arXiv:cond-mat/0703270 · Physical Review B

Non-perturbative treatment of charge and spin fluctuations in the two-dimensional extended Hubbard model: Extended two-particle self-consistent approach

B. Davoudi, A.–M. S. Tremblay

Published Aug 13, 2007• 36 citations

Abstract

We study the spin and charge fluctuations of the extended Hubbard model with on-site interaction $U$ and first neighbor interaction $V$ on the two-dimensional square lattice in the weak to intermediate coupling regime. We propose an extension of the two-particle self-consistent approximation that includes the effect of functional derivatives of the pair-correlation functions on irreducible spin and charge vertices. These functional derivatives were ignored in our previous work. We evaluate them assuming particle-hole symmetry. The resulting theory satisfies conservation laws and the Mermin-Wagner theorem. Our current results are in much better agreement with benchmark quantum Monte Carlo results. This theory allows us to reliably determine the crossover temperatures toward renormalized-classical regimes, and hence, the dominant instability as a function of $U$ and $V$. We have considered fillings $n=1$ and $n=0.75$. Either spin or charge fluctuations can dominate. The wave vector is self-determined by the approach.

DOI: 10.1103/physrevb.76.085115

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arXiv:cond-mat/0703270 · Physical Review B

Non-perturbative treatment of charge and spin fluctuations in the two-dimensional extended Hubbard model: Extended two-particle self-consistent approach

B. Davoudi, A.–M. S. Tremblay

Published Aug 13, 2007• 36 citations

arXiv HTML PDF

Abstract

We study the spin and charge fluctuations of the extended Hubbard model with on-site interaction

U

and first neighbor interaction

V

on the two-dimensional square lattice in the weak to intermediate coupling regime. We propose an extension of the two-particle self-consistent approximation that includes the effect of functional derivatives of the pair-correlation functions on irreducible spin and charge vertices. These functional derivatives were ignored in our previous work. We evaluate them assuming particle-hole symmetry. The resulting theory satisfies conservation laws and the Mermin-Wagner theorem. Our current results are in much better agreement with benchmark quantum Monte Carlo results. This theory allows us to reliably determine the crossover temperatures toward renormalized-classical regimes, and hence, the dominant instability as a function of

U

and

V

. We have considered fillings

n=1

and

n=0.75

. Either spin or charge fluctuations can dominate. The wave vector is self-determined by the approach.

DOI: 10.1103/physrevb.76.085115

Related papers

Theory of spin and charge fluctuations in the Hubbard model

Yu. M. Vil’k, Liang Chen, A.–M. S. Tremblay

A self-consistent theory of both spin and charge fluctuations in the Hubbard model is presented. It is in quantitative agreement with Monte Carlo data at least up to intermediate coupling (U\ensuremath{\sim}8t). It includes both short-wavelength quantum renormalization effects, and long-wavelength thermal fluctuations, which can destroy long-range order in two dimensions. The last effect leads to a small energy scale, as often observed in high-temperature superconductors. The theory is conserving, satisfies the Pauli principle, and includes three-particle correlations necessary to account for the incipient Mott transition.A self-consistent theory of both spin and charge fluctuations in the Hubbard model is presented. It is in quantitative agreement with Monte Carlo data at least up to intermediate coupling (U\ensuremath{\sim}8t). It includes both short-wavelength quantum renormalization effects, and long-wavelength thermal fluctuations, which can destroy long-range order in two dimensions. The last effect leads to a small energy scale, as often observed in high-temperature superconductors. The theory is conserving, satisfies the Pauli principle, and includes three-particle correlations necessary to account for the incipient Mott transition.

