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Dynamics of strong-coupling models for cuprate superconductors: Exact results on finite lattices

H. B. Schuettler, C.-X. Chen, A. J. Fedro

Published Jan 1, 1989• 0 citations
OpenAlex

Abstract

We discuss recent applications of exact numerical continued fraction expansion (CFE) techniques to calculate dynamical correlation functions of various strong-coupling models related to the high-temperature cuprate superconductors. For the two-dimensional square-lattice spin-1/2 Heisenberg antiferromagnet, we present exact results for the zero-temperature dynamical structure factor on finite-sized lattices and compare them to approximate results from a Schwinger boson mean-field theory, recently proposed by Arovas and Auerbach. We find that the mean-field theory represents a very good approximation to the exact spin excitation spectra and to the static spin correlations. We then investigation the dynamical spin-spin structure factor and the single-particle spectral function for finite model clusters with dopant induced hole-type charge carriers, in the strong-coupling limit of both the single-band Hubbard model (t-J-model) and the three-band Hubbard model (Kondo-Heisenberg model). Our results are consistent with the physical picture recently proposed by Zhang and Rice which implies an approximate mapping of the low-energy states in the three-band model onto an effective single-band theory. 33 refs., 4 figs.

Related papers

Hole excitation spectra in cuprate superconductors: A comparative study of single- and multiple-band strong-coupling theories

C.-H. Chen, H.-B. Schüttler, A. J. Fedro

We compare ground-state properties as well as spin and single-particle excitation spectra of a single dopant-induced hole carrier in strong-coupling-limit single-orbital and three-orbital Hubbard model lattices with up to 16 square lattice sites. Our results indicate that there is a close correspondence in their ground-state and low-lying excited-state properties, consistent with the recent proposal by Zhang and Rice (ZR). For the three-orbital model, we also find direct spectroscopic evidence for the singlet-triplet excitation gap associated with the dopant-induced ZR Cu-O spin-hole singlet state.

Physical review. B, Condensed matter•Feb 1, 1990•43 citations

Correlation strength, gaps, and particle-hole asymmetry in high- T c cuprates: A dynamical mean field study of the three-band copper-oxide model

Luca de’ Medici, Xin Wang, Massimo Capone +1 more

The three-band model relevant to high-temperature copper-oxide superconductors is solved using single-site dynamical mean field theory and a tight-binding parametrization of the copper and oxygen bands. For a band filling of one hole per unit cell the metal/charge-transfer-insulator phase diagram is determined. The electron spectral function, optical conductivity, and quasiparticle mass enhancement are computed as functions of electron and hole doping for parameters such that at one hole per cell the paramagnetic phase is insulating and for parameters such that at one hole per unit cell the paramagnetic phase is metallic. The optical conductivity is computed using the Peierls phase approximation for the optical matrix elements. The calculation includes the physics of ``Zhang-Rice singlets.'' The effects of antiferromagnetism on the magnitude of the gap and the relation between correlation strength and doping-induced changes in state density are determined. Three-band and one-band models are compared. The two models are found to yield quantitatively consistent results for all energies less than about 4 eV including energies in the vicinity of the charge-transfer gap. Parameters on the insulating side of the metal/charge-transfer insulator phase boundary lead to gaps which are too large and near-gap conductivities which are too small relative to data. The results place the cuprates clearly in the intermediate correlation regime, on the paramagnetic metal side of the metal/charge-transfer insulator phase boundary.

arXiv:0804.2250•Physical Review B•Aug 4, 2009•58 citations

Physics of cuprates with the two-band Hubbard model: The validity of the one-band Hubbard model

Alexandru Macridin, Mark Jarrell, Thomas Maier +1 more

We calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respect to electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen in one-particle spectral functions and in the charge, spin, and $d$-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhang and Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, we have found similar low-energy physics provided that the next-nearest-neighbor hopping term ${t}^{\ensuremath{'}}$ has a significant value $({t}^{\ensuremath{'}}∕t\ensuremath{\approx}0.3)$. The parameter ${t}^{\ensuremath{'}}$ is the main culprit for the electron-hole asymmetry. However, a significant value of ${t}^{\ensuremath{'}}$ cannot be provided in a strict Zhang and Rice [Phys. Rev. B 37, R3759 (1988); 41, 7243 (1990)] picture where the extra holes added into the system bind to the existing Cu holes forming local singlets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of ${t}^{\ensuremath{'}}$, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromagnetism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describe physical properties involving energy scales larger than about 0.5 eV.

