Clustering theorem in 1D long-range interacting systems at arbitrary temperatures
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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2024
- DOI:
- 10.48550/arXiv.2403.11431
- arXiv:
- arXiv:2403.11431
- Bibcode:
- 2024arXiv240311431K
- Keywords:
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- Quantum Physics;
- Condensed Matter - Statistical Mechanics;
- Mathematical Physics
- E-Print:
- 32 pages, 5 figures,