<p>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 <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5242_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">$r^{-2}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>r</mi> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="220_2025_5242_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="48" /> </InlineMediaObject> <EquationSource Format="TEX">$e^{\mathrm{const.} \beta }$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>e</mi> <mrow> <mrow> <mi mathvariant="normal">const</mi> <mo>.</mo> </mrow> <mi>β</mi> </mrow> </msup> </math></EquationSource> </InlineEquation>, 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.</p>

Clustering Theorem in 1D Long-Range Interacting Systems at Arbitrary Temperatures

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2025年2月20日

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Yusuke Kimura,
Tomotaka Kuwahara

摘要

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 $r^{-2}$ $r^{- 2}$ 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 $e^{\mathrm{const.} \beta }$ $e^{const . β}$ , 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.