arXiv:cond-mat/0411092•Physical Review B•Apr 29, 2005•95 citations

One- and Two-Particle Correlation Functions in the Dynamical Quantum Cluster Approach

S. Hochkeppel

This thesis is dedicated to a theoretical study of the 1-band Hubbard model in the strong coupling limit. The investigation is based on the Dynamical Cluster Approximation (DCA) which systematically restores non-local corrections to the Dynamical Mean Field approximation (DMFA). The DCA is formulated in momentum space and is characterised by a patching of the Brillouin zone where momentum conservation is only recovered between two patches. The approximation works well if k-space correlation functions show a weak momentum dependence. In order to study the temperature and doping dependence of the spin- and charge excitation spectra, we explicitly extend the Dynamical Cluster Approximation to two-particle response functions. The full irreducible two-particle vertex with three momenta and frequencies is approximated by an effective vertex dependent on the momentum and frequency of the spin and/or charge excitations. The effective vertex is calculated by using the Quantum Monte Carlo method on the finite cluster whereas the analytical continuation of dynamical quantities is performed by a stochastic version of the maximum entropy method. A comparison with high temperature auxiliary field quantum Monte Carlo data serves as a benchmark for our approach to two-particle correlation functions. Our method can reproduce basic characteristics of the spin- and charge excitation spectrum. Near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin and single-particle excitations: a collective spin mode emerges at optimal doping and sufficiently low temperatures in the spin response spectrum and exhibits the energy scale of the magnetic exchange interaction J. Simultaneously, the low energy single-particle excitations are characterised by a coherent quasiparticle with bandwidth J. The origin of the quasiparticle can be quite well understood in a picture of a more or less antiferromagnetic ordered background in which holes are dressed by spin-excitations to allow for a coherent motion. By increasing doping, all features which are linked to the spin-polaron vanish in the single-particle as well as two-particle spin response spectrum. In the second part of the thesis an analysis of superconductivity in the Hubbard model is presented. The superconducting instability is implemented within the Dynamical Cluster Approximation by essentially allowing U(1) symmetry breaking baths in the QMC calculations for the cluster. The superconducting transition temperature T_c is derived from the d-wave order parameter which is directly estimated on the Monte Carlo cluster. The critical temperature T_c is in astonishing agreement with the temperature scale estimated by the divergence of the pair-field susceptibility in the paramagnetic phase. A detailed study of the pseudo and superconducting gap is continued by the investigation of the local and angle-resolved spectral function.

