In this study the Hubbard Model is investigated up to a system of 12 x 12 lattice sites. This study shows that antiferromagnetismus, stripes and superconductivity can coexist together in the doped case. The approach is new, a diagrammatic selfconsistent Feynman Approch. In this study the graphs selfconsitent up to second order in irreducible selfenergy, where the symmetry is broken in charge, spin and superconductivity (Nambu Formalism). The self-consistent dynamical Nambu approach reveals a short-range anisotropic pairing state associated with the charge stripes. The anomalous pairing function is predominantly localized within the stripes, with a larger delocalization along the stripe direction than perpendicular to it. Furthermore, the pairing correlations on the two parallel stripes exhibit the same phase, indicating a phase-aligned inter-stripe coupling. The symmetry of the superconducting phase is a mix of s-wave and d-wave. The ratio (d/s)-wave is (1/2). The coherence length parallel to the stripe has nearly the same quantity than vertical to it 8. That means the superconducting phase is as large as the lattice size. The stripes are not independent in superconductivity. In this study the transition temerature can be calculated. So the complete set of correlation functions, single particle density of states are presented. The result is checked on a 4 x 4 Hubbard Cluster, where exact results are possible by Lanczos Diagonalization. With more CPU-power larger clusters are possible to determine, and it is in principle possible to determine the transition temerature in a Emery Model. So one could make a phase diagramm for every U. It is possible to calculate transition temerature depending on the hole concentration.
In this study the Hubbard Model is investigated up to a system of 12 x 12 lattice sites. This study shows that antiferromagnetismus, stripes and superconductivity can coexist together in the doped case. The approach is new, a diagrammatic selfconsistent Feynman Approch. In this study the graphs selfconsitent up to second order in irreducible selfenergy, where the symmetry is broken in charge, spin and superconductivity (Nambu Formalism). The self-consistent dynamical Nambu approach reveals a short-range anisotropic pairing state associated with the charge stripes. The anomalous pairing function is predominantly localized within the stripes, with a larger delocalization along the stripe direction than perpendicular to it. Furthermore, the pairing correlations on the two parallel stripes exhibit the same phase, indicating a phase-aligned inter-stripe coupling. The symmetry of the superconducting phase is a mix of s-wave and d-wave. The ratio (d/s)-wave is (1/2). The coherence length parallel to the stripe has nearly the same quantity than vertical to it 8. That means the superconducting phase is as large as the lattice size. The stripes are not independent in superconductivity. In this study the transition temerature can be calculated. So the complete set of correlation functions, single particle density of states are presented. The result is checked on a 4 x 4 Hubbard Cluster, where exact results are possible by Lanczos Diagonalization. With more CPU-power larger clusters are possible to determine, and it is in principle possible to determine the transition temerature in a Emery Model. So one could make a phase diagramm for every U. It is possible to calculate transition temerature depending on the hole concentration.
We perform a renormalized mean-field study of the two-dimensional repulsive Hubbard model, focusing on the intricate interplay and possible coexistence of magnetic, charge, and superconducting orders. We improve on conventional mean-field theory by utilizing a renormalization group framework that captures high-energy fluctuations. This method generates effective magnetic and d -wave pairing interactions, and allows for an unbiased exploration of coexisting phases at weak and moderate interaction strengths. Unrestricted mean-field calculations of the effective Hamiltonian on large finite lattices are combined with analyses in the thermodynamic limit, revealing a rich phase diagram with extensive regions of coexisting orders. We find that d -wave superconductivity coexists with Néel order on the electron-doped side. On the hole-doped side, superconductivity is found to coexist with spiral or stripe magnetic orders. Within the stripe-ordered region, the superconducting order parameter is spatially modulated, with a period that follows the charge modulation of the stripes. Below Van Hove filling, pairing provides the primary energy gain, while the stripe order yields only a small, and hence fragile, additional energy lowering.
The long-studied Hubbard model is one of the simplest models of copper-oxide superconductors. However, the connection between the model and the experimental phase diagram is still under debate, in particular regarding the existence and extent of the d-wave superconducting phase. Recent rapid progress in improving the accuracy of numerical solvers has opened a way to answer this question reliably. Here, we study the hole-doping concentration ($\ensuremath{\delta}$) dependence of the Hubbard model in the ground states on a square lattice at strong coupling U/t=10, for the on-site interaction U and the transfer t, using a variational Monte Carlo method. The method, which combines tensor network and Lanczos methods on top of Pfaffian wave functions, reveals a rich phase diagram, in which many orders compete severely and degenerate within the energy range of 0.01t. We have identified distinct phases including a uniform d-wave superconducting phase for $0.17\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.22$ and a stripe charge/spin ordered phase for $\ensuremath{\delta}\ensuremath{\lesssim}0.17$ with the stripe period depending on $\ensuremath{\delta}$, together with presumable spatially coexisting antiferromagnetic and stripe order for $\ensuremath{\delta}\ensuremath{\lesssim}0.07$ and coexisting stripe and d-wave superconductivity for $0.07\ensuremath{\lesssim}\ensuremath{\delta}\ensuremath{\lesssim}0.17$. The present, improved method revealed a wider region of a charge uniform superconducting phase than the previous studies and shows a qualitative similarity to the phase diagram of the cuprate superconductors. The superconducting order parameter is largest at doping of around $\ensuremath{\delta}=0.17$ in the ground state, which undergoes phase transitions from an inhomogeneous to a uniform state.
