Gapped 1D Systems

Key Concepts

1. Gapped System: A quantum system where the ground state is separated from excited states by a finite energy gap \(\Delta > 0\).
2. Area Law: In 1D gapped systems, entanglement entropy typically satisfies \(S \sim \text{constant}\) (rather than scaling with system size).

Discrete Example: Spin-1/2 Chain with Nearest-Neighbor Coupling

Consider a 4-site spin chain with Hamiltonian: \(H = -J\sum_{i=1}^{3}\sigma_i^z\sigma_{i+1}^z\) where \(J > 0\) and \(\sigma^z\) is the Pauli matrix.

Properties:

1. Ground State: $$

   \psi_0\rangle =

   \uparrow\uparrow\uparrow\uparrow\rangle\(or\)

   \downarrow\downarrow\downarrow\downarrow\rangle$$

2. Energy Gap: \(\Delta = 2J\) (to flip any single spin)
3. Entanglement: For any bipartition, the entanglement entropy \(S = 0\) because the ground state is separable.

Why Low Entanglement?

• The gap \(\Delta\) prevents long-range quantum correlations
• Local perturbations cannot generate extensive entanglement
• Satisfies the 1D area law: \(S \leq \text{constant}\) independent of system size

This demonstrates how gapped 1D systems naturally suppress entanglement growth.