Field theory of composite fermions Dam Thanh Son (University of Chicago) Les Houches, 09/06/2019 Part 1: FQHE and field theoretic duality, Dirac composite fermion Part 2: Hydrodynamics of the composite fermions, “emergent graviton” Fractional quantum Hall effect and field-theoretic dualities Plan • Fractional quantum Hall effect • Composite fermion • Duality The quantum Hall effect The microscopic theory of the quantum Hall effect 2D electrons in a magnetic field At the microscopic level: a known Hamiltonian (p + eA )2 e2 H = a a + 2m xa xb a a,b X hXi | − | + impurities Landau levels and IQHE • Ignore interactions between electrons • Energy levels of charged particle in magnetic field in 2D: Landau levels • When some Landau level fully occupied: integer quantum Hall effect (IQHE) von Klitzing et al 1980 Landau levels and IQHE • Ignore interactions between electrons • Energy levels of charged particle in magnetic field in 2D: Landau levels • When some Landau level fully occupied: integer quantum Hall effect (IQHE) von Klitzing et al 1980 n=2 B ∆ = n=1 m BA n=0 degeneracy 2⇡ The fractional quantum Hall effect (FQHE) n=2 n=1 n=0 The fractional quantum Hall effect (FQHE) filling factor number of electrons ⌫ = <latexit sha1_base64="Y48WDBW+Xx4eF9rf0IYihV+70Wg=">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</latexit>sha1_base64="RnD/0P8J+f+ENJA6SRREledA0Ps=">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</latexit>sha1_base64="ye6i+Bdr028MoBEwHHB/D3tfM20=">AAACPXicbZA7SwQxFIUzvt+uWtoEF8FqmbHRRhBFsLBQcFXYWZZM9s4azGNI7ojLML/IH+FvsFQrCzuxtTWzLuLrQuBwzgk3+ZJMCodh+BCMjI6NT0xOTc/Mzs0vLNaWls+cyS2HJjfS2IuEOZBCQxMFSrjILDCVSDhPrvar/PwarBNGn2I/g7ZiPS1SwRl6q1M7iHW+Q+PUMl7ECDdoVaFzlYClJqUggaM12pXlV9qFHmjw/X7VYPToqCw7tXrYCAdDv0T0W9TJcI47tee4a3iuQCOXzLlWFGbYLphFwSWUM3HuIGP8ivWg5aVmCly7GHy3pOve6dLUWH800oH7/UbBlHN9lfimYnjpfmeV+V/WyjHdbhdCZzmC5p+L0lxSNLRiR7vCeh6y7wXjVvi3Un7JPAr0hH9sSVTF5A+Bv+JssxGFjegkrO/uDelMkVWyRjZIRLbILjkkx6RJOLkl9+SRPAV3wUvwGrx9VkeC4Z0V8mOC9w+Um7EW</latexit> degeneracy of a LL n=2 n=1 n=0 The fractional quantum Hall effect (FQHE) filling factor number of electrons ⌫ = <latexit