Theory of Robust Quantum Many-Body Scars in Long-Range Interacting Systems

Quantum many-body scars (QMBS) are exceptional energy eigenstates of quantum many-body systems associated with violations of thermalization for special nonequilibrium initial states. Their various systematic constructions require fine-tuning of local Hamiltonian parameters. In this work, we demonstrate that long-range interacting quantum spin systems generically host robust QMBS. We analyze spectral properties upon raising the power-law decay exponent <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>α</a:mi></a:math> of spin-spin interactions from the solvable permutationally symmetric limit <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>α</c:mi><c:mo>=</c:mo><c:mn>0</c:mn></c:math>. First, we numerically establish that, despite the fact that spectral signatures of chaos appear for infinitesimal <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>α</e:mi></e:math>, the towers of <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>α</g:mi><g:mo>=</g:mo><g:mn>0</g:mn></g:math> energy eigenstates with large collective spin are smoothly deformed as <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>α</i:mi></i:math> is increased and exhibit characteristic QMBS features. To elucidate the nature and fate of these states in larger systems, we introduce an analytical approach based on mapping the spin Hamiltonian onto a relativistic quantum rotor nonlinearly coupled to an extensive set of bosonic modes. We analytically solve for the eigenstates of this interacting impurity model by means of a novel polaron-type canonical transformation and show their self-consistent localization in large-spin sectors of the original Hamiltonian for <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mn>0</k:mn><k:mo>&lt;</k:mo><k:mi>α</k:mi><k:mo>&lt;</k:mo><k:mi>d</k:mi></k:math> (with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>d</m:mi><m:mo>=</m:mo><m:mtext>spatial dimension of the lattice</m:mtext></m:math>). Our theory unveils the stability mechanism of such QMBS for an arbitrary system size and predicts instances of its breakdown, e.g., near dynamical critical points or in the presence of semiclassical chaos, which we verify numerically in long-range quantum Ising chains. As a by-product, we find a predictive criterion for the presence or absence of heating under periodic driving for <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mn>0</o:mn><o:mo>&lt;</o:mo><o:mi>α</o:mi><o:mo>&lt;</o:mo><o:mi>d</o:mi></o:math>, beyond existing Floquet-prethermalization theorems.