD produces accurate density of states and self-energies at a variety of values of $U/t$ and filling fractions. However, the low cost allows for the use of many bath sites, which we defi…
We compute the spectral functions for the two-site dynamical cluster theory and for the two-orbital dynamical mean-field theory in the density-matrix renormalization group (DMRG) framework using Chebyshev expansions represented with matrix product states (MPS). We obtain quantitatively precise results at modest computational effort through technical improvements regarding the truncation scheme and the Chebyshev rescaling procedure. We furthermore establish the relation of the Chebyshev iteration to real-time evolution, and discuss technical aspects as computation time and implementation in det…
We extend a previously proposed rotation and truncation scheme to optimize quantum Anderson impurity calculations with exact diagonalization [PRB 90, 085102 (2014)] to density-matrix renormalization group (DMRG) calculations. The method reduces the solution of a full impurity problem with virtually unlimited bath sites to that of a small subsystem based on a natural impurity orbital basis set. The later is solved by DMRG in combination with a restricted-active-space truncation scheme. The method allows one to compute Green's functions directly on the real frequency or time axis. We critically …
We present a quantum impurity solver based on a pseudo-particle framework, which combines diagrammatic resummations for a three-point vertex with diagrammatic Monte Carlo sampling of a four-point vertex. This recently proposed approach [A. J. Kim et al., arXiv:2112.15549] is generalized here to fermionic impurity problems and we discuss the technical details of the implementation, including the time-stepping approach, the Monte Carlo updates, and the routines for checking the two-particle irreducibility of the four-point vertex. We also explain how the vertex information can be efficiently sto…
We derive an exact mapping from the action of nonequilibrium dynamical mean-field theory (DMFT) to a single-impurity Anderson model (SIAM) with time-dependent parameters, which can be solved numerically by exact diagonalization. The representability of the nonequilibrium DMFT action by a SIAM is established as a rather general property of nonequilibrium Green functions. We also obtain the nonequilibrium DMFT equations using the cavity method alone. We show how to numerically obtain the SIAM parameters using Cholesky or eigenvector matrix decompositions. As an application, we use a Krylov-based…
Unifying Variational and Dynamical Quantum Embedding: From Ghost Gutzwiller Approximation to Dynamical Mean-Field Theory
Center for Computational Quantum Physics, Flatiron Institute（平顶山研究所计算量子物理中心）；Department of Physics, National Chung Cheng University（国立中正大学物理系）；Ames National Laboratory（艾姆斯国家实验室）；Department of Physics and Astronomy, Iowa State University（爱荷华州立大学物理与天文学系）；Physics and Astronomy Department, Rutgers University（罗格斯大学物理与天文学系）；Condensed Matter Physics and Materials Science Department, Brookhaven National Laboratory（布鲁克海文国家实验室凝聚态物理与材料科学系）；Physics Department, Boston University（波士顿大学物理系）；Université Paris-Saclay, CNRS, CEA, Institut de physique théorique（巴黎萨克雷大学、法国国家科学研究中心、法国原子能和替代能源委员会、理论物理研究所）；School of Physics and Astronomy, Rochester Institute of Technology（罗切斯特理工学院物理与天文学院）
We introduce a representation of electron operators as a product of a spin-carry ing fermion and of a phase variable dual to the total charge (slave quantum rotor). Based on this representation, a new method is proposed for solving multi-orbital Anderson quantum impurity models at finite interaction strength U. It consists in a set of coupled integral equations for the auxiliary field Green's functions, which can be derived from a controlled saddle-point in the limit of a large number of field components. In contrast to some finite-U extensions of the non-crossing approximation, the new method…