•Jan 1, 2008•0 citations

Correlation effects in low dimensional quantum systems

Chun Yang

This dissertation discussed four low-dimensional systems: Hubbard ladder and its 2D extension, anisotropic Hubbard ladder, one-dimensional two-orbital t-J model, and Kondo model. All of them are of strongly correlated. Doped cuprates, as strongly correlated systems, have attracted a lot of attention in the condensed matter physics community, not just from the purely theoretical perspective, but also due to the novel applications beyond the conventional semiconductor paradigm. They display complicated phase diagram, defying the conventional rigid band theory, shaking our common sense and driving us to wrestle with many-body physics. With the aid of advanced experimental techniques such as ARPES, its various exotic electronic structure, such as waterfalls, hole pockets, Fermi arcs and kinks have been unveiled and remain open to interpretation. To attack such complicated problems, people usually build various models to mimic the CuO2 planes, which is the common feature of various cuprate compounds and is believed to be mainly responsible for their complicated properties, especially the high temperature superconductivity. In Chapter 3, we simulated the spectral properties of the 2- leg single-band Hubbard ladder and its 2-dimensional extension at and away from half-lling using the numerical tool: the combination of Cluster Perturbation Theory (CPT) with the time-dependent density matrix renormalization group method (tDMRG) as a solver. The main features of the spectrum can be described with a mean-eld dispersion, with kinks and pseudogap traced back to scattering between spin and charge degrees of freedom. As a bonus, we nd that the spectrum of hubbard ladder involves spin-polaron, the bound state of spinon and holon. Interacting one-dimensional electron systems generically behave dierent from higher dimensional systems in the sense that its low-energy excitations are described by Luttinger liquid theory as collective bosonic modes carrying spin and charge separately, the so-called spinons and holons, rather than the coherent quasi-particles in Fermi liquid theory. This phenomenon is known as spin-charge separation, an intrinsic property of 1D system. It can be thought of as a particular case of electron fractionalization. When multiple 1D chains get coupled, how the picture changes? In Chapter 4, we study the spectral function of the anisotropic 2-leg Hubbard ladder, one of the simplest multiple-chain systems. The spectrum can be understood as the scattering resultants of spinon and holons: As the inter-chain hopping is increased, the continuum in the bonding channel moves to higher energies and spinon and holon branches merge into a single coherent quasi-particle band. Simultaneously, the spectrum undergoes a crossover from a regime with two minima at incommensurate values of kx (a Mott insulator), to one with a single minimum at kx = (a band insulator). In addition, the spectrum contains a continuum of scattering states consisting of a triplon and a polaron, accompanied by the formation of bound states with higher spin 3 2 , ferromagnetic \spin bags" in which the hole can move freely. To analyze the formation of quasi-particle, we study the time-evolution of charge and spin degrees of freedom in real space after a hole is injected, which indicates that incoherent holons and spinons can emerge into polarons, that can propagate coherently, after a characterstic time . Multiorbital systems can involve Mott and Hund physics, and are rich in various phenomena, including orbital selective Mott transitions, spin-orbital separations, and pairing density wave due to the complex interplay between spin, charge and orbital degrees of freedom. Therefore, it can capture many unique features in real strongly correlated systems, such as iron pnictides and ruthenates, and thus attract a lot of investigations. In Chapter 5, we comprehensively study a one-dimensional two-orbital model at and below quarter lling, which realizes a number of unconventional phases, by means of DMRG and some analytical calculations. In particular, we nd that the system supports an extensive density wave in which excitons quasicondense with nite center-of-mass momentum. In this phase, excitons behave as hardcore bosons without charge order. In addition, excitons can pair to form biexcitons in a state that is close to a charge density wave instability. Finally, a genuine orbital-selective paired state (metallic plus spin-gapped superconductor) can be realized when pairing dominates over the interorbital repulsion. Kondo problem, as one of the most studied and best understood correlated systems, has been studied by various techniques, but the internal structure of its ground state has two deeply con icting pictures: At strong coupling limit, the system is essentially composed of a scattering center, which is a tightly bound singlet formed by the impurity and a localized electron at r0, and the remaining conduction electrons that are decoupled to the impurity and remain almost \free". At weak enough couplings, the singlet extends to the entire volume and the impurity will couple mostly to one electron at the Fermi level. In chapter 6, we provide a coherent picture to unveil the puzzle in the context of quantum information theory by deconstructing the Kondo singlet, both in real and energy space, and measuring the spin correlations between the impurity and the invididual free electron eigenstate based on two dierent measures. As a bonus, we can identify \projected natural orbitals" that can be used as a \good" approximate wavefunctions for simulations, and a quasi-order parameter that determines the crossover from weak to strong coupling regimes and furthermore oers the means to measure the Kondo screening length K. This screening length might be used to characterize the \size" of the Kondo cloud.

•Jan 1, 2018•0 citations

Towards effective models for low-dimensional cuprates: From ground-state Hamiltonian reconstruction to spectral functions

Hannah Lange, Tizian Blatz, Ulrich Schollwöck +2 more

Understanding which minimal effective model captures the essential physics of cuprates is a key step towards unraveling the mechanism behind high- T c superconductivity. Recent measurements of the dynamical spin structure factor (DSF) in cuprate ladder compounds have indicated the presence of an additional, attractive term in the single-band Hubbard model, potentially originating from electron-phonon interactions. Here, we demonstrate that similar DSF features can also be captured by t − J descriptions with a smaller attractive term. Motivated by this observation, we systematically investigate the strength and origin of different contributions to the single-band Hamiltonians by downfolding either from the three-band Emery model or the electron-phonon coupled Hubbard-Holstein model. For one-dimensional systems, we find that the extended versions of both single-band descriptions can reproduce the experimentally observed DSF signatures. Finally, we extend our analysis to two dimensions by comparing two-hole correlation functions for the different single-band models. Our results provide new insights into the long-standing question of which single-band Hamiltonian can capture the essential physics of cuprates.

