Gapped 1D Systems and Area Law

Created in July 07, 2025

2025   ·   Quantum   Physics,   Entanglement,   Area   Law,   Condensed   Matter

1. Why Gapped 1D Systems Have Low Entanglement

Gapped 1D quantum systems (e.g., spin chains) typically exhibit low entanglement because their ground states avoid long-range quantum correlations. A discrete example is the Majumdar-Ghosh spin chain:

  • Consider a 1D spin-1/2 chain with Hamiltonian:
    \(H = \sum_i \left( \vec{S}_i \cdot \vec{S}_{i+1} + \frac{1}{2} \vec{S}_i \cdot \vec{S}_{i+2} \right)\)
  • The ground state is a dimerized product state (e.g., spins form singlets on alternating bonds):
    \(|\psi\rangle = (|01\rangle - |10\rangle) \otimes (|01\rangle - |10\rangle) \otimes \dots\)
  • The entanglement entropy of any bipartition is zero (no entanglement across cuts) because the state is a product of local singlets.

Gapped systems generally satisfy the area law, meaning entanglement entropy scales with the boundary size (here, a constant in 1D).

2. Area Law Explained with a Simple Example

The area law states that the entanglement entropy \(S_A\) of a subsystem \(A\) scales with its boundary size (not volume).

Example: 1D Spin Chain with Nearest-Neighbor Coupling

Hamiltonian: \(H = -J \sum_i \sigma_i^z \sigma_{i+1}^z\) (Ising model in its gapped phase).
Ground state (for \(J > 0\)): \(|\psi\rangle = |\uparrow \uparrow \dots \uparrow\rangle\) or \(|\downarrow \downarrow \dots \downarrow\rangle\).
Bipartition the chain into \(A\) (first \(L\) spins) and \(B\) (rest).
Reduced density matrix \(\rho_A\) is pure (e.g., \(\rho_A = |\uparrow \dots \uparrow\rangle \langle \uparrow \dots \uparrow|\)), so \(S_A = 0\).

For slightly entangled states (e.g., perturbed ground states), \(S_A\) remains constant (independent of \(L\)), obeying the area law. In higher dimensions, \(S_A \sim \text{boundary area of } A\).

Violations of Area Law

  • Critical (gapless) systems in 1D: \(S_A \sim \log L\) (logarithmic divergence).
  • Highly excited states: Volume-law scaling (\(S_A \sim L\)).

The area law is a hallmark of low-complexity quantum states, relevant for efficient numerical simulations (e.g., DMRG, tensor networks).