Dynamical mean-field theory for the Bose-Hubbard model

formula omitted]QIST v0.7: An open source continuous-time quantum Monte Carlo impurity solver toolkit

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 define by a discretization of the hybridization directly on the real frequency axis. We observe convergence of dynamical quantities using approximately 30 bath sites per impurity site, with our largest 4-site cluster DMFT calculation using 120 bath sites. We suggest coupled cluster impurity solvers will be attractive in ab initio formulations of dynamical mean-field theory.

We formulate a bosonic dynamical mean-field theory (B-DMFT) which provides a comprehensive, thermodynamically consistent framework for the theoretical investigation of correlated lattice bosons. The B-DMFT is applicable for arbitrary values of the coupling parameters and temperature and becomes exact in the limit of high spatial dimensions $d$ or coordination number $Z$ of the lattice. In contrast to its fermionic counterpart, the construction of the B-DMFT requires different scalings of the hopping amplitudes with $Z$ depending on whether the bosons are in their normal state or in the Bose-Einstein condensate (BEC). A detailed discussion of how this conceptual problem can be overcome by performing the scaling in the action rather than in the Hamiltonian itself is presented. The B-DMFT treats normal and condensed bosons on equal footing and thus includes the effects caused by their dynamic coupling. It reproduces all previously investigated limits in parameter space, such as the Beliaev-Popov and Hartree-Fock-Bogoliubov approximations, and generalizes the existing mean-field theories of interacting bosons. The self-consistency equations of the B-DMFT are those of a bosonic single-impurity coupled to two reservoirs corresponding to bosons in the condensate and in the normal state, respectively. We employ the B-DMFT to solve a model of itinerant and localized, interacting bosons analytically. The local correlations are found to enhance the condensate density and the BEC transition temperature ${T}_{\text{BEC}}$. This effect may be used experimentally to increase ${T}_{\text{BEC}}$ of bosonic atoms in optical lattices.

Pyqcm is a Python/C++ library that implements a few quantum cluster methods with an exact diagonalization impurity solver. Quantum cluster methods are used in the study of strongly correlated electrons to provide an approximate solution to Hubbard-like models. The methods covered by this library are Cluster Perturbation Theory (CPT), the Variational Cluster Approach (VCA) and Cellular (or Cluster) Dynamical Mean Field Theory (CDMFT). The impurity solver (the technique used to compute the cluster’s interacting Green function) is exact diagonalization from sparse matrices, using the Lanczos algorithm and variants thereof. The core library is written in C++ for performance, but the interface is in Python, for ease of use and inter-operability with the numerical Python ecosystem. The library is distributed under the GPL license.

Pyqcm is a Python/C++ library that implements a few quantum cluster methods with an exact diagonalization impurity solver. Quantum cluster methods are used in the study of strongly correlated electrons to provide an approximate solution to Hubbard-like models. The methods covered by this library are Cluster Perturbation Theory (CPT), the Variational Cluster Approach (VCA) and Cellular (or Cluster) Dynamical Mean Field Theory (CDMFT). The impurity solver (the technique used to compute the cluster’s interacting Green function) is exact diagonalization from sparse matrices, using the Lanczos algorithm and variants thereof. The core library is written in C++ for performance, but the interface is in Python, for ease of use and inter-operability with the numerical Python ecosystem. The library is distributed under the GPL license.

We apply a recently introduced hybridization-flow functional renormalization group scheme for Anderson-like impurity models as an impurity solver in a dynamical mean-field theory (DMFT) approach to lattice Hubbard models. We present how this scheme is capable of reproducing metallic and insulating solutions of the lattice model. Our setup also offers a numerically rather inexpensive method to calculate two-particle correlation functions. For the paramagnetic Hubbard-model on the Bethe lattice in infinite dimensions we calculate the local two-particle-vertex for the metallic and the insulating phase. Then we go to a two-site cluster-DMFT-scheme for the two-dimensional Hubbard-model that includes short-range antiferromagnetic fluctuations and obtain the local and non-local two-particle-vertex-functions. We discuss the rich frequency structures of these vertices and compare with the vertex in the single-site solution.

