New model-independent compact representations of imaginary-time data are presented in terms of the intermediate representation (IR) of analytical continuation. We demonstrate the efficiency of the IR through continuous-time quantum Monte Carlo calculations of an Anderson impurity model. We find that the IR yields a significantly compact form of various types of correlation functions. This allows the direct quantum Monte Carlo measurement of Green's functions in a compressed form, which considerably reduces the computational cost and memory usage. Furthermore, the present framework will provide general ways to boost the power of cutting-edge diagrammatic/quantum Monte Carlo treatments of many-body systems.

The dual-fermion approach offers a way to perform diagrammatic expansion around the dynamical mean field theory. Using this formalism, the influence of antiferromagnetic fluctuations on the self-energy is taken into account through ladder-type diagrams in the particle-hole channel. The resulting phase diagram for the (quasi-) two-dimensional Hubbard model exhibits antiferromagnetism and d-wave superconductivity. Furthermore, a uniform charge instability, i.e., phase separation, is obtained in the low-doping regime around the Mott insulator. We also examine spin/charge density wave fluctuations including d-wave symmetry. The model exhibits a tendency towards an unconventional charge density wave, but no divergence of the susceptibility is found.

An impurity solver based on a continuous-time quantum Monte Carlo method is developed for the Coqblin-Schrieffer model. The Monte Carlo simulation does not encounter a sign problem for antiferromagnetic interactions, and accurately reproduces the Kondo effect. Our algorithm can deal with an arbitrary number N of local degrees of freedom, becomes more efficient for larger values of N , and is hence suitable for models with orbital degeneracy. The dynamical susceptibility and the impurity t-matrix are derived with the aid of the Padé approximation for various values of N , and good agreement is found with other methods and available exact results. We point out that the Korringa-Shiba relation needs correction for a finite value of the exchange interaction.

A new approach of solving the ill-conditioned inverse problem for analytical continuation is proposed. The root of the problem lies in the fact that even tiny noise of imaginary-time input data has a serious impact on the inferred real-frequency spectra. By means of a modern regularization technique, we eliminate redundant degrees of freedom that essentially carry the noise, leaving only relevant information unaffected by the noise. The resultant spectrum is represented with minimal bases and thus a stable analytical continuation is achieved. This framework further provides a tool for analyzing to what extent the Monte Carlo data need to be accurate to resolve details of an expected spectral function.

The single-particle spectrum of the Kondo lattice model is derived with the use of the continuous-time quantum Monte Carlo method, combined with the dynamical mean-field theory. Crossover behavior is traced quantitatively either to a heavy Fermi-liquid state or to a magnetically ordered state from the local-moment state at high temperatures. The momentum distribution in the low-temperature limit acquires a discontinuity at the location that involves the local-spin degrees of freedom. Even without the charge degrees of freedom for local electrons, the excitation spectra exhibit hybridized bands similar to those in the Anderson lattice. Temperature dependence in the zero-energy component of the self-energy is crucial in forming the Fermi-liquid state with the large Fermi surface.