Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Europhysics Letters (EPL)

Holes and Magnetic Textures in the Two-Dimensional Hubbard Model

A. R. Bishop, F. Guinea, Peter S. Lomdahl, Enrique Louis, José Antonio Vergés

Published Jan 15, 1991• 39 citations
Publisher

Abstract

The Hubbard model in a two-dimensional square lattice is studied within the unrestricted Hartree-Fock approximation. A wide variety of solutions are found, as a function of filling and value of the Coulomb repulsion. Their properties range from those of magnetic polarons to a behavior similar to electrons in an external magnetic field (flux phases). The main features of the vortex solution are studied. It is shown that its energy increases logarithmically with system size, and that it becomes more stable for high values of the Coulomb repulsion and doping levels. Also, multivortex configurations are shown to exist. The implications of our results in the qualitative behavior of the Hubbard model as function of doping and strength of the interaction are discussed.

DOI
10.1209/0295-5075/14/2/011

Related papers

Instability of vortex textures in the two-dimensional Hubbard model

Kenji Yonemitsu, A. R. Bishop

The Hubbard model on a two-dimensional square lattice is investigated within an unrestricted, inhomogeneous Hartree-Fock approximation. Vortexlike states discussed by Verge\ifmmode\acute\else\textasciiacute\fi{}s et al. [Phys. Rev. B 43, 6099 (1991)] are studied in detail. They are found to be unstable against transverse spin fluctuations. Doping of holes does not suppress these fluctuations strongly enough to stabilize the vortex states.

Physical review. B, Condensed matter•Sep 1, 1993•1 citation

Analysis of the New Unrestricted Hartree‐Fock Vortex Solution of the Hubbard Hamiltonian in Two‐Dimensional Systems A Small‐Cluster Study

Enrique Louis, J. Galán, F. Guinea +2 more

Abstract The properties of a recently discovered unrestricted Hartree‐Fock solution of the Hubbard Hamiltonian is analysed on the square lattice that shows a really two‐dimensional spin polarization of a vortex‐like character. This and other several mean‐field solutions are compared with the exact ground‐state energy and wave function of two and four electrons on a four‐site system (square). Results for two correlated electrons in large 3 × 3 and 4 × 4 clusters are also given and discussed. The vortex mean‐field solution provides the best solution in almost all systems that deviate from half‐filling.

physica status solidi (b)•Oct 1, 1992•5 citations

Helical States vs Soliton and Vortex Lattice Statesin the Two-Dimensional Hubbard Model

Mitsutaka Fujita, Masahiko Ichimura, Kenji Nakao

We study the spin and charge textures of the two-dimensional Hubbard model for a wide range of electron filling at the ground state both on a square lattice and on a triangular lattice within a mean field approximation. We complete the overall phase diagram of the single-Q helical spin density wave state, which spreads widely in the electron filling and the Coulomb interaction plane, and discuss the relative stability of the soliton and the vortex lattice states on it. It is shown that the imperfect nesting condition, which is the intrinsic feature of two-dimensional systems, induces a rich variety of self-consistent solutions including collinear and noncollinear spin textures.

Journal of the Physical Society of Japan•Sep 15, 1991•26 citations

Magnetism in the two-dimensional Hubbard model (abstract)

J. E. Hirsch, Hsueh‐Ju Lin

The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.

Journal of Applied Physics•Apr 15, 1985•0 citations

Spin- and charge-density waves in the Hartree–Fock ground state of the two-dimensional Hubbard model

Jie Xu, Chia-Chen Chang, Eric J Walter +1 more

The ground states of the two-dimensional repulsive Hubbard model are studied within the unrestricted Hartree-Fock (UHF) theory. Magnetic and charge properties are determined by systematic, large-scale, exact numerical calculations, and quantified as a function of electron doping, h. In the solution of the self-consistent UHF equations, multiple initial configurations and simulated annealing are used to facilitate convergence to the global minimum. New approaches are employed to minimize finite-size effects in order to reach the thermodynamic limit. At low to moderate interacting strengths and low doping, the UHF ground state is a linear spin-density wave (l-SDW), with antiferromagnetic order and a modulating wave. The wavelength of the modulating wave is 2/h. Corresponding charge order exists but is substantially weaker than the spin order, hence holes are mobile. As the interaction is increased, the l-SDW states evolve into several different phases, with the holes eventually becoming localized. A simple pairing model is presented with analytic calculations for low interaction strength and small doping, to help understand the numerical results and provide a physical picture for the properties of the SDW ground state. By comparison with recent many-body calculations, it is shown that, for intermediate interactions, the UHF solution provides a good description of the magnetic correlations in the true ground state of the Hubbard model.

arXiv:1107.0976•Journal of Physics Condensed Matter•Nov 30, 2011•39 citations

The zoo of states in the Hubbard model

Robin Scholle

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.

