# 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

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## Abstract

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## 1. Introduction

## 2. The Model, a Suitable Rotated-Electron Description, and Relation to the Global $SO(3)\times \phantom{\rule{3.33333pt}{0ex}}SO(3)\times \phantom{\rule{3.33333pt}{0ex}}U(1)$ Symmetry

## 3. Three Elementary Quantum Objects and Corresponding c, $\eta $-Spin, and Spin Effective Lattices

#### 3.1. Elementary Quantum Objects and Their Operators

#### 3.2. Interplay of the Global Symmetry with the Transformation Laws Under the Operator $\widehat{V}$: Three Basic Effective Lattices and the Theory Vacua

- (i)
- The occupancy configurations of the c fermions associated with the operators ${f}_{{\overrightarrow{r}}_{j},c}^{\u2020}={({f}_{{\overrightarrow{r}}_{j},c})}^{\u2020}$ of Equation (11) where $j=1,\dots ,{N}_{a}^{D}$ correspond to the state representations of the hidden global $U(1)$ symmetry found in Reference [6]. Such c fermions live on the c effective lattice. It is identical to the original lattice. Its occupancies are related to those of the rotated electrons: The number of c fermion occupied and unoccupied sites is given by ${N}_{c}={N}_{{a}_{s}}^{D}=2{S}_{c}$ and ${N}_{c}^{h}={N}_{{a}_{\eta}}^{D}=[{N}_{a}^{D}-2{S}_{c}]$, respectively. Indeed, the c fermions occupy the sites singly occupied by the rotated electrons. In turn, the rotated-electron doubly-occupied and unoccupied sites are those unoccupied by the c fermions. Hence the c fermion occupancy configurations describe the relative positions in the original lattice of the ${N}_{{a}_{\eta}}^{D}=[{N}_{a}^{D}-2{S}_{c}]$ sites of the η-spin effective lattice and ${N}_{{a}_{s}}^{D}=2{S}_{c}$ sites of the spin effective lattice.
- (ii)
- The remaining degrees of freedom of rotated-electron occupancies of the sets of ${N}_{{a}_{\eta}}^{D}=[{N}_{a}^{D}-2{S}_{c}]$ and ${N}_{{a}_{s}}^{D}=2{S}_{c}$ original-lattice sites correspond to the occupancy configurations associated with the $\eta $-spin $SU(2)$ symmetry and spin $SU(2)$ symmetry state representations, respectively. The occupancy configurations of the set of ${N}_{{a}_{\eta}}^{D}=[{N}_{a}^{D}-2{S}_{c}]$ sites of the $\eta $-spin effective lattice and set of ${N}_{{a}_{s}}^{D}=2{S}_{c}$ sites of the spin effective lattice are independent. The former configurations refer to the operators ${p}_{{\overrightarrow{r}}_{j}}^{l}$ of Equation (12), which act only onto the ${N}_{{a}_{\eta}}^{D}=[{N}_{a}^{D}-2{S}_{c}]$ sites of the $\eta $-spin effective lattice. The latter configurations correspond to the operators ${s}_{{\overrightarrow{r}}_{j}}^{l}$ given in the same equation, which act onto the ${N}_{{a}_{s}}^{D}=2{S}_{c}$ sites of the spin effective lattice. This is assured by the operators $(1-{n}_{{\overrightarrow{r}}_{j},c})$ and ${n}_{{\overrightarrow{r}}_{j},c}$ in their expressions provided in that equation, which play the role of projectors onto the $\eta $-spin and spin effective lattice, respectively.

#### 3.3. Important Energy Scales

#### 3.4. Spacing and Occupied and Unoccupied Sites of the $\eta $-Spin and Spin Effective Lattices

- (1)
- The representation associated with the present description contains full information about the relative positions of the sites of the $\eta $-spin and spin effective lattices in the original lattice. For each energy-eigenstate rotated-electron real-space occupancy configuration, that information is stored in the corresponding occupancy configurations of the c fermions in their c effective lattice. The latter lattice is identical to the original lattice. Such configurations correspond to the state representations of the $U(1)$ symmetry in the subspaces spanned by states with fixed values of ${S}_{c}$, ${S}_{\eta}$, and ${S}_{s}$. Indeed, the sites of the $\eta $-spin (and spin) effective lattice have in the original lattice the same real-space coordinates as the sites of the c effective lattice unoccupied (and occupied) by c fermions.
- (2)
- Within the ${N}_{a}^{D}\gg 1$ limit that the rotated-electron related operational description refers to, provided that the electronic density $n=(1-x)$ (and hole concentration x) is finite, the dominant c effective lattice occupancy configurations of an energy eigenstate of the Hubbard model on the square lattice refer to a nearly uniform distribution of the c fermions occupied sites (and unoccupied sites). Hence due to the spinon and $\eta $ spinon distribution “average order” emerging for the model on the square lattice in the ${N}_{a}^{D}\to \infty $ limit, the spin and $\eta $-spin effective lattices may be represented by square lattices. Moreover, the chain order invariance occurring for the 1D model both for the finite system and in that limit justifies why such effective lattices are 1D lattices. For both models the corresponding spin effective lattice spacing ${a}_{s}$ and $\eta $-spin effective lattice spacing ${a}_{\eta}$ refers to the average spacing between the c effective lattice occupied sites and between such a lattice unoccupied sites, respectively, given in Equation (30). Such spin and $\eta $-spin effective lattices obey the physical requirement condition that in the $x\to 0$ and $x\to \pm 1$ limit, respectively, equal the original lattice. Note that in the $x\to 0$ (and $x\to \pm 1$) limit one has that ${N}_{{a}_{s}}^{D}={N}_{a}^{D}$ and the $\eta $-spin effective lattice does not exist (and ${N}_{{a}_{\eta}}^{D}={N}_{a}^{D}$ and the spin effective lattice does not exist.)

