# Tomonaga–Luttinger Spin Liquid and Kosterlitz–Thouless Transition in the Spin-1/2 Branched Chains: The Study of Topological Phase Transition

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## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Magnetic Properties

- ❖
- Condition (I): We assume ferromagnetic coupling $|{J}_{2}|$ is the energy unit (${J}_{2}<0$) where the magnetization of the models is examined for different antiferromagnetic interaction ratios ${J}_{1}/\left|{J}_{2}\right|>0$.
- ❖
- Condition (II): We assume antiferromagnetic coupling ${J}_{2}>0$ is the energy unit, and the magnetization is investigated for various fixed values of the ferromagnetic interaction ratio ${J}_{1}/{J}_{2}<0$.

#### 3.1. The Heisenberg Spin-1/2 Branched Chains under Condition (I)

#### 3.2. The Heisenberg Spin-1/2 Branched Chains under Condition (II)

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic structure of the spin-1/2 XXX Heisenberg model on the branched chains (

**a**) ${{\rm Y}}_{1}$ with one, (

**b**) ${{\rm Y}}_{2}$ with two, and (

**c**) ${{\rm Y}}_{3}$ with three intra-chain interactions ${J}_{2}$. ${J}_{1}$ denotes inter-chain interaction. In each case, the dotted rectangle indicates a unit cell that uniformly repeats throughout the chains. Solid balls labeled as A, B, and C represent spin-1/2 particles in a unit block, and the balls marked with ${\mathrm{C}}_{i}$ indicate the side spins.

**Figure 2.**QMC results obtained for the magnetization curve of the three Heisenberg branched chains at low temperatures ${k}_{B}T/\left|{J}_{2}\right|=0.02$ under the condition (I) where three different fixed values of the inter-chain interaction ${J}_{1}/\left|{J}_{2}\right|=\{0.5,0.75,1.0\}$ are considered. (

**a**) The magnetization of chain ${{\rm Y}}_{1}$. (

**b**) The magnetization of chain ${{\rm Y}}_{2}$. (

**c**) The magnetization of chain ${{\rm Y}}_{3}$.

**Figure 3.**QMC results obtained for the magnetization curve of the three Heisenberg branched chains at low temperatures ${k}_{B}T/{J}_{2}=0.02$ under the condition (II), assuming a few fixed values of the inter-chain interaction ${J}_{1}/{J}_{2}$. (

**a**) Chain ${{\rm Y}}_{1}$. (

**b**) Chain ${{\rm Y}}_{2}$. (

**c**) Chain ${{\rm Y}}_{3}$.

**Figure 4.**The low-temperature susceptibility $\chi $ versus magnetic field for (

**a**) the Heisenberg branched chain ${{\rm Y}}_{2}$ and (

**b**) chain ${{\rm Y}}_{3}$, at ${k}_{B}T/{J}_{2}=0.02$ under the condition (II). Three different values of the inter-chain interaction ${J}_{1}/{J}_{2}=\{-2.0,-3.0,-3.5\}$ are considered for the both panels.

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**MDPI and ACS Style**

Arian Zad, H.; Zoshki, A.; Ananikian, N.; Jaščur, M.
Tomonaga–Luttinger Spin Liquid and Kosterlitz–Thouless Transition in the Spin-1/2 Branched Chains: The Study of Topological Phase Transition. *Materials* **2022**, *15*, 4183.
https://doi.org/10.3390/ma15124183

**AMA Style**

Arian Zad H, Zoshki A, Ananikian N, Jaščur M.
Tomonaga–Luttinger Spin Liquid and Kosterlitz–Thouless Transition in the Spin-1/2 Branched Chains: The Study of Topological Phase Transition. *Materials*. 2022; 15(12):4183.
https://doi.org/10.3390/ma15124183

**Chicago/Turabian Style**

Arian Zad, Hamid, Azam Zoshki, Nerses Ananikian, and Michal Jaščur.
2022. "Tomonaga–Luttinger Spin Liquid and Kosterlitz–Thouless Transition in the Spin-1/2 Branched Chains: The Study of Topological Phase Transition" *Materials* 15, no. 12: 4183.
https://doi.org/10.3390/ma15124183