Symmetry

An Application of the Extended Global SO(3) × SO(3) × U(1) Symmetry of the Hubbard Model on a Square Lattice: The Spinon, η-Spinon, and c Fermion Description

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

Published Dec 12, 2011• 1 citation

Abstract

In this paper we review recent results on the preliminary applications of the new-found extended global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on a bipartite lattice. Our results refer to the particular case of the bipartite square lattice. Specifically, we review a general description for such a model with nearest-neighbor transfer integral t and on-site repulsion U on a square lattice with N2a 1 sites consistent with its extended global symmetry. It refers to three types of elementary objects whose occupancy configurations generate the state representations of the model extended global symmetry. Such objects emerge from a suitable electron-rotated-electron unitary transformation. An application to the spin spectrum of the parent compound La2CuO4 is shortly reviewed.

DOI: 10.3390/sym3040780

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

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arXiv:0804.2358Apr 15, 20080 citations

The Hubbard model on a square lattice in the one- and two-electron subspace

J. M. P. Carmelo

arXiv:0804.2379Apr 15, 20081 citation

Superconductivity and magnetism in strongly correlated electronic systems

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Sep 11, 20090 citations

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

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arXiv:0802.2146Feb 15, 20080 citations

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

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arXiv:1003.4619Mar 24, 20100 citations

Finite-temperature charge and spin transport in the one-dimensional Hubbard model accounting for its global [ SU ( 2 ) × SU ( 2 ) × U ( 1 ) ]/ Z 2 2 symmetry

J. M. P. Carmelo, J. E. C. Carmelo

arXiv:2503.21573Physical review. B./Physical review. BMar 26, 20257 citations

Symmetry

An Application of the Extended Global SO(3) × SO(3) × U(1) Symmetry of the Hubbard Model on a Square Lattice: The Spinon, η-Spinon, and c Fermion Description

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

Published Dec 12, 2011• 1 citation

Publisher

Abstract

DOI: 10.3390/sym3040780

Related papers

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

The Hubbard model on a square lattice in the one- and two-electron subspace

J. M. P. Carmelo

In this paper the spin configurations of the ground state and one- and two-electron excited states of the Hubbard model on the square lattice are studied. We profit from a general rotated-electron description, which is consistent with the model global $SO(3)\times SO(3)\times U(1)$ symmetry. For rotated electrons, doubly and single occupancy are good quantum numbers for on-site repulsion $U>0$ . The above states are within that description generated by occupancy configurations of charge $c$ fermions and spin-singlet two-spinon $s1$ bond particles. Those describe the charge and spin degrees of freedom, respectively, of the rotated electrons that singly occupy sites. While the $c$ fermions have no internal structure, that of the spin-neutral $s1$ bond-particle occupancy configurations is here described in terms of spinon occupancies of a well-defined effective spin lattice. In reference \cite{companion} it is confirmed that our results contribute to the further understanding of the role of electronic correlations in the spin spectrum of the parent compound La $_2$ CuO $_4$ . They are also of interest for studies of ultra-cold fermionic atoms on an optical lattice.In this paper the spin configurations of the ground state and one- and two-electron excited states of the Hubbard model on the square lattice are studied. We profit from a general rotated-electron description, which is consistent with the model global $SO(3)\times SO(3)\times U(1)$ symmetry. For rotated electrons, doubly and single occupancy are good quantum numbers for on-site repulsion $U>0$. The above states are within that description generated by occupancy configurations of charge $c$ fermions and spin-singlet two-spinon $s1$ bond particles. Those describe the charge and spin degrees of freedom, respectively, of the rotated electrons that singly occupy sites. While the $c$ fermions have no internal structure, that of the spin-neutral $s1$ bond-particle occupancy configurations is here described in terms of spinon occupancies of a well-defined effective spin lattice. In reference \cite{companion} it is confirmed that our results contribute to the further understanding of the role of electronic correlations in the spin spectrum of the parent compound La$_2$CuO$_4$. They are also of interest for studies of ultra-cold fermionic atoms on an optical lattice.

