Locally Interacting Lattice Bosons: Clustering Theorem, Low-Density Condition and Their Applications

TL;DR

This work establishes a bosonic clustering theorem for locally interacting lattice bosons at high temperature, focusing on the canonical Bose-Hubbard model. The authors develop an imaginary-time interaction-picture cluster expansion to overcome the unbounded norms intrinsic to bosons, enabling rigorous bounds on local observables, two-point correlations, and thermodynamic quantities. They prove a high-temperature clustering bound with a finite correlation length $\xi(β)$ and provide a rigorous low-boson-density condition, both pivotal for further results such as a quasi Dulong–Petit bound on the specific heat and a thermal area law for mutual information. The framework generalizes to a broad Bose-Hubbard class and offers a robust toolbox for analyzing locality, correlations, and thermodynamics in infinite-dimensional bosonic systems.

Abstract

In statistical and quantum many-body physics, the correlation function is a fundamental quantity, especially for lattice models described by local Hamiltonians. Away from the phase-transition point, correlation function typically satisfies the clustering property, meaning that the correlation concentrates at short ranges while decays rapidly (exponentially) at long distances. Though the clustering property has been extensively studied for spin and fermion systems, whether a similar result holds for boson systems remains a long-standing open problem. The essential difficulty lies in the infinite Hilbert-space dimension of a boson, in stark contrast to the finite dimension of a spin or fermion. This work is devoted to establishing the boson counterpart of the clustering of correlations at high temperatures, focusing primarily on the canonical Bose-Hubbard model. As a byproduct, we rigorously justify the low-boson-density assumption for the Gibbs state of the Bose-Hubbard type. This assumption is often invoked as a preliminary requirement for proving various rigorous results, including the boson Lieb-Robinson bound. Building on the results above, we show that, at high temperature, the specific heat density can be bounded from above by a constant, and the boson thermal area law holds true. Our achievement is based on the imaginary-time interaction picture, which is expected to have much broader applications to other open problems concerning bosons in statistical and quantum many-body physics.

Figures (6)

• Figure 1: A mindmap of the main results, methodological framework and manuscript structure of this work. We present a comprehensive study on the high-temperature Gibbs state of the Bose-Hubbard model from various angles.
• Figure 2: A 2D square lattice illustrating the relation $\qty(V_{G})^{\mathrm{c}}=V_{\overline{G}^{\mathrm{c}}}$. For clarity we only explicitly draw out the edges of size $2$.
• Figure 3: A $2$D square lattice as an illustartion of $\mathcal{C}_{\geq L}^{k}(G)$ and $\mathcal{C}(G)$ with the different thickness represents the multiplicities. For clarity we only explicitly draw out the edges of size $2$.
• Figure 4: A $2$D square lattice as an illustartion of bipartition of the system into vertex subsets and their boundaries.
• Figure 5: A $2$D square lattice as an illustartion of the relation $V_{X}^{\mathrm{c}}\equiv V_{G_{X}^{\mathrm{c}}}$. For clarity we only explicitly draw out the edges of size $2$.

Theorems & Definitions (48)

• Theorem 1
• Corollary 1: low-boson-density assumption
• Lemma 1
• proof : Proof of Corollary \ref{['c1']}
• Theorem 2: boson clustering property at high temperatures
• Theorem 3: fermionic clustering property at high temperatures
• Corollary 2: quasi Dulong-Petit law
• proof : Proof of Corollary \ref{['c2']}
• Corollary 3: boson thermal area law
• proof : Proof of Corollary \ref{['c3']}