sha1_base64="Y48WDBW+Xx4eF9rf0IYihV+70Wg=">AAACPXicbZBLSxxBFIVvG42PaJwky2wKJeBq6M4mbkRRAlm4mEBmFKaHobrm9lhYj6bqtjg0Dfk/+RH5DS4TF5JFdiFbt1bPiPi6UHA45xS36ssKJT3F8a9o7sX8wsvFpeWVV6trr9dbb972vC2dwK6wyrrjjHtU0mCXJCk8LhxynSk8yk4PmvzoDJ2X1nyjSYEDzcdG5lJwCtaw9Tk15Q5Lc8dFlRKek9OVKXWGjtmcoUJBzhpf13fpCMdoMPQnTYOzw8O6HrY243Y8HXYnksdic2/36uI7AHSGrT/pyIpSoyGhuPf9JC5oUHFHUiisV9LSY8HFKR9jP0jDNfpBNf1uzT4EZ8Ry68IxxKbu/RsV195PdBaamtOJf5w15nNZv6R8e1BJU5SERswW5aViZFnDjo2kCzzUJAgunAxvZeKEBxQUCD/YkumGyRMCT0XvYzuJ28nXAGcfZrME72EDtiCBT7AHX6ADXRDwAy7gN1xGP6O/0b/o/6w6F93eeQcPJrq+AVyMs8U=</latexit>sha1_base64="RnD/0P8J+f+ENJA6SRREledA0Ps=">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</latexit>sha1_base64="ye6i+Bdr028MoBEwHHB/D3tfM20=">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</latexit> degeneracy of a LL n=2 n=1 n=0 ⌫<latexit sha1_base64="VsEgriGm/ZwWg/ckepm5BlGRjm4=">AAAB+3icbVA9SwNBEJ2LXzF+RS1tFoNgFe5sTCEYtLGM4CWB5Ih7m71kye7esbsnhON+g63WdmLrj7H0Z9i5+ShM4oOBx3szzMwLE860cd0vp7C2vrG5Vdwu7ezu7R+UD4+aOk4VoT6JeazaIdaUM0l9wwyn7URRLEJOW+HoduK3nqjSLJYPZpzQQOCBZBEj2FjJ78r0yuuVK27VnQKtEm9OKtc/UfQIAI1e+bvbj0kqqDSEY607npuYIMPKMMJpXuqmmiaYjPCAdiyVWFAdZNNjc3RmlT6KYmVLGjRV/05kWGg9FqHtFNgM9bI3Ef/zOqmJakHGZJIaKslsUZRyZGI0+Rz1maLE8LElmChmb0VkiBUmxuazsCUUuc3EW05glTQvqp5b9e7dSv0GZijCCZzCOXhwCXW4gwb4QIDBM7zAq5M7b8678zFrLTjzmWNYgPP5C2ZMl04=</latexit>sha1_base64="Y3OXJQg5XDLqHllaXg66BdSQAFs=">AAAB+3icbVA9SwNBEJ2LXzF+RS1tFoNgFe5stBAM2lhG8JJAcoS9zV6yZHfv2N0TwnG/wVZrO7H1x1j6M+zcXFKYxAcDj/dmmJkXJpxp47pfTmltfWNzq7xd2dnd2z+oHh61dJwqQn0S81h1QqwpZ5L6hhlOO4miWISctsPx3dRvP1GlWSwfzSShgcBDySJGsLGS35Pptdev1ty6WwCtEm9Oajc/UYFmv/rdG8QkFVQawrHWXc9NTJBhZRjhNK/0Uk0TTMZ4SLuWSiyoDrLi2BydWWWAoljZkgYV6t+JDAutJyK0nQKbkV72puJ/Xjc10VWQMZmkhkoyWxSlHJkYTT9HA6YoMXxiCSaK2VsRGWGFibH5LGwJRW4z8ZYTWCWti7rn1r0Ht9a4hRnKcAKncA4eXEID7qEJPhBg8Awv8Orkzpvz7nzMWkvOfOYYFuB8/gIZjpiG</latexit>sha1_base64="w+2djzQHowBJAyGU/JAQ9zYlBhE=">AAAB+3icbVA9T8MwEL2Ur1K+AowsFhUSU5WwwMBQwcJYJNJWaqPKcZ3Wqu1EtoNURfkNrDCzIVZ+DCP/BLfNQFuedNLTe3e6uxelnGnjed9OZWNza3unulvb2z84PHKPT9o6yRShAUl4oroR1pQzSQPDDKfdVFEsIk470eR+5neeqdIskU9mmtJQ4JFkMSPYWCnoy+zWH7h1r+HNgdaJX5I6lGgN3J/+MCGZoNIQjrXu+V5qwhwrwwinRa2faZpiMsEj2rNUYkF1mM+PLdCFVYYoTpQtadBc/TuRY6H1VES2U2Az1qveTPzP62UmvglzJtPMUEkWi+KMI5Og2edoyBQlhk8twUQxeysiY6wwMTafpS2RKGwm/moC66R91fC9hv/o1Zt3ZTpVOINzuAQfrqEJD9CCAAgweIFXeHMK5935cD4XrRWnnDmFJThfvwbKlOo=</latexit> < 1 The fractional quantum Hall effect (FQHE) filling factor number of electrons ⌫ = <latexit sha1_base64="Y48WDBW+Xx4eF9rf0IYihV+70Wg=">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</latexit>sha1_base64="RnD/0P8J+f+ENJA6SRREledA0Ps=">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</latexit>sha1_base64="ye6i+Bdr028MoBEwHHB/D3tfM20=">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</latexit> degeneracy of a LL n=2 n=1 n=0 ⌫<latexit sha1_base64="VsEgriGm/ZwWg/ckepm5BlGRjm4=">AAAB+3icbVA9SwNBEJ2LXzF+RS1tFoNgFe5sTCEYtLGM4CWB5Ih7m71kye7esbsnhON+g63WdmLrj7H0Z9i5+ShM4oOBx3szzMwLE860cd0vp7C2vrG5Vdwu7ezu7R+UD4+aOk4VoT6JeazaIdaUM0l9wwyn7URRLEJOW+HoduK3nqjSLJYPZpzQQOCBZBEj2FjJ78r0yuuVK27VnQKtEm9OKtc/UfQIAI1e+bvbj0kqqDSEY607npuYIMPKMMJpXuqmmiaYjPCAdiyVWFAdZNNjc3RmlT6KYmVLGjRV/05kWGg9FqHtFNgM9bI3Ef/zOqmJakHGZJIaKslsUZRyZGI0+Rz1maLE8LElmChmb0VkiBUmxuazsCUUuc3EW05glTQvqp5b9e7dSv0GZijCCZzCOXhwCXW4gwb4QIDBM7zAq5M7b8678zFrLTjzmWNYgPP5C2ZMl04=</latexit>sha1_base64="Y3OXJQg5XDLqHllaXg66BdSQAFs=">AAAB+3icbVA9SwNBEJ2LXzF+RS1tFoNgFe5stBAM2lhG8JJAcoS9zV6yZHfv2N0TwnG/wVZrO7H1x1j6M+zcXFKYxAcDj/dmmJkXJpxp47pfTmltfWNzq7xd2dnd2z+oHh61dJwqQn0S81h1QqwpZ5L6hhlOO4miWISctsPx3dRvP1GlWSwfzSShgcBDySJGsLGS35Pptdev1ty6WwCtEm9Oajc/UYFmv/rdG8QkFVQawrHWXc9NTJBhZRjhNK/0Uk0TTMZ4SLuWSiyoDrLi2BydWWWAoljZkgYV6t+JDAutJyK0nQKbkV72puJ/Xjc10VWQMZmkhkoyWxSlHJkYTT9HA6YoMXxiCSaK2VsRGWGFibH5LGwJRW4z8ZYTWCWti7rn1r0Ht9a4hRnKcAKncA4eXEID7qEJPhBg8Awv8Orkzpvz7nzMWkvOfOYYFuB8/gIZjpiG</latexit>sha1_base64="w+2djzQHowBJAyGU/JAQ9zYlBhE=">AAAB+3icbVA9T8MwEL2Ur1K+AowsFhUSU5WwwMBQwcJYJNJWaqPKcZ3Wqu1EtoNURfkNrDCzIVZ+DCP/BLfNQFuedNLTe3e6uxelnGnjed9OZWNza3unulvb2z84PHKPT9o6yRShAUl4oroR1pQzSQPDDKfdVFEsIk470eR+5neeqdIskU9mmtJQ4JFkMSPYWCnoy+zWH7h1r+HNgdaJX5I6lGgN3J/+MCGZoNIQjrXu+V5qwhwrwwinRa2faZpiMsEj2rNUYkF1mM+PLdCFVYYoTpQtadBc/TuRY6H1VES2U2Az1qveTPzP62UmvglzJtPMUEkWi+KMI5Og2edoyBQlhk8twUQxeysiY6wwMTafpS2RKGwm/moC66R91fC9hv/o1Zt3ZTpVOINzuAQfrqEJD9CCAAgweIFXeHMK5935cD4XrRWnnDmFJThfvwbKlOo=</latexit>