Journal of Physics Condensed Matter

The spin reversal symmetry of the ground state of the half-filled Hubbard model

Z. Song, H Shen

Published Feb 14, 1994• 0 citations

Abstract

Using a method based on Lieb's theorems and SU(2) symmetry for the half-filled Hubbard model on a bipartite lattice with mod A mod = mod B mod =N (where mod A mod ( mod B mod ) is the number of sites in the A (B) sublattice), we show that the ground-state wave function has even (odd) symmetry with respect to the spin reversal operator when N is even (odd) for both U>0 and U<0 cases. These rigorous results hold in all dimensions without the necessity for a periodic boundary condition.

DOI: 10.1088/0953-8984/6/7/010

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Two theorems on the Hubbard model

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Journal of Physics Condensed Matter

The spin reversal symmetry of the ground state of the half-filled Hubbard model

Z. Song, H Shen

Published Feb 14, 1994• 0 citations

Publisher

Abstract

DOI: 10.1088/0953-8984/6/7/010

Related papers

Two theorems on the Hubbard model

Élliott H. Lieb

In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.In the attractive Hubbard Model (and some extended versions of it), the ground state is proved to have spin angular momentum S=0 for every (even) electron filling. In the repulsive case, and with a bipartite lattice and a half-filled band, the ground state has S=(1/2\ensuremath{\parallel}B\ensuremath{\Vert}-\ensuremath{\Vert}A\ensuremath{\Vert}\ensuremath{\Vert}, where \ensuremath{\Vert}B\ensuremath{\Vert} (\ensuremath{\Vert}A\ensuremath{\Vert}) is the number of sites in the B (A) sublattice. In both cases the ground state is unique. The second theorem confirms an old, unproved conjecture in the \ensuremath{\Vert}B\ensuremath{\Vert}=\ensuremath{\Vert}A\ensuremath{\Vert} case and yields, with \ensuremath{\Vert}B\ensuremath{\Vert}\ensuremath{\ne}\ensuremath{\Vert}A\ensuremath{\Vert}, the first provable example of itinerant-electron ferromagnetism. The theorems hold in all dimensions without even the necessity of a periodic lattice structure.

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We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.

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In this paper, we show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half filling on some bipartite lattices. When ${\mathit{N}}_{\mathit{A}}$ > ${\mathit{N}}_{\mathit{B}}$ ( ${\mathit{N}}_{\mathit{A}}$ and ${\mathit{N}}_{\mathit{B}}$ are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201 (1989)], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U>0 and is independent of the dimension of the lattices.In this paper, we show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half filling on some bipartite lattices. When ${\mathit{N}}_{\mathit{A}}$>${\mathit{N}}_{\mathit{B}}$(${\mathit{N}}_{\mathit{A}}$ and ${\mathit{N}}_{\mathit{B}}$ are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201 (1989)], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U>0 and is independent of the dimension of the lattices.

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Ferromagnetism beyond Lieb's theorem

Natanael C. Costa, Tiago Mendes-Santos, Thereza Paiva +2 more

The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction $U$ is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a `` ${\mathrm{CuO}}_{2}$ lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which `` $d$ orbitals'' occupy the vertices of the squares, while `` $p$ orbitals'' lie halfway between two $d$ orbitals; both $d$ and $p$ orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of $U$ and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, ${U}_{d}={U}_{p}$ , originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all $U$ . For the IH system at half-filling, we argue that the case ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases ${U}_{p}=0,{U}_{d}=U$ and ${U}_{p}=U,{U}_{d}=0$ . We found that the different environments of $d$ and $p$ sites lead to a ferromagnetic insulator when ${U}_{d}=0$ ; by contrast, ${U}_{p}=0$ leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each $d$ site.The noninteracting electronic structures of tight-binding models on bipartite lattices with unequal numbers of sites in the two sublattices have a number of unique features, including the presence of spatially localized eigenstates and flat bands. When a uniform on-site Hubbard interaction $U$ is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a ``${\mathrm{CuO}}_{2}$ lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which ``$d$ orbitals'' occupy the vertices of the squares, while ``$p$ orbitals'' lie halfway between two $d$ orbitals; both $d$ and $p$ orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of $U$ and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case, ${U}_{d}={U}_{p}$, originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all $U$. For the IH system at half-filling, we argue that the case ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases ${U}_{p}=0,{U}_{d}=U$ and ${U}_{p}=U,{U}_{d}=0$. We found that the different environments of $d$ and $p$ sites lead to a ferromagnetic insulator when ${U}_{d}=0$; by contrast, ${U}_{p}=0$ leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each $d$ site.