Dynamical mean-field theory is used to study the magnetic instabilities and phase diagram of the double-exchange (DE) model with Hund's coupling JH > 0 in infinite dimensions. In addition to ferromagnetic (FM) and antiferromagnetic (AF) phases, the DE model also supports a broad class of short-range ordered (SRO) states with extensive entropy and short-range magnetic order. For any site on the Bethe lattice, the correlation parameter q of an SRO state is given by the average q = ⟨ sin2 (θi/2)⟩, where θi is the angle between any spin and its neighbours. Unlike the FM (q = 0) and AF (q = 1) transitions, the transition temperature of an SRO state with 0 < q < 1 cannot be obtained from the magnetic susceptibility. But a solution of the coupled Green's functions in the weak-coupling limit indicates that an SRO state always has a higher transition temperature than the AF for all fillings p below 1 and even has a higher transition temperature than the FM for 0.26 ⩽ p ⩽ 0.39. For 0.39 < p < 0.73, where both the FM and AF phases are unstable for small JH, an SRO phase has a nonzero transition temperature except close to p = 0.5. As JH increases, the SRO transition temperature eventually vanishes and the FM phase dominates the phase diagram. For small JH, the T = 0 phase diagram of the DE model is greatly simplified by the presence of the SRO phase. An SRO phase is found to have lower energy than either the FM or AF phases for 0.26 ⩽ p < 1. Phase separation (PS) disappears as JH → 0 but appears for any nonzero coupling. For fillings near p = 1, PS occurs between an AF with p = 1 and either an SRO or a FM phase. The stability of an SRO state at T = 0 can be understood by examining the interacting density-of-states, which is gapped for any nonzero JH in an AF but only when JH exceeds a critical value in an SRO state.

In condensed matter physics, the idea of localized magnetic moments and the scattering of conduction electrons by those local moments are of central importance not only for explaining the transport of electrons at low temperature region but also for developing the related models and theories to extend our understanding of novel physics like non-conventional superconductor, quantum phase transition and non-Fermi liquid behavior. In the first chapter of this work, I will look back at the discovery of Kondo scattering, the introduction of Anderson impurity scenario, the extension to lattice version of the Anderson impurities, the periodic Anderson model, and how these ideas are used to the heavy fermion materials. In order to attack those problems and models numerically, in the second chapter, I will discuss the algorithm of dynamical mean-field theory, where interaction expansion continuous time quantum Monte Carlo is used as impurity solver. The introduction of dynamical mean-field theory and its related methods are one of the largest victory of computational condensed matter physics in the past three decades. By mapping the lattice problem to an impurity problem and solving it self-consistently, we can get the static and dynamic outputs in the thermal dynamical limit from dynamical mean-field theory simulation by solving a single site problem. In the infinite coordinate number limit, the dynamical mean-field theory results converge to the exact results. Quantum Monte Carlo is widely used in computational physics. Compared with the conventional quantum Monte Carlo, the recently introduced interaction expansion continuous time quantum Monte Carlo is free of decomposition error in imaginary time. In the third chapter, the effect of phonon coupling to the conduction band of the periodic Anderson model is discussed. In the periodic Anderson model at low temperature region, the local moments on f -band are screened by the conduction electrons and form the coherent Kondo singlet states. When the lattice oscillation degree of freedom is introduced to the conduction band, the conduction electrons bond with each other because of the retarded attractive interaction incurred by the phonons. As the electron-phonon interaction is large, the competition of electronic bonding on the c -band will compete with the Kondo singlet bonding between the conduction electrons and the local moments and finally lead to a Kondo collapse. In the fourth chapter, the pressure induced volume collapse problem of Cerium is explored. It has been discovered that Cerium will experience a 15% volume collapse from a

The cluster perturbation theory is presented in the 2D Hubbard model constructed using X operators in the Hubbard-I approximation. The short-range magnetic order is taken into account by dividing the entire lattice into individual 2 × 2 clusters and solving the eigenvalue problem in an individual cluster using exact diagonalization taking into account all excited levels. The case of half-filling taking into account jumps between nearest neighbors is considered. As a result of numerical solution, a shadow zone is discovered in the quasiparticle spectrum. It is also found that a gap in the density of states in the quasiparticle spectrum at zero temperature exists for indefinitely small values of Coulomb repulsion parameter U and increases with this parameter. It is found that the presence of this gap in the spectrum is due to the formation of a short-range antiferromagnetic order. An analysis of the temperature evolution of the density of states shows that the metal-insulator transition occurs continuously. The existence of two characteristic energy scales at finite temperatures is demonstrated, the larger scale is associated with the formation of a pseudogap in the vicinity of the Fermi level, and the smaller scale is associated with the metal-insulator transition temperature. A peak in the density of states at the Fermi level, which is predicted in the dynamic mean field theory in the vicinity of the metal-insulator transition, is not observed.

Dynamical mean-field theory (DMFT) is one of the most widely-used methods to treat accurately electron correlation effects in ab-initio real material calculations. Many modern large-scale implementations of DMFT in electronic structure codes involve solving a quantum impurity model with a Continuous-Time Quantum Monte Carlo (CT-QMC) solver. The main advantage of CT-QMC is that, unlike standard quantum Monte Carlo approaches, it is able to generate the local Green's functions of the correlated system on an arbitrarily fine imaginary time grid, and is free of any systematic errors. In this work, we extend a hybrid QMC solver proposed by Khatami et al. and Rost et al. to a multi-orbital context. This has the advantage of enabling impurity solver QMC calculations to scale linearly with inverse temperature and permit its application to d and f band materials. In addition, we present a novel Green's function processing scheme which generates accurate quasi-continuous imaginary time solutions of the impurity problem which overcome errors inherent to standard QMC approaches. This solver and processing scheme are incorporated into a full DFT+DMFT calculation using the CASTEP DFT code. Benchmark calculations for strontium vanadate properties are presented.

