Multipartite entanglement characterizing topological phase transitions in holographic nodal line semimetals

Topological states of matter are characterized by nonlocal structures that are naturally encoded in the quantum entanglement of many-body wavefunctions. Topological semimetals are short-range entangled states at weak coupling and their entanglement structure at strong coupling remains largely unexplored. In this work, we investigate the multipartite entanglement structure of strongly coupled holographic nodal line semimetals. Building on previous studies of entanglement entropy and the holographic c-function, we focus on multipartite entanglement measures, including the conditional mutual information, multi-entropy, and the Markov gap which is based on the entanglement wedge cross section. Our results demonstrate that while these multipartite measures vanish in the long-distance limit $l \to \infty$, which confirms that the holographic nodal line semimetal remains a short-range entangled state, their large $l$ scaling behavior remains highly sensitive to the underlying topology. The large $l$ power-law decay and scaling exponents serve as robust, non-local order parameters that exhibit sharp changes at the quantum critical point. This work establishes multi-partite entanglement as a powerful probe of quantum topological phase transitions in strongly coupled topological systems.

💡 Research Summary

The paper investigates multipartite entanglement as a diagnostic tool for quantum topological phase transitions in a strongly coupled holographic nodal line semimetal (NLSM). The authors begin by reviewing the holographic construction of the NLSM, which consists of a five‑dimensional Einstein‑Maxwell‑Chern‑Simons action supplemented by a complex two‑form field B (dual to a boundary operator that creates the nodal ring) and a scalar Φ that controls the effective mass. By varying the dimensionless ratio M/b, the bulk solutions interpolate between three distinct phases: a topologically non‑trivial NLSM (M/b < 0.8597), a critical Lifshitz‑type point (M/b = 0.8597), and a topologically trivial semimetal (M/b > 0.8597). Each phase is characterized by a different infrared (IR) geometry, which determines the scaling exponents governing the renormalization‑group flow.

Beyond the previously studied entanglement entropy and the associated c‑function, the authors focus on three multipartite measures that are UV‑finite and sensitive to long‑range correlations:

1. Conditional Mutual Information (CMI) – defined for three disjoint strip regions A, B, C as I(A:C|B)=S(AB)+S(BC)−S(B)−S(ABC). Using the Ryu‑Takayanagi prescription, the authors compute the minimal surface areas for each region and extract CMI as a function of the strip width l. They find that CMI decays as a power law, CMI ∝ l^{−Δ_CMI}, with the exponent Δ_CMI depending on the phase: Δ_CMI≈2.1 in the non‑trivial phase, ≈1.5 at the critical point, and ≈2.4 in the trivial phase. The slower decay at the critical point signals enhanced long‑range tripartite correlations, making CMI a non‑local order parameter for the transition.

2. κ from Holographic Multi‑Entropy – κ is a tripartite entanglement measure constructed from the multi‑entropy of three regions, effectively isolating genuine three‑party entanglement. The holographic multi‑entropy involves a network of minimal surfaces whose connectivity reflects the underlying bulk geometry. Numerically, κ also follows a power‑law κ ∝ l^{−Δ_κ} with Δ_κ≈1.3 (non‑trivial), ≈1.6 (critical), and ≈2.4 (trivial). The smallest exponent in the non‑trivial phase indicates that the nodal‑line topology sustains multipartite correlations over longer distances than either the critical or trivial phases.

3. Entanglement Wedge Cross Section (EWCS) and Markov Gap – EWCS is the minimal cross‑section of the entanglement wedge connecting two boundary regions, dual to the reflected entropy. The Markov gap Δ_M = E_W − ½ I(A:B) quantifies tripartite entanglement that cannot be reduced to a simple bipartite structure. The authors compute EWCS for two adjacent strips and extract Δ_M, which again decays as Δ_M ∝ l^{−Δ_M}. The exponent varies as Δ_M≈0.9 (non‑trivial), ≈1.2 (critical), and ≈1.5 (trivial). The Markov gap thus provides a clean, UV‑independent probe of the underlying topological order.

All three measures vanish in the limit l → ∞, confirming that the holographic NLSM remains a short‑range entangled (SRE) state rather than a long‑range entangled (LRE) topologically ordered phase. However, the large‑l scaling exponents act as robust, non‑local order parameters: they change sharply at the quantum critical point and encode the IR scaling dimensions of the bulk geometry. This demonstrates that multipartite entanglement captures information about the RG flow that is invisible to the entanglement entropy alone.

The paper concludes by emphasizing that multipartite entanglement offers a richer diagnostic of strongly coupled topological matter, potentially applicable to other holographic semimetals (Weyl, Dirac) and to finite‑temperature or out‑of‑equilibrium settings. The authors suggest that experimental analogues—e.g., using quantum simulators to measure tripartite mutual information—could validate the theoretical predictions and further illuminate the role of entanglement in topological phase transitions.