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Two-dimensional Hubbard model at finite temperature: Weak, strong, and long correlation regimes

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We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as $T=0.067$ and interactions up to $U=7$ , while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.We investigate the momentum-resolved spin and charge susceptibilities, as well as the chemical potential and double occupancy in the two-dimensional Hubbard model as functions of doping, temperature, and interaction strength. Through these quantities, we identify a weak-coupling regime, a strong-coupling regime with short-range correlations and an intermediate-coupling regime with long magnetic correlation lengths. In the spin channel, we observe an additional crossover from commensurate to incommensurate correlations. In contrast, we find charge correlations to be only short ranged for all studied temperatures, which suggests that the spin and charge responses are decoupled. These findings were obtained by a connected determinant diagrammatic Monte Carlo algorithm for the computation of double expansions, which we introduce in this paper. This permits us to obtain numerically exact results at temperatures as low as $T=0.067$ and interactions up to $U=7$, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.

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Comparison of Monte Carlo and diagrammatic calculations for the two-dimensional Hubbard model

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Monte Carlo results for a two-dimensional Hubbard model in the intermediate-coupling regime U=4t are compared with a diagrammatic spin-fluctuation approximation. Simulations on an 8\ifmmode\times\else\texttimes\fi{}8 lattice doped away from half filling were carried out down to temperatures of order 140 of the bandwidth. Results for the spin susceptibility \ensuremath{\chi}(q,i\ensuremath{\omega} ${}_{\mathrm{m}}$ ), the electron self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega} ${}_{\mathrm{n}}$ ), various pair-field susceptibilities, and the irreducible particle-particle scattering vertex \ensuremath{\Gamma}(p\ensuremath{'},i\ensuremath{\omega}${}_{\mathrm{n}\ensuremath{'}}$|p,i\ensuremath{\omega} ${}_{\mathrm{n}}$ ) were obtained. A random-phase approximation for \ensuremath{\chi}(q,i\ensuremath{\omega} ${}_{\mathrm{m}}$ ) with a renormalized Coulomb coupling U\ifmmode\bar\else\textasciimacron\fi{} is shown to provide a fit to the Monte Carlo data. A similar approximation for the Berk-Schrieffer spin-fluctuation interaction also provides a reasonable fit to the self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega} ${}_{\mathrm{n}}$ ) in the region explored by the Monte Carlo data. However, a similar approximation for the irreducible particle-particle interaction failed to reproduce the Monte Carlo results. Higher-order vertex corrections were calculated, but significant discrepancies with Monte Carlo results for \ensuremath{\Gamma} remain.Monte Carlo results for a two-dimensional Hubbard model in the intermediate-coupling regime U=4t are compared with a diagrammatic spin-fluctuation approximation. Simulations on an 8\ifmmode\times\else\texttimes\fi{}8 lattice doped away from half filling were carried out down to temperatures of order 140 of the bandwidth. Results for the spin susceptibility \ensuremath{\chi}(q,i\ensuremath{\omega}${}_{\mathrm{m}}$), the electron self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}${}_{\mathrm{n}}$), various pair-field susceptibilities, and the irreducible particle-particle scattering vertex \ensuremath{\Gamma}(p\ensuremath{'},i\ensuremath{\omega}${}_{\mathrm{n}\ensuremath{'}}$|p,i\ensuremath{\omega}${}_{\mathrm{n}}$) were obtained. A random-phase approximation for \ensuremath{\chi}(q,i\ensuremath{\omega}${}_{\mathrm{m}}$) with a renormalized Coulomb coupling U\ifmmode\bar\else\textasciimacron\fi{} is shown to provide a fit to the Monte Carlo data. A similar approximation for the Berk-Schrieffer spin-fluctuation interaction also provides a reasonable fit to the self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}${}_{\mathrm{n}}$) in the region explored by the Monte Carlo data. However, a similar approximation for the irreducible particle-particle interaction failed to reproduce the Monte Carlo results. Higher-order vertex corrections were calculated, but significant discrepancies with Monte Carlo results for \ensuremath{\Gamma} remain.