arXiv:2509.06947•Physical review. B./Physical review. B•Jan 26, 2026•0 citations
We discuss recent applications of exact numerical continued fraction expansion (CFE) techniques to calculate dynamical correlation functions of various strong-coupling models related to the high-temperature cuprate superconductors. For the two-dimensional square-lattice spin-1/2 Heisenberg antiferromagnet, we present exact results for the zero-temperature dynamical structure factor on finite-sized lattices and compare them to approximate results from a Schwinger boson mean-field theory, recently proposed by Arovas and Auerbach. We find that the mean-field theory represents a very good approximation to the exact spin excitation spectra and to the static spin correlations. We then investigation the dynamical spin-spin structure factor and the single-particle spectral function for finite model clusters with dopant induced hole-type charge carriers, in the strong-coupling limit of both the single-band Hubbard model (t-J-model) and the three-band Hubbard model (Kondo-Heisenberg model). Our results are consistent with the physical picture recently proposed by Zhang and Rice which implies an approximate mapping of the low-energy states in the three-band model onto an effective single-band theory. 33 refs., 4 figs.
We compare ground-state properties as well as spin and single-particle excitation spectra of a single dopant-induced hole carrier in strong-coupling-limit single-orbital and three-orbital Hubbard model lattices with up to 16 square lattice sites. Our results indicate that there is a close correspondence in their ground-state and low-lying excited-state properties, consistent with the recent proposal by Zhang and Rice (ZR). For the three-orbital model, we also find direct spectroscopic evidence for the singlet-triplet excitation gap associated with the dopant-induced ZR Cu-O spin-hole singlet state.
The three-band model relevant to high-temperature copper-oxide superconductors is solved using single-site dynamical mean field theory and a tight-binding parametrization of the copper and oxygen bands. For a band filling of one hole per unit cell the metal/charge-transfer-insulator phase diagram is determined. The electron spectral function, optical conductivity, and quasiparticle mass enhancement are computed as functions of electron and hole doping for parameters such that at one hole per cell the paramagnetic phase is insulating and for parameters such that at one hole per unit cell the paramagnetic phase is metallic. The optical conductivity is computed using the Peierls phase approximation for the optical matrix elements. The calculation includes the physics of ``Zhang-Rice singlets.'' The effects of antiferromagnetism on the magnitude of the gap and the relation between correlation strength and doping-induced changes in state density are determined. Three-band and one-band models are compared. The two models are found to yield quantitatively consistent results for all energies less than about 4 eV including energies in the vicinity of the charge-transfer gap. Parameters on the insulating side of the metal/charge-transfer insulator phase boundary lead to gaps which are too large and near-gap conductivities which are too small relative to data. The results place the cuprates clearly in the intermediate correlation regime, on the paramagnetic metal side of the metal/charge-transfer insulator phase boundary.
We calculate the properties of the two-band Hubbard model using the dynamical cluster approximation. The phase diagram resembles the generic phase diagram of the cuprates, showing a strong asymmetry with respect to electron- and hole-doped regimes, in agreement with experiment. Asymmetric features are also seen in one-particle spectral functions and in the charge, spin, and ddd-wave pairing-susceptibility functions. We address the possible reduction of the two-band model to a low-energy single-band one, as it was suggested by Zhang and Rice. Comparing the two-band Hubbard model properties with the single-band Hubbard model ones, we have found similar low-energy physics provided that the next-nearest-neighbor hopping term ${t}^{\ensuremath{'}}$ has a significant value $({t}^{\ensuremath{'}}∕t\ensuremath{\approx}0.3)$. The parameter ${t}^{\ensuremath{'}}$ is the main culprit for the electron-hole asymmetry. However, a significant value of ${t}^{\ensuremath{'}}$ cannot be provided in a strict Zhang and Rice [Phys. Rev. B 37, R3759 (1988); 41, 7243 (1990)] picture where the extra holes added into the system bind to the existing Cu holes forming local singlets. We note that by considering approximate singlet states, such as plaquette states, reasonable values of ${t}^{\ensuremath{'}}$, which capture qualitatively the physics of the two-band model, can be obtained. We conclude that a single-band $t\text{\ensuremath{-}}{t}^{\ensuremath{'}}\text{\ensuremath{-}}U$ Hubbard model captures the basic physics of the cuprates concerning superconductivity, antiferromagnetism, pseudogap, and electron-hole asymmetry, but is not suitable for a quantitative analysis or to describe physical properties involving energy scales larger than about 0.5 eV.
