Dynamical quantum-cluster approaches, such as different cluster extensions of the dynamical mean-field theory (cluster DMFT) or the variational cluster approximation (VCA), combined with efficient cluster solvers, such as the quantum Monte Carlo (QMC) method, provide controlled approximations of the single-particle Green's function for lattice models of strongly correlated electrons. To access the thermodynamics, however, a thermodynamical potential is needed. We present an efficient numerical algorithm to compute the grand potential within cluster-embedding approaches that are based on novel continuous-time QMC schemes. It is shown that the numerically exact cluster grand potential can be obtained from a quantum Wang-Landau technique to reweight the coefficients in the expansion of the partition function. The lattice contributions to the grand potential are computed by a proper infinite summation over Matsubara frequencies. A proof of principle is given by applying the VCA to antiferromagnetic (short-range) order in the two-dimensional Hubbard model at finite temperatures.
We formulate a finite-temperature scheme of the variational cluster approximation (VCA) particularly suitable for an exact-diagonalization cluster solver. Based on the analytical properties of the single-particle Green's function matrices, we explicitly show the branch-cut structure of logarithm of the complex determinant functions appearing in the self-energy-functional theory (SFT) and whereby construct an efficient scheme for the finite-temperature VCA. We also derive the explicit formulas for entropy and specific heat within the framework of the SFT. We first apply the method to explore the antiferromagnetic order in a half-filled Hubbard model by calculating the entropy, specific heat, and single-particle excitation spectrum for different values of on-site Coulomb repulsion
U and temperature
T. We also calculate the
T dependence of the single-particle excitation spectrum in the strong coupling region, and discuss the overall similarities to and the fine differences from the spectrum obtained by the spin-density-wave mean-field theory at low temperatures and the Hubbard-I approximation at high temperatures. Moreover, we show a necessary and sufficient condition for the third law of thermodynamics in the SFT. On the basis of the thermodynamic properties, such as the entropy and the double occupancy, calculated via the
T and/or
U derivative of the grand potential, we obtain a crossover diagram in the
(U,T) plane, which separates a Slater-type insulator and a Mott-type insulator. Next, we demonstrate the finite-temperature scheme in the cluster-dynamical-impurity approximation (CDIA), i.e., the VCA with noninteracting bath orbitals attached to each cluster, and study the paramagnetic Mott metal-insulator transition in the half-filled Hubbard model. Formulating the finite-temperature CDIA, we first address a subtle issue regarding the treatment of the artificially introduced bath degrees of freedom which are absent in the originally considered Hubbard model. We then apply the finite-temperature CDIA to calculate the finite-temperature phase diagram in the
(U,T) plane. Metallic, insulating, coexistence, and crossover regions are distinguished from the bath-cluster hybridization-variational-parameter dependence of the grand-potential functional. We find that the Mott transition at low temperatures is discontinuous, and the coexistence region of the metallic and insulating states persists down to zero temperature. The result obtained here by the finite-temperature CDIA is complementary to the previously reported zero-temperature CDIA phase diagram.
The investigation of quantum many-particle systems is a central goal of modern condensed-matter physics. Complex physical phenomena such as high-temperature superconductivity, heavy fermion behavior, and correlation-driven metal-insulator transitions are central aspects of this particular field of science. The Hubbard model is a paradigm of correlated-electron physics. It is one of the simplest quantum-mechanical lattice models capable of capturing relevant physical aspects of strongly correlated electron systems. The Hubbard model also plays an important role in the explanation of many-body phenomena observed in ultra-cold atoms trapped in optical lattices. In this thesis, we study the three-dimensional Hubbard model using quantum Monte Carlo simulations (QMC). Our aim is a precise numerical study of its properties in the thermodynamic limit. However, a direct investigation of lattice models in the thermodynamic limit is generally impossible because QMC simulations cannot treat infinite lattice sizes. Therefore, approximative schemes that allow calculations directly in the thermodynamic limit are advantageous. A widely applied approximation of this kind is the dynamical mean-field theory (DMFT) which maps the infinite-lattice problem onto a single-site impurity model embedded in a mean field. We apply cluster mean-field theories which represent a systematic extension of the DMFT by expanding the single impurity to a finite cluster. The second focus of this thesis is the calculation of spectral properties of the Hubbard model. QMC methods map the quantum-mechanical lattice model on a classical one at the expense of an additional imaginary time dimension. For this reason, QMC algorithms can only provide dynamical data on the imaginary time or frequency axis. The necessary analytic continuation to the physically relevant real axis has proven to be a difficult problem that has to be approached by advanced data analysis tools. The maximum-entropy method (MEM) is the standard technique to handle this problem. We apply the MEM to the problem of extracting momentum-resolved single particle spectra from QMC simulations of the 3D Hubbard model. Furthermore, we discuss and expand algorithms that were recently proposed as alternatives to the standard MEM approach.