arXiv:0804.2379Apr 15, 20081 citation

Superconductivity and magnetism in strongly correlated electronic systems

M. J. Sampaio

In this thesis, symmetry and related transformation laws under a suitable unitary transformation are used in the study of the superconducting and magnetic properties of strongly correlated electronic systems. Such a study profits from a recently found global SO(3)×SO(3)×U(1) = [SO(4)×U(1)]/Z2 symmetry in the Hubbard model on a bipartite lattice. The introduction of the square-lattice quantum liquid relies on a general description of the Hubbard model consistent with its exact global SO(3) × SO(3) × U(1) symmetry. Such a quantum liquid is expected to play the same role for many-electron problems with short-range interactions on a square lattice, as a Fermi liquid for three-dimensional isotropic metals. It refers to the Hubbard model on the square lattice in the subspace spanned by the ground state, and the excited states generated by application onto it of one- and two-electron operators. For such a square-lattice quantum liquid only the charge c fermions and spin-neutral two-spinon s1 fermions play an active role. Part of the studies of this thesis use an effective superconductivity theory according to which, for zero spin density m = 0 and the range 0 < x < x∗ of finite hole concentrations of a short-range spin ordered phase, the ground state of the square-lattice quantum liquid of c and s1 fermions with residual interactions is superconducting. (For U/4t > ui ≈ 3/4 the critical hole concentration x∗ increases continuously upon increasing the U/4t value from x∗ = 1/√2 ≈ 0.225 at U/4t = ui to x∗ = 1/ ≈ 0.318 for U/4t → ∞.) Superconductivity emerges naturally from the effects of the residual interactions of the c and s1 fermions and as a side product of the short-range spin correlations. The phase of the c fermion pairs reads = 0 + 1 where the fluctuations of the phases 0 and 1 become large for x → 0 and x → x∗, respectively. For 0 < x < x∗ and x = 0, the symmetry of the action for the phase is a global superconducting U(1) symmetry and a local compact gauge symmetry, respectively. The former U(1) symmetry is identified with the U(1) symmetry of the Hubbard-model global SO(3) × SO(3) × U(1) = [SO(4) × U(1)]/Z2 symmetry. The amplitudes g = g0 g1, g0 = |heiθ0i|, and g1 = |heiθ1i|, their zero-temperature values ˘g ≈ (x/x∗)(1−x/x∗), ˘g0 ≈ (x/x∗), and ˘g1 ≈ (1−x/x∗) and the fluctuations of the phase , play an important role in the physics of the square-lattice quantum liquid. Together with the energy scale 2 0, which gives the maximum magnitude of the s1 fermion spinon pairing energy, they fully control the order parameters of both the short-range spin and long-range superconducting correlations. Indeed, for temperatures T < Tc, the short-range spin ordered phase has both a macroscopic quantum phase-coherent pair superconducting order so that |heiθi| > 0, and a short-range spin order, whereas for Tc < T < T∗ only the latter order prevails yet there remain c fermion pairing correlations such that |heiθ1i| > 0 but heiθ0i = 0. Here the critical temperature Tc and the crossover pseudogap temperature T∗ are fully controlled by the absolute value of the order parameters of the two above orders, respectively. However, the square-lattice quantum liquid alone does not describe quantitatively the unusual properties of the hole-doped cuprates. To fulfill such a goal we profit from a suitable theory according to which the interplay of electronic correlations described by the square-lattice quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions with intrinsic disorder, is behind the unusual properties of the holedoped hight-temperature cuprate superconductors. By profiting from such an interplay, an excellent quantitative agreement with experiments for the physical quantities of several families of hole-doped cuprates is reached. That includes universal properties such as a dome-like critical-temperature hole-concentration dependence and normal-state linear-T resistivity near optimal doping, extending over a wide temperature window. The results confirm a coexisting two-gap scenario associated with an incoherent two-spinon pairing and a coherent twoc- fermion pairing, respectively. While most of our studies focus on the superconducting and magnetic properties of the square-lattice quantum liquid associated with the Hubbard model on the square lattice, the magnetic properties of a related model – the periodic