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arXiv:2301.08176 · Physical review. B./Physical review. B

Pairing susceptibility of the two-dimensional Hubbard model in the thermodynamic limit

Rayan Farid, Maxence Grandadam, J. P. F. LeBlanc

Published May 22, 2023• 7 citations
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Abstract

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 $\mathbf{q}$ 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 $\mathbf{q}=(0,0)$ pairs and present the relative weights of these modes for variation in temperature, doping, and interaction strength.

DOI
10.1103/physrevb.107.195138

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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 dx2\ensuremath−y2{d}_{{x}^{2}\ensuremath{-}{y}^{2}}dx2\ensuremath−y2​-wave pairing is expected in the model in the U/t\ensuremath→0+U/t\ensuremath{\rightarrow}{0}^{+}U/t\ensuremath→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 (\ensuremathπ,\ensuremathπ)(\ensuremath{\pi},\ensuremath{\pi})(\ensuremathπ,\ensuremathπ) that can be well fit as separate bosonic modes. We extract amplitudes and correlation length scales where we find a predominantly local (\ensuremathπ,\ensuremathπ)(\ensuremath{\pi},\ensuremath{\pi})(\ensuremathπ,\ensuremathπ) 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.
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 present a systematic study of the interaction, doping, and layer dependence of the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-wave pairing susceptibility of the Hubbard model for a stacked 2D square lattice. We perform a multi-index perturbative expansion up to fourth order to obtain coefficients in powers of the Hubbard UUU, the interlayer VVV, and the pair-hopping JJJ interactions. We evaluate the vertex diagrams that contribute to the pairing susceptibility for $\ensuremath{\ell}=2,\phantom{\rule{4pt}{0ex}}3,\phantom{\rule{4pt}{0ex}}4$ layered models in the $U\text{\ensuremath{-}}V\text{\ensuremath{-}}J$ interaction space. This provides unprecedented access to the pairing amplitudes, allowing us to identify the processes that enhance or reduce pairing. We distinguish pairing within the diagonal channel, ${P}_{d}^{\ensuremath{\parallel}}$, and off-diagonal channel, ${P}_{d}^{\ensuremath{\perp}}$, and find that, in the absence of JJJ, the qualitative behavior of the layered system is equivalent to the single-layer model. In the presence of JJJ, we show that pairing is enhanced sublinearly with increasing $\ensuremath{\ell}$ and is primarily mediated by the ${P}_{d}^{\ensuremath{\perp}}$ component and find which coefficients and diagram sets are responsible. Finally, we construct a generalized $\ensuremath{\ell}$-dependent equation for ${P}_{d}^{\ensuremath{\perp}}$ to speculate pairing beyond $\ensuremath{\ell}=4$.
A dynamic cluster quantum Monte Carlo algorithm is used to study a spin-susceptibility representation of the pairing interaction for the two-dimensional Hubbard model with an on-site Coulomb interaction equal to the bandwidth for various doping levels. We find that the pairing interaction is well approximated by $\frac{3}{2}\overline{U}{(T)}^{2}\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$, with an effective temperature and doping dependent coupling U‾(T)\overline{U}(T)U(T) and the numerically calculated spin susceptibility $\ensuremath{\chi}(K\ensuremath{-}{K}^{\ensuremath{'}})$. We show that at low temperatures, U‾\overline{U}U may be accurately determined from a corresponding spin-susceptibility based calculation of the single-particle self-energy. We conclude that the strength of the ddd-wave pairing interaction, characterized by the mean-field transition temperature, can be determined from a knowledge of the dressed spin susceptibility and the nodal quasiparticle spectral weight. This has important implications with respect to the questions of whether spin fluctuations are responsible for pairing in the high-Tc{T}_{c}Tc​ cuprates.
We study the finite-temperature Fermi-liquid to non-Fermi-liquid crossover in the two-dimensional Hubbard model for a range of dopings using the self-consistent ladder dual fermion method. We consider relatively high temperatures where by comparing the single-particle self-energy and the two-particle spin susceptibility we identify a suppression of the density of states near the Fermi level caused by a quasi-antiferromagnetic behavior that is itself characterized by a long, but finite, correlation length scale which we extract. We compute the single-particle self-energy, and the charge/spin susceptibilities for the entire doping range at relatively high temperature, and also for the half-filled weak-coupling case at relatively low temperatures. We perform fluctuation diagnostics to decompose the single-particle self-energy into scattering qqq-vector and bosonic frequency contributions in the spin channel. Within this framework we find that the key contributions to the single-particle self-energy that give non-Fermi-liquid character show complicated dependence on zero, positive, and negative frequency bosonic modes. We find that for a nominal choice of ${t}^{\ensuremath{'}}$ that electron- and hole-doped systems with similar Fermi-liquid character have distinct spin and charge susceptibility and distinct spin-fluctuation diagnostics spectra.
We present the exact solution of the one-dimensional extended Hubbard model in the atomic limit within the Green's function and equations of motion formalism. We provide a comprehensive and systematic analysis of the model by considering all the relevant response and correlation functions as well as thermodynamic quantities in the whole parameters space. At zero temperature we identify four phases in the plane (U,n) ( U is the on-site potential and n is the filling) and relative phase transitions as well as different types of charge ordering. These features are endorsed by investigating at T=0 the chemical potential and pertinent local correlators, the particle and double occupancy correlation functions, the entropy, and by studying the behavior in the limit T-->0 of the charge and spin susceptibilities. A detailed study of the thermodynamic quantities is also presented at finite temperature. This study evidences that a finite-range order persists for a wide range of the temperature, as shown by the behavior of the correlation functions and by the two-peak structure exhibited by the charge susceptibility and by the entropy. Moreover, the equations of motion formalism, together with the use of composite operators, allows us to exactly determine the set of elementary excitations. As a result, the density of states can be determined and a detailed analysis of the specific heat allows for identifying the excitations and for ascribing its two-peak structure to a redistribution of the charge density.
Dynamic cluster Monte Carlo calculations for the doped two-dimensional Hubbard model are used to study the irreducible particle-particle vertex responsible for dx2-y2 pairing in this model. This vertex increases with increasing momentum transfer and decreases when the energy transfer exceeds a scale associated with the Q=(pi, pi) spin susceptibility. Using an exact decomposition of this vertex into a fully irreducible two-fermion vertex and charge and magnetic exchange channels, the dominant part of the effective pairing interaction is found to come from the magnetic, spin S=1 exchange channel.