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

TL;DR

This work proves a clustering theorem for 1D quantum systems with long-range interactions, showing absence of thermal phase transitions at arbitrary temperatures for decays faster than $r^{-2}$. The authors develop an approach that blends interaction-truncated Hamiltonians, cluster expansion, and quantum belief propagation to overcome the divergence of the imaginary-time Lieb–Robinson bound and to establish explicit, temperature-dependent bounds on correlations that scale as $\xi_\beta = e^{\Theta(\beta)}$. For subexponential decays, the paper derives a decay bound ${\rm Cor}_{\rho_\beta}(O_X,O_Y) \le c_1 e^{-R_\beta^{\kappa/(\kappa+1)}}$, while slower decays yield bounds that involve the decay function $\bar{J}$ evaluated at a rescaled distance, with the correlation length growing at most exponentially with inverse temperature. The results generalize prior findings to broader interaction classes without requiring translation invariance or infinite systems, and they highlight both the potential and current limits (e.g., suboptimal decay rates for certain regimes and lack of partition-function analyticity results) of the approach, pointing to future work on zero-free regions and sharper decay bounds via refined analytical techniques.

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 $e^{{\rm const.} β}$, 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.

Figures (5)

• Figure 1: The image illustrates an approximation of the time-evolved operator $O_{L_0}(t)$ by $O_{L_0}(H_L,t)$, where $L_0$ and $L^{\rm c}$ are separated by a distance $\ell$ (Corollary \ref{['corol_time_evo_approx']}). The approximation error is obtained by applying the Lieb--Robinson bound and Lemma \ref{['lem:Lieb-Robinson_start']}, with the boundary interactions $h_Z$ such that $Z$ satisfies $Z \cap L \neq \emptyset$ and $Z\cap L^{\rm c} \neq \emptyset$.
• Figure 2: We truncate the Hamiltonian's interactions by introducing subregions $\{B_s\}_{s=0}^{q+1}$ across the system. This results in partitionings of the system into $(q+2)$ blocks, with $q=6$, as depicted in the figure. Each block in the sequence of blocks $\{B_s\}_{s=1}^q$ has a length of $l_0$, while the terminal blocks $B_0$ and $B_{q+1}$ extend to the left and right extremities of the system, respectively. Following this, we truncate all interactions between these distinct blocks, which ensures the interaction length of at most $2l_0$. Then, the truncated Hamiltonian, denoted as $H_{{\rm t}}$ in Eq. \ref{['def:truncated_Hamiltonian']}, remains substantially similar to the original Hamiltonian $H$, as established in Lemma \ref{['thm:locality_exp_effectiveHam']}. We here consider the open boundary condition, but the same procedure can be applied under periodic boundary conditions (see Ref. kuwahara2019area).
• Figure 3: The quantum belief propagation operator $\Phi_s$ characterizes the transformation from $e^{\beta H_{\le s}} \otimes e^{\beta H_{> s}}$ to $e^{\beta H}$. It is defined using time evolution with respect to $H_\tau:=H_{\le s} + H_{> s} + \tau h_s$. The Lieb--Robinson bound ensures the quasi-locality of $\Phi_s$ near the boundary, as shown in Proposition \ref{['lem:belief_error']}.
• Figure 4: An image displaying the decomposition of the collection of sets $\{X, Z_1, Z_2, \ldots, Z_8, Y\}$ into two disjoint sub-collections, $\{X, Z_1, Z_2, Z_3, Z_5, Z_8 \}$ and $\{Z_4, Z_6,Z_7, R \}$, is shown. When the sets $Z_1$, $Z_2$, $\ldots$, $Z_8$ cannot connect the distant regions $L$ and $R$, the trace vanishes.
• Figure 5: Approximating $\{\Phi_j\}_{j=0}^{m+1}$ by $\{\tilde{\Phi}_j\}_{j=0}^{m+1}$, the dependence on $h_{s_1}, \ldots, h_{s_m}$ are removed from $\{\tilde{\Phi}_j\}_{j=0}^{m+1}$. These approximations allow for a simple analysis to remove the zeroth-order terms for $h_{s_1}, \ldots, h_{s_m}$. The center site in the region $L_j$ is denoted by $i_{s_j}$, and the operator $h_{s_j}$ characterizes the interaction between the left half and the right half of the block $L_j$.

Theorems & Definitions (15)

• Lemma 1
• Theorem 1
• Lemma 2: Finite-range interaction
• Lemma 3: Infinite interaction length
• Lemma 4: Supplementary Lemma 14 in Ref. PhysRevLett.127.070403
• Corollary 5
• Lemma 6
• Lemma 7
• Corollary 8
• Lemma 9