Anderson model – are also addressed, on the cubic lattice.In this thesis, symmetry and related transformation laws under a suitable unitary transformation are used in the study of the superconducting and magnetic properties of strongly correlated electronic systems. Such a study profits from a recently found global SO(3)×SO(3)×U(1) = [SO(4)×U(1)]/Z2 symmetry in the Hubbard model on a bipartite lattice. The introduction of the square-lattice quantum liquid relies on a general description of the Hubbard model consistent with its exact global SO(3) × SO(3) × U(1) symmetry. Such a quantum liquid is expected to play the same role for many-electron problems with short-range interactions on a square lattice, as a Fermi liquid for three-dimensional isotropic metals. It refers to the Hubbard model on the square lattice in the subspace spanned by the ground state, and the excited states generated by application onto it of one- and two-electron operators. For such a square-lattice quantum liquid only the charge c fermions and spin-neutral two-spinon s1 fermions play an active role. Part of the studies of this thesis use an effective superconductivity theory according to which, for zero spin density m = 0 and the range 0 < x < x∗ of finite hole concentrations of a short-range spin ordered phase, the ground state of the square-lattice quantum liquid of c and s1 fermions with residual interactions is superconducting. (For U/4t > ui ≈ 3/4 the critical hole concentration x∗ increases continuously upon increasing the U/4t value from x∗ = 1/√2 ≈ 0.225 at U/4t = ui to x∗ = 1/ ≈ 0.318 for U/4t → ∞.) Superconductivity emerges naturally from the effects of the residual interactions of the c and s1 fermions and as a side product of the short-range spin correlations. The phase of the c fermion pairs reads = 0 + 1 where the fluctuations of the phases 0 and 1 become large for x → 0 and x → x∗, respectively. For 0 < x < x∗ and x = 0, the symmetry of the action for the phase is a global superconducting U(1) symmetry and a local compact gauge symmetry, respectively. The former U(1) symmetry is identified with the U(1) symmetry of the Hubbard-model global SO(3) × SO(3) × U(1) = [SO(4) × U(1)]/Z2 symmetry. The amplitudes g = g0 g1, g0 = |heiθ0i|, and g1 = |heiθ1i|, their zero-temperature values ˘g ≈ (x/x∗)(1−x/x∗), ˘g0 ≈ (x/x∗), and ˘g1 ≈ (1−x/x∗) and the fluctuations of the phase , play an important role in the physics of the square-lattice quantum liquid. Together with the energy scale 2 0, which gives the maximum magnitude of the s1 fermion spinon pairing energy, they fully control the order parameters of both the short-range spin and long-range superconducting correlations. Indeed, for temperatures T < Tc, the short-range spin ordered phase has both a macroscopic quantum phase-coherent pair superconducting order so that |heiθi| > 0, and a short-range spin order, whereas for Tc < T < T∗ only the latter order prevails yet there remain c fermion pairing correlations such that |heiθ1i| > 0 but heiθ0i = 0. Here the critical temperature Tc and the crossover pseudogap temperature T∗ are fully controlled by the absolute value of the order parameters of the two above orders, respectively. However, the square-lattice quantum liquid alone does not describe quantitatively the unusual properties of the hole-doped cuprates. To fulfill such a goal we profit from a suitable theory according to which the interplay of electronic correlations described by the square-lattice quantum liquid of charge c fermions and spin-neutral two-spinon s1 fermions with intrinsic disorder, is behind the unusual properties of the holedoped hight-temperature cuprate superconductors. By profiting from such an interplay, an excellent quantitative agreement with experiments for the physical quantities of several families of hole-doped cuprates is reached. That includes universal properties such as a dome-like critical-temperature hole-concentration dependence and normal-state linear-T resistivity near optimal doping, extending over a wide temperature window. The results confirm a coexisting two-gap scenario associated with an incoherent two-spinon pairing and a coherent twoc- fermion pairing, respectively. While most of our studies focus on the superconducting and magnetic properties of the square-lattice quantum liquid associated with the Hubbard model on the square lattice, the magnetic properties of a related model – the periodic Anderson model – are also addressed, on the cubic lattice.

