Sublattice scars and beyond in two-dimensional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>U</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math> quantum link lattice gauge theories
Indrajit Sau, Paolo Stornati, Debasish Banerjee, Arnab Sen
Abstract
In this article, we elucidate the structure and properties of a class of anomalous high-energy states of matter-free <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>U</a:mi><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:mo stretchy="false">)</a:mo></a:math> quantum link gauge theory Hamiltonians using numerical and analytical methods. Such anomalous states, known as quantum many-body scars in the literature, have generated a lot of interest due to their athermal nature. Our starting Hamiltonian is <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>H</e:mi><e:mo>=</e:mo><e:msub><e:mi mathvariant="script">O</e:mi><e:mi>kin</e:mi></e:msub><e:mo>+</e:mo><e:mi>λ</e:mi><e:msub><e:mi mathvariant="script">O</e:mi><e:mi>pot</e:mi></e:msub></e:math>, where <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>λ</i:mi></i:math> is a real-valued coupling, and <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi mathvariant="script">O</k:mi><k:mi>kin</k:mi></k:msub></k:math> (<n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"><n:msub><n:mi mathvariant="script">O</n:mi><n:mi>pot</n:mi></n:msub></n:math>) are summed local off-diagonal (diagonal) operators in the electric flux basis acting on the elementary plaquette <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mo>□</q:mo></q:math>. The spectrum of the model in its spin-<s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mfrac><s:mn>1</s:mn><s:mn>2</s:mn></s:mfrac></s:math> representation on <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>L</u:mi><u:mi>x</u:mi></u:msub><u:mo>×</u:mo><u:msub><u:mi>L</u:mi><u:mi>y</u:mi></u:msub></u:math> lattices reveal the existence of sublattice scars, <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mo stretchy="false">|</w:mo><w:msub><w:mi>ψ</w:mi><w:mi>s</w:mi></w:msub><w:mo stretchy="false">⟩</w:mo></w:math>, which satisfy <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi mathvariant="script">O</ab:mi><ab:mrow><ab:mi>pot</ab:mi><ab:mo>,</ab:mo><ab:mo>□</ab:mo></ab:mrow></ab:msub><ab:mo stretchy="false">|</ab:mo><ab:msub><ab:mi>ψ</ab:mi><ab:mi>s</ab:mi></ab:msub><ab:mo stretchy="false">⟩</ab:mo><ab:mo>=</ab:mo><ab:mo stretchy="false">|</ab:mo><ab:msub><ab:mi>ψ</ab:mi><ab:mi>s</ab:mi></ab:msub><ab:mo stretchy="false">⟩</ab:mo></ab:math> for all elementary plaquettes on one sublattice and <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML" display="inline"><hb:msub><hb:mi mathvariant="script">O</hb:mi><hb:mrow><hb:mi>pot</hb:mi><hb:mo>,</hb:mo><hb:mo>□</hb:mo></hb:mrow></hb:msub><hb:mo stretchy="false">|</hb:mo><hb:msub><hb:mi>ψ</hb:mi><hb:mi>s</hb:mi></hb:msub><hb:mo stretchy="false">⟩</hb:mo><hb:mo>=</hb:mo><hb:mn>0</hb:mn></hb:math> on the other, while being simultaneous zero modes or nonzero integer-valued eigenstates of <mb:math xmlns:mb="http://www.w3.org/1998/Math/MathML" display="inline"><mb:msub><mb:mi mathvariant="script">O</mb:mi><mb:mi>kin</mb:mi></mb:msub></mb:math>. We demonstrate a “triangle relation” connecting the sublattice scars with nonzero integer eigenvalues of <pb:math xmlns:pb="http://www.w3.org/1998/Math/MathML" display="inline"><pb:msub><pb:mi mathvariant="script">O</pb:mi><pb:mi>kin</pb:mi></pb:msub></pb:math> to particular sublattice scars with <sb:math xmlns:sb="http://www.w3.org/1998/Math/MathML" display="inline"><sb:msub><sb:mi mathvariant="script">O</sb:mi><sb:mi>kin</sb:mi></sb:msub><sb:mo>=</sb:mo><sb:mn>0</sb:mn></sb:math> eigenvalues. A fraction of the sublattice scars have a simple description in terms of emergent short singlets, on which we place analytic bounds. We further construct a long-ranged parent Hamiltonian for which all sublattice scars in the null space of <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" display="inline"><vb:msub><vb:mi mathvariant="script">O</vb:mi><vb:mi>kin</vb:mi></vb:msub></vb:math> become unique ground states and elucidate some of the properties of its spectrum. In particular, zero energy states of this parent Hamiltonian turn out to be exact scars of another <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:mi>U</yb:mi><yb:mo stretchy="false">(</yb:mo><yb:mn>1</yb:mn><yb:mo stretchy="false">)</yb:mo></yb:math> quantum link model with a staggered short-ranged diagonal term. Published by the American Physical Society 2024