This thesis will describe efforts to enhance our ability to simulate the 2D Hubbard model. Chapter 2 provides a background on the main computational techniques used throughout the work, including quantum Monte Carlo (QMC), dynamical mean field theory (DMFT), the dynamical cluster approximation (DCA), the continuous time auxiliary field algorithm (CTAUX), and the Maximum Entropy Method (MEM) for numeric analytic continuation. Chapter 3 presents new work on applying Twisted Boundary Conditions to the DCA framework. This method is applied in a effort to access thermodynamic (i.e. large system) information about the Hubbard model without the immense computational expense required to simulate large lattices directly. Chapter 4 describes efforts to study the extended 2D Hubbard model, which re-introduces non-local interactions between electrons that drive the formation of charge ordered phases. The work includes a thorough analysis of the phase diagram of the model away from half-filling, as well as analysis of the effect of non-local interactions on anti-ferromagnetic fluctuations and competition between the charge order and AFM states. Chapter 5 includes a derivation of the dual fermions diagrammatic expansion and the dual fermions ladder approximation used to compute corrections to single site DMFT calculations. The method is applied to the 2D Hubbard model in order to study the evolution of its spectral function, Fermi surface, and momentum dependent mass renormalization as the temperature, interaction strength, and doping are changed. The great advantage of this method is that we are able to access arbitrary momenta throughout the Brillouin zone, in contrast to the limitation to a few cluster momenta that characterize cluster methods like DCA.
Many phenomena of strongly correlated materials are encapsulated in the Fermi-Hubbard model whose thermodynamic properties can be computed from its grand-canonical potential. In general, there is no closed-form expression of the grand-canonical potential for lattices of more than one spatial dimension, but solutions can be numerically approximated using cluster methods. To model long-range effects such as order parameters, a powerful method to compute the cluster's Green's function consists of finding its self-energy through a variational principle. This allows the possibility of studying various phase transitions at finite temperature in the Fermi-Hubbard model. However, a classical cluster solver quickly hits an exponential wall in the memory (or computation time) required to store the computation variables. Here it is shown theoretically that the cluster solver can be mapped to a subroutine on a quantum computer whose quantum memory usage scales linearly with the number of orbitals in the simulated cluster and the number of measurements scales quadratically. A quantum computer with a few tens of qubits could therefore simulate the thermodynamic properties of complex fermionic lattices inaccessible to classical supercomputers.
Classic numerical approaches to lattice models such as the Hubbard model are usually based on a solution of the model on a periodic lattice with a small number of sites. For instance, Exact Diagonalizations (ED) calculations are performed on periodic systems with no more than ∼ 20 sites and Quantum Monte Carlo (QMC) calculations are limited in practice on systems with . 100 sites. Some extrapolation is then needed to make statements about the thermodynamic (e.g. infinite-size) limit. One advantage of such approaches is their relative simplicity and lack of ambiguity. A disadvantage is that broken symmetry states need careful analysis to be identified, as they are fully revealed only in the thermodynamic limit. Quantum cluster methods are a set of closely related approaches that instead consider a finite cluster of sites embedded in the infinite lattice. The embedding is done by adding to the cluster additional fields or bath degrees of freedom such as to best represent the effect of the surrounding infinite lattice. Variational or self-consistency principles are used to set the values of these extra parameters. In these approaches, broken symmetry states can appear even for the smallest clusters used, somewhat like ordinary mean-field theory. However, unlike mean field theory, these approaches are dynamical and retain the full effect of strong correlations. These methods are usually known by their acronyms:
Even after decades of intense research, the single band Hubbard model representing the fundamental model for interacting quantum systems and electronic correlations remains an unsolved cornerstone paradigm in theoretical solid state physics. Within the non-perturbative dynamical mean-field theory (DMFT), the lattice problem is mapped to a self-consistent auxiliary quantum impurity model leading to a local approximation of the self-energy. Nonlocal correlations can be included by the real-space cluster extension of the DMFT (CDMFT). Enlarging the unit cell makes the calculation of a numerically exact solution in CDMFT very challenging. We here use a quantum Monte-Carlo approach in the imaginary-time space in order to solve the self-consistency equations for different cluster sizes. In order to restore the broken translational symmetry re-periodization schemes for the Green function, the self-energy, or its cumulants have been introduced. However, these suffer from ambiguity and may even lead to convergence problems when attempted inside the self-consistency loop. The comparison to numerically exact diagrammatic quantum Monte Carlo calculations shows that introducing a so-called centerfocused extrapolation (CFE) to approximate the lattice self-energy yields very accurate results. Moreover, the CFE converges faster with the cluster size than previous periodization schemes. We here perform a detailed CDMFT analysis for the single-band Hubbard model on the 2D square lattice, reaching real-space cluster sizes of up to 9x9 sites. In addition to spectral properties, we also compute two-particle correlation functions which are not accessible in DMFT. Using benchmarks against diagrammatic Monte Carlo at high temperature, we show that the cluster spin susceptibility can be extrapolated to the exact result at large cluster size. In particular, the exponential decay of spin-spin correlations is very well captured by CDMFT calculations, even when correlations extend beyond the size of the cluster. We further present results at lower temperature T and larger
U than the range currently accessible with diagrammatic Monte Carlo methods techniques. The CDMFT+CFE represents therefore a powerful computational tool to access the physics of non-local correlations beyond dynamical mean-field theory.