•Jan 1, 2026•0 citations
The Hubbard model in a two-dimensional square lattice is studied within the unrestricted Hartree-Fock approximation. A wide variety of solutions are found, as a function of filling and value of the Coulomb repulsion. Their properties range from those of magnetic polarons to a behavior similar to electrons in an external magnetic field (flux phases). The main features of the vortex solution are studied. It is shown that its energy increases logarithmically with system size, and that it becomes more stable for high values of the Coulomb repulsion and doping levels. Also, multivortex configurations are shown to exist. The implications of our results in the qualitative behavior of the Hubbard model as function of doping and strength of the interaction are discussed.
Abstract The properties of a recently discovered unrestricted Hartree‐Fock solution of the Hubbard Hamiltonian is analysed on the square lattice that shows a really two‐dimensional spin polarization of a vortex‐like character. This and other several mean‐field solutions are compared with the exact ground‐state energy and wave function of two and four electrons on a four‐site system (square). Results for two correlated electrons in large 3 × 3 and 4 × 4 clusters are also given and discussed. The vortex mean‐field solution provides the best solution in almost all systems that deviate from half‐filling.
We study the spin and charge textures of the two-dimensional Hubbard model for a wide range of electron filling at the ground state both on a square lattice and on a triangular lattice within a mean field approximation. We complete the overall phase diagram of the single-Q helical spin density wave state, which spreads widely in the electron filling and the Coulomb interaction plane, and discuss the relative stability of the soliton and the vortex lattice states on it. It is shown that the imperfect nesting condition, which is the intrinsic feature of two-dimensional systems, induces a rich variety of self-consistent solutions including collinear and noncollinear spin textures.
The Hubbard model has been widely used to describe magnetism due to itinerant electrons. However, except for the one-dimensional case, no exact solution of the model exists. We have studied the two-dimensional Hubbard model on a square lattice using a Monte Carlo simulation technique. Simulations were performed on lattices of up to 8×8 in spatial size. We will discuss results for thermodynamic properties and correlation functions as a function of temperature and interaction strength. For the half-filled band case, our results indicate that the system exhibits long-range antiferromagnetic order in the ground state for any nonzero interaction, although substantially reduced from the classical Néel state due to both charge and spin fluctuations. For the non-half-filled band case, our results suggest that no long-range magnetic order exists. We also don’t find any indication of short-ranged ferromagnetic correlations in the parameter range studied. We have also studied the predictions of various approximate treatments for this model, particularly the static approximation and the coherent potential approximation (CPA). By comparison with the Monte Carlo results, we discuss the validity of these schemes. Finally, we discuss the effect of changing the band structure on the magnetic properties of the system.
The ground states of the two-dimensional repulsive Hubbard model are studied within the unrestricted Hartree-Fock (UHF) theory. Magnetic and charge properties are determined by systematic, large-scale, exact numerical calculations, and quantified as a function of electron doping, h. In the solution of the self-consistent UHF equations, multiple initial configurations and simulated annealing are used to facilitate convergence to the global minimum. New approaches are employed to minimize finite-size effects in order to reach the thermodynamic limit. At low to moderate interacting strengths and low doping, the UHF ground state is a linear spin-density wave (l-SDW), with antiferromagnetic order and a modulating wave. The wavelength of the modulating wave is 2/h. Corresponding charge order exists but is substantially weaker than the spin order, hence holes are mobile. As the interaction is increased, the l-SDW states evolve into several different phases, with the holes eventually becoming localized. A simple pairing model is presented with analytic calculations for low interaction strength and small doping, to help understand the numerical results and provide a physical picture for the properties of the SDW ground state. By comparison with recent many-body calculations, it is shown that, for intermediate interactions, the UHF solution provides a good description of the magnetic correlations in the true ground state of the Hubbard model.
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.