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Hironobu Yoshida, Hosho Katsura

We study the attractive SU(N) Hubbard model with particle-hole symmetry. The model is defined on a bipartite lattice with the number of sites N_{A} (N_{B}) in the A (B) sublattice. We prove three theorems that allow us to identify the basic ground-state properties: the degeneracy, the fermion number, and the SU(N) quantum number. We also show that the ground state exhibits charge density wave order when |N_{A}-N_{B}| is macroscopically large. The theorems hold for a bipartite lattice in any dimension, even without translation invariance.We study the attractive SU(N) Hubbard model with particle-hole symmetry. The model is defined on a bipartite lattice with the number of sites N_{A} (N_{B}) in the A (B) sublattice. We prove three theorems that allow us to identify the basic ground-state properties: the degeneracy, the fermion number, and the SU(N) quantum number. We also show that the ground state exhibits charge density wave order when |N_{A}-N_{B}| is macroscopically large. The theorems hold for a bipartite lattice in any dimension, even without translation invariance.

arXiv:2011.02296Physical Review LettersMar 12, 202117 citations

Ground-state quantum numbers of the half-filled Hubbard model

Adriana Moreo, Elbio R. Dagotto

Working on a square lattice with an even number of sites ${\mathit{N}}^{2}$ and assuming a smooth connection between strong and weak coupling, we show that the ground state of the two-dimensional half-filled Hubbard model has momentum ${\mathit{k}}_{\mathit{x}}$ - ${\mathit{k}}_{\mathit{y}}$ =0, it is even under reflections, and transforms as an s wave under rotations when N/2 is even and as a d wave when N/2 is odd.Working on a square lattice with an even number of sites ${\mathit{N}}^{2}$ and assuming a smooth connection between strong and weak coupling, we show that the ground state of the two-dimensional half-filled Hubbard model has momentum ${\mathit{k}}_{\mathit{x}}$-${\mathit{k}}_{\mathit{y}}$=0, it is even under reflections, and transforms as an s wave under rotations when N/2 is even and as a d wave when N/2 is odd.

Physical review. B, Condensed matterMay 1, 199016 citations

In this paper, we show rigorously that there exists the ferrimagnetic long-range order in the ground state of the positive-U Hubbard model at half filling on some bipartite lattices. When

{\mathit{N}}_{\mathit{A}}

{\mathit{N}}_{\mathit{B}}

(

{\mathit{N}}_{\mathit{A}}

and

{\mathit{N}}_{\mathit{B}}

are the total site numbers of two sublattices A and B), except for the ferromagnetism which was found by Lieb [Phys. Rev. Lett. 62, 1201 (1989)], there also exists the antiferromagnetic long-range order in the ground state. This result only requires U>0 and is independent of the dimension of the lattices.

U

is turned on, Lieb proved rigorously that at half-filling ($\ensuremath{\rho}=1$) the ground state has a nonzero spin. In this paper we consider a ``

{\mathrm{CuO}}_{2}

lattice'' (also known as ``Lieb lattice,'' or as a decorated square lattice), in which ``

d

orbitals'' occupy the vertices of the squares, while ``

p

orbitals'' lie halfway between two

d

orbitals; both

d

and

p

orbitals can accommodate only up to two electrons. We use exact determinant quantum Monte Carlo (DQMC) simulations to quantify the nature of magnetic order through the behavior of correlation functions and sublattice magnetizations in the different orbitals as a function of

U

and temperature; we have also calculated the projected density of states, and the compressibility. We study both the homogeneous (H) case,

{U}_{d}={U}_{p}

, originally considered by Lieb, and the inhomogeneous (IH) case, ${U}_{d}\ensuremath{\ne}{U}_{p}$. For the H case at half-filling, we found that the global magnetization rises sharply at weak coupling, and then stabilizes towards the strong-coupling (Heisenberg) value, as a result of the interplay between the ferromagnetism of like sites and the antiferromagnetism between unlike sites; we verified that the system is an insulator for all

U

. For the IH system at half-filling, we argue that the case ${U}_{p}\ensuremath{\ne}{U}_{d}$ falls under Lieb's theorem, provided they are positive definite, so we used DQMC to probe the cases

{U}_{p}=0,{U}_{d}=U

and

{U}_{p}=U,{U}_{d}=0

. We found that the different environments of

d

and

p

sites lead to a ferromagnetic insulator when

{U}_{d}=0

; by contrast,

{U}_{p}=0

leads to to a metal without any magnetic ordering. In addition, we have also established that at density $\ensuremath{\rho}=1/3$, strong antiferromagnetic correlations set in, caused by the presence of one fermion on each

d

site.

Working on a square lattice with an even number of sites

{\mathit{N}}^{2}

and assuming a smooth connection between strong and weak coupling, we show that the ground state of the two-dimensional half-filled Hubbard model has momentum

{\mathit{k}}_{\mathit{x}}

{\mathit{k}}_{\mathit{y}}

=0, it is even under reflections, and transforms as an s wave under rotations when N/2 is even and as a d wave when N/2 is odd.