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arXiv:2406.02656 · Physical review. B./Physical review. B

Pairing susceptibility in the weakly interacting multilayer Hubbard model evaluated by direct perturbative expansion

Rayan Farid, J. P. F. LeBlanc

Published Oct 11, 2024• 3 citations
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Abstract

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 $U$, the interlayer $V$, and the pair-hopping $J$ 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 $J$, the qualitative behavior of the layered system is equivalent to the single-layer model. In the presence of $J$, 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 $\ensuremath{\ell}=4$.

DOI
10.1103/physrevb.110.144511

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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 ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$-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.
Using numerical diagonalization of a $4\ifmmode\times\else\texttimes\fi{}4$ cluster, we calculate on-site s, extended-s,s,s, and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ pairing correlation functions (PCF's) in an effective generalized Hubbard model for the cuprates, with nearest-neighbor correlated hopping and next-nearest-neighbor hopping ${t}^{\ensuremath{'}}.$ The vertex contributions to the PCF's are significantly enhanced, relative to the ${t\ensuremath{-}t}^{\ensuremath{'}}\ensuremath{-}U$ model. The behavior of the PCF's and their vertex contributions, and signatures of anomalous flux quantization, indicate superconductivity in the d-wave channel for moderate doping and in the s-wave channel for high doping and small U.
Pairing operators of electrons with an arbitrary distance on a square lattice are defined in each possible wave channel, s, p, dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2, dxy{\mathit{d}}_{\mathit{x}\mathit{y}}dxy​, and gx3{\mathit{g}}_{\mathit{x}}^{3}gx3​y-xy3{\mathit{xy}}^{3}xy3. Exact identities among pairing amplitudes in the Hubbard model are presented, giving powerful constraints on the nature of pairings. At half filling, some of the pairing amplitudes, including those of the second-shortest pairing distance, vanish identically in each wave channel except the p wave. There are correlations among pairing amplitudes away from half filling, as well as nonidentically vanishing ones at half filling. Even if the pairing amplitudes vanish up to a certain distance, those of larger distances can still possibly be nonzero. The order of magnitude is estimated for the pairine amplitudes, indicating that those of large distances can by no means be neglected. The scheme is also used in the extended Hubbard model including nearest-neighbor density-density interaction and that including next-nearest-neighbor hopping. In that including nearest-neighbor interaction, the features of pairing amplitudes are just the same as in the original Hubbard model at half filling, and are also similar away from half filling, but with the difference that the amplitude is now dependent on the nearest-neighbor interaction. In the model including next-nearest-neighbor hopping, the features are quite different, the constraint identities are more underconstrained, and the amplitude has no monotonic dependence upon the pairing distance if the next-nearest-neighbor hopping has an opposite sign to the nearest-neighbor hopping as required physically. The same scheme is also used to discuss the possible pairings in the coupled two-plane Hubbard model and in the infinite-layer Hubbard model.
We have studied the pair susceptibilities for all possible pair wave functions that fit on a two-dimensional (2D) eight-site Hubbard cluster by exact diagonalization of the Hamiltonian. Band fillings corresponding to four and six electrons were studied (two or four holes in the half-filled band) for a wide range of Hubbard interaction strengths and temperatures. Our results show that all pairing susceptibilities are suppressed by the Hubbard repulsion. We have also carried out perturbation-theory calculations which show that the leading-order U2{U}^{2}U2 contributions to the d-wave pair susceptibility suppresses d-wave pairing over a significant temperature range. These results are consistent with recent Monte Carlo results and provide further evidence suggesting that the 2D Hubbard model does not exhibit superconductivity.
We investigate the Hubbard model on a two-dimensional square lattice by the perturbation expansion to the fourth order in the on-site Coulomb repulsion U. Numerically calculating all diagrams up to the fourth order in self-energy, we examine the convergence of perturbation series in the lattice system. We indicate that the coefficient of each order term rapidly decreases as in the impurity Anderson model for T > 0.1t in the half-filled case, but it holds in the doped case even at lower temperatures. Thus, we can expect that the convergence of perturbation expansion in U is very good in a wide parameter region also in the lattice system, except for T < 0.1t in the half-filled case. We next calculate the density of states in the fourth-order perturbation. In the half-filled case, the shape in a moderate correlation regime is quite different from the three peak structure in the second-order perturbation. Remarkable upper and lower Hubbard bands locate at w = +(-)U/2, and a pseudogap appears at the Fermi level w=0. This is considered as the precursor of the Mott-Hubbard antiferromagnetic structure. In the doped case, quasiparticles with very heavy mass are formed at the Fermi level. Thus, we conclude that the fourth-order perturbation theory overall well explain the asymptotic behaviors in a strong correlation regime.
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.