Sep 11, 20090 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

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

Finite-temperature charge and spin transport in the one-dimensional Hubbard model accounting for its global [ SU ( 2 ) × SU ( 2 ) × U ( 1 ) ]/ Z 2 2 symmetry

J. M. P. Carmelo, J. E. C. Carmelo

Most studies on the one-dimensional (1D) Hubbard model with transfer integral $t$ and on-site repulsion $U$ have assumed that for $u=U/4t>0$ and $h=\ensuremath{\mu}=0$, where $h$ denotes the magnetic field and $\ensuremath{\mu}$ the chemical potential, its global symmetry is $\mathrm{SO}(4)=[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/{Z}_{2}$. However, both in 1D and for the $U>0$ Hubbard model on any bipartite lattice it is actually larger and given by $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}^{2}=[\mathrm{SO}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}$. The studies of this paper account for the irreducible representations of that symmetry to identify both the spin and charge carriers that populate each of the energy eigenstates that span that model full Hilbert space. The global $\ensuremath{\tau}$-translational U(1) symmetry beyond $\mathrm{SO}(4)$ is found to describe the relative translational degrees of freedom of such spin and charge carriers, respectively. We show that important finite-temperature transport quantities as the $h>0$ spin and $\ensuremath{\mu}>0$ charge stiffnesses and the $h=0$ spin and $\ensuremath{\mu}=0$ charge diffusion constants are controlled by microscopic processes associated with the spin and charge elementary currents carried by these spin and charge carriers, respectively. We describe the model finite-temperature transport properties in terms of the microscopic processes associated with the spin and charge elementary currents carried by the corresponding carriers. We expect that the general $\ensuremath{\tau}$- and physical $\ensuremath{\alpha}$-spin representation used in our study is that suitable to address the open problems on finite-temperature transport in the 1D Hubbard model also discussed in this paper.Most studies on the one-dimensional (1D) Hubbard model with transfer integral $t$ and on-site repulsion $U$ have assumed that for $u=U/4t>0$ and $h=\ensuremath{\mu}=0$, where $h$ denotes the magnetic field and $\ensuremath{\mu}$ the chemical potential, its global symmetry is $\mathrm{SO}(4)=[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/{Z}_{2}$. However, both in 1D and for the $U>0$ Hubbard model on any bipartite lattice it is actually larger and given by $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}^{2}=[\mathrm{SO}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}$. The studies of this paper account for the irreducible representations of that symmetry to identify both the spin and charge carriers that populate each of the energy eigenstates that span that model full Hilbert space. The global $\ensuremath{\tau}$-translational U(1) symmetry beyond $\mathrm{SO}(4)$ is found to describe the relative translational degrees of freedom of such spin and charge carriers, respectively. We show that important finite-temperature transport quantities as the $h>0$ spin and $\ensuremath{\mu}>0$ charge stiffnesses and the $h=0$ spin and $\ensuremath{\mu}=0$ charge diffusion constants are controlled by microscopic processes associated with the spin and charge elementary currents carried by these spin and charge carriers, respectively. We describe the model finite-temperature transport properties in terms of the microscopic processes associated with the spin and charge elementary currents carried by the corresponding carriers. We expect that the general $\ensuremath{\tau}$- and physical $\ensuremath{\alpha}$-spin representation used in our study is that suitable to address the open problems on finite-temperature transport in the 1D Hubbard model also discussed in this paper.

arXiv:2503.21573Physical review. B./Physical review. BMar 26, 20257 citations

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

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

symmetry. For rotated electrons, doubly and single occupancy are good quantum numbers for on-site repulsion

U>0

. The above states are within that description generated by occupancy configurations of charge

c

fermions and spin-singlet two-spinon

s1

bond particles. Those describe the charge and spin degrees of freedom, respectively, of the rotated electrons that singly occupy sites. While the

c

fermions have no internal structure, that of the spin-neutral

s1

bond-particle occupancy configurations is here described in terms of spinon occupancies of a well-defined effective spin lattice. In reference \cite{companion} it is confirmed that our results contribute to the further understanding of the role of electronic correlations in the spin spectrum of the parent compound La

_2

CuO

_4

. They are also of interest for studies of ultra-cold fermionic atoms on an optical lattice.

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.

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

Most studies on the one-dimensional (1D) Hubbard model with transfer integral

t

and on-site repulsion

U

have assumed that for

u=U/4t>0

and $h=\ensuremath{\mu}=0$, where

h

denotes the magnetic field and $\ensuremath{\mu}$ the chemical potential, its global symmetry is $\mathrm{SO}(4)=[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)]/{Z}_{2}$. However, both in 1D and for the

U>0

Hubbard model on any bipartite lattice it is actually larger and given by $[\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{SU}(2)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}^{2}=[\mathrm{SO}(4)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)]/{Z}_{2}$. The studies of this paper account for the irreducible representations of that symmetry to identify both the spin and charge carriers that populate each of the energy eigenstates that span that model full Hilbert space. The global $\ensuremath{\tau}$-translational U(1) symmetry beyond

\mathrm{SO}(4)

is found to describe the relative translational degrees of freedom of such spin and charge carriers, respectively. We show that important finite-temperature transport quantities as the

h>0

spin and $\ensuremath{\mu}>0$ charge stiffnesses and the

h=0

spin and $\ensuremath{\mu}=0$ charge diffusion constants are controlled by microscopic processes associated with the spin and charge elementary currents carried by these spin and charge carriers, respectively. We describe the model finite-temperature transport properties in terms of the microscopic processes associated with the spin and charge elementary currents carried by the corresponding carriers. We expect that the general $\ensuremath{\tau}$- and physical $\ensuremath{\alpha}$-spin representation used in our study is that suitable to address the open problems on finite-temperature transport in the 1D Hubbard model also discussed in this paper.