The Magnetism of Li doped La₂CuO₄: the antiferromagnetic spin-shard state

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We study the dynamics of a single hole in Li and Sr doped La₂CuO₄ and its extension to a finite hole concentration. We compare the physics of La₂₋ₓSrₓCuO₄ and La₂Cu₁₋ₓLiₓO₄ and explain why these systems are remarkably different. We demonstrate that holes in La₂Cu₁₋ₓLiₓO₄ are always localized and that there is a critical concentration, xc≈ 0.03, above which the holes break the global antiferromagnetic state into an array of weakly coupled antiferromagnetic clusters (antiferromagnetic shards). We show that the spin-shard state provides a description of the magnetic and electric properties of La₂Cu₁₋ₓLiₓO₄. Two experiments that can test our theory are proposed.