Physics of composite fermions in the fractional quantum Hall effect Dam Thanh Son (University of Chicago) Solvay workshop on Quantum Simulation Brussels, 18-20 February 2019
Plan • Fractional quantum Hall effect • Theories of the composite fermion: HLR and Dirac • Bosonized composite Fermi surface • Structure factor from composite fermions Dung Xuan Nguyen and DTS, to be published
<latexit sha1_base64="7c9tmZ3xDlEUzN/apRhOzrHjscE=">ACH3icbZDLSgMxFIbPeK31VnXpJlhFQSgzbnQjFEVwWcFeoC1DJk3b2GRmSDLSYegj+BA+g+BKwZ07cdulD+He9ALa1h8CP985h3PyeyFnSt235qbX1hcWk6tpFfX1jc2M1vbJRVEktAiCXgKx5WlDOfFjXTnFZCSbHwOC17nctBvXxPpWKBf6vjkNYFbvmsyQjWBrmZwzu3i85RTbGWwG7S7fbQlSHvyQekNjNZO2cPRSaNc7YZP7389vAFBwM1+1RkAiQX1NOFaq6tihridYakY47aVrkaIhJh3colVjfSyoqifD/XQgSEN1Aykeb5GQ/p3IsFCqVh4plNg3VbTtQH8r1aNdPOsnjA/jDT1yWhRM+JIB2iQDmowSYnmsTGYSGZuRaSNJSbaZDixRM9k4kzncCsKZ3kHDvn3JhwLmCkFOzCHhyBA6eQh2soQBEIPMATvMCr9Wi9Wx/W56h1zhrP7MCErP4P4mWlnQ=</latexit> <latexit sha1_base64="JIBG5LXc4zqDI8VUWsiRYbLg62U=">ACH3icbZDLSgMxFIYz9VZbL1WXboJVFIQy40Y3QlElxXsBdoyZNJMG5vMDEmdBjmEXwIn0FwpSsX7sRtlz6Erk0voG39IfDznXM4J78TMCqVaQ6M1MLi0vJKejWTXVvf2MxtbVekHwpMythnvqg5SBJGPVJWVDFSCwRB3Gk6nQvh/VqjwhJfe9WRQFpctT2qEsxUhrZucM7uw/PYUPSNkd23O8n8EqT418SDUlk5/JmwRwJzhtrYvLF/a+n172u2TnPhstH4eceAozJGXdMgPVjJFQFDOSZBqhJAHCXdQmdW09xIlsxqMPJfBAkxZ0faGfp+CI/p2IEZcy4o7u5Eh15GxtCP+r1UPlnjVj6gWhIh4eL3JDBpUPh+nAFhUEKxZpg7Cg+laIO0grHSGU1scnuhMrNkE5k3lpGCZBetGh3MBxkqDXbAHjoAFTkERXIMSKAM7sEjeAYvxoPxZrwbH+PWlDGZ2QFTMgY/8SunFw=</latexit> <latexit sha1_base64="JIBG5LXc4zqDI8VUWsiRYbLg62U=">ACH3icbZDLSgMxFIYz9VZbL1WXboJVFIQy40Y3QlElxXsBdoyZNJMG5vMDEmdBjmEXwIn0FwpSsX7sRtlz6Erk0voG39IfDznXM4J78TMCqVaQ6M1MLi0vJKejWTXVvf2MxtbVekHwpMythnvqg5SBJGPVJWVDFSCwRB3Gk6nQvh/VqjwhJfe9WRQFpctT2qEsxUhrZucM7uw/PYUPSNkd23O8n8EqT418SDUlk5/JmwRwJzhtrYvLF/a+n172u2TnPhstH4eceAozJGXdMgPVjJFQFDOSZBqhJAHCXdQmdW09xIlsxqMPJfBAkxZ0faGfp+CI/p2IEZcy4o7u5Eh15GxtCP+r1UPlnjVj6gWhIh4eL3JDBpUPh+nAFhUEKxZpg7Cg+laIO0grHSGU1scnuhMrNkE5k3lpGCZBetGh3MBxkqDXbAHjoAFTkERXIMSKAM7sEjeAYvxoPxZrwbH+PWlDGZ2QFTMgY/8SunFw=</latexit> <latexit sha1_base64="46nCIeBu8mBtVM37B59pxwthgJE=">ACH3icbZDLSsNAFIYn9VbrLerSzWARBaEkbnQjFEVwWcFeoA1hMp20ozNJmJlIQ8gj+BA+g1tduxO3XfomTtqAtvWHgZ/vnM583sRo1JZ1tgoLS2vrK6V1ysbm1vbO+buXkuGscCkiUMWio6HJGE0IE1FSOdSBDEPUba3uN1Xm8/ESFpGNyrJCIOR4OA+hQjpZFrHj+4I3gJe5IOHLT0SiDN5qc/pIkJ4lrVq2aNRFcNHZhqBQwzW/e/0Qx5wECjMkZde2IuWkSCiKGckqvViSCOFHNCBdbQPEiXTSyYcyeKRJH/qh0C9QcEL/TqSIS5lwT3dypIZyvpbD/2rdWPkXTkqDKFYkwNFfsygCmGeDuxTQbBiTYIC6pvhXiIBMJKZzizxeOZzsSeT2DRtM5qtlWz76xq/apIpwOwCE4ATY4B3VwCxqgCTB4Bq/gDbwbL8aH8Wl8TVtLRjGzD2ZkjH8A7dSizg=</latexit> Quantum Hall Effect 2D electrons in a magnetic field j x = σ xx E x + σ xy E y ( p a + e A a ) 2 e 2 X X H = + 2 m | x a − x b | a h a,b i
Integer quantum Hall effect • Electrons completely fill n Landau levels • gapped ground state • interaction can be treated perturbatively n=2 ∆ = B m n=1 degeneracy BA n=0 2 π
Fractional QHE Electrons fill a fraction ν of a Landau level Interactions cannot be treated perturbatively FQHE exists in the limit of a single Landau level for example, the lowest Landau level
Lowest Landau level limit ( p a + e A a ) 2 e 2 X X H = + 2 m | x a − x b | a h a,b i n= 1 B m n=0
Lowest Landau level limit ( p a + e A a ) 2 e 2 X X H = + 2 m | x a − x b | a h a,b i m → 0 n= 1 B → ∞ m n=0
Lowest Landau level limit ( p a + e A a ) 2 e 2 X X H = + 2 m | x a − x b | a h a,b i e 2 m → 0 X H = P LLL | x a − x b | a,b n= 1 B → ∞ m Projection to n=0 lowest Landau level
Jain’s sequences ν = n + 1 n energy gap at ν = 2 n + 1 2 n + 1
Composite fermion • Halperin-Lee-Read 1993: low-energy quasiparticle of half-filled Landau level: a “composite fermion” • “attaching 2 flux quanta to an electrons” • End result: an effective field theory of the composite fermion
HLR field theory L = i † ( @ 0 − iA 0 + ia 0 ) − 1 2 m | ( @ i − iA i + ia i ) | 2 + 1 1 4 ⇡ ✏ µ νλ a µ @ ν a λ 2 b = r ⇥ a = 2 ⇥ 2 πψ † ψ “flux attachment” mean field: B e ff = B − b = B − 4 π n ν = 1 B e ff = 0 2 In half filled Landau level CFs form a Fermi surface away from half filling: CFs in a magnetic field
Particle-hole symmetr • One problem with HLR theory: lack of particle- hole symmetry • Solving this problem: the Dirac composite fermion theory • Modification to the HLR theory: CF has a Berry phase of π around the Fermi disk • Composite fermion: particle-vortex dual of the electron
Particle-vortex duality <latexit sha1_base64="J7/cSH25RYmW3QnHzDQ/79k8BY=">ACWnicbZDLixQxEMZr2te+1PFx8xJcBE9DWhYUYWHQi8cVnd2