# Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry

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## Abstract

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## 1. Introduction

**Figure 1.**(Color online) Physical analogies between quantum Hall effects (QHE) and valence bond solid (VBS) state. The bulk excitation on QHE is gapful, while the edge mode is a gapless (chiral) mode. Meanwhile, the bulk excitation on the VBS state is gapful while the motion of edge spins is a freely rotating gapless mode.

**Figure 2.**Close relations among fuzzy geometry, QHE and VBS. They are “transformed” to each other with appropriate translations.

## 2. Fuzzy Geometry and Valence Bond Solid States

#### 2.1. Fuzzy Two-Spheres and the Lowest Landau Level Physics

**Figure 3.**(Color online) One-body-level relationship among fuzzy two-sphere (upper left), Haldane sphere (upper right) and local spin states of the VBS state (lower middle). The fuzzy two-sphere consists of a finite number of patches, i.e., the basis states, with width, $2R/(n+1)$. The Haldane sphere is a two-sphere with a Dirac monopole at its center. The $S=n/2$ is the monopole charge quantized as a half-integer or an integer by the Dirac quantization condition. In the local spin state of the VBS state (lower center), each blob denotes spin-$1/2$ degrees of freedom, and n blobs amount to $S=n/2$ local spin by a large Hund coupling on each site.

#### 2.2. Valence Bond Solid States

**Figure 4.**(Color online) VBS states on 1D and 2D lattices. Filled circles denote auxiliary spin-1/2 objects, which are finally symmetrized to form $S=Mz$ physical spins at each site. Solid lines stand for singlet valence bonds between the spin-1/2s.

**Figure 5.**(Color online) For the $S=1$ VBS state on a finite open chain, there exist spin-1/2 degrees of freedom at each edge. By construction, the VBS state is the ground state of the VBS Hamiltonian, regardless of the spin states at the edges.

**Table 1.**Correspondences between physical quantities of many-body wavefunctions of QHE and quantum anti-ferromagnets (QAFM).

QHE | QAFM | |
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Many-body state | Laughlin-Haldane wave function | VBS state |

${\Phi}_{\text{LH}}={\prod}_{i<j}^{N}{({u}_{i}{v}_{j}-{v}_{i}{u}_{j})}^{m}$ | $|\Phi \rangle ={\prod}_{\langle ij\rangle}^{z}{({a}_{i}^{\u2020}{b}_{j}^{\u2020}-{b}_{i}^{\u2020}{a}_{j}^{\u2020})}^{M}|\text{vac}\rangle $ | |

Power | m: inverse of filling factor | M: number of VBs between neighboring sites |

Charge | $S=mN/2$: monopole charge | $S=Mz/2$: local spin magnitude |

## 3. Fuzzy Two-Supersphere

#### 3.1. $\mathcal{N}=1$

**Figure 6.**(Color online) $\mathcal{N}=1$ fuzzy supersphere is a “compound” of two fuzzy two-spheres with radii, $\frac{R}{d}n$ and $\frac{R}{d}(n-1)$. This figure corresponds to $n=2$.

#### 3.2. $\mathcal{N}=2$

**s**($k=2{n}_{2}$, $k=2{l}_{2}+2$) correspond to the bosonic basis states, (78a) and (78b), while odd ks ($k=2{m}_{2}+1$, $k=2{m}_{2}^{\prime}+1$) to the fermionic basis states, (78c) and (78d) (Figure 7). Except for non-degenerate states at the north and south poles ${X}_{3}=\pm n$, all the other eigenvalues of ${X}_{3}$ (82) are doubly-degenerate.

**Figure 7.**(Color online) ${S}_{\mathrm{f}}^{2|4}$ is a “compound” made of four fuzzy two-spheres that are considered as $\mathcal{N}=2$ superpartners. The above picture corresponds to $n=2$.

## 4. Supersymmetric Valence Bond Solid States

#### 4.1. Construction of SVBS States

#### 4.1.1. $\mathcal{N}=1$

**Table 2.**The physical interpretation of the local states made by the Schwinger operators. ${f}^{\u2020}$ denotes the hole degrees of freedom.

Schwinger operator | $\mathit{SU}(2)$ quantum number | Spin state |
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${a}^{\u2020}$ | 1/2 | $|\phantom{\rule{-0.166667em}{0ex}}\uparrow \rangle ={a}^{\u2020}|\text{vac}\rangle $ |

${b}^{\u2020}$ | $-1/2$ | $|\phantom{\rule{-0.166667em}{0ex}}\downarrow \rangle ={b}^{\u2020}|\text{vac}\rangle $ |

${f}^{\u2020}$ | 0 | $|h\rangle ={f}^{\u2020}|\text{vac}\rangle $ |

**Figure 10.**(Color online) The type-I SVBS is a superposed state of hole-pair-doped VBS states. With the finite hole-doping parameter, r, all of the hole-pair-doped VBS states are superposed to form the SVBS state, and the SVBS state exhibits the SC property. At $r=0$, the SVBS state is reduced to the original VBS state (depicted as the first chain), while for $r\to \infty $, the SVBS state is reduced to the MG dimer state (depicted as the last two chains). The figure and caption are taken from [9].

#### 4.1.2. $\mathcal{N}=2$

#### 4.2. Superconducting Properties

**Figure 11.**(Color online) Like the type-I SVBS chain, the type-II SVBS chain is also expressed as a superposition of the hole-pair doped VBS chains. What is different to the type-I SVBS chain is the appearance of the spinless sites, depicted by the large white circles with double holes. The MG states are realized in the “middle” of the sequence. The original VBS state and the hole-VBS state are respectively realized in the first and last lines. The figure and caption are taken from [9].

**Figure 12.**The SVBS states exhibit a superconducting property in the charge sector with finite r in addition to a quantum AFM property in the spin sector.

#### 4.2.1. $\mathcal{N}=1$

**Figure 13.**(Color online) Plot of ${\mathcal{O}}_{\text{sc}}=\langle \Delta \rangle $, the hole density, $\langle {n}_{\mathrm{f}}\rangle $, and the hole-number fluctuation, $\delta {n}_{\mathrm{f}}^{2}$, as a function of r. Here,the bulk values are plotted. Inset: pProfile of the hole density ($r=0.5$) for a finite system ($L=20$) with different left edge states (↑, ↓ and “hole”). Only the left edge state is changed with the right one fixed to ${s}_{\text{R}}=\uparrow $. The hole density approaches exponentially to the bulk value as we move away from the edge. The figure and caption are taken from [9].

#### 4.2.2. $\mathcal{N}=2$

**Figure 14.**(Color online) Plot of ${\mathcal{O}}_{\text{sc}}=\langle {\Delta}_{i}\rangle $, the hole density, $\langle {n}_{\text{hole}}(i)\rangle =\langle {f}_{i}^{\u2020}{f}_{i}\rangle $, and the hole-number fluctuation, $\delta {n}_{\text{hole}}^{2}$, as a function of r. Inset: profile of the hole density ($r=0.5$) for a finite system ($L=20$). Only the left edge state is changed with the right one fixed to ${s}_{\text{R}}=\uparrow $. The figure and caption are taken from [9].