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arXiv:2307.02681 · Physical review. B./Physical review. B

Superconductivity in the two-dimensional Hubbard model with cellular dynamical mean-field theory: A quantum impurity model analysis

C. Walsh, M. Charlebois, P. Sémon, A.–M. S. Tremblay, G. Sordi

Published Aug 28, 2023• 5 citations
arXivHTMLPDF

Abstract

Doping a Mott insulator gives rise to unconventional superconducting correlations. Here we address the interplay between $d$-wave superconductivity and Mott physics using the two-dimensional Hubbard model with cellular dynamical mean-field theory on a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette. Our approach is to study superconducting correlations from the perspective of a cluster quantum impurity model embedded in a self-consistent bath. At the level of the cluster, we calculate the probabilities of the possible cluster electrons configurations. Upon condensation, we find an increased probability that cluster electrons occupy a four-electron singlet configuration, enabling us to identify this type of short-range spin correlation as key to superconducting pairing. The increased probability of this four-electron singlet comes at the expense of a reduced probability of a four-electron triplet with no significant probability redistribution of fluctuations of charges. This allows us to establish that superconductivity at the level of the cluster primarily involves a reorganization of short-range spin correlations rather than charge correlations. We gain information about the bath by studying the spectral weight of the hybridization function. Upon condensation, we find a transfer of spectral weight leading to the opening of a superconducting gap. We use these insights to interpret the signatures of superconducting correlations in the density of states of the system and in the zero-frequency spin susceptibility.

