Topological metals Anton Burkov CATS Kickoff Meeting Ag Online Theoretical Physics Colloquium, April 29, 2020
Where do gapless excitations come from?
<latexit sha1_base64="jF4d1tejJM2Vw6iFzNce2wqWalU=">ACIHicbVDLSgMxFM34rPVdekmWIS6sExbsboQim5cVrAP6LQlk2ba0CQzJBmhDPMpbvwVNy4U0Z1+jZl2ELUeuHA4517uvcNGFXatj+shcWl5ZXVzFp2fWNzazu3s9tUfigxaWCf+bLtIkUYFaShqWakHUiCuMtIyx1fJX7rjkhFfXGrJwHpcjQU1KMYaSP1c1WHBIoyXxQix/XgOD6CF9DxJMKRM3KR7JVhavTKcVSGPIbH0OFhP5e3i/YUcJ6UpIHKer93Lsz8HIidCYIaU6JTvQ3QhJTEjcdYJFQkQHqMh6RgqECeqG0fjOGhUQbQ86UpoeFU/TkRIa7UhLumkyM9Un+9RPzP64TaO+tGVAShJgLPFnkhg9qHSVpwQCXBmk0MQVhScyvEI2TS0SbT7DSE8wSn3y/Pk2a5WKoUKzcn+dplGkcG7IMDUAlUAU1cA3qoAEwuAeP4Bm8WA/Wk/Vqvc1aF6x0Zg/8gvX5BXKmogU=</latexit> Where do gapless excitations come from? • Liquid of interacting bosons, e.g. liquid helium. ✏ ( k ) = ~ 2 k 2 2 m − µ
<latexit sha1_base64="zl46iCeSlhqVijXuqw6w7Tm21Aw=">ACIHicbZDLSgMxFIYz9VbrbdSlm2ARKkiZkUrdCEVBXEmF3qBTSibNtKGZC8kZoQzKG58FTcuFNGdPo3pBdTWHwIf/zknyfndSHAFlvVpZJaWV1bXsu5jc2t7R1zd6+hwlhSVqehCGXLJYoJHrA6cBCsFUlGfFewpju8Gteb90wqHgY1GEWs45N+wD1OCWira5YdP8YX2BHMg4LjSUITJyISOBH4Ov3h2xQ7kvcHcNxNaieNtGvmraI1EV4EewZ5NFO1a34vZDGPguACqJU27Yi6CTj6lgac6JFYsIHZI+a2sMiM9UJ5ksmOIj7fSwF0p9AsAT9/dEQnylRr6rO30CAzVfG5v/1doxeOedhAdRDCyg04e8WGAI8Tgt3OSURAjDYRKrv+K6YDolEBnmtMh2PMrL0LjtGiXimd3pXzlchZHFh2gQ1RANiqjCrpBVRHFD2gJ/SCXo1H49l4M96nrRljNrOP/sj4+gYuf6MB</latexit> <latexit sha1_base64="vI3oQKPO1yHf4raXHBXcRgXJb8Y=">AB73icbVDLSgNBEJyNrxhfUY9eBoPgKWyM+LgFvXiMYB6QLGF20psMmZ1Z2aFsOQnvHhQxKu/482/cXaziBoLGoqbrq7/IgzbVz30yksLa+srhXSxubW9s75d29tpaxotCikvV9YkGzgS0DMcupECEvocOv7kOvU7D6A0k+LOTCPwQjISLGCUGCt1+xBpxqUYlCtu1c2AF0ktJxWUozkof/SHksYhCEM50bpXcyPjJUQZRjnMSv1YQ0TohIygZ6kgIWgvye6d4SOrDHEglS1hcKb+nEhIqPU09G1nSMxY/VS8T+vF5vgwkuYiGIDgs4XBTHRuL0eTxkCqjhU0sIVczeiumYKEKNjaiUhXCZ4uz75UXSPqnW6tX67WmlcZXHUQH6BAdoxo6Rw10g5qohSji6BE9oxfn3nlyXp23eWvByWf20S84719lmZBX</latexit> <latexit sha1_base64="YbSkHxKvCGSr7ROEIAHs5yOhUck=">AB8nicbVBNSwMxEJ31s9avqkcvwSJ4sexKRS9CURSPFfsF26Vk07QNzSZLkhXK0p/hxYMiXv013vw3pu0etPXBwO9GWbmhTFn2rjut7O0vLK6tp7byG9ube/sFvb2G1omitA6kVyqVog15UzQumG01asKI5CTpvh8GbiN5+o0kyKmhnFNIhwX7AeI9hYyb9DV+gWnaIaeuwUim7JnQItEi8jRchQ7RS+2l1JkogKQzjW2vfc2AQpVoYRTsf5dqJpjMkQ96lvqcAR1UE6PXmMjq3SRT2pbAmDpurviRHWo+i0HZG2Az0vDcR/P8xPQug5SJODFUkNmiXsKRkWjyP+oyRYnhI0swUczeisgAK0yMTSlvQ/DmX14kjbOSVy6dP5SLlesjhwcwhGcgAcXUIF7qEIdCEh4hld4c4z4rw7H7PWJSebOYA/cD5/ABIsjyw=</latexit> <latexit