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arXiv:2305.01882 · Materials

One- and Two-Particle Correlation Functions in the Cluster Perturbation Theory for Cuprates

V. I. Kuz’min, Sergey V. Nikolaev, M. M. Korshunov, Sergey G. Ovchinnikov

Published Jun 27, 2023• 6 citations
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Abstract

The physics of high-Tc superconducting cuprates is obscured by the effect of strong electronic correlations. One way to overcome this problem is to seek an exact solution at least within a small cluster and expand it to the whole crystal. Such an approach is at the heart of cluster perturbation theory (CPT). Here, we developed CPT for the dynamic spin and charge susceptibilities (spin-CPT and charge-CPT), with the correlation effects explicitly taken into account by the exact diagonalization. We applied spin-CPT and charge-CPT to the effective two-band Hubbard model for the cuprates obtained from the three-band Emery model and calculated one- and two-particle correlation functions, namely, a spectral function and spin and charge susceptibilities. The doping dependence of the spin susceptibility was studied within spin-CPT and CPT-RPA, that is, the CPT generalization of the random phase approximation (RPA). In the underdoped region, both our methods resulted in the signatures of the upper branch of the spin excitation dispersion with the lowest excitation energy at the (π,π) wave vector and no presence of low-energy incommensurate excitations. In the high doping region, both methods produced a low energy response at four incommensurate wave vectors in qualitative agreement with the results of the inelastic neutron scattering experiments on overdoped cuprates.

