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In recent decades, the field of strongly correlated quantum spin systems has undergone rapid development, yielding the discovery of new exotic phases of matter and phase transitions that challenge Landau's paradigm for describing phases and transitions based on symmetry breaking. Moreover, the investigation of systems with long-range interactions has garnered significant attention due to experimental advancements in two-dimensional quantum moire materials and Rydberg atom arrays. To keep up with these developments and provide insights into theoretical explorations in this field, we conducted numerical examinations of several recent proposals for characterizing and identifying various quantum states of matter.

Stochastic series expansion quantum Monte Carlo methods are utilized to investigate 1-form symmetry breaking at Ising transitions by measuring the scaling of its disorder operator, entanglement properties of various spin systems via studying the scaling forms of Renyi entanglement entropies, and the low-energy excitations of systems with long-range interactions by fitting to the imaginary time correlation functions. We confirm the spontaneous breaking of 1-form symmetry at the (2+1)d quantum critical point of the Ising model, obtain the correct topological entanglement entropies of Z₂ topological ordered states, observe the non-unitary scaling form of Renyi entanglement entropies at the Neel-valance bond solid deconfined quantum critical point, and explore the dynamic spectra of long-range Heisenberg models at three different regimes, namely the standard Goldstone, the anomalous Goldstone, and the gapped Higg regimes. The research presented in this dissertation provides valuable insights into the properties of exotic phases of matter and critical points beyond the conventional paradigm of symmetry breaking, as well as the novel behavior of systems with long-range interactions. These findings have the potential to advance the field of condensed matter physics and pave the way for future investigations in these areas.

References: 

1. Jiarui Zhao, Zheng Yan, Meng Cheng, and Zi Yang Meng, Phys. Rev. Research 3, 033024 (2021)

2. Jiarui Zhao, Yan-Cheng Wang, Zheng Yan, Meng Cheng, and Zi Yang Meng, Phys. Rev. Lett. 128, 010601 (2022)

3. Jiarui Zhao, Bin-Bin Chen, Yan-Cheng Wang, Zheng Yan, Meng Cheng, Zi Yang Meng, npj Quantum Materials 7, 69 (2022) 

4. Menghan Song, Jiarui Zhao, Yan Zheng, Zi Yang Meng, Phys. Rev. Research 5, 033046 (2023)