DOI
10.1103/physrevb.108.075163

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Doping a Mott insulator gives rise to unconventional superconducting correlations. Here we address the interplay between ddd-wave superconductivity and Mott physics using the two-dimensional Hubbard model with cellular dynamical mean-field theory on a 2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2 plaquette. Our approach is to study superconducting correlations from the perspective of a cluster quantum impurity model embedded in a self-consistent bath. At the level of the cluster, we calculate the probabilities of the possible cluster electrons configurations. Upon condensation, we find an increased probability that cluster electrons occupy a four-electron singlet configuration, enabling us to identify this type of short-range spin correlation as key to superconducting pairing. The increased probability of this four-electron singlet comes at the expense of a reduced probability of a four-electron triplet with no significant probability redistribution of fluctuations of charges. This allows us to establish that superconductivity at the level of the cluster primarily involves a reorganization of short-range spin correlations rather than charge correlations. We gain information about the bath by studying the spectral weight of the hybridization function. Upon condensation, we find a transfer of spectral weight leading to the opening of a superconducting gap. We use these insights to interpret the signatures of superconducting correlations in the density of states of the system and in the zero-frequency spin susceptibility.
We use cluster dynamical mean-field theory to study the simplest models of correlated electrons, the Hubbard model and the $t\text{\ensuremath{-}}J$ model. We use a plaquette embedded in a medium as a reference frame to compute and interpret the physical properties of these models. We study various observables such as electronic lifetimes, one electron spectra, optical conductivities, superconducting stiffness, and the spin response in both the normal and the superconducting state in terms of correlation functions of the embedded cluster. We find that the shortest electron lifetime occurs near optimal doping where the superconducting critical temperature is maximal. A second critical doping connected to the change of topology of the Fermi surface is also identified. The mean-field theory provides a simple physical picture of three doping regimes, the underdoped, the overdoped, and the optimally doped regime, in terms of the physics of the quantum plaquette impurity model. We compare the plaquette dynamical mean-field theory results with earlier resonating valence bond mean-field theories, noting the improved description of the momentum space anisotropy of the normal state properties and the doping dependence of the coefficient of the linear temperature dependence of the superfluid density in the superconducting state.
Abstract We investigate the interplay between the pseudogap state and d-wave superconductivity in the two-dimensional doped Hubbard model by employing an eight-site cluster dynamical mean-field theory method. By tuning electron hopping parameters, the strong-coupling pseudogap in the two-dimensional Hubbard model can be either enhanced or suppressed in the doped Mott insulator regime. We find that in underdoped cases, the closing of pseudogap leads to a significant enhancement of superconductivity, indicating competition between the two in the underdoped regime. In contrast, at large dopings, suppressing the pseudogap is accompanied by a concurrent decrease in the superconducting transition temperature T c, which can be attributed to a reduction in antiferromagnetic correlations behind both the pseudogap and superconductivity. We elucidate this evolving relationship between pseudogap and superconductivity across different doping regimes.
We study the competition of antiferromagnetism and ddd-wave superconductivity at zero temperature in the two-dimensional Hubbard model using cellular dynamical mean-field theory for a $2\ifmmode\times\else\texttimes\fi{}2$ plaquette, and solve the associated cluster impurity model at zero temperature by means of exact diagonalization. The interplay between the two phases depends strongly on the strength of the correlation. At strong coupling $(U\ensuremath{\gtrsim}8t)$ the two phases do not mix, and a first-order transition takes place as a function of doping between two pure phases. At weak coupling $(U\ensuremath{\lesssim}8t)$ the two order parameters coexist within the same solution in a range of doping and the system smoothly evolves from the antiferromagnet to the superconductor. When the transition between the superconducting and the antiferromagetic phases is of the first-order, it is accompanied by a phase separation.
Understanding the impact of electronic correlations onto the properties of cuprate superconductors is one of the most prominent tasks of solid state physics. In particular, the intermingling of short range strong correlations, driving the Mott or charge-transfer insulator in the parent compound, and the long range antiferromagnetic scattering processes in the highly doped regime poses severe difficulties to the theoretical description. This thesis studies the impact of short and intermediate ranged strong correlations on the electronic properties of the 2D Hubbard model in the half-filled and doped cases by means of cellular dynamical mean-field theory (CDMFT), i.e. real-space clusters embedded in a dynamical mean-field. Specifically, the effect of incorporating gradually longer ranged correlations onto the metal-insulator transition is analysed and dynamic next-neighbour spin correlations are identified as the cause of the insulating state. Furthermore, by introducing the Bethe-Salpeter scheme for embedded clusters, spin, charge and pairing susceptibilities are computed, incorporating the short ranged strong correlations exactly via the effective interaction of CDMFT and at the same time treating long-ranged fluctuations in a perturbative manner. Firstly, this allows to study the charge response at the Mott transition, identifying the non-local extended s-wave character of the transition. Secondly, in the hole-doped case, the pseudogap, the Knight-shift, a d-wave superconducting instability and unilateral charge instabilities with a periodicity of 4 lattice sites in the s-wave and d-wave channel are observed, all of which strongly depend on next-neighbour spin singlet-like correlations contained in the effective interaction of CDMFT. This work paves the way for computing the conductivity at the boundary between the pseudogap regime and the Fermi liquid. This doctoral thesis was conducted at the Max Planck Institute for Solid State Research and the University of Stuttgart.
An intricate interplay between superconductivity, pseudogap, and Mott transition, either bandwidth driven or doping driven, occurs in materials. Layered organic conductors and cuprates offer two prime examples. We provide a unified perspective of this interplay in the two-dimensional Hubbard model within cellular dynamical mean-field theory on a 2×2 plaquette and using the continuous-time quantum Monte Carlo method as impurity solver. Both at half filling and at finite doping, the metallic normal state close to the Mott insulator is unstable to d-wave superconductivity. Superconductivity can destroy the first-order transition that separates the pseudogap phase from the overdoped metal, yet that normal state transition leaves its marks on the dynamic properties of the superconducting phase. For example, as a function of doping one finds a rapid change in the particle-hole asymmetry of the superconducting density of states. In the doped Mott insulator, the dynamical mean-field superconducting transition temperature T(c)(d) does not scale with the order parameter when there is a normal-state pseudogap. T(c)(d) corresponds to the local pair formation temperature observed in tunneling experiments and is distinct from the pseudogap temperature.
In this thesis we study the strongly-correlated-electron physics of the longstanding H-Tc-superconductivity problem using a non-perturbative method, the Dynamical Mean Field Theory (DMFT), capable to go beyond standard perturbation-theory techniques. DMFT is by construction a local theory which neglects spatial correlation. Experiments have however shown that the latter is a fundamental property of cuprate materials. In a first step, we approach the problem of spatial correlation in the normal state of cuprate materials using a phenomenological Fermi-Liquid-Boltzmann model. We then introduce and develop in detail an extension to DMFT, the Cellular Dynamical Mean Field Theory (CDMFT), capable of considering short-ranged spatial correlation in a system, and we implement it using the exact diagonalization algorithm . After benchmarking CDMFT with the exact one-dimensional solution of the Hubbard Model, we employ it to study the density-driven Mott metal-insulator transition in the two-dimensional Hubbard Model, focusing in particular on the anomalous properties of the doped normal state close to the Mott insulator. We finally study the superconducting state. We show that within CDMFT the one-band Hubbard Model supports a d-wave superconductive state, which strongly departs from the standard BCS theory. We conjecture a link between the instabilities found in the normal state and the onset of superconductivity.