Modern Physics Letters B

GLOBAL AND GAUGE SYMMETRIES OF THE HUBBARD MODEL

Elisa Ercolessi, G. Morandi, F. Ortolani

Published Jan 20, 1992• 2 citations

Abstract

A remarkable global SO(4) symmetry of the Hubbard model, when there is on the average one electron per site (“weak” half-filling condition) has been recently uncovered by Yang and Zhang. We discuss here its relationship to the SU(2) gauge symmetry of the model that holds when there is exactly one electron per site (“strong” half-filling condition) as well as its possible generalizations to lattice structures more general than those considered by Yang and Zhang.

DOI: 10.1142/s0217984992000107

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SO(4) Symmetry of the Hubbard Model and its Experimental Consequences

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Symmetry and spontaneous symmetry breaking of the Hubbard model on a square lattice ground state

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

Z. Song, H Shen

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Modern Physics Letters B

GLOBAL AND GAUGE SYMMETRIES OF THE HUBBARD MODEL

Elisa Ercolessi, G. Morandi, F. Ortolani

Published Jan 20, 1992• 2 citations

Publisher

Abstract

DOI: 10.1142/s0217984992000107

Related papers

SO(4) Symmetry of the Hubbard Model and its Experimental Consequences

Shoucheng Zhang

The one band Hubbard model on a bi-partite lattice possess a global SO(4) symmetry group which is a direct product of the spin SU(2) and the pseudospin SU(2) symmetry group. Recently, the group theorectical consequence of this symmetry property has been investigated systematically. It is shown that this symmetry property can be used to reduce the complexity of diagonalizing the Hubbard Hamiltonian, and classify all the eigenstates according to the representation of the SO(4) group. It also leads to an exact one-to-one corespondence of the eigenstates at half-filling to that away from half-filling. Most importantly however, it is shown that this symmetry property leads to a direct experimental prediction for the high Tc superconductivity. If the high Tc superconductivity is correctly described by the Hubbard model, then the SO(4) symmetry dictates the existence of a collective mode that only appears below the superconducting transition temperature. Such a mode can be detected in an electron energy loss experiment as a sharp resonance feature. The location of the peak measure the parameters in the Hubbard model, the intensity is proportional to the superconducting order parameter and the width measures the departure of the Hubbard model from reality.The one band Hubbard model on a bi-partite lattice possess a global SO(4) symmetry group which is a direct product of the spin SU(2) and the pseudospin SU(2) symmetry group. Recently, the group theorectical consequence of this symmetry property has been investigated systematically. It is shown that this symmetry property can be used to reduce the complexity of diagonalizing the Hubbard Hamiltonian, and classify all the eigenstates according to the representation of the SO(4) group. It also leads to an exact one-to-one corespondence of the eigenstates at half-filling to that away from half-filling. Most importantly however, it is shown that this symmetry property leads to a direct experimental prediction for the high Tc superconductivity. If the high Tc superconductivity is correctly described by the Hubbard model, then the SO(4) symmetry dictates the existence of a collective mode that only appears below the superconducting transition temperature. Such a mode can be detected in an electron energy loss experiment as a sharp resonance feature. The location of the peak measure the parameters in the Hubbard model, the intensity is proportional to the superconducting order parameter and the width measures the departure of the Hubbard model from reality.

International Journal of Modern Physics BJan 1, 199128 citations

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Physical Review LettersApr 20, 19922 citations

Symmetry and spontaneous symmetry breaking of the Hubbard model on a square lattice ground state

