Vertex-Based Diagrammatic Treatment of Light-Matter-Coupled Systems.

We propose a diagrammatic Monte Carlo approach for quantum impurity models, which can be regarded as a generalization of the strong-coupling expansion for fermionic impurity models. The algorithm is based on a self-consistently computed three-point vertex and a stochastically sampled four-point vertex, and it allows one to obtain numerically exact results in a wide parameter regime. The performance of the algorithm is demonstrated with applications to a spin-boson model representing an emitter in a waveguide. As a function of the coupling strength, the spin exhibits a delocalization-localization crossover at low temperatures, signaling a qualitative change in the real-time relaxation. In certain parameter regimes, the response functions of the emitter coupled to the electromagnetic continuum can be described by an effective Rabi model with appropriately defined parameters. We also discuss the spatial distribution of the photon density around the emitter.

Evaluating Second-Order Phase Transitions with Diagrammatic Monte Carlo: Néel Transition in the Doped Three-Dimensional Hubbard Model.

Diagrammatic Monte Carlo-the technique for the numerically exact summation of all Feynman diagrams to high orders-offers a unique unbiased probe of continuous phase transitions. Being formulated directly in the thermodynamic limit, the diagrammatic series is bound to diverge and is not resummable at the transition due to the nonanalyticity of physical observables. This enables the detection of the transition with controlled error bars from an analysis of the series coefficients alone, avoiding the challenge of evaluating physical observables near the transition. We demonstrate this technique by the example of the Néel transition in the 3D Hubbard model. At half filling and higher temperatures, the method matches the accuracy of state-of-the-art finite-size techniques, but surpasses it at low temperatures and allows us to map the phase diagram in the doped regime, where finite-size techniques struggle from the fermion sign problem. At low temperatures and sufficient doping, the transition to an incommensurate spin density wave state is observed.

Entropy in the Non-Fermi-Liquid Regime of the Doped 2D Hubbard Model.

We study thermodynamic properties of the doped Hubbard model on the square lattice in the regime of strong charge and spin fluctuations at low temperatures near the metal-to-insulator crossover and obtain results with controlled accuracy using the diagrammatic Monte Carlo method directly in the thermodynamic limit. The behavior of the entropy reveals a non-Fermi-liquid state at sufficiently high interactions near half filling: A maximum in the entropy at nonzero doping develops as the coupling strength is increased, along with an inflection point, evidencing a metal to non-Fermi-liquid crossover. The specific heat exhibits additional distinctive features of a non-Fermi-liquid state. Measurements of the entropy can, therefore, be used as a probe of the state of the system in quantum simulation experiments with ultracold atoms in optical lattices.

Homotopic Action: A Pathway to Convergent Diagrammatic Theories.

The major obstacle preventing Feynman diagrammatic expansions from accurately solving many-fermion systems in strongly correlated regimes is the series slow convergence or divergence problem. Several techniques have been proposed to address this issue: series resummation by conformal mapping, changing the nature of the starting point of the expansion by shifted action tools, and applying the homotopy analysis method to the Dyson-Schwinger equation. They emerge as dissimilar mathematical procedures aimed at different aspects of the problem. The proposed homotopic action offers a universal and systematic framework for unifying the existing-and generating new-methods and ideas to formulate a physical system in terms of a convergent diagrammatic series. It eliminates the need for resummation, allows one to introduce effective interactions, enables a controlled ultraviolet regularization of continuous-space theories, and reduces the intrinsic polynomial complexity of the diagrammatic Monte Carlo method. We illustrate this approach by an application to the Hubbard model.

Spin and Charge Correlations across the Metal-to-Insulator Crossover in the Half-Filled 2D Hubbard Model.

The 2D Hubbard model with nearest-neighbor hopping on the square lattice and an average of one electron per site is known to undergo an extended crossover from metallic to insulating behavior driven by proliferating antiferromagnetic correlations. We study signatures of this crossover in spin and charge correlation functions and present results obtained with controlled accuracy using the diagrammatic Monte Carlo approach in the range of parameters amenable to experimental verification with ultracold atoms in optical lattices. The qualitative changes in charge and spin correlations associated with the crossover are observed at well-separated temperature scales, which encase the intermediary regime of non-Fermi-liquid character, where local magnetic moments are formed and nonlocal fluctuations in both channels are essential.

Extended Crossover from a Fermi Liquid to a Quasiantiferromagnet in the Half-Filled 2D Hubbard Model.

The ground state of the Hubbard model with nearest-neighbor hopping on the square lattice at half filling is known to be that of an antiferromagnetic (AFM) band insulator for any on-site repulsion. At finite temperature, the absence of long-range order makes the question of how the interaction-driven insulator is realized nontrivial. We address this problem with controlled accuracy in the thermodynamic limit using self-energy diagrammatic determinant Monte Carlo and dynamical cluster approximation methods and show that development of long-range AFM correlations drives an extended crossover from Fermi liquid to insulating behavior in the parameter regime that precludes a metal-to-insulator transition. The intermediate crossover state is best described as a non-Fermi liquid with a partially gapped Fermi surface.

J Freezing and Hund's Rules in Spin-Orbit-Coupled Multiorbital Hubbard Models.

We investigate the phase diagram of the spin-orbit-coupled three orbital Hubbard model at arbitrary filling by means of dynamical mean-field theory combined with the continuous-time quantum Monte Carlo method. We find that the spin-freezing crossover occurring in the metallic phase of the nonrelativistic multiorbital Hubbard model can be generalized to a J-freezing crossover, with J=L+S, in the spin-orbit-coupled case. In the J-frozen regime the correlated electrons exhibit a nontrivial flavor selectivity and energy dependence. Furthermore, in the regions near n=2 and n=4 the metallic states are qualitatively different from each other, which reflects the atomic Hund's third rule. Finally, we explore the appearance of magnetic order from exciton condensation at n=4 and discuss the relevance of our results for real materials.