Project Grant 2340394

CAREER: GEOMETRY AND TOPOLOGY OF QUANTUM MATERIALS -NONTECHNICAL SUMMARY THIS CAREER AWARD SUPPORTS RESEARCH AND EDUCATION IN THE RAPIDLY EVOLVING FIELD OF QUANTUM MATERIALS. QUANTUM MATERIALS REFER TO THOSE IN WHICH THE COLLECTIVE BEHAVIOR OF ELECTRONS GIVES RISE TO EXTRAORDINARY PROPERTIES WITH POTENTIAL APPLICATIONS IN QUANTUM TECHNOLOGIES FROM SPINTRONICS TO QUANTUM COMPUTING. IN A PARADIGM SHIFTING DISCOVERY, TOPOLOGY, THE FIELD IN MATHEMATICS THAT DESCRIBES PROPERTIES WHICH REMAIN UNCHANGED WHEN OBJECTS ARE DEFORMED, WAS RECOGNIZED TO RESULT IN REMARKABLE MATERIALS SUCH AS THOSE THAT ARE INSULATORS IN THE INTERIOR BUT POSSESS DISSIPATIONLESS ELECTRIC CONDUCTION THROUGH STATES ON THEIR SURFACES AND EDGES THAT ARE REQUIRED TO EXIST BY TOPOLOGY. LESS EXPLORED BUT EQUALLY REMARKABLE IS THE CRUCIAL ROLE THAT TOPOLOGY PLAYS IN DETERMINING WHICH PHASES OF ELECTRONIC MATTER ARE FOUND IN EACH MATERIAL, AND THE POSSIBILITY THEY MIGHT BE LONG SOUGHT-AFTER EXOTIC ELECTRONIC STATES OF MATTER. EXAMPLES ARE SUPERCONDUCTIVITY WHERE ELECTRONS FLOW WITH EXACTLY ZERO RESISTANCE IN THE ENTIRE MATERIAL OR PHASES WHERE ELECTRONS SEEM TO DISSOCIATE INTO SMALLER ENTITIES WITH UNCONVENTIONAL PROPERTIES, CALLED FRACTIONALIZED PHASES. THE PI WILL STUDY HOW TOPOLOGY AND QUANTUM GEOMETRY, THE GEOMETRIC PROPERTIES OF AN ABSTRACT REPRESENTATION OF QUANTUM STATES, CAN INFLUENCE THE NATURE OF ELECTRONS IN MATTER, PARTICULARLY WHEN UNAVOIDABLE IMPERFECTIONS OR ?DIRT? ARE PRESENT IN MATERIALS. CAREFUL CONTROL OF DEFECTS MAY ENABLE PHASES OF ELECTRONIC MATTER WITH DESIRED PROPERTIES TO BE REALIZED. THROUGH THIS RESEARCH, THE PI AIMS TO GAIN INSIGHTS INTO THE STABILITY OF UNIQUE QUANTUM PHENOMENA, AS WELL AS OFFER NEW MATHEMATICAL TOOLS FOR THE CHARACTERIZATION OF TOPOLOGICAL QUANTUM MATERIALS AND FOR THE PREDICTION AND, WORKING WITH EXPERIMENT, THE DISCOVERY OF NEW ONES. IN THIS PROJECT, RESEARCH AND EDUCATION ARE INTEGRATED THROUGH MULTIPLE EFFORTS FOCUSING ON GRADUATE STUDENTS THAT WILL ENHANCE AND DIVERSIFY THEIR TRAINING IN QUANTUM PROPERTIES OF MATTER. THE ACTIVITY WILL INCLUDE A BOOTCAMP COVERING ESSENTIALS OF TOPOLOGICAL MATERIALS AND COMPUTATIONAL TECHNIQUES FOR STUDYING THEIR ELECTRONIC PROPERTIES. THE PI WILL ALSO ORGANIZE A WORKSHOP TO SHOWCASE TALENTED YOUNG RESEARCHERS FROM DIVERSE BACKGROUNDS WORKING ON QUANTUM MATERIALS. TECHNICAL SUMMARY THIS CAREER AWARD SUPPORTS THEORETICAL RESEARCH AND EDUCATION AIMED TO STUDY THE INFLUENCE OF QUANTUM GEOMETRY IN THE COLLECTIVE PROPERTIES OF ELECTRONS IN THE SOLID STATE. THE PI WILL EXPLORE HOW THE MOMENTUM SPACE TEXTURES OF ELECTRON WAVEFUNCTIONS AFFECT ELECTRON BEHAVIOR IN BOTH CLEAN AND DIRTY MATERIALS, INFLUENCING LONG-RANGE COHERENCE, TRANSPORT, AND THE EMERGENCE OF EXOTIC EXCITATIONS. THE PROJECT FOCUSES ON A NEW PERSPECTIVE ON QUANTUM MATERIALS, WITH THE AIM TO UNIFY QUANTUM GEOMETRIC PHENOMENA FROM THE POINT OF VIEW OF THE STRUCTURE OF THE GREEN?S FUNCTION OPERATOR, IN PARTICULAR ITS TOPOLOGICALLY ROBUST ZEROS WHEN PROJECTED TO VARIOUS SPATIAL DEFECTS. THE GOAL OF THIS APPROACH IS TO CONSTRUCT TOOLS THAT CAN BE EFFICIENTLY APPLIED TO IDENTIFY NONTRIVIAL GEOMETRY BOTH IN SYSTEMS WITH AND WITHOUT TRANSLATIONAL SYMMETRY, THEREFORE OPENING THE POSSIBILITY TO 1) CHARACTERIZE THE BEHAVIOR OF DISORDERED TOPOLOGICAL CRYSTALLINE MATTER; 2) OFFER GUIDELINES FOR THE SEARCH OF NEW MATERIALS WITH EXCEPTIONAL PHYSICAL PROPERTIES; 3) IDENTIFY ROBUST PHYSICAL RESPONSES THAT STEM FROM THE NONTRIVIAL GEOMETRY OF THE GROUND STATE. THIS APPROACH IS SET TO CONTRIBUTE SIGNIFICANTLY TO THE BURGEONING FIELD OF TOPOLOGICAL MATTER, WITH APPLICATIONS IN ELECTRONIC STRUCTURE THEORY, CHEMISTRY, AND MATERIALS SCIENCE. THIS ACTIVITY ALSO INCLUDES ESTABLISHING A BOOTCAMP FOR THE COMPUTATION OF MATERIALS TOPOLOGICAL PROPERTIES COVERING THE ESSENTIALS OF BAND THEORY, GROUP THEORY, AND DENSITY FUNCTIONAL THEORY, AND ALLOWING STUDENTS TO GAIN HANDS-ON EXPERIENCE SIMULATING VARIOUS QUANTITIES OF EXPERIMENTAL AND TECHNOLOGICAL RELEVANCE. THE PI ALSO AIMS TO ESTABLISH A NEW YORK CITY BASED WORKSHOP TO SPOTLIGHT EXCELLENT YOUNG RESEARCHERS FROM DIVERSE BACKGROUNDS. THIS AWARD REFLECTS NSF'S STATUTORY MISSION AND HAS BEEN DEEMED WORTHY OF SUPPORT THROUGH EVALUATION USING THE FOUNDATION'S INTELLECTUAL MERIT AND BROADER IMPACTS REVIEW CRITERIA.- SUBAWARDS ARE NOT PLANNED FOR THIS AWARD.