J. M. P. Carmelo

In this paper we consider the simplified form that a recently introduced general operator description of the Hubbard model on the square lattice with $N_a^2\gg 1$ sites, effective transfer integral $t$ , and onsite repulsion $U$ has in a suitable one- and two-electron subspace. Such an operator description is that consistent with the model exact global symmetry recently extended to $SO(3)\times SO(3)\times U(1)$ . There is a large consensus that in the thermodynamic limit $N_a^2\rightarrow\infty$ long-range antiferromagnetic order occurs in the spin-density $m=0$ ground state of the half-filled Hubbard model on the square lattice. Here we find that the corresponding spontaneous symmetry breaking lowers the symmetry of such a state from $SO(3)\times SO(3)\times U(1)$ for $N_a^2\gg 1$ large but finite to $[U(2)\times U(1)]/Z_2^2 \equiv [SO(3)\times U(1)\times U(1)]/Z_2$ for $N_a^2\rightarrow\infty$ . Moreover, we argue that the spin effective lattice being identical to the original lattice is a necessary condition for the occurrence of ground-state long-range antiferromagnetic order in the limit $N_a^2\rightarrow\infty$ . Consistently, strong evidence is provided that for very small hole concentration $0In this paper we consider the simplified form that a recently introduced general operator description of the Hubbard model on the square lattice with $N_a^2\gg 1$ sites, effective transfer integral $t$, and onsite repulsion $U$ has in a suitable one- and two-electron subspace. Such an operator description is that consistent with the model exact global symmetry recently extended to $SO(3)\times SO(3)\times U(1)$. There is a large consensus that in the thermodynamic limit $N_a^2\rightarrow\infty$ long-range antiferromagnetic order occurs in the spin-density $m=0$ ground state of the half-filled Hubbard model on the square lattice. Here we find that the corresponding spontaneous symmetry breaking lowers the symmetry of such a state from $SO(3)\times SO(3)\times U(1)$ for $N_a^2\gg 1$ large but finite to $[U(2)\times U(1)]/Z_2^2 \equiv [SO(3)\times U(1)\times U(1)]/Z_2$ for $N_a^2\rightarrow\infty$. Moreover, we argue that the spin effective lattice being identical to the original lattice is a necessary condition for the occurrence of ground-state long-range antiferromagnetic order in the limit $N_a^2\rightarrow\infty$. Consistently, strong evidence is provided that for very small hole concentration $0

arXiv:1003.4619Mar 24, 20100 citations

A description of the Hubbard model on a square lattice consistent with its global $SO(3)\times SO(3)\times U(1)$ symmetryA description of the Hubbard model on a square lattice consistent with its global $SO(3)\times SO(3)\times U(1)$ symmetry

J. M. P. Carmelo, M. J. Sampaio

In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral $t$ , on-site repulsion $U$ , and $N_a^2\gg 1$ sites consistent with its exact global $SO(3)\times SO(3)\times U(1)$ symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For $U/4t>0$ the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge $c$ fermions, spin-1/2 spinons, and $η$ -spin-1/2 $η$ -spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the $η$ -spinon (and spinons) that are not invariant under that transformation have $η$ spin 1/2 (and spin 1/2) but are part of $η$ -spin-neutral (and spin-neutral) $2ν$ - $η$ -spinon (and $2ν$ -spinon) composite $ην$ fermions (and $sν$ fermions) where $ν=1,2,...$ is the number of $η$ -spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at $x=0$ hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations $x>0$ .In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral $t$, on-site repulsion $U$, and $N_a^2\gg 1$ sites consistent with its exact global $SO(3)\times SO(3)\times U(1)$ symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For $U/4t>0$ the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge $c$ fermions, spin-1/2 spinons, and $η$-spin-1/2 $η$-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the $η$-spinon (and spinons) that are not invariant under that transformation have $η$ spin 1/2 (and spin 1/2) but are part of $η$-spin-neutral (and spin-neutral) $2ν$-$η$-spinon (and $2ν$-spinon) composite $ην$ fermions (and $sν$ fermions) where $ν=1,2,...$ is the number of $η$-spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at $x=0$ hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations $x>0$.

arXiv:0804.2358Apr 15, 20080 citations

Global $SO(3)\times SO(3)\times U(1)$ symmetry of the Hubbard model on bipartite latticesGlobal $SO(3)\times SO(3)\times U(1)$ symmetry of the Hubbard model on bipartite lattices

J. M. P. Carmelo, Stellan S. Ostlund, M. J. Sampaio

It is found that for on-site interaction $U\neq 0$ the local $SU(2)\times SU(2) \times U(1)$ gauge symmetry of the Hubbard model on a bipartite lattice with vanishing transfer integral $t=0$ can be lifted to a global $[SU(2)\times SU(2)\times U(1)]/Z_2^2=SO(3)\times SO(3)\times U(1)$ symmetry in the presence of the kinetic-energy hopping term of the Hamiltonian with $t>0$ . The generator of the new found hidden independent charge global U(1) symmetry is one half the rotated-electron number of singly-occupied sites operator. It is confirmed elsewhere that our results have important physical consequences concerning the further understanding of the unusual properties of the hole-doped cuprates.It is found that for on-site interaction $U\neq 0$ the local $SU(2)\times SU(2) \times U(1)$ gauge symmetry of the Hubbard model on a bipartite lattice with vanishing transfer integral $t=0$ can be lifted to a global $[SU(2)\times SU(2)\times U(1)]/Z_2^2=SO(3)\times SO(3)\times U(1)$ symmetry in the presence of the kinetic-energy hopping term of the Hamiltonian with $t>0$. The generator of the new found hidden independent charge global U(1) symmetry is one half the rotated-electron number of singly-occupied sites operator. It is confirmed elsewhere that our results have important physical consequences concerning the further understanding of the unusual properties of the hole-doped cuprates.