Dynamical mean-field theory calculation with the dynamical density-matrix renormalization group

The metal-insulator transition (MIT) remains among the most thoroughly studied phenomena in solid state physics, but the complexity of the phenomena, which usually involves cooperation of many degrees of freedom including orbitals, fluctuating local moments, magnetism, and the crystal structure, have resisted predictive ab-initio treatment. Here we develop ab-initio theoretical method for correlated electron materials, based on Dynamical Mean Field Theory, which can predict the change of the crystal structure across the MIT at finite temperature. This allows us to study the coupling between electronic, magnetic and orbital degrees of freedom with the crystal structure across the MIT in rare-earth nickelates. We predict the electronic free energy profile of the competing states, and the theoretical magnetic ground state configuration, which is in agreement with neutron scattering data, but is different from the magnetic models proposed before. The resonant elastic X-ray response at the K-edge, which was argued to be a probe of the charge order, is theoretically modelled within the Dynamical Mean Field Theory, including the core-hole interaction. We show that the line-shape of the measured resonant elastic X-ray response can be explained with the “site-selective” Mott scenario without real charge order on Ni sites.

: The broad goal of the project is the identification of the factors controlling the nontrivial many-body physics in artificially nanostructured materials based on transition metal oxides, thereby enabling the design of materials with desired correlated electron properties. The means of achieving this goal is the implementation and development of the hybrid methodology of density functional theory and dynamical mean-field theory (DFT+DMFT). We achieved a vastly improved understanding of the physics of the correlation-driven metal-insulator transition in real materials based on the identification of the d-valence as a critical parameter controlling the transition, and identifying that this finding is insensitive to the choice of orbital provided it is well localized. Secondly, we discovered a new mechanism for correlation-driven metal-insulator transitions (the site-selective Mott transition) and demonstrated that this mechanism accounts for the main features of the previously mysterious metal-insulator phase diagram of the rare-earth nickelates. Thirdly, we demonstrated the ability to design new Mott insulators via heterostructuring. Finally, we developed a fully charge self-consistent implementation of DFT+DMFT, which allows us to compute total energies. This major technical development allowed us to compute the pressure vs. rare earth phase diagram in the nickelates.

The temperature‐dependent behavior emerging in the vicinity of the superfluid (SF) to Mott‐insulator (MI) transition of interacting bosons in a 2D optical lattice, described by the Bose–Hubbard model is investigated. The equilibrium phase diagram at finite‐temperature is computed using the cluster mean‐field (CMF) theory including a finite‐cluster‐size‐scaling. The SF, MI, and normal fluid (NF) phases are characterized as well as the transition or crossover temperatures between them are estimated by computing physical quantities such as the superfluid fraction, compressibility and sound velocity using the CMF method. It is found that the nonlocal correlations included in a finite cluster, when extrapolated to infinite size, leads to quantitative agreement of the phase boundaries with quantum Monte Carlo (QMC) results as well as with experiments. Moreover, it is shown that the von Neumann entanglement entropy within a cluster corresponds to the system's entropy density and that it is enhanced near the SF–MI quantum critical point (QCP) and at the SF–NF boundary. The behavior of the transition lines near this QCP, at and away from the particle‐hole (p–h) symmetric point located at the Mott‐tip, is also discussed. The results obtained by using the CMF theory can be tested experimentally using the quantum gas microscopy method.

Modeling many-body quantum systems is widely regarded as one of the most promising applications for near-term noisy quantum computers. However, in the near term, system size limitation will remain a severe barrier for applications in materials science or strongly correlated systems. A promising avenue of research is to combine many-body physics with machine learning for the classification of distinct phases. We present a workflow that synergizes quantum computing, many-body theory, and quantum machine learning (QML) for studying strongly correlated systems. In particular, it can capture a putative quantum phase transition of the stereotypical strongly correlated system, the Hubbard model. Following the recent proposal of the hybrid quantum-classical algorithm for the two-site dynamical mean-field theory (DMFT), we present a modification that allows the self-consistent solution of the single bath site DMFT. The modified algorithm can be generalized for multiple bath sites. This approach is used to generate a database of zero-temperature wavefunctions of the Hubbard model within the DMFT approximation. We then use a QML algorithm to distinguish between the metallic phase and the Mott insulator phase to capture the metal-to-Mott insulator phase transition. We train a recently proposed quantum convolutional neural network (QCNN) and then utilize the QCNN as a quantum classifier to capture the phase transition region. This work provides a recipe for application to other phase transitions in strongly correlated systems and represents an exciting application of small-scale quantum devices realizable with near-term technology.