## 4. The Composite $\alpha \nu $ Fermions

#### 4.1. The ${M}_{\eta}^{co}$-$\eta $-Spinon and ${M}_{s}^{co}$-Spinon Configuration Partitions

#### 4.2. The $\alpha \nu $ Fermion Operators

#### 4.3. Ranges of the c and $\alpha \nu $ Fermion Energies, Their Transformation Laws, and the Ground-State Occupancies

#### The $\eta \nu $ Fermion Energy Range

#### The $s\nu $ Fermion Energy Range

#### The c Fermion Energy Range

#### Transformation Laws of $\alpha \nu $ Fermions and c Fermions under the Electron-Rotated-Electron Unitary Transformation

#### Ground State Occupancies

#### 4.4. The Site Numbers and Spacing of the $\alpha \nu $ Effective Lattices

## 5. The Square-Lattice Quantum Liquid: A Two-Component Fluid of Charge c Fermions and Spin-Neutral Two-Spinon $s1$ Fermions

#### 5.1. The One- and Two-Electron Subspace

#### General $\mathcal{N}$-Electron Subspaces

#### The One- and Two-Electron Subspace

#### 5.2. The Spin and $s1$ Effective Lattices for the One- and Two-Electron Subspace

#### 5.3. The Square-Lattice Quantum Liquid of c and $s1$ Fermions

#### 5.4. A Preliminary Application: The Inelastic Neutron Scattering of LCO

## 6. Concluding Remarks

## Acknowledgments

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**Figure 1.**The $\eta 1$ fermion band for the 1D Hubbard model in units of t plotted for electronic densities $n=(1-x)=1/2,5/6$, spin density $m=0$, and a set of $U/t$ values. The band $U/t\to \infty $ limit corresponds to the horizontal line chosen as zero-energy level. [26]

**Figure 3.**The $s1$ fermion band for the 1D Hubbard model in units of t plotted for electronic densities $n=(1-x)=1/2,5/6$, spin density $m=0$, and a set of $U/t$ values. The ground-state energy level was chosen to correspond to zero energy and overlaps and overlaps the energy band at the $s1$ Fermi points $q=\pm {q}_{Fs1}=\pm {k}_{F}=\pm [\pi /2]\phantom{\rule{0.166667em}{0ex}}n=\pm [\pi /2]\phantom{\rule{0.166667em}{0ex}}(1-x)$. For 1D there is no spin short-range spin order, so that $2|\mathsf{\Delta}|=0$. [26]

**Figure 4.**The c fermion band for the 1D Hubbard model in units of t plotted for electronic densities $n=(1-x)=1/2,5/6$, spin density $m=0$, and a set of $U/t$ values. The ground-state energy level was chosen to correspond to zero energy. It is marked by a horizontal line which overlaps the c band at the c Fermi points $q=\pm {q}_{Fc}=\pm 2{k}_{F}=\pi \phantom{\rule{0.166667em}{0ex}}n=\pi \phantom{\rule{0.166667em}{0ex}}(1-x)$. [26]

**Figure 5.**The theoretical spin spectrum Equation (65) (solid lines) plotted for the high symmetry directions in the second Brillouin zone for ${\mu}^{0}=565.6$ meV and ${W}_{s1}^{0}=49.6$ meV and the experimental data of Reference [13] (circles) in meV. Such theoretical magnitudes correspond to $t\approx 0.295$ eV and $U\approx 1.800$ eV, so that $U/4t\approx 1.525$. The momentum is given in units of $2\pi $. The corresponding theoretical lines plotted in Figure 5 of Reference [14] are very similar to those plotted here yet are obtained within the standard formalism of many-body physics by summing up an infinite number of ladder diagrams. [10]

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Carmelo, J.M.P.; Sampaio, M.J.
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. *Symmetry* **2011**, *3*, 780-827.
https://doi.org/10.3390/sym3040780

**AMA Style**

Carmelo JMP, Sampaio MJ.
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. *Symmetry*. 2011; 3(4):780-827.
https://doi.org/10.3390/sym3040780

**Chicago/Turabian Style**

Carmelo, Jose M. P., and Maria J. Sampaio.
2011. "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" *Symmetry* 3, no. 4: 780-827.
https://doi.org/10.3390/sym3040780