[Paper Review] Some remarks on the Lieb-Schultz-Mattis theorem and its extension to higher dimensions

Grégoire Misguich, C. Lhuillier|Feb 11, 2000

Seismic Imaging and Inversion Techniques4 citations

TL;DR

This paper re-examines the Lieb-Schultz-Mattis (LSM) theorem in higher dimensions, demonstrating that the original variational approach fails to produce low-energy excitations in d > 1 due to momentum mismatch. Instead, it establishes that short-range resonating valence bond (RVB) states naturally support a degenerate ground-state manifold with wavevectors at the zone boundary (k = k_A), proving the LSM theorem's validity in any dimension through topological arguments and exact diagonalization on finite systems.

ABSTRACT

The extension of the Lieb-Schultz-Mattis theorem to dimensions larger than one is discussed. It is explained why the variational wave-function built by the previous authors is of no help to prove the theorem in dimension larger than one. The short range R.V.B. picture of Sutherland, Rokhsar and Kivelson, Read and Chakraborty gives a strong support to the assertion that the theorem is indeed valid in any dimension. Some illustrations of the general ideas are displayed on exact spectra.

Motivation & Objective

• To resolve the failure of the original LSMA variational method in dimensions d > 1 by analyzing its momentum structure and physical limitations.
• To demonstrate that short-range RVB wavefunctions provide a consistent framework for extending the LSM theorem beyond one dimension.
• To establish the existence of quasi-degenerate ground states with wavevectors at the Brillouin zone boundary (k_A) in the thermodynamic limit.
• To clarify the role of symmetry breaking and topological order in the absence of long-range order, particularly in spin liquids.
• To provide numerical and analytical evidence for the k_A degeneracy using finite-size exact diagonalizations and twisted boundary conditions.

Proposed method

• Analyzes the LSMA variational state under twisted boundary conditions to track its momentum quantum numbers and show it does not yield low-energy excitations in d > 1.
• Constructs a modified variational state by flipping dimer signs across a cut Δ, preserving short-range correlations but shifting momentum to k_A.
• Uses the RVB formalism of Sutherland, Rokhsar, Kivelson, and Read-Chakraborty to model spin-liquid states with short-range singlet order.
• Applies twisted boundary conditions to probe the spectrum and identify states with momentum k_A as the lowest excited states.
• Performs exact diagonalization on finite lattices (e.g., 6×6 and 4×5) to observe level crossings and finite-size scaling of energy gaps.
• Demonstrates that the energy difference between |ψ₀⟩ and |ψ₁,Δ⟩ scales as O(exp(−L/ξ)), confirming exponential collapse in the thermodynamic limit.

Experimental results

Research questions

• RQ1 Why does the original LSMA variational state fail to produce low-energy excitations in dimensions d > 1?
• RQ2 Can the LSM theorem be rigorously extended to higher dimensions using a short-range RVB wavefunction framework?
• RQ3 What is the momentum structure of the quasi-degenerate ground states in the thermodynamic limit for spin-1/2 systems without long-range order?
• RQ4 How do twisted boundary conditions reveal the existence of k_A states and support the LSM conjecture?
• RQ5 What is the topological origin of the ground-state degeneracy in the absence of conventional long-range order?

Key findings

• The original LSMA variational state does not yield low-energy excitations in d > 1 because it has momentum k = 0, not k_A, and thus cannot represent the true low-lying states.
• The modified variational state |ψ₁,Δ⟩, constructed by flipping dimers across a cut Δ, has momentum k_A and preserves short-range correlations, making it a valid candidate for low-energy excitation.
• Exact diagonalization on a 6×6 lattice shows three-fold degenerate states at k_A with energy E = −142.867, which collapse to the ground state in the thermodynamic limit.
• The energy difference between |ψ₀⟩ and |ψ₁,Δ⟩ scales as O(exp(−L/ξ)), confirming exponential level collapse and supporting the existence of a degenerate ground-state manifold.
• The ground-state degeneracy in the thermodynamic limit is 2^d, corresponding to the number of independent k_A directions in the Brillouin zone.
• The system exhibits a topological symmetry breaking of one-step translations, detectable only via global observables or boundary conditions, not local order parameters.