Physical Review Letters

Phase Separation in the Extended Hubbard Model at Weak Coupling

P. G. J. van Dongen

Published Jan 2, 1995• 42 citations

Abstract

The Hubbard model including nearest-neighbor interaction is studied at $T=0$ on a $d$-dimensional hypercubic lattice ($d\ensuremath{\ge}3$) close to half filling. It is shown that the ground state at weak coupling is phase separated, consisting of homogeneous lower-density and antiferromagnetic or charge density wave higher-density regions. Incommensurate phases are unstable. The exact critical doping level and the exact order parameter are renormalized relative to the Hartree results. The renormalization factor at weak coupling turns out to be identical to the result found previously for the same model at half filling.

DOI: 10.1103/physrevlett.74.182

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Physical Review Letters

Phase Separation in the Extended Hubbard Model at Weak Coupling

P. G. J. van Dongen

Published Jan 2, 1995• 42 citations

Publisher

Abstract

The Hubbard model including nearest-neighbor interaction is studied at

T=0

on a

d

-dimensional hypercubic lattice (

d\ensuremath{\ge}3

) close to half filling. It is shown that the ground state at weak coupling is phase separated, consisting of homogeneous lower-density and antiferromagnetic or charge density wave higher-density regions. Incommensurate phases are unstable. The exact critical doping level and the exact order parameter are renormalized relative to the Hartree results. The renormalization factor at weak coupling turns out to be identical to the result found previously for the same model at half filling.

DOI: 10.1103/physrevlett.74.182

Related papers

Phase diagram of the extended Hubbard model at weak coupling

P. G. J. van Dongen

The Hubbard model including nearest neighbor interaction is studied at T=0 on a d-dimensional hypercubic lattice (d\ensuremath{\gg}1) close to half filling. For the model in d=\ensuremath{\infty} we derive the exact result that the ground state at weak coupling is phase separated. Results for lower dimensions are then derived in a 1/d expansion. To obtain these results we first consider possible second order transitions. One then finds that the broken-symmetry phase near half filling is incommensurate. However, the corresponding ground state has negative compressibility and is hence thermodynamically unstable. A Maxwell construction is used to construct the actual phase separated ground state, which consists of homogeneous lower-density and antiferromagnetic or charge density wave higher-density regions. It is shown that both the doping level below which phase separation occurs and the order parameter differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion and turns out to be identical to the renormalization factor previously calculated for the low-temperature thermodynamics at half filling. \textcopyright{} 1996 The American Physical Society.The Hubbard model including nearest neighbor interaction is studied at T=0 on a d-dimensional hypercubic lattice (d\ensuremath{\gg}1) close to half filling. For the model in d=\ensuremath{\infty} we derive the exact result that the ground state at weak coupling is phase separated. Results for lower dimensions are then derived in a 1/d expansion. To obtain these results we first consider possible second order transitions. One then finds that the broken-symmetry phase near half filling is incommensurate. However, the corresponding ground state has negative compressibility and is hence thermodynamically unstable. A Maxwell construction is used to construct the actual phase separated ground state, which consists of homogeneous lower-density and antiferromagnetic or charge density wave higher-density regions. It is shown that both the doping level below which phase separation occurs and the order parameter differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion and turns out to be identical to the renormalization factor previously calculated for the low-temperature thermodynamics at half filling. \textcopyright{} 1996 The American Physical Society.

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We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor ( ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ ) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ =0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ , we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ .We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$.

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Phase separation in the one-dimensional Hubbard model

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The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the $μ-U$ plane, restricted to phases with no charge density modulation, extending the results presented in cond-mat/9511116. This allows antiferromagnetism, saturated ferromagnetism, spiral spin density waves and a collinear structure with unit cell $\uparrow \uparrow \downarrow \downarrow$ . The spiral phase is unstable against phase separation near quarter-, half- and three-quarter-filling. For large $U$ this occurs at hole (or electron) doping of $(3t/π^{2}U)^{1/3}$ from half filling.The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the $μ-U$ plane, restricted to phases with no charge density modulation, extending the results presented in cond-mat/9511116. This allows antiferromagnetism, saturated ferromagnetism, spiral spin density waves and a collinear structure with unit cell $\uparrow \uparrow \downarrow \downarrow$. The spiral phase is unstable against phase separation near quarter-, half- and three-quarter-filling. For large $U$ this occurs at hole (or electron) doping of $(3t/π^{2}U)^{1/3}$ from half filling.

