Clustering theorem in 1D long-range interacting systems at arbitrary temperatures

• URL: http://arxiv.org/abs/2403.11431v1
• Date: Mon, 18 Mar 2024 02:54:55 GMT
• Title: Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
• Authors: Yusuke Kimura, Tomotaka Kuwahara,
• Abstract summary: This paper delves into a fundamental aspect of quantum statistical mechanics -- the absence of thermal phase transitions in one-dimensional (1D) systems. We successfully derive a clustering theorem applicable to a wide range of interaction decays at arbitrary temperatures. Our findings indicate the absence of phase transitions in 1D systems with super-polynomially decaying interactions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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 $e^{{\rm const.} \beta}$, 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.

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