DOI
10.3390/ma16134640

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The physics of high-Tc superconducting cuprates is obscured by the effect of strong electronic correlations. One way to overcome this problem is to seek an exact solution at least within a small cluster and expand it to the whole crystal. Such an approach is at the heart of cluster perturbation theory (CPT). Here, we developed CPT for the dynamic spin and charge susceptibilities (spin-CPT and charge-CPT), with the correlation effects explicitly taken into account by the exact diagonalization. We applied spin-CPT and charge-CPT to the effective two-band Hubbard model for the cuprates obtained from the three-band Emery model and calculated one- and two-particle correlation functions, namely, a spectral function and spin and charge susceptibilities. The doping dependence of the spin susceptibility was studied within spin-CPT and CPT-RPA, that is, the CPT generalization of the random phase approximation (RPA). In the underdoped region, both our methods resulted in the signatures of the upper branch of the spin excitation dispersion with the lowest excitation energy at the (π,π) wave vector and no presence of low-energy incommensurate excitations. In the high doping region, both methods produced a low energy response at four incommensurate wave vectors in qualitative agreement with the results of the inelastic neutron scattering experiments on overdoped cuprates.
In this thesis, we use the dynamical cluster approximation to study strongly correlated electron systems, especially from the angle of two-particle quantities, such as dynamical susceptibilities and vertex functions. The thesis starts with an introduction to the strongly correlated systems, including their definitions, prominent features, applications, difficulties in explaining them theoretically and some numerical approaches developed. The following section is an introduction to one family of strongly correlated systems we focus on in this thesis, the high temperature cuprates. The salient features in their general phase diagrams are described and discussed. Then an overview of the model we used to study high Tc cuprates is provided, with its limitations and extensions. To solve this model, the numerical method we employ for our study is the dynamical mean-field theory, the dynamical cluster approximation and the continuous time auxiliary field impurity solver. the last part of Chap.1 contains brief derivations for these algorithms. In Chap. 2, we apply dynamical cluster approximation to solve the one-band 2D Hub- bard model. The physical quantities of interest are two-particle quantitites, such as the dynamical susceptibility, irreducible vertex functions and full vertex functions. In this chapter, we describe how to obtain these susceptibilities via linear response theory and write down a detailed example for superconducting susceptibility. Finally we show how to calculate two particle quantities within DCA and obtain phase boundary with them. Chap. 3 is based on one of our publications. In this work, we specifically address the problem of optimizing the superconducting transition temperature in the 2D Hubbard model by analyzing wide regions of parameter space in density, interaction, and second- nearest-neighbor hopping strength.We mainly focus on d-wave superconductivity but show results of other symmetries in the last section. Chap. 4 follows another publication of ours. We study the temperature and doping evolution of NMR response in the normal state of the 2D Hubbard model using cluster dynamical mean-field theory. We simulate the Knight shift, the spin-echo decay rate and the spin-lattice relaxation time, and compare them to the cuprates experimental results. The last chapter extends the calculation of the NMR response to the superconducting state with the Nambu formalism. We show the detailed formulas and diagrams to calculate two-particle quantities, including the Dyson-Schwinger equation of motion.
Cluster perturbation theory (CPT) is a computationally economic method commonly used to estimate the momentum- and energy-resolved single-particle Green's function. It has been used extensively in direct comparisons with experiments that effectively measure the single-particle Green's function, e.g., angle-resolved photoemission spectroscopy. However, many experimental observables are given by two-particle correlation functions. CPT can be extended to compute two-particle correlation functions by approximately solving the Bethe-Salpeter equation. We implement this method and focus on the transverse spin susceptibility, measurable via inelastic neutron scattering or with optical probes of atomic gases in optical lattices. We benchmark the method with the one-dimensional Fermi-Hubbard model by comparing with known results.
To shed light on how electronic correlations vary across the phase diagram of the cuprate superconductors, we examine the doping evolution of spin and charge excitations in the single-band Hubbard model using determinant quantum Monte Carlo (DQMC). In the single-particle response, we observe that the effects of correlations weaken rapidly with doping, such that one may expect the random phase approximation (RPA) to provide an adequate description of the two-particle response. In contrast, when compared to RPA, we find that significant residual correlations in the two-particle excitations persist up to 40% hole and 15% electron doping (the range of dopings achieved in the cuprates). These fundamental differences between the doping evolution of single- and multiparticle renormalizations show that conclusions drawn from single-particle processes cannot necessarily be applied to multiparticle excitations. Eventually, the system smoothly transitions via a momentum-dependent crossover into a weakly correlated metallic state where the spin and charge excitation spectra exhibit similar behavior and where RPA provides an adequate description.
A nonzero-temperature technique for strongly correlated electron lattice systems, combining elements of both variational wave function (VWF) approach and expansion in the inverse number of fermionic flavors (1/Nf1/{\mathcal{N}}_{f}1/Nf​), is developed. The departure point, VWF method, goes beyond the renormalized mean-field theory and provides semiquantitative description of principal equilibrium properties of high-Tc{T}_{c}Tc​ superconducting cuprates. The developed here scheme of VWF+1/Nf\text{VWF}+1/{\mathcal{N}}_{f}VWF+1/Nf​, in the leading order provides dynamical spin and charge responses around the VWF solution, generalizing the weak-coupling spin-fluctuation theory to the regime of strong correlations. Thermodynamic corrections to the correlated saddle-point state arise systematically at consecutive orders. Explicitly, VWF+1/Nf\text{VWF}+1/{\mathcal{N}}_{f}VWF+1/Nf​ is applied to evaluate dynamical response functions for the hole-doped Hubbard model and compared with available determinant quantum Monte Carlo data, yielding a good overall agreement in the regime of coherent collective-mode dynamics. The emergence of well-defined spin and charge excitations from the incoherent continua is explicitly demonstrated and a nonmonotonic dependence of the charge-excitation energy on the interaction magnitude is found. The charge mode energy saturates slowly when approaching the strong-coupling limit, which calls for a reevaluation of the $t\text{\ensuremath{-}}J$-model approach to the charge dynamics in favor of more general $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U$ and $t\text{\ensuremath{-}}J\text{\ensuremath{-}}U\text{\ensuremath{-}}V$ models. The results are also related to recent inelastic resonant x-ray and neutron-scattering experiments for the high-Tc{T}_{c}Tc​ cuprates.
We study the three-band Hubbard model for the copper oxide plane of the high-temperature superconducting cuprates using determinant quantum Monte Carlo and the dynamical cluster approximation (DCA) and provide a comprehensive view of the pairing correlations in this model using these methods. Specifically, we compute the pair-field susceptibility and study its dependence on temperature, doping, interaction strength, and charge-transfer energy. Using the DCA, we also solve the Bethe-Salpeter equation for the two-particle Green's function in the particle-particle channel to determine the transition temperature to the superconducting phase on smaller clusters. Our calculations reproduce many aspects of the cuprate phase diagram and indicate that there is an ``optimal'' value of the charge-transfer energy for the model where Tc{T}_{c}Tc​ is largest. These results have implications for our understanding of superconductivity in both the cuprates and other doped charge-transfer insulators.
The generic phase diagram of lightly hole doped high-Tc{T}_{c}Tc​ cuprates hosts an antiferromagnetic insulating phase with well-defined spin-wave excitations. Contrary to the weak-coupling prediction, these modes persist up to the overdoped metallic regime as intense and dispersive paramagnons. Here we report on our study of the low-energy magnetic and charge excitations within the extended Hubbard model at strong coupling, using a modified 1/N1/N1/N expansion method with a variational state serving as the saddle point solution. Despite clear separation of magnetic and Hubbard-UUU energy scales, we find that incipient Mottness affects qualitatively dispersions and widths of magnetic modes throughout the entire phase diagram. The obtained magnetic and charge dynamical structure factors agree semiquantitatively with recent resonant x-ray and neutron scattering data for ${\mathrm{La}}_{2\ensuremath{-}\mathit{x}}{\mathrm{Sr}}_{\mathit{x}}{\mathrm{CuO}}_{4}$ and ${(\mathrm{Bi},\mathrm{Pb})}_{2}{(\mathrm{Sr},\mathrm{La})}_{2}{\mathrm{CuO}}_{6+\ensuremath{\delta}}$ at all available doping levels. The weak-coupling random-phase approximation fails already for underdoped samples, pointing to the nontrivial intertwining of distinct energy scales in cuprate superconductors. The existence of a discrete charge mode which splits off the electron-hole continuum is also predicted.