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We investigate numerically the ground state phase diagram of the one-dimensional extended Hubbard model, including an on-site interaction U and a nearest-neighbor interaction V. We focus on the ground state phases of the model in the $V\ensuremath{\gg}U$ region, where previous studies have suggested the possibility of dominant superconducting pairing fluctuations before the system phase separates at a critical value ${V=V}_{\mathrm{PS}}.$ Using quantum Monte Carlo methods on lattices much larger than in previous Lancz\"os diagonalization studies, we determine the boundary of phase separation, the Luttinger-liquid correlation exponent ${K}_{\ensuremath{\rho}},$ and other correlation functions in this region. We find that phase separation occurs for V significantly smaller than previously reported. In addition, for negative U, we find that a uniform state reenters from phase separation as the electron density is increased towards half filling. For $V<{V}_{\mathrm{PS}},$ our results show that superconducting fluctuations are not dominant. The system behaves asymptotically as a Luttinger liquid with ${K}_{\ensuremath{\rho}}<1,$ but we also find strong low-energy (but gapped) charge-density fluctuations at a momentum not expected for a standard Luttinger liquid.We investigate numerically the ground state phase diagram of the one-dimensional extended Hubbard model, including an on-site interaction U and a nearest-neighbor interaction V. We focus on the ground state phases of the model in the $V\ensuremath{\gg}U$ region, where previous studies have suggested the possibility of dominant superconducting pairing fluctuations before the system phase separates at a critical value ${V=V}_{\mathrm{PS}}.$ Using quantum Monte Carlo methods on lattices much larger than in previous Lancz\"os diagonalization studies, we determine the boundary of phase separation, the Luttinger-liquid correlation exponent ${K}_{\ensuremath{\rho}},$ and other correlation functions in this region. We find that phase separation occurs for V significantly smaller than previously reported. In addition, for negative U, we find that a uniform state reenters from phase separation as the electron density is increased towards half filling. For $V<{V}_{\mathrm{PS}},$ our results show that superconducting fluctuations are not dominant. The system behaves asymptotically as a Luttinger liquid with ${K}_{\ensuremath{\rho}}<1,$ but we also find strong low-energy (but gapped) charge-density fluctuations at a momentum not expected for a standard Luttinger liquid.

arXiv:cond-mat/9810010Physical review. B, Condensed matterFeb 15, 199965 citations

Phase separation and competition of superconductivity and magnetism in the two-dimensional Hubbard model: From strong to weak coupling

