[Paper Review] Exact symmetry breaking ground states for quantum spin chains

S. Alipour, Sima Baghbanzadeh+1 more|ArXiv.org|Jan 8, 2008

Quantum many-body systems2 references3 citations

TL;DR

This paper introduces a family of spin-1/2 quantum spin chains with four-site interactions that exhibit exact ground states breaking both translational and rotational symmetries of the Hamiltonian. Using a projection-based construction, the authors derive closed-form expressions for the ground state, excited states, and correlation functions, and extend the method to construct a spin-3/2 chain with nearest-neighbor interactions whose ground state also breaks rotational symmetry, all with analytically tractable correlation functions.

ABSTRACT

We introduce a family of spin-1/2 quantum chains, and show that their exact ground states break the rotational and translational symmetries of the original Hamiltonian. We also show how one can use projection to construct a spin-3/2 quantum chain with nearest neighbor interaction, whose exact ground states break the rotational symmetry of the Hamiltonian. Correlation functions of both models are determined in closed form. Although we confine ourselves to examples, the method can easily be adapted to encompass more general models.

Motivation & Objective

• To construct exactly solvable quantum spin chains whose ground states spontaneously break rotational and translational symmetries of the Hamiltonian.
• To develop a systematic method for generating such symmetry-breaking ground states using local projection operators on entangled spin configurations.
• To derive exact expressions for the ground state, maximum energy state, and selected excited states in closed analytical form.
• To extend the construction to spin-3/2 chains with nearest-neighbor interactions, preserving exact solvability.
• To compute correlation functions explicitly using matrix product state representations and validate results against known models.

Proposed method

• Define a target ground state |φ₀⟩ as a periodic array of singlets and polarized spins: |S₁₂, +₃, S₄₅, +₆, ..., S₃ₙ₋₂,₃ₙ₋₁, +₃ₙ⟩.
• Construct a local Hamiltonian h as a sum of projectors: h = 2J P₂ + P₀, where P₂ projects onto total spin-2 and P₀ = |χ⟩⟨χ| projects onto a four-spin singlet state.
• Ensure h annihilates all local configurations in |φ₀⟩ by proving ⟨χ|S₂₃⟩ = 0 and similar identities using Clebsch-Gordan decomposition.
• Use the resulting local Hamiltonian h to build a global Hamiltonian H = ∑ₖ hₖ,ₖ₊₁,ₖ₊₂,ₖ₊₃ with periodic boundary conditions.
• For the spin-3/2 chain, map the spin-1/2 ground state to a matrix product state (MPS) representation using operators A₊, A₋, B₊, and derive effective spin-3/2 matrices.
• Compute correlation functions using the MPS representation, yielding analytical expressions for Gᶻ(1,r) and Gᵗ(1,r) with exponential decay and oscillatory behavior.

Experimental results

Research questions

• RQ1 Can a quantum spin chain with rotational and translational symmetry in its Hamiltonian possess an exact ground state that breaks these symmetries?
• RQ2 What local Hamiltonian structure ensures that a given entangled many-body state is the exact ground state?
• RQ3 Can such symmetry-breaking ground states be constructed with only nearest-neighbor interactions in higher-spin chains?
• RQ4 What are the exact correlation functions in such symmetry-breaking ground states, and how do they decay?
• RQ5 Is it possible to analytically derive not only the ground state but also the maximum energy and other excited states?

Key findings

• The spin-1/2 chain with four-site interactions has an exact ground state |φ₀⟩ that breaks both translational and rotational symmetries, despite the Hamiltonian being symmetric.
• The local Hamiltonian h = 2J P₂ + P₀ annihilates all configurations in |φ₀⟩, ensuring it is the ground state for any J > 0.
• The maximum energy state and several low-lying excited states are derived in closed analytical form, a feature not previously achieved in AKLT or matrix product models.
• A spin-3/2 chain with nearest-neighbor interactions is constructed whose ground state breaks rotational symmetry and admits an exact matrix product state representation.
• Correlation functions are computed explicitly: Gᶻ(1,r) = Aₗ (−1)ʳ⁻¹ e⁻ʳ/ξₗ and Gᵗ(1,r) = Aₜ (−1)ʳ⁻¹ e⁻ʳ/ξₜ with Aₗ = (12 + 6√3)/4, Aₜ = (12 + 7√3)/4, ξₗ = 1/ln(2+√3), ξₜ = 1/ln(1+√3).
• The ground state of the spin-3/2 chain has a total spin multiplet of size N/2, with the highest weight state explicitly constructed via MPS formalism.