Physical review. B, Condensed matterFeb 1, 1993103 citations

Competition between antiferromagnetic and charge density wave fluctuations in the extended Hubbard model

Petra Pudleiner, Anna Kauch, Karsten Held +1 more

By extending our victory implementation of the parquet approach to include nonlocal Coulomb interactions, we study the extended Hubbard model on the two-dimensional square lattice with a particular focus on the competition of the nonlocal charge and spin fluctuations. Surprisingly, we find that their competition, as the mechanism driving the phase transition towards the charge density wave, dominates only in a very narrow parameter regime in the immediate vicinity of the phase transition. Due to the special geometry and the Fermi surface topology of the square lattice, antiferromagnetic fluctuations dominate even for sizable next-nearest-neighbor interactions. Our conclusions are based on the consistent observations in both the single- and two-particle quantities, including the self-energy, the single-particle spectral function, the two-particle susceptibility, the density-density vertex function, and the optical conductivity. Our work unbiasedly establishes the connection of these quantities to the charge fluctuations, and the way of interpretation can be readily applied to any many-body method with access to the two-particle vertex.By extending our victory implementation of the parquet approach to include nonlocal Coulomb interactions, we study the extended Hubbard model on the two-dimensional square lattice with a particular focus on the competition of the nonlocal charge and spin fluctuations. Surprisingly, we find that their competition, as the mechanism driving the phase transition towards the charge density wave, dominates only in a very narrow parameter regime in the immediate vicinity of the phase transition. Due to the special geometry and the Fermi surface topology of the square lattice, antiferromagnetic fluctuations dominate even for sizable next-nearest-neighbor interactions. Our conclusions are based on the consistent observations in both the single- and two-particle quantities, including the self-energy, the single-particle spectral function, the two-particle susceptibility, the density-density vertex function, and the optical conductivity. Our work unbiasedly establishes the connection of these quantities to the charge fluctuations, and the way of interpretation can be readily applied to any many-body method with access to the two-particle vertex.

arXiv:1905.03065Physical review. B./Physical review. BAug 5, 201917 citations

Néel order in the Hubbard model within a spin-charge rotating reference frame approach: Crossover from weak to strong coupling

T. A. Zaleski, T. K. Kopeć

The antiferromagnetic phase of two-dimensional (2D) and three-dimensional (3D) Hubbard model with nearest neighbor hopping is studied on a bipartite cubic lattice by means of the quantum $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ rotor approach that yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of the model and satisfy the Mermin-Wagner theorem. The collective variables for charge and spin are isolated in the form of the space-time fluctuating U(1) phase field and rotating spin-quantization axis governed by the SU(2) symmetry, respectively. As a result interacting electrons appear as composite objects consisting of bare fermions with attached U(1) and SU(2) gauge fields. An effective action consisting of a spin-charge rotor and a fermionic field is derived as a function of the Coulomb repulsion $U$ and hopping parameter $t$ . At zero temperature, our theory describes the evolution from a Slater $(U⪡t)$ to a Mott-Heisenberg $(U⪢t)$ antiferromagnet. The results for zero-temperature sublatice magnetization (2D) and finite temperature (3D) phase diagram of the antiferromagnetic Hubbard model as a function of the crossover parameter $U∕t$ are presented and the role of the spin Berry phase in the interaction driven crossover is analyzed.The antiferromagnetic phase of two-dimensional (2D) and three-dimensional (3D) Hubbard model with nearest neighbor hopping is studied on a bipartite cubic lattice by means of the quantum $\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ rotor approach that yields a fully self-consistent treatment of the antiferromagnetic state that respects the symmetry properties of the model and satisfy the Mermin-Wagner theorem. The collective variables for charge and spin are isolated in the form of the space-time fluctuating U(1) phase field and rotating spin-quantization axis governed by the SU(2) symmetry, respectively. As a result interacting electrons appear as composite objects consisting of bare fermions with attached U(1) and SU(2) gauge fields. An effective action consisting of a spin-charge rotor and a fermionic field is derived as a function of the Coulomb repulsion $U$ and hopping parameter $t$. At zero temperature, our theory describes the evolution from a Slater $(U⪡t)$ to a Mott-Heisenberg $(U⪢t)$ antiferromagnet. The results for zero-temperature sublatice magnetization (2D) and finite temperature (3D) phase diagram of the antiferromagnetic Hubbard model as a function of the crossover parameter $U∕t$ are presented and the role of the spin Berry phase in the interaction driven crossover is analyzed.