Markus Aichhorn, Enrico Arrigoni, Michael Potthoff +1 more

Cooperation and competition between the antiferromagnetic, $d$ -wave superconducting, and Mott-insulating states are explored for the two-dimensional Hubbard model including nearest and next-nearest-neighbor hoppings at zero temperature. Using the variational cluster approach with clusters of different shapes and sizes up to 10 sites, it is found that the doping-driven transition from a phase with microscopic coexistence of antiferromagnetism and superconductivity to a purely superconducting phase is discontinuous for strong interaction and accompanied by phase separation. At half-filling the system is in an antiferromagnetic Mott-insulating state with vanishing charge compressibility. Upon decreasing the interaction strength $U$ below a certain critical value of roughly ${U}_{c}\ensuremath{\sim}4$ (in units of the nearest-neighbor hopping), however, the filling-dependent magnetic transition changes its character and becomes continuous. Phase separation or, more carefully, the tendency towards the formation of inhomogeneous states disappears. This critical value is in contrast to previous studies, where a much larger value was obtained. Moreover, we find that the system at half-filling undergoes the Mott transition from an insulator to a state with a finite charge compressibility at essentially the same value. The weakly correlated state at half-filling exhibits superconductivity microscopically admixed to the antiferromagnetic order. This scenario suggests a close relation between phase separation and the Mott-insulator physics.Cooperation and competition between the antiferromagnetic, $d$-wave superconducting, and Mott-insulating states are explored for the two-dimensional Hubbard model including nearest and next-nearest-neighbor hoppings at zero temperature. Using the variational cluster approach with clusters of different shapes and sizes up to 10 sites, it is found that the doping-driven transition from a phase with microscopic coexistence of antiferromagnetism and superconductivity to a purely superconducting phase is discontinuous for strong interaction and accompanied by phase separation. At half-filling the system is in an antiferromagnetic Mott-insulating state with vanishing charge compressibility. Upon decreasing the interaction strength $U$ below a certain critical value of roughly ${U}_{c}\ensuremath{\sim}4$ (in units of the nearest-neighbor hopping), however, the filling-dependent magnetic transition changes its character and becomes continuous. Phase separation or, more carefully, the tendency towards the formation of inhomogeneous states disappears. This critical value is in contrast to previous studies, where a much larger value was obtained. Moreover, we find that the system at half-filling undergoes the Mott transition from an insulator to a state with a finite charge compressibility at essentially the same value. The weakly correlated state at half-filling exhibits superconductivity microscopically admixed to the antiferromagnetic order. This scenario suggests a close relation between phase separation and the Mott-insulator physics.

arXiv:0707.3557Physical Review BDec 11, 200763 citations

The Hubbard model including nearest-neighbor interaction is studied at

T=0

on a

d

-dimensional hypercubic lattice (

d\ensuremath{\ge}3

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the

μ-U

plane, restricted to phases with no charge density modulation, extending the results presented in cond-mat/9511116. This allows antiferromagnetism, saturated ferromagnetism, spiral spin density waves and a collinear structure with unit cell

\uparrow \uparrow \downarrow \downarrow

. The spiral phase is unstable against phase separation near quarter-, half- and three-quarter-filling. For large

U

this occurs at hole (or electron) doping of

(3t/π^{2}U)^{1/3}

from half filling.

{V=V}_{\mathrm{PS}}.

Using quantum Monte Carlo methods on lattices much larger than in previous Lancz\"os diagonalization studies, we determine the boundary of phase separation, the Luttinger-liquid correlation exponent ${K}_{\ensuremath{\rho}},$ and other correlation functions in this region. We find that phase separation occurs for V significantly smaller than previously reported. In addition, for negative U, we find that a uniform state reenters from phase separation as the electron density is increased towards half filling. For

V<{V}_{\mathrm{PS}},

our results show that superconducting fluctuations are not dominant. The system behaves asymptotically as a Luttinger liquid with ${K}_{\ensuremath{\rho}}<1,$ but we also find strong low-energy (but gapped) charge-density fluctuations at a momentum not expected for a standard Luttinger liquid.

Cooperation and competition between the antiferromagnetic,

d

-wave superconducting, and Mott-insulating states are explored for the two-dimensional Hubbard model including nearest and next-nearest-neighbor hoppings at zero temperature. Using the variational cluster approach with clusters of different shapes and sizes up to 10 sites, it is found that the doping-driven transition from a phase with microscopic coexistence of antiferromagnetism and superconductivity to a purely superconducting phase is discontinuous for strong interaction and accompanied by phase separation. At half-filling the system is in an antiferromagnetic Mott-insulating state with vanishing charge compressibility. Upon decreasing the interaction strength

U

below a certain critical value of roughly ${U}_{c}\ensuremath{\sim}4$ (in units of the nearest-neighbor hopping), however, the filling-dependent magnetic transition changes its character and becomes continuous. Phase separation or, more carefully, the tendency towards the formation of inhomogeneous states disappears. This critical value is in contrast to previous studies, where a much larger value was obtained. Moreover, we find that the system at half-filling undergoes the Mott transition from an insulator to a state with a finite charge compressibility at essentially the same value. The weakly correlated state at half-filling exhibits superconductivity microscopically admixed to the antiferromagnetic order. This scenario suggests a close relation between phase separation and the Mott-insulator physics.