This thesis is dedicated to a theoretical study of the 1-band Hubbard model in the strong coupling limit. The investigation is based on the Dynamical Cluster Approximation (DCA) which systematically restores non-local corrections to the Dynamical Mean Field approximation (DMFA). The DCA is formulated in momentum space and is characterised by a patching of the Brillouin zone where momentum conservation is only recovered between two patches. The approximation works well if k-space correlation functions show a weak momentum dependence. In order to study the temperature and doping dependence of the spin- and charge excitation spectra, we explicitly extend the Dynamical Cluster Approximation to two-particle response functions. The full irreducible two-particle vertex with three momenta and frequencies is approximated by an effective vertex dependent on the momentum and frequency of the spin and/or charge excitations. The effective vertex is calculated by using the Quantum Monte Carlo method on the finite cluster whereas the analytical continuation of dynamical quantities is performed by a stochastic version of the maximum entropy method. A comparison with high temperature auxiliary field quantum Monte Carlo data serves as a benchmark for our approach to two-particle correlation functions. Our method can reproduce basic characteristics of the spin- and charge excitation spectrum. Near and beyond optimal doping, our results provide a consistent overall picture of the interplay between charge, spin and single-particle excitations: a collective spin mode emerges at optimal doping and sufficiently low temperatures in the spin response spectrum and exhibits the energy scale of the magnetic exchange interaction J. Simultaneously, the low energy single-particle excitations are characterised by a coherent quasiparticle with bandwidth J. The origin of the quasiparticle can be quite well understood in a picture of a more or less antiferromagnetic ordered background in which holes are dressed by spin-excitations to allow for a coherent motion. By increasing doping, all features which are linked to the spin-polaron vanish in the single-particle as well as two-particle spin response spectrum. In the second part of the thesis an analysis of superconductivity in the Hubbard model is presented. The superconducting instability is implemented within the Dynamical Cluster Approximation by essentially allowing U(1) symmetry breaking baths in the QMC calculations for the cluster. The superconducting transition temperature T_c is derived from the d-wave order parameter which is directly estimated on the Monte Carlo cluster. The critical temperature T_c is in astonishing agreement with the temperature scale estimated by the divergence of the pair-field susceptibility in the paramagnetic phase. A detailed study of the pseudo and superconducting gap is continued by the investigation of the local and angle-resolved spectral function.
This dissertation discussed four low-dimensional systems: Hubbard ladder and its 2D extension, anisotropic Hubbard ladder, one-dimensional two-orbital t-J model, and Kondo model. All of them are of strongly correlated. Doped cuprates, as strongly correlated systems, have attracted a lot of attention in the condensed matter physics community, not just from the purely theoretical perspective, but also due to the novel applications beyond the conventional semiconductor paradigm. They display complicated phase diagram, defying the conventional rigid band theory, shaking our common sense and driving us to wrestle with many-body physics. With the aid of advanced experimental techniques such as ARPES, its various exotic electronic structure, such as waterfalls, hole pockets, Fermi arcs and kinks have been unveiled and remain open to interpretation. To attack such complicated problems, people usually build various models to mimic the CuO2 planes, which is the common feature of various cuprate compounds and is believed to be mainly responsible for their complicated properties, especially the high temperature superconductivity. In Chapter 3, we simulated the spectral properties of the 2- leg single-band Hubbard ladder and its 2-dimensional extension at and away from half-lling using the numerical tool: the combination of Cluster Perturbation Theory (CPT) with the time-dependent density matrix renormalization group method (tDMRG) as a solver. The main features of the spectrum can be described with a mean-eld dispersion, with kinks and pseudogap traced back to scattering between spin and charge degrees of freedom. As a bonus, we nd that the spectrum of hubbard ladder involves spin-polaron, the bound state of spinon and holon. Interacting one-dimensional electron systems generically behave dierent from higher dimensional systems in the sense that its low-energy excitations are described by Luttinger liquid theory as collective bosonic modes carrying spin and charge separately, the