Entanglement area law in interacting bosons: from Bose-Hubbard, $φ$4, and beyond

Abstract

The entanglement area law is a universal principle that characterizes the information structure in quantum many-body systems and serves as the foundation for modern algorithms based on tensor network representations. Historically, the area law has been well understood under two critical assumptions: short-range interactions and bounded local energy. However, extending the area law beyond these assumptions has been a long-sought goal in quantum many-body theory. This challenge is especially pronounced in interacting boson systems, where the breakdown of the bounded energy assumption is universal and poses significant difficulties. In this work, we prove the area law for one-dimensional interacting boson systems including the long-range interactions. Our model encompasses the Bose-Hubbard class and the $\phi4$ class, two of the most fundamental models in quantum condensed matter physics, statistical mechanics, and high-energy physics. This result achieves the resolution of the area law that incorporates both the challenges of unbounded local energy and long-range interactions in a unified manner. Additionally, we establish an efficiency-guaranteed approximation of the quantum ground states using Matrix Product States (MPS). These results significantly advance our understanding of quantum complexity by offering new insights into how bosonic parameters and interaction decay rates influence entanglement. Our findings provide crucial theoretical foundations for simulating long-range interacting cold atomic systems, which are central to modern quantum technologies, and pave the way for more efficient simulation techniques in future quantum applications.

Figures (5)

• Figure S.1: Results of boson number concentration for the $\phi$4 model $H = \pi^{2} + \phi^{2} + \phi^{4}$ on a single site. The blue circles represent the computed values of $\langle \Omega | \Pi_{>N} | \Omega \rangle$ with the Hilbert space dimension limited to 10,000, and the red line is its subexponential fit given by $\langle \Omega | \Pi_{ >N} | \Omega \rangle = 0.3806 \, e^{-2.25 N^{0.6917}}$.
• Figure S.2: Interaction truncation in the Hamiltonian. The system is decomposed into $(q+2)$ blocks ($q=6$ shown above). Blocks $\{B_s\}_{s=1}^q$ each have length $l$, and edge blocks $B_0$ and $B_{q+1}$ extend to the system’s left and right ends. We truncate all interactions between separated blocks, so the truncated Hamiltonian $H_{\rm t}$ [Eq. \ref{['def:truncated_Hamiltonian']}] remains close to the original Hamiltonian $H$, as shown in Lemma \ref{['lemm:error_int_truncation']}.
• Figure S.3: Schematic of the effective Hamiltonian $\tilde{H}_{\rm t}$. We modify the energy spectrum in each $\{h_s\}_{s=0}^{q+1}$ so that energies above $\tau_s$ are constant, while $\{h_{s,s+1}\}_{s=0}^q$ remains the same as the original Hamiltonian. The low-energy spectrum is approximately preserved. The accuracy improves exponentially with the cut-off energy $\tau$ (Theorem \ref{['Effective Hamiltonian_multi_truncation']}).
• Figure S.4: Schematic picture of boson number truncation. For the purpose of applying the method to high-dimensional systems, we here consider the two-dimensional lattice. The truncation of the boson numbers poly-logarithmically increases as the site separates from the boundary between $L$ and $R$.
• Figure S.5: Block-block interaction $\overline{V}_{X,Y}$. We denote the right-end site of $X$ and the left-end site of $Y$ as $x$ and $y$, respectively. The interaction $\overline{V}_{X,Y}$ picks up all the interaction norm of $\left \| \bar{h}_Z \right \|$ that acts on both $X$ and $Y$.

Theorems & Definitions (47)

• Definition 1: Parameters $g$ and $k$
• Definition 2: Ground state
• Definition 3
• Definition 4
• Lemma 3
• Proposition 4
• Lemma 5
• Proposition 6
• Corollary 7
• Lemma 8