sha1_base64="zv3kJuY3ZtWtS+rMPCYFhqa5pA=">AB7HicbVBNS8NAEJ3Ur1q/qh69LBbBU0lU/LgVvXisYNpCG8pmu2mXbjZhdyOUtL/BiwdFvPqDvPlv3KRB1Ppg4PHeDPz/JgzpW370yotLa+srpXKxubW9s71d29loSahLIh7Jjo8V5UxQVzPNaSeWFIc+p21/fJP57QcqFYvEvZ7E1AvxULCAEayN5E57YTLtV2t23c6BFolTkBoUaParH71BRJKQCk04Vqr2LH2Uiw1I5zOKr1E0RiTMR7SrqECh1R5aX7sDB0ZYCSJoSGuXqz4kUh0pNQt90hliP1F8vE/zuokOLr2UiTjRVJD5oiDhSEco+xwNmKRE84khmEhmbkVkhCUm2uRTyUO4ynD+/fIiaZ3UndP6d1ZrXFdxFGAziEY3DgAhpwC01wgQCDR3iGF0tYT9ar9TZvLVnFzD78gvX+BTskjxk=</latexit> <latexit sha1_base64="jF4d1tejJM2Vw6iFzNce2wqWalU=">ACIHicbVDLSgMxFM34rPVdekmWIS6sExbsboQim5cVrAP6LQlk2ba0CQzJBmhDPMpbvwVNy4U0Z1+jZl2ELUeuHA4517uvcNGFXatj+shcWl5ZXVzFp2fWNzazu3s9tUfigxaWCf+bLtIkUYFaShqWakHUiCuMtIyx1fJX7rjkhFfXGrJwHpcjQU1KMYaSP1c1WHBIoyXxQix/XgOD6CF9DxJMKRM3KR7JVhavTKcVSGPIbH0OFhP5e3i/YUcJ6UpIHKer93Lsz8HIidCYIaU6JTvQ3QhJTEjcdYJFQkQHqMh6RgqECeqG0fjOGhUQbQ86UpoeFU/TkRIa7UhLumkyM9Un+9RPzP64TaO+tGVAShJgLPFnkhg9qHSVpwQCXBmk0MQVhScyvEI2TS0SbT7DSE8wSn3y/Pk2a5WKoUKzcn+dplGkcG7IMDUAlUAU1cA3qoAEwuAeP4Bm8WA/Wk/Vqvc1aF6x0Zg/8gvX5BXKmogU=</latexit> <latexit sha1_base64="eAvSGX1XnPW2HrQoS6t4JHudNTo=">AB7nicbVDLSsNAFL2pr1pfVZduBovgqiS2+NgV3bisYB/QhjKZTtohk0mYmQgl9CPcuFDErd/jzr9xkgZR64ELh3Pu5d57vJgzpW370yqtrK6tb5Q3K1vbO7t71f2DroSWiHRDySfQ8rypmgHc0p/1YUhx6nPa84Cbzew9UKhaJez2LqRviWA+I1gbqZcOPR8F81G1ZtftHGiZOAWpQYH2qPoxHEckCanQhGOlBo4dazfFUjPC6bwyTBSNMQnwhA4MFTikyk3zc+foxChj5EfSlNAoV39OpDhUahZ6pjPEeqr+epn4nzdItH/pkzEiaCLBb5CUc6QtnvaMwkJZrPDMFEMnMrIlMsMdEmoUoewlWG8+Xl0n3rO406o27Zq1XcRhiM4hlNw4AJacAt6ACBAB7hGV6s2HqyXq23RWvJKmYO4Res9y80EY+e</latexit> Where do gapless excitations come from? • Chemical potential is large and negative at high T, thus all bosons have high energy relative to it, no low energy excitations. ✏ ✏ ( k ) = ~ 2 k 2 2 m − µ ✓ ∂ F | µ | ◆ µ = ∂ N T,V k F = E − TS
