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

arXiv:cond-mat/9610030 · Physical review. B, Condensed matter

Mean-field approach to antiferromagnetic domains in the doped Hubbard model

Edwin Langmann A, Mats Wallin

Published Apr 15, 1997• 11 citations
arXivHTMLPDF

Abstract

We present a restricted path integral approach to the two-dimensional and three-dimensional repulsive Hubbard model. In this approach the partition function is approximated by restricting the summation over all states to a (small) subclass that is chosen so as to well represent the important states. This procedure generalizes mean-field theory and can be systematically improved by including more states or fluctuations. We analyze in detail the simplest of these approximations, which corresponds to summing over states with local antiferromagnetic (AF) order. If in the states considered the AF order changes sufficiently little in space and time, the path integral becomes a finite-dimensional integral for which the saddle point evaluation is exact. This leads to generalized mean-field equations allowing for the possibility of more than one relevant saddle point. In a big parameter regime (both in temperature and filling), we find that this integral has two relevant saddle points, one corresponding to finite AF order and the other without. These degenerate saddle points describe a phase of AF ordered fermions coexisting with free, metallic fermions. We argue that this mixed phase is a simple mean-field description of a variety of possible inhomogeneous states, appropriate on length scales where these states appear homogeneous. We sketch systematic refinements of this approximation, which can give more detailed descriptions of the system.

DOI
10.1103/physrevb.55.9439

Related papers

Restricted path integral approach to the doped Hubbard model

Edwin Langmann A, Mats Wallin

We present a restricted path integral approach to the 2D repulsive Hubbard model. It amounts to evaluating the partition function by restricting the summation over states to a small subclass and generalizes mean-field theory. Analyzing the simplest of these approximations, we find a mixed phase of fermions with local antiferromagnetic correlations coexisting with conventional metallic fermions. This mixed phase persists in a big parameter regime. We also discuss a refinement of our approximation which suggests simple explanations of several peculiar experimental features of HTSC.

Europhysics Letters (EPL)•Jan 20, 1997•6 citations

Antiferromagnetism and single-particle properties in the two-dimensional half-filled Hubbard model: A nonlinear sigma model approach

Karol Borejsza, N. Dupuis

We describe a low-temperature approach to the two-dimensional half-filled Hubbard model which allows us to study both antiferromagnetism and single-particle properties. This approach ignores amplitude fluctuations of the antiferromagnetic (AF) order parameter and is valid below a crossover temperature ${T}_{X}$ which marks the onset of AF short-range order. Directional fluctuations (spin waves) are described by a nonlinear sigma model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ that we derive from the Hubbard model. The parameters of the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$---the spin stiffness and spin-wave velocity---are calculated as a function of the Coulomb repulsion U. The $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is solved by a saddle-point approximation within the ${\mathrm{CP}}^{1}$ representation where the N\'eel field is parametrized by two Schwinger bosons. At zero temperature, there is always Bose condensation of the Schwinger bosons, which signals AF long-range order for any value of the Coulomb repulsion. At finite temperature, the AF long-range order is suppressed (in agreement with the Mermin-Wagner theorem), but the AF correlation length remains exponentially large. In the ${\mathrm{CP}}^{1}$ representation, the fermion field is naturally expressed as the product of a Schwinger boson and a pseudofermion whose spin is quantized along the (fluctuating) N\'eel field. This allows us to write the fermion Green's function as the product (in direct space) of the Schwinger boson propagator (which is derived from the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ and the pseudofermion propagator. At zero temperature and weak coupling, our results are typical of a Slater antiferromagnet. The AF gap is exponentially small; there are well-defined Bogoliubov quasiparticles (QP's) (carrying most of the spectral weight) coexisting with a high-energy incoherent excitation background. As U increases, the Slater antiferromagnet progressively becomes a Mott-Heisenberg antiferromagnet. The Bogoliubov bands evolve into Mott-Hubbard bands separated by a large AF gap. A significant fraction of spectral weight is transferred from the Bogoliubov QP's to incoherent excitations. At finite temperature, there is a metal-insulator transition between a pseudogap phase at weak coupling and a Mott-Hubbard insulator at strong coupling. Finally, we point out that our results straightforwardly translate to the half-filled attractive Hubbard model, where the $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge and $\mathbf{q}=0$ pairing fluctuations combine to form an order parameter with SO(3) symmetry.

