Scars from protected zero modes and beyond in $U(1)$ quantum link and quantum dimer models

Abstract

We demonstrate the presence of anomalous high-energy eigenstates, or many-body scars, in U(1)U(1) quantum link and quantum dimer models on square and rectangular lattices. In particular, we consider the paradigmatic Rokhsar-Kivelson Hamiltonian H=\mathcal{O}_{\mathrm{kin}} + \lambda \mathcal{O}_{\mathrm{pot}}H=𝒪kin+λ𝒪pot where \mathcal{O}_{\mathrm{pot}}𝒪pot (\mathcal{O}_{\mathrm{kin}}𝒪kin) is defined as a sum of terms on elementary plaquettes that are diagonal (off-diagonal) in the computational basis. Both these interacting models possess an exponentially large number of mid-spectrum zero modes in system size at \lambda=0λ=0 that are protected by an index theorem preventing any mixing with the nonzero modes at this coupling. We classify different types of scars for |\lambda| \lesssim \mathcal{O}(1)|λ|≲𝒪(1) both at zero and finite winding number sectors complementing and significantly generalizing our previous work [Banerjee and Sen, Phys. Rev. Lett. 126, 220601 (2021)]. The scars at finite \lambdaλ show a rich variety with those that are composed solely from the zero modes of \mathcal{O}_{\mathrm{kin}}𝒪kin, those that contain an admixture of both the zero and the nonzero modes of \mathcal{O}_{\mathrm{kin}}𝒪kin, and finally those composed solely from the nonzero modes of \mathcal{O}_{\mathrm{kin}}𝒪kin. These scars have tell-tale energies such as (non-zero) integers and irrationals like \pm \sqrt{2}±2 at \lambda=0λ=0 or n_1 \lambda \pm n_2n1λ±n2 at \lambda \neq 0λ≠0 where both n_1, n_2n1,n2 are integers. We give analytic expressions for certain ``lego scars’’ for the quantum dimer model on rectangular lattices where one of the linear dimensions can be made arbitrarily large, with the building blocks (legos) being composed of emergent singlets and other more complicated entangled structures.