arXiv:0802.2146Feb 15, 20080 citations

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

Z. Song, H Shen

Journal of Physics Condensed MatterFeb 14, 19940 citations

In this paper we consider the simplified form that a recently introduced general operator description of the Hubbard model on the square lattice with

N_a^2\gg 1

sites, effective transfer integral

t

, and onsite repulsion

U

has in a suitable one- and two-electron subspace. Such an operator description is that consistent with the model exact global symmetry recently extended to

SO(3)\times SO(3)\times U(1)

. There is a large consensus that in the thermodynamic limit

N_a^2\rightarrow\infty

long-range antiferromagnetic order occurs in the spin-density

m=0

ground state of the half-filled Hubbard model on the square lattice. Here we find that the corresponding spontaneous symmetry breaking lowers the symmetry of such a state from

SO(3)\times SO(3)\times U(1)

for

N_a^2\gg 1

large but finite to

[U(2)\times U(1)]/Z_2^2 \equiv [SO(3)\times U(1)\times U(1)]/Z_2

for

N_a^2\rightarrow\infty

. Moreover, we argue that the spin effective lattice being identical to the original lattice is a necessary condition for the occurrence of ground-state long-range antiferromagnetic order in the limit

N_a^2\rightarrow\infty

. Consistently, strong evidence is provided that for very small hole concentration $0

A description of the Hubbard model on a square lattice consistent with its global

SO(3)\times SO(3)\times U(1)

symmetry

In this paper a description of the Hubbard model on the square lattice with nearest-neighbor transfer integral

t

, on-site repulsion

U

, and

N_a^2\gg 1

sites consistent with its exact global

SO(3)\times SO(3)\times U(1)

symmetry is constructed. Our studies profit from the interplay of that recently found global symmetry of the model on any bipartite lattice with the transformation laws under a suitable electron - rotated-electron unitary transformation of a well-defined set of operators and quantum objects. For

U/4t>0

the occupancy configurations of these objects generate the energy eigenstates that span the one- and two-electron subspace. Such a subspace as defined in this paper contains nearly the whole spectral weight of the excitations generated by application onto the zero-spin-density ground state of one- and two-electron operators. Our description involves three basic objects: charge

c

fermions, spin-1/2 spinons, and

η

-spin-1/2

η

-spinons. Alike in chromodynamics the quarks have color but all quark-composite physical particles are color-neutral, the

η

-spinon (and spinons) that are not invariant under that transformation have

η

spin 1/2 (and spin 1/2) but are part of

η

-spin-neutral (and spin-neutral)

2ν

η

-spinon (and

2ν

-spinon) composite

ην

fermions (and

sν

fermions) where

ν=1,2,...

is the number of

η

-spinon (and spinon) pairs.The description introduced here is consistent with a Mott-Hubbard insulating ground state with antiferromagnetic long-range order for half filling at

x=0

hole concentration and a ground state with short-range spin order for a well-defined range of finite hole concentrations

x>0

Global

SO(3)\times SO(3)\times U(1)

symmetry of the Hubbard model on bipartite lattices

It is found that for on-site interaction

U\neq 0

the local

SU(2)\times SU(2) \times U(1)

gauge symmetry of the Hubbard model on a bipartite lattice with vanishing transfer integral

t=0

can be lifted to a global

[SU(2)\times SU(2)\times U(1)]/Z_2^2=SO(3)\times SO(3)\times U(1)

symmetry in the presence of the kinetic-energy hopping term of the Hamiltonian with

t>0

. The generator of the new found hidden independent charge global U(1) symmetry is one half the rotated-electron number of singly-occupied sites operator. It is confirmed elsewhere that our results have important physical consequences concerning the further understanding of the unusual properties of the hole-doped cuprates.