arXiv:cond-mat/0307238•Physical Review B•Feb 27, 2004•77 citations

ALLOY ANALOG APPROACH AND THE GUTZWILLER APPROXIMATION AS PARTICULAR CASES OF A MEAN-FIELD THEORY

Armando A. Aligia, M. Avignon

We develop a new mean field-theory for systems with on-site correlations, like the Hubbard and Anderson models. It consists in a generalization of the saddle-point approximation to the functional integral representation proposed by Kotliar and Ruckenstein to more than one saddle point. It contains also the alloy analog approximation proposed by Hubbard as a particular case. For a half-filled Hubbard model and a model density of states appropriate for three dimensions, we obtain a metal to insulator transition as U increases. The effective Hamiltonian for the insulating phase contains two split bands (rather than only one with an extremely heavy mass) which reproduce correctly the Green’s functions of the atomic limit.

International Journal of Modern Physics B•Oct 20, 1992•3 citations

Renormalized mean-field analysis of antiferromagnetism and d -wave superconductivity in the two-dimensional Hubbard model

J. Reiss, D. Rohe, Walter Metzner

We analyze the competition between antiferromagnetism and superconductivity in the two-dimensional Hubbard model by combining a functional renormalization group flow with a mean-field theory for spontaneous symmetry breaking. Effective interactions are computed by integrating out states above a scale ${\ensuremath{\Lambda}}_{\mathrm{MF}}$ in one-loop approximation, which captures in particular the generation of an attraction in the $d$-wave Cooper channel from fluctuations in the particle-hole channel. These effective interactions are then used as an input for a mean-field treatment of the remaining low-energy states, with antiferromagnetism, singlet superconductivity, and triplet $\ensuremath{\pi}$ pairing as the possible order parameters. Antiferromagnetism and superconductivity suppress each other, leaving only a small region in parameter space where both orders can coexist with a sizable order parameter for each. Triplet $\ensuremath{\pi}$ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.

arXiv:cond-mat/0611164•Physical Review B•Feb 20, 2007•92 citations

Electronic and magnetic features of twisted spin-density-wave states in the two-dimensional Hubbard model

Sajeev John, Puru Voruganti, William E. Goff

Using a path-integral formalism and decomposing the Hubbard interaction into charge and spin order parameters, we develop a 1/S loop expansion of the two-dimensional Hubbard model. At the saddle-point level, we present a detailed study of the energetics of twisted antiferromagnetic order. Doping the antiferromagnetic state leads to a rich mean-field phase diagram at zero temperature, which includes spiral phases, a column phase, and ferromagnetism. The on-site magnetization, the density of states, the plasma frequency, and the quasiparticle weights are evaluated for the twisted spin-density-wave (SDW) states as a function of doping and U/t. Doping-induced frustration remarkably leads to a closure of the Mott-Hubbard band gap for any U/t. At one-loop order, we delineate the collective spin and charge fluctuations of the mean-field SDW state. The action for the two transverse spin modes at half-filling maps to a nonlinear \ensuremath{\sigma} model, which tends to disorder upon doping. Twisted magnetic ground states exhibit a third Goldstone mode due to the complete breaking of rotational invariance. For the commensurate SDW, the charge fluctuations couple only to longitudinal spin fluctuations, whereas, for the incommensurate metallic SDW states, the charge modes are coupled nontrivially to the massless spin waves. This suggests a metallic state quite unlike traditional Fermi liquids.

