The Hubbard model represents the fundamental model for interacting quantum systems and electronic correlations. Using the two-dimensional half-filled Hubbard model at weak coupling as testing grounds, we perform a comparative study of a comprehensive set of state of the art quantum many-body methods. Upon cooling into its insulating antiferromagnetic ground-state, the model hosts a rich sequence of distinct physical regimes with crossovers between a high-temperature incoherent regime, an intermediate temperature metallic regime and a low-temperature insulating regime with a pseudogap created by antiferromagnetic fluctuations. We assess the ability of each method to properly address these physical regimes and crossovers through the computation of several observables probing both quasiparticle properties and magnetic correlations, with two numerically exact methods (diagrammatic and determinantal quantum Monte Carlo) serving as a benchmark. By combining computational results and analytical insights, we elucidate the nature and role of spin fluctuations in each of these regimes and explain, in particular, how quasiparticles can coexist with increasingly long-range antiferromagnetic correlations in the metallic regime. We also critically discuss whether imaginary time methods are able to capture the non-Fermi liquid singularities of this fully nested system.

Tracking the Footprints of Spin Fluctuations: A MultiMethod, MultiMessenger Study of the Two-Dimensional Hubbard Model

A new diagrammatic technique is developed to describe nonlocal effects (e.g., pseudogap formation) in the Hubbard-like models. In contrast to cluster approaches, this method utilizes an exact transition to the dual set of variables, and it therefore becomes possible to treat the irreducible vertices of an effective {\it single-impurity} problem as small parameters. This provides a very efficient interpolation between weak-coupling (band) and atomic limits. The antiferromagnetic pseudogap formation in the Hubbard model is correctly reproduced by just the lowest-order diagrams.

Dual fermion approach to nonlocal correlations in the Hubbard model

While the breakdown of the perturbation expansion for the many-electron problem has several formal consequences, here we unveil its physical effect: flipping the sign of the effective electronic interaction in specific scattering channels. By decomposing local and uniform susceptibilities of the Hubbard model via their spectral representations, we prove how entering the nonperturbative regime causes an enhancement of the charge response, ultimately responsible for the phase-separation instabilities close to the Mott metal-insulator transition. Our analysis opens a new route for understanding phase transitions in the nonperturbative regime and clarifies why attractive effects emerging from a strong repulsion can induce phase separations but not s-wave pairing or charge-density wave instabilities.

Attractive Effect of a Strong Electronic Repulsion: The Physics of Vertex Divergences.

The quantum circuit model is the de-facto way of designing quantum algorithms. Yet any level of abstraction away from the underlying hardware incurs overhead. In the era of near-term, noisy, intermediate-scale quantum (NISQ) hardware with severely restricted resources, this overhead may be unjustifiable. In this work, we develop quantum algorithms for Hamiltonian simulation "one level below" the circuit model, exploiting the underlying control over qubit interactions available in principle in most quantum hardware implementations. We then analyse the impact of these techniques under the standard error model where errors occur per gate, and an error model with a constant error rate per unit time. 
To quantify the benefits of this approach, we apply it to a canonical example: time-dynamics simulation of the 2D spin Fermi-Hubbard model. We derive analytic circuit identities for efficiently synthesising multi-qubit evolutions from two-qubit interactions. Combined with new error bounds for Trotter product formulas tailored to the non-asymptotic regime and a careful analysis of error propagation under the aforementioned per-gate and per-time error models, we improve upon the previous best methods for Hamiltonian simulation by multiple orders of magnitude. By our calculations, for a 5$\mathbf\times$5 Fermi-Hubbard lattice we reduce the circuit depth from 800,160 to 1460 in the per-gate error model, or the circuit-depth-equivalent to 440 in the per-time error model. This brings Hamiltonian simulation, previously beyond reach of current hardware for non-trivial examples, significantly closer to being feasible in the NISQ era.

/pdf/hamiltonian-simulation-algorithms-for-near-term-quantum-3qb95438cd.pdf

Hamiltonian Simulation Algorithms for Near-Term Quantum Hardware.

We propose a formalism to take account of the correction of the spatial fluctuations to the local self-energy obtained by the dynamical mean-field approximation. For this purpose, the approximate dynamical susceptibility in the framework of the iterated perturbation theory is proposed and examined. Using the formalism, it is demonstrated that the one-particle spectral intensity in the two-dimensional Hubbard model at half-filling exhibits the pseudo-gap behavior in the central coherent quasiparticle peak due to the critical antiferromagnetic fluctuation. The specific heat is considerably enhanced by the short-range order, which assists a tendency of the Mott localization showing the reduction of the double occupancy. We briefly discuss a formulation for the superconducting transition temperature in the present approximation.

Influence of Spatial Correlations in Strongly Correlated Electron Systems : Extension to Dynamical Mean Field Approximation(Condensed Matter: Electronic Structure, Electrical, Magnetic and Optical Properties)

We study a simple model system for the conjugated $\ensuremath{\pi}$ bonds in benzene, the Pariser-Parr-Pople (PPP) model, within the parquet approximation (PA), exemplifying the prospects of the PA for molecules. Advantages of the PA are its polynomial scaling with the number of orbitals, and the natural calculation of one- and two-particle spectral functions as well as of response and correlation functions. We find large differences in the electronic correlations in the PPP model compared to a Hubbard model with only local interactions. The quasiparticle renormalization (or mass enhancement) is much weaker in the PPP than in the Hubbard model, but the static part of the self-energy enhances the band gap of the former. Furthermore, the vertex corrections to the optical conductivity are much more important in the PPP model. Because nonlocal interactions strongly alter the self-energy, we conclude that the PA is more suitable for calculating conjugated $\ensuremath{\pi}$ bonds in molecules than single site dynamical mean-field theory.

Parquet approximation for molecules: Spectrum and optical conductivity of the Pariser-Parr-Pople model

The interaction of light with solids gives rise to new bosonic quasiparticles, with the exciton being---undoubtedly---the most famous of these polaritons. While excitons are the generic polaritons of semiconductors, we show that for strongly correlated systems another polariton is prevalent---originating from the dominant antiferromagnetic or charge density wave fluctuations in these systems. As these are usually associated with a wave vector $(\ensuremath{\pi},\ensuremath{\pi},\dots{})$ or close to it, we propose to call the derived polaritons $\ensuremath{\pi}$-tons. These $\ensuremath{\pi}$-tons yield the leading vertex correction to the optical conductivity in all correlated models studied: the Hubbard, the extended Hubbard model, the Falicov-Kimball, and the Pariser-Parr-Pople model, both in the insulating and in the metallic phase.

Generic Optical Excitations of Correlated Systems: π-tons.

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Parquet approximation for molecules: Spectrum and optical conductivity of the Pariser-Parr-Pople model

Generic Optical Excitations of Correlated Systems: π-tons.