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arXiv:2110.05863 · Physical Review Research

Two-dimensional Hubbard model at finite temperature: Weak, strong, and long correlation regimes

Fedor Šimkovic, Riccardo Rossi, Michel Ferrero

Published Dec 20, 2022• 45 citations
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Abstract

We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as $T=0.067$ and interactions up to $U=7$, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.

DOI
10.1103/physrevresearch.4.043201

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We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as T=0.067T=0.067T=0.067 and interactions up to U=7U=7U=7, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.
We compute the diagrammatic expansion of the particle-particle susceptibility via algorithmic Matsubara integration and compute the correlated pairing susceptibility in the thermodynamic limit of the two-dimensional Hubbard model. We study the static susceptibility and its dependence on the pair momentum q\mathbf{q}q for a range of temperature, interaction strength, and chemical potential. We show that ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing is expected in the model in the $U/t\ensuremath{\rightarrow}{0}^{+}$ limit from direct perturbation theory. From this, we identify key second- and third-order diagrams that support pairing processes and note that the diagrams responsible are not a part of charge or spin susceptibility expansions. We find two key components for pairing at momenta (0,0) and $(\ensuremath{\pi},\ensuremath{\pi})$ that can be well fit as separate bosonic modes. We extract amplitudes and correlation length scales where we find a predominantly local $(\ensuremath{\pi},\ensuremath{\pi})$ pairing and nonlocal q=(0,0)\mathbf{q}=(0,0)q=(0,0) pairs and present the relative weights of these modes for variation in temperature, doping, and interaction strength.
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 a testing ground, 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 methods) 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.Based on this analysis, we explain how quasiparticles can coexist with increasingly longrange antiferromagnetic correlations and why dynamical mean-field theory is found to provide a remarkably accurate approximation of local quantities in the metallic regime.We also critically
The relationship between the pseudogap and underlying ground-state phases has not yet been rigorously established. We investigated the doped two-dimensional Hubbard model at finite temperature using controlled diagrammatic Monte Carlo calculations, allowing for the computation of spectral properties in the infinite-size limit and with arbitrary momentum resolution. We found three distinct regimes as a function of doping and interaction strength: a weakly correlated metal, a correlated metal with strong interaction effects, and a pseudogap regime at low doping. We show that the pseudogap forms both at weak coupling, when the magnetic correlation length is large, and at strong coupling, when it is shorter. As the temperature goes to zero, the pseudogap regime extrapolates precisely to the ordered stripe phase found by ground-state methods.
We study the spin and charge fluctuations of the extended Hubbard model with on-site interaction UUU and first neighbor interaction VVV on the two-dimensional square lattice in the weak to intermediate coupling regime. We propose an extension of the two-particle self-consistent approximation that includes the effect of functional derivatives of the pair-correlation functions on irreducible spin and charge vertices. These functional derivatives were ignored in our previous work. We evaluate them assuming particle-hole symmetry. The resulting theory satisfies conservation laws and the Mermin-Wagner theorem. Our current results are in much better agreement with benchmark quantum Monte Carlo results. This theory allows us to reliably determine the crossover temperatures toward renormalized-classical regimes, and hence, the dominant instability as a function of UUU and VVV. We have considered fillings n=1n=1n=1 and n=0.75n=0.75n=0.75. Either spin or charge fluctuations can dominate. The wave vector is self-determined by the approach.
Abstract. Magnetic and charge susceptibilities of the two-dimensional repulsive Hubbard model are investigated applying a strong coupling diagram technique in which the expansion in powers of the hopping constants is used. For small lattices and high temperatures results are in agreement with Monte Carlo simulations. With the departure from half-filling x the low-frequency magnetic susceptibility becomes incommensurate and the incommensurability parameter grows with x. The incommensurability, its dependence on frequency and on x resemble experimental results in lanthanum cuprates. Also for finite x sharp maxima appear in the static charge susceptibility. The maxima are finite which points to the absence of the long-range charge ordering (static stripes). However, for x ≈ 0.12 the maxima are located near the momenta (0, ±π/2), (±π/2, 0). In this case an interaction of carriers with tetragonal distortions can stabilize stripes with the wavelength of four lattice spacings, as observed in the low-temperature tetragonal phase of cuprates. As follows from the obtained results, the magnetic incommensurability is not a consequence of the stripes. The Hubbard model is thought to be appropriate to describe the main features of electron correlations in narrow energy bands, leading to collective effects such as magnetism and
We utilize numerical linked-cluster expansions (NLCEs) and the determinantal quantum Monte Carlo algorithm to study pairing correlations in the square-lattice Hubbard model. To benchmark the NLCE, we first locate the finite-temperature phase transition of the attractive model to a superconducting state away from half filling. We then explore the superconducting properties of the repulsive model for the ddd-wave and extended sss-wave pairing symmetries. The pairing structure factor shows a strong tendency to ddd-wave pairing and peaks at an interaction strength comparable to the bandwidth. The extended sss-wave structure factor and correlation length are larger at higher temperatures but clearly saturate as temperature is lowered, whereas the ddd-wave counterparts, which start off lower at high temperatures, continue to rise near half filling. This rise is even more dramatic in the ddd-wave susceptibility. The convergence of NLCEs breaks down as the susceptibilities and correlation lengths become large, so we are unable to determine the onset of long-range order. However, our results extend the conclusion, previously restricted to only magnetic and charge correlations, that NLCEs offer a unique window into pairing in the Hubbard model at strong coupling.