<latexit sha1_base64="ng2CxYv0EW+MD0vVr0nr0w5DKTc=">AB6HicbVDLSsNAFL2pr1pfVZduBovgqiRWfOyKbly20Be0oUymk3bsZBJmJkIJ/QI3LhRx6ye582+cpEHUeuDC4Zx7ufceL+JMadv+tAorq2vrG8XN0tb2zu5ef+go8JYEtomIQ9lz8OKciZoWzPNaS+SFAcep1vepv63QcqFQtFS8i6gZ4LJjPCNZGaraG5YpdtTOgZeLkpAI5GsPyx2AUkjigQhOleo7dqTdBEvNCKfz0iBWNMJkise0b6jAVukh06RydGSE/lKaERpn6cyLBgVKzwDOdAdYT9dLxf+8fqz9KzdhIo1FWSxyI850iFKv0YjJinRfGYIJpKZWxGZYImJNtmUshCuU1x8v7xMOmdVp1atNc8r9Zs8jiIcwTGcgOXUIc7aEAbCFB4hGd4se6tJ+vVelu0Fqx85hB+wXr/AshxjQ8=</latexit> <latexit sha1_base64="HC8609G9521Bs7mUQtDSY9Zk+i4=">AB6nicbVDLSgNBEOyNrxhfUY9eBoPgKWyM+LgFvXiMaB6QLGF2MpsMmZldZmaFsOQTvHhQxKtf5M2/cXaziBoLGoqbrq7/IgzbVz30yksLa+srhXSxubW9s75d29tg5jRWiLhDxUXR9rypmkLcMp91IUSx8Tjv+5Dr1Ow9UaRbKezONqCfwSLKAEWysdNcX8aBcatuBrRIajmpQI7moPzRH4YkFlQawrHWvZobGS/ByjDC6azUjzWNMJngEe1ZKrGg2kuyU2foyCpDFITKljQoU39OJFhoPRW+7RTYjPVfLxX/83qxCS68hMkoNlS+aIg5siEKP0bDZmixPCpJZgoZm9FZIwVJsamU8pCuEx9v3yImfVGv1av32tNK4yuMowgEcwjHU4BwacANaAGBETzCM7w43HlyXp23eWvByWf24Rec9y92Jo4N</latexit> <latexit sha1_base64="dFRuiNUdS7yZT0uVLYLiFRE8LA=">AB6nicbVDLSsNAFL2pr1pfVZduBovgqiS2+NgV3bis2Be0oUymk3boZBJmJkIJ/Q3LhRx6xe582+cpEHUeuDC4Zx7ufceL+JMadv+tAorq2vrG8XN0tb2zu5ef+go8JYEtomIQ9lz8OKciZoWzPNaS+SFAcep1vepP63QcqFQtFS8i6gZ4LJjPCNZGum8NybBcsat2BrRMnJxUIEdzWP4YjEISB1RowrFSfceOtJtgqRnhdF4axIpGmEzxmPYNFTigyk2yU+foxCgj5IfSlNAoU39OJDhQahZ4pjPAeqL+eqn4n9ePtX/pJkxEsaCLBb5MUc6ROnfaMQkJZrPDMFEMnMrIhMsMdEmnVIWwlWK8+Xl0nrOrUqrW7eqVxncdRhCM4hlNw4AIacAtNaAOBMTzCM7xY3HqyXq23RWvBymcO4Res9y85aI3l</latexit> Where do gapless excitations come from? • Transition to superfluid at low T. µ T c T
<latexit sha1_base64="vI3oQKPO1yHf4raXHBXcRgXJb8Y=">AB73icbVDLSgNBEJyNrxhfUY9eBoPgKWyM+LgFvXiMYB6QLGF20psMmZ1Z2aFsOQnvHhQxKu/482/cXaziBoLGoqbrq7/IgzbVz30yksLa+srhXSxubW9s75d29tpaxotCikvV9YkGzgS0DMcupECEvocOv7kOvU7D6A0k+LOTCPwQjISLGCUGCt1+xBpxqUYlCtu1c2AF0ktJxWUozkof/SHksYhCEM50bpXcyPjJUQZRjnMSv1YQ0TohIygZ6kgIWgvye6d4SOrDHEglS1hcKb+nEhIqPU09G1nSMxY/VS8T+vF5vgwkuYiGIDgs4XBTHRuL0eTxkCqjhU0sIVczeiumYKEKNjaiUhXCZ4uz75UXSPqnW6tX67WmlcZXHUQH6BAdoxo6Rw10g5qohSji6BE9oxfn3nlyXp23eWvByWf20S84719lmZBX</latexit> <latexit sha1_base64="eAvSGX1XnPW2HrQoS6t4JHudNTo=">AB7nicbVDLSsNAFL2pr1pfVZduBovgqiS2+NgV3bisYB/QhjKZTtohk0mYmQgl9CPcuFDErd/jzr9xkgZR64ELh3Pu5d57vJgzpW370yqtrK6tb5Q3K1vbO7t71f2DroSWiHRDySfQ8rypmgHc0p/1YUhx6nPa84Cbzew9UKhaJez2LqRviWA+I1gbqZcOPR8F81G1ZtftHGiZOAWpQYH2qPoxHEckCanQhGOlBo4dazfFUjPC6bwyTBSNMQnwhA4MFTikyk3zc+foxChj5EfSlNAoV39OpDhUahZ6pjPEeqr+epn4nzdItH/pkzEiaCLBb5CUc6QtnvaMwkJZrPDMFEMnMrIlMsMdEmoUoewlWG8+Xl0n3rO406o27Zq1XcRhiM4hlNw4AJacAt6ACBAB7hGV6s2HqyXq23RWvJKmYO4Res9y80EY+e</latexit> <latexit