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 $e^{const.\beta }$, 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.

35 pages, 5 figures,

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• Spectral Gap and Exponential Decay of Correlations
• Lieb-Robinson Bounds and the Exponential Clustering Theorem
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Cited by in corpus (6)

• Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
• Clustering of conditional mutual information and quantum Markov structure at arbitrary temperatures
• Provably Efficient Simulation of 1D Long-Range Interacting Systems at Any Temperature
• Stability of thermal equilibrium in long-range quantum systems
• High-temperature partition functions and classical simulatability of long-range quantum systems
• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains