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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>U</mml:mi> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> 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}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>𝒪</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>k</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:mstyle> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>𝒪</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>p</mml:mi> <mml:mi>o</mml:mi> <mml:mi>t</mml:mi> </mml:mstyle> </mml:msub> </mml:mrow> </mml:math> where \mathcal{O}_{\mathrm{pot}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>𝒪</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>p</mml:mi> <mml:mi>o</mml:mi> <mml:mi>t</mml:mi> </mml:mstyle> </mml:msub> </mml:math> ( \mathcal{O}_{\mathrm{kin}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>𝒪</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>k</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:mstyle> </mml:msub> </mml:math> ) 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=0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> 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) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mi>λ</mml:mi> <mml:mo stretchy="false" form="prefix">|</mml:mo> <mml:mo>≲</mml:mo> <mml:mstyle mathvariant="script"> <mml:mi>𝒪</mml:mi> </mml:mstyle> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> 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 \lambda <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>λ</mml:mi> </mml:math> show a rich variety with those that are composed solely from the zero modes of \mathcal{O}_{\mathrm{kin}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>𝒪</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>k</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:mstyle> </mml:msub> </mml:math> , those that contain an admixture of both the zero and the nonzero modes of \mathcal{O}_{\mathrm{kin}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>𝒪</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>k</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:mstyle> </mml:msub> </mml:math> , and finally those composed solely from the nonzero modes of \mathcal{O}_{\mathrm{kin}} <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mstyle mathvariant="script"> <mml:mi>𝒪</mml:mi> </mml:mstyle> <mml:mstyle mathvariant="normal"> <mml:mi>k</mml:mi> <mml:mi>i</mml:mi> <mml:mi>n</mml:mi> </mml:mstyle> </mml:msub> </mml:math> </jats:inl

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