sha1_base64="41YKZte0cfHUeoxehpUOj03/mQw=">ACAHicbVDLSsNAFJ34rPUVdeHCzWAR6qakVnwshKIblxXsA5pQJtNJO2QyCTOTQgnZ+CtuXCji1s9w5984SYOo9cCFwzn3cu89bsSoVJb1aSwsLi2vrJbWyusbm1vb5s5uR4axwKSNQxaKnoskYZSTtqKkV4kCApcRrquf5P53QkRkob8Xk0j4gRoxKlHMVJaGpj7NokZSGvJrbrQT89hldwAv2BWbFqVg4T+oFqYACrYH5YQ9DHAeEK8yQlP26FSknQUJRzEhatmNJIoR9NCJ9TkKiHS/IEUHmlCL1Q6OIK5urPiQFUk4DV3cGSI3lXy8T/P6sfIunITyKFaE49kiL2ZQhTBLAw6pIFixqSYIC6pvhXiMBMJKZ1bOQ7jMcPb98jzpnNTqjVrj7rTSvC7iKIEDcAiqoA7OQRPcghZoAwxS8AiewYvxYDwZr8brHXBKGb2wC8Y718fiJWU</latexit> Where do gapless excitations come from? • Excitations are phonons with a gapless linear dispersion. ✏ ✏ ( k ) = vk k
<latexit sha1_base64="Y1pfLfEAmsQPRmNtU3zKtZEYuRs=">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</latexit> Where do gapless excitations come from? • The origin of phonons is spontaneous symmetry breaking. h φ ( r , t ) i = hp ρ e i θ ( r ,t ) i 6 = 0
Gapless excitations in crystals • Crystal breaks translational symmetry of space, has gapless acoustic phonons.
Where do gapless excitations come from? • Lesson 1: gapless excitations do not arise accidentally, there must be a reason. • Lesson 2: their existence is associated with interesting physics, such as superfluidity, crystallinity, etc.
Gapless fermions • Energy spectrum of electrons in solids has form of bands separated by bandgaps. • Electrons obey Pauli principle. • Number of states in a band is always 2 times the number of unit cells.
<latexit sha1_base64="eSYGZzyFhlnoaEb9h/vQAwaz8Js=">AB8XicbVDLSgNBEOyNrxhfUY9eBoPgKeyq+LgFBfEYwTwWcLsZDYZMjuzMwKYclfePGgiFf/xpt/42SziBoLGoqbrq7gpgzbVz30yksLC4trxRXS2vrG5tb5e2dpaJIrRBJeqHWBNORO0YZjhtB0riqOA01Ywupr6rQeqNJPizoxj6kd4IFjICDZWu/SWDMuRe+6V64VTcDmideTiqQo94rf3T7kiQRFYZwrHXHc2Pjp1gZRjidlLqJpjEmIzygHUsFjqj20+ziCTqwSh+FUtkSBmXqz4kUR1qPo8B2RtgM9V9vKv7ndRITnvspE3FiqCzRWHCkZFo+j7qM0WJ4WNLMFHM3orIECtMjA2plIVwMcXp98vzpHlU9Y6rx7cnldplHkcR9mAfDsGDM6jBDdShAQEPMIzvDjaeXJenbdZa8HJZ3bhF5z3L7CtkRA=</latexit> Gapless fermions • Even integer number of electrons per unit cell: insulator, no gapless excitations. ✏ F
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