arXiv:0811.0248Physical Review BMar 14, 200814 citations

Nonequilibrium two-particle self-consistent approach

Olivier Simard, Philipp Werner

We present the nonequilibrium implementation of the two-particle self-consistent (TPSC) approach, which has been shown to provide a reliable equilibrium description of interacting lattice systems in the weak- and intermediate-correlation regime. This method captures the effects of local and nonlocal correlations in two- and higher-dimensional systems and satisfies the Mermin-Wagner theorem. We demonstrate the versatility of nonequilibrium TPSC with calculations of the time-dependent spin and charge response functions and the evolution of effective temperatures extracted from different correlation functions, after interaction ramps in the two-dimensional Hubbard model.We present the nonequilibrium implementation of the two-particle self-consistent (TPSC) approach, which has been shown to provide a reliable equilibrium description of interacting lattice systems in the weak- and intermediate-correlation regime. This method captures the effects of local and nonlocal correlations in two- and higher-dimensional systems and satisfies the Mermin-Wagner theorem. We demonstrate the versatility of nonequilibrium TPSC with calculations of the time-dependent spin and charge response functions and the evolution of effective temperatures extracted from different correlation functions, after interaction ramps in the two-dimensional Hubbard model.

arXiv:2205.13813Physical review. B./Physical review. BDec 22, 202216 citations

We study the spin and charge fluctuations of the extended Hubbard model with on-site interaction

U

and first neighbor interaction

V

U

and

V

. We have considered fillings

n=1

and

n=0.75

. Either spin or charge fluctuations can dominate. The wave vector is self-determined by the approach.

T=0.067

and interactions up to

U=7

, while working on arbitrarily large lattices. Our method also allows us to gain physical insights from investigating the analytic structure of perturbative series. We connect to previous work by studying smaller lattice geometries and report substantial finite-size effects.

{}_{\mathrm{m}}

), the electron self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}

{}_{\mathrm{n}}

), various pair-field susceptibilities, and the irreducible particle-particle scattering vertex \ensuremath{\Gamma}(p\ensuremath{'},i\ensuremath{\omega}${}_{\mathrm{n}\ensuremath{'}}$|p,i\ensuremath{\omega}

{}_{\mathrm{n}}

) were obtained. A random-phase approximation for \ensuremath{\chi}(q,i\ensuremath{\omega}

{}_{\mathrm{m}}

) with a renormalized Coulomb coupling U\ifmmode\bar\else\textasciimacron\fi{} is shown to provide a fit to the Monte Carlo data. A similar approximation for the Berk-Schrieffer spin-fluctuation interaction also provides a reasonable fit to the self-energy \ensuremath{\Sigma}(p,i\ensuremath{\omega}

{}_{\mathrm{n}}

) in the region explored by the Monte Carlo data. However, a similar approximation for the irreducible particle-particle interaction failed to reproduce the Monte Carlo results. Higher-order vertex corrections were calculated, but significant discrepancies with Monte Carlo results for \ensuremath{\Gamma} remain.

U

and hopping parameter

t

. At zero temperature, our theory describes the evolution from a Slater

(U⪡t)

to a Mott-Heisenberg

(U⪢t)

antiferromagnet. The results for zero-temperature sublatice magnetization (2D) and finite temperature (3D) phase diagram of the antiferromagnetic Hubbard model as a function of the crossover parameter

U∕t

are presented and the role of the spin Berry phase in the interaction driven crossover is analyzed.