Physical review. B, Condensed matter•Jun 1, 1991•23 citations

Dual fermion approach to the two-dimensional Hubbard model: Antiferromagnetic fluctuations and Fermi arcs

Alexey N. Rubtsov, Mikhail I. Katsnelson, Alexander I. Lichtenstein +1 more

We present an efficient diagrammatic method to describe nonlocal correlation effects in lattice fermion Hubbard-type models, which is based on a change of variables in the Grassmann path integrals. The new fermions are dual to the original ones and correspond to weakly interacting quasiparticles in the case of strong local correlations in the Hubbard model. The method starts with dynamical mean-field theory as a zeroth-order approximation and includes nonlocal effects in a perturbative way. In contrast to cluster approaches, this method utilizes an exact transition to a dual set of variables. It therefore becomes possible to treat vertices of an effective single-impurity problem as small parameters. This provides a very efficient interpolation between bandlike weak-coupling and atomic limits. The method is illustrated on the two-dimensional Hubbard model. The antiferromagnetic pseudogap, Fermi-arc formations, and non-Fermi-liquid effects due to the Van Hove singularity are correctly reproduced by the lowest-order diagrams. Extremum properties of the dual fermion approach are discussed in terms of the Feynman variational principle.

arXiv:0810.3819•Physical Review B•Jan 30, 2009•144 citations
We present a restricted path integral approach to the two-dimensional and three-dimensional repulsive Hubbard model. In this approach the partition function is approximated by restricting the summation over all states to a (small) subclass that is chosen so as to well represent the important states. This procedure generalizes mean-field theory and can be systematically improved by including more states or fluctuations. We analyze in detail the simplest of these approximations, which corresponds to summing over states with local antiferromagnetic (AF) order. If in the states considered the AF order changes sufficiently little in space and time, the path integral becomes a finite-dimensional integral for which the saddle point evaluation is exact. This leads to generalized mean-field equations allowing for the possibility of more than one relevant saddle point. In a big parameter regime (both in temperature and filling), we find that this integral has two relevant saddle points, one corresponding to finite AF order and the other without. These degenerate saddle points describe a phase of AF ordered fermions coexisting with free, metallic fermions. We argue that this mixed phase is a simple mean-field description of a variety of possible inhomogeneous states, appropriate on length scales where these states appear homogeneous. We sketch systematic refinements of this approximation, which can give more detailed descriptions of the system.
We present a restricted path integral approach to the 2D repulsive Hubbard model. It amounts to evaluating the partition function by restricting the summation over states to a small subclass and generalizes mean-field theory. Analyzing the simplest of these approximations, we find a mixed phase of fermions with local antiferromagnetic correlations coexisting with conventional metallic fermions. This mixed phase persists in a big parameter regime. We also discuss a refinement of our approximation which suggests simple explanations of several peculiar experimental features of HTSC.
We describe a low-temperature approach to the two-dimensional half-filled Hubbard model which allows us to study both antiferromagnetism and single-particle properties. This approach ignores amplitude fluctuations of the antiferromagnetic (AF) order parameter and is valid below a crossover temperature TX{T}_{X}TX​ which marks the onset of AF short-range order. Directional fluctuations (spin waves) are described by a nonlinear sigma model $(\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ that we derive from the Hubbard model. The parameters of the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$---the spin stiffness and spin-wave velocity---are calculated as a function of the Coulomb repulsion U. The $\mathrm{NL}\ensuremath{\sigma}\mathrm{M}$ is solved by a saddle-point approximation within the CP1{\mathrm{CP}}^{1}CP1 representation where the N\'eel field is parametrized by two Schwinger bosons. At zero temperature, there is always Bose condensation of the Schwinger bosons, which signals AF long-range order for any value of the Coulomb repulsion. At finite temperature, the AF long-range order is suppressed (in agreement with the Mermin-Wagner theorem), but the AF correlation length remains exponentially large. In the CP1{\mathrm{CP}}^{1}CP1 representation, the fermion field is naturally expressed as the product of a Schwinger boson and a pseudofermion whose spin is quantized along the (fluctuating) N\'eel field. This allows us to write the fermion Green's function as the product (in direct space) of the Schwinger boson propagator (which is derived from the $\mathrm{NL}\ensuremath{\sigma}\mathrm{M})$ and the pseudofermion propagator. At zero temperature and weak coupling, our results are typical of a Slater antiferromagnet. The