The two-dimensional Hubbard model is one of the most fundamental models in condensed matter physics. It captures many properties of cuprate high-temperature superconductors. Despite its apparent simplicity, solving the Hubbard model is a challenging task. In this work, we employ the Hartree-Fock approximation, as well as functional renormalization group methods, to construct phase diagrams of the Hubbard model and to shed light on the zoo of magnetic, charge ordered, and superconducting states. The Hartree-Fock approximation can self-consistently determine the expectation values of local charge and magnetic order parameters. Mean-field theories like this can lead to a deeper understanding of the competition, coexistence and transition of different orders. We start by employing the Hartree-Fock approximation in a completely unrestricted form on finite-size lattices and complement the results with calculations in the thermodynamic limit. We construct mean-field phase diagrams of the two-dimensional Hubbard model in the thermodynamic limit for different values of the next-nearest neighbor hopping as a function of filling and temperature. We find that the dominant magnetic orders at moderate coupling strength are Néel order around half-filling, followed by spiral order, and stripe order as the hole doping is increased. We also clarify the nature of the transition from magnetic spiral to stripe order. We then extend the Hartree-Fock scheme by a functional renormalization group flow, which generates new effective interactions and allows us to also access the superconducting phase of the Hubbard model. We present an improved phase diagram, featuring Néel, spiral, and stripe order coexisting with d-wave superconductivity at low temperatures. We analyze the coexistence and competition between these orders. Finally, we perform unrestricted Hartree-Fock calculations of a Hubbard model on the graphene lattice and the Bernal bilayer graphene lattice. We also construct magnetic phase diagrams for these geometries, which feature Néel order around half-filling and various stripe orders in a secondary magnetic dome.
The dualism between superconductivity and charge/spin modulations (the so-called stripes) dominates the phase diagram of many strongly-correlated systems. A prominent example is given by the Hubbard model, where these phases compete and possibly coexist in a wide regime of electron dopings for both weak and strong couplings. Here, we investigate this antagonism within a variational approach that is based upon Jastrow-Slater wave functions, including backflow correlations, which can be treated within a quantum Monte Carlo procedure. We focus on clusters having a ladder geometry with M M legs (with M M ranging from 2 2 to 10 10 ) and a relatively large number of rungs, thus allowing us a detailed analysis in terms of the stripe length. We find that stripe order with periodicity \lambda=8 λ = 8 in the charge and 2\lambda=16 2 λ = 16 in the spin can be stabilized at doping \delta=1/8 δ = 1 / 8 . Here, there are no sizable superconducting correlations and the ground state has an insulating character. A similar situation, with \lambda=6 λ = 6 , appears at \delta=1/6 δ = 1 / 6 . Instead, for smaller values of dopings, stripes can be still stabilized, but they are weakly metallic at \delta=1/12 δ = 1 / 12 and metallic with strong superconducting correlations at \delta=1/10 δ = 1 / 10 , as well as for intermediate (incommensurate) dopings. Remarkably, we observe that spin modulation plays a major role in stripe formation, since it is crucial to obtain a stable striped state upon optimization. The relevance of our calculations for previous density-matrix renormalization group results and for the two-dimensional case is also discussed.
We study the ground state properties of the hole-doped three-band Hubbard (Emery) model, describing the copper-oxygen planes of the cuprates, using large-scale two-dimensional tensor network calculations. Our simulations reveal a period 4 stripe state with spin and weak charge order over an extended doping range beyond $\ensuremath{\delta}\ensuremath{\sim}0.12$ and stripes with larger periods at smaller doping. The period 4 stripe exhibits coexisting d-wave superconductivity in the doping range $0.15\ensuremath{\lesssim}\ensuremath{\delta}<0.25$, while at smaller doping around $\ensuremath{\delta}\ensuremath{\sim}1/8$ we find a strong competition between superconducting and nonsuperconducting stripes, including also pair density-wave states with an alternating sign structure on neighboring stripes, suggesting that the fate of superconductivity around 1/8 doping may be sensitive on the model parameters.
We determine the spin and charge orders in the ground state of the doped two-dimensional (2D) Hubbard model in its simplest form, namely with only nearest-neighbor hopping and on-site repulsion. At half-filling, the ground state is known to be an antiferromagnetic Mott insulator. Doping Mott insulators is believed to be relevant to the superconductivity observed in cuprates. A variety of candidates have been proposed for the ground state of the doped 2D Hubbard model. A recent work employing a combination of several state-of-the-art numerical many-body methods established the stripe order as the ground state near 1/8 doping at strong interactions. In this paper, we apply one of these methods, the cutting-edge constrained-path auxiliary field quantum Monte Carlo (AFQMC) method with self-consistently optimized gauge constraints, to systematically study the model as a function of doping and interaction strength. With careful finite size scaling based on large-scale computations, we map out the ground state phase diagram in terms of its spin and charge order. We find that modulated antiferromagnetic order persists from near half-filling to about 1/5 doping. At lower interaction strengths or larger doping, these ordered states are best described as spin-density waves, with essentially delocalized holes and modest oscillations in charge correlations. When the charge correlations are stronger (large interaction or small doping), they are best described as stripe states, with the holes more localized near the node in the antiferromagnetic spin order. In both cases, we find that the wavelength in the charge correlations is consistent with so-called filled stripes.