so-called spinons and holons, rather than the coherent quasi-particles in Fermi liquid theory. This phenomenon is known as spin-charge separation, an intrinsic property of 1D system. It can be thought of as a particular case of electron fractionalization. When multiple 1D chains get coupled, how the picture changes? In Chapter 4, we study the spectral function of the anisotropic 2-leg Hubbard ladder, one of the simplest multiple-chain systems. The spectrum can be understood as the scattering resultants of spinon and holons: As the inter-chain hopping is increased, the continuum in the bonding channel moves to higher energies and spinon and holon branches merge into a single coherent quasi-particle band. Simultaneously, the spectrum undergoes a crossover from a regime with two minima at incommensurate values of kx (a Mott insulator), to one with a single minimum at kx = (a band insulator). In addition, the spectrum contains a continuum of scattering states consisting of a triplon and a polaron, accompanied by the formation of bound states with higher spin 3 2 , ferromagnetic \spin bags" in which the hole can move freely. To analyze the formation of quasi-particle, we study the time-evolution of charge and spin degrees of freedom in real space after a hole is injected, which indicates that incoherent holons and spinons can emerge into polarons, that can propagate coherently, after a characterstic time . Multiorbital systems can involve Mott and Hund physics, and are rich in various phenomena, including orbital selective Mott transitions, spin-orbital separations, and pairing density wave due to the complex interplay between spin, charge and orbital degrees of freedom. Therefore, it can capture many unique features in real strongly correlated systems, such as iron pnictides and ruthenates, and thus attract a lot of investigations. In Chapter 5, we comprehensively study a one-dimensional two-orbital model at and below quarter lling, which realizes a number of unconventional phases, by means of DMRG and some analytical calculations. In particular, we nd that the system supports an extensive density wave in which excitons quasicondense with nite center-of-mass momentum. In this phase, excitons behave as hardcore bosons without charge order. In addition, excitons can pair to form biexcitons in a state that is close to a charge density wave instability. Finally, a genuine orbital-selective paired state (metallic plus spin-gapped superconductor) can be realized when pairing dominates over the interorbital repulsion. Kondo problem, as one of the most studied and best understood correlated systems, has been studied by various techniques, but the internal structure of its ground state has two deeply con icting pictures: At strong coupling limit, the system is essentially composed of a scattering center, which is a tightly bound singlet formed by the impurity and a localized electron at r0, and the remaining conduction electrons that are decoupled to the impurity and remain almost \free". At weak enough couplings, the singlet extends to the entire volume and the impurity will couple mostly to one electron at the Fermi level. In chapter 6, we provide a coherent picture to unveil the puzzle in the context of quantum information theory by deconstructing the Kondo singlet, both in real and energy space, and measuring the spin correlations between the impurity and the invididual free electron eigenstate based on two dierent measures. As a bonus, we can identify \projected natural orbitals" that can be used as a \good" approximate wavefunctions for simulations, and a quasi-order parameter that determines the crossover from weak to strong coupling regimes and furthermore oers the means to measure the Kondo screening length K. This screening length might be used to characterize the \size" of the Kondo cloud.
Understanding which minimal effective model captures the essential physics of cuprates is a key step towards unraveling the mechanism behind high- T c superconductivity. Recent measurements of the dynamical spin structure factor (DSF) in cuprate ladder compounds have indicated the presence of an additional, attractive term in the single-band Hubbard model, potentially originating from electron-phonon interactions. Here, we demonstrate that similar DSF features can also be captured by t − J descriptions with a smaller attractive term. Motivated by this observation, we systematically investigate the strength and origin of different contributions to the single-band Hamiltonians by downfolding either from the three-band Emery model or the electron-phonon coupled Hubbard-Holstein model. For one-dimensional systems, we find that the extended versions of both single-band descriptions can reproduce the experimentally observed DSF signatures. Finally, we extend our analysis to two dimensions by comparing two-hole correlation functions for the different single-band models. Our results provide new insights into the long-standing question of which single-band Hamiltonian can capture the essential physics of cuprates.