AF gap is exponentially small; there are well-defined Bogoliubov quasiparticles (QP's) (carrying most of the spectral weight) coexisting with a high-energy incoherent excitation background. As U increases, the Slater antiferromagnet progressively becomes a Mott-Heisenberg antiferromagnet. The Bogoliubov bands evolve into Mott-Hubbard bands separated by a large AF gap. A significant fraction of spectral weight is transferred from the Bogoliubov QP's to incoherent excitations. At finite temperature, there is a metal-insulator transition between a pseudogap phase at weak coupling and a Mott-Hubbard insulator at strong coupling. Finally, we point out that our results straightforwardly translate to the half-filled attractive Hubbard model, where the $\mathbf{q}=(\ensuremath{\pi},\ensuremath{\pi})$ charge and q=0\mathbf{q}=0q=0 pairing fluctuations combine to form an order parameter with SO(3) symmetry.
We develop a new mean field-theory for systems with on-site correlations, like the Hubbard and Anderson models. It consists in a generalization of the saddle-point approximation to the functional integral representation proposed by Kotliar and Ruckenstein to more than one saddle point. It contains also the alloy analog approximation proposed by Hubbard as a particular case. For a half-filled Hubbard model and a model density of states appropriate for three dimensions, we obtain a metal to insulator transition as U increases. The effective Hamiltonian for the insulating phase contains two split bands (rather than only one with an extremely heavy mass) which reproduce correctly the Green’s functions of the atomic limit.
We analyze the competition between antiferromagnetism and superconductivity in the two-dimensional Hubbard model by combining a functional renormalization group flow with a mean-field theory for spontaneous symmetry breaking. Effective interactions are computed by integrating out states above a scale ${\ensuremath{\Lambda}}_{\mathrm{MF}}$ in one-loop approximation, which captures in particular the generation of an attraction in the ddd-wave Cooper channel from fluctuations in the particle-hole channel. These effective interactions are then used as an input for a mean-field treatment of the remaining low-energy states, with antiferromagnetism, singlet superconductivity, and triplet $\ensuremath{\pi}$ pairing as the possible order parameters. Antiferromagnetism and superconductivity suppress each other, leaving only a small region in parameter space where both orders can coexist with a sizable order parameter for each. Triplet $\ensuremath{\pi}$ pairing appears generically in the coexistence region, but its feedback on the other order parameters is very small.
Using a path-integral formalism and decomposing the Hubbard interaction into charge and spin order parameters, we develop a 1/S loop expansion of the two-dimensional Hubbard model. At the saddle-point level, we present a detailed study of the energetics of twisted antiferromagnetic order. Doping the antiferromagnetic state leads to a rich mean-field phase diagram at zero temperature, which includes spiral phases, a column phase, and ferromagnetism. The on-site magnetization, the density of states, the plasma frequency, and the quasiparticle weights are evaluated for the twisted spin-density-wave (SDW) states as a function of doping and U/t. Doping-induced frustration remarkably leads to a closure of the Mott-Hubbard band gap for any U/t. At one-loop order, we delineate the collective spin and charge fluctuations of the mean-field SDW state. The action for the two transverse spin modes at half-filling maps to a nonlinear \ensuremath{\sigma} model, which tends to disorder upon doping. Twisted magnetic ground states exhibit a third Goldstone mode due to the complete breaking of rotational invariance. For the commensurate SDW, the charge fluctuations couple only to longitudinal spin fluctuations, whereas, for the incommensurate metallic SDW states, the charge modes are coupled nontrivially to the massless spin waves. This suggests a metallic state quite unlike traditional Fermi liquids.
We present an efficient diagrammatic method to describe nonlocal correlation effects in lattice fermion Hubbard-type models, which is based on a change of variables in the Grassmann path integrals. The new fermions are dual to the original ones and correspond to weakly interacting quasiparticles in the case of strong local correlations in the Hubbard model. The method starts with dynamical mean-field theory as a zeroth-order approximation and includes nonlocal effects in a perturbative way. In contrast to cluster approaches, this method utilizes an exact transition to a dual set of variables. It therefore becomes possible to treat vertices of an effective single-impurity problem as small parameters. This provides a very efficient interpolation between bandlike weak-coupling and atomic limits. The method is illustrated on the two-dimensional Hubbard model. The antiferromagnetic pseudogap, Fermi-arc formations, and non-Fermi-liquid effects due to the Van Hove singularity are correctly reproduced by the lowest-order diagrams. Extremum properties of the dual fermion approach are discussed in terms of the Feynman variational principle.