Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916624562388992 |
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| author | Kimura, Yusuke Kuwahara, Tomotaka |
| author_facet | Kimura, Yusuke Kuwahara, Tomotaka |
| contents | This paper delves into a fundamental aspect of quantum statistical mechanics -- the absence of thermal phase transitions in one-dimensional (1D) systems. Originating from Ising's analysis of the 1D spin chain, this concept has been pivotal in understanding 1D quantum phases, especially those with finite-range interactions as extended by Araki. In this work, we focus on quantum long-range interactions and successfully derive a clustering theorem applicable to a wide range of interaction decays at arbitrary temperatures. This theorem applies to any interaction forms that decay faster than $r^{-2}$ and does not rely on translation invariance or infinite system size assumptions. Also, we rigorously established that the temperature dependence of the correlation length is given by $e^{{\rm const.} β}$, which is the same as the classical cases. Our findings indicate the absence of phase transitions in 1D systems with super-polynomially decaying interactions, thereby expanding upon previous theoretical research. To overcome significant technical challenges originating from the divergence of the imaginary-time Lieb-Robinson bound, we utilize the quantum belief propagation to refine the cluster expansion method. This approach allowed us to address divergence issues effectively and contributed to a deeper understanding of low-temperature behaviors in 1D quantum systems. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_11431 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Clustering theorem in 1D long-range interacting systems at arbitrary temperatures Kimura, Yusuke Kuwahara, Tomotaka Quantum Physics Statistical Mechanics Mathematical Physics This paper delves into a fundamental aspect of quantum statistical mechanics -- the absence of thermal phase transitions in one-dimensional (1D) systems. Originating from Ising's analysis of the 1D spin chain, this concept has been pivotal in understanding 1D quantum phases, especially those with finite-range interactions as extended by Araki. In this work, we focus on quantum long-range interactions and successfully derive a clustering theorem applicable to a wide range of interaction decays at arbitrary temperatures. This theorem applies to any interaction forms that decay faster than $r^{-2}$ and does not rely on translation invariance or infinite system size assumptions. Also, we rigorously established that the temperature dependence of the correlation length is given by $e^{{\rm const.} β}$, which is the same as the classical cases. Our findings indicate the absence of phase transitions in 1D systems with super-polynomially decaying interactions, thereby expanding upon previous theoretical research. To overcome significant technical challenges originating from the divergence of the imaginary-time Lieb-Robinson bound, we utilize the quantum belief propagation to refine the cluster expansion method. This approach allowed us to address divergence issues effectively and contributed to a deeper understanding of low-temperature behaviors in 1D quantum systems. |
| title | Clustering theorem in 1D long-range interacting systems at arbitrary temperatures |
| topic | Quantum Physics Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2403.11431 |