We present a possible definition of a mobility gap for a many-body quantum system, in analogy to definitions of dynamical localization for single particle systems. Using this definition, we construct "corrected" quasi-adiabatic continuation operators. Under an appropriate definition of a unique ground state, we show how to introduce virtual fluxes. Armed with these results, we can directly carry over previous results in the case of a spectral gap. We present a proof of decay of correlation functions and we present a proof of Hall conductance quantization under very mild density-of-states assumptions defined later. We also generalize these definitions to the case of a "bulk mobility gap", in the case of a system with boundaries, and present a proof of Hall conductance quantization on an annulus under appropriate assumptions.
Further, we present a new "optimized" quasi-adiabatic continuation operator which simplifies previous estimates and tightens bounds in certain cases. This is presented in an appendix which can be read independently of the rest of the paper as it also improves estimates in the case of systems with a spectral gap. This filter function used decays in time at least as fast as ${\cal O}(\exp(-t^α))$ for all $α<1$, a class of decay called subexponential (a tighter description of what is possible is below). Using this function it is possible to tighten recent estimates of the Hall conductance quantization for gapped systems\cite{hall} to a decay which is subexponential in system size.
Modulated symmetries from generalized Lieb-Schultz-Mattis anomalies
Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；Interdisciplinary Theoretical and Mathematical Sciences Program (iTHEMS) RIKEN（理化学研究所跨学科理论与数学科学项目）；Institute for Theoretical Physics, University of Cologne（科隆大学理论物理研究所）；Center for Quantum Mathematics at IMADA, Southern Denmark University（南丹麦大学IMADA量子数学中心）；Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen（哥本哈根大学尼尔斯·玻尔研究所尼尔斯·玻尔国际学院）
Matrix Product States for Modulated Topological Phases: Crystalline Equivalence Principle and Lieb-Schultz-Mattis Constraints
Hong Kong University of Science and Technology（香港科技大学）；The Chinese University of Hong Kong（香港中文大学）；RIKEN（理化学研究所）；Massachusetts Institute of Technology（麻省理工学院）；Stanford University（斯坦福大学）；University of Southern California（南加州大学）
Entanglement area law and Lieb-Schultz-Mattis theorem in long-range interacting systems, and symmetry-enforced long-range entanglement
Dalhousie University（达尔豪斯大学）；Perimeter Institute for Theoretical Physics（珀蒂默理论物理研究所）；University of Waterloo（滑铁卢大学）；National University of Singapore（新加坡国立大学）
State Key Laboratory of Low-Dimensional Quantum Physics, Department of Physics, Tsinghua University（清华大学物理系低维量子物理国家重点实验室）；Qiuzhen College, Tsinghua University（清华大学邱成通数学科学中心）；Frontier Science Center for Quantum Information（量子信息前沿科学中心）；Hefei National Laboratory（合肥国家实验室）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Lieb-Mattis ordering theorem of electronic energy levels in the thermodynamic limit
University of Granada（格拉纳达大学）；Institute Carlos I of Theoretical and Computational Physics (iC1), University of Granada（格拉纳达大学理论计算物理卡洛斯一世研究所）；University CEU Fernando III（塞乌费尔南多三世大学）；Higher Polytechnic School, University CEU Fernando III（塞乌费尔南多三世大学高等理工学院）；University of Jaen（哈恩大学）；Department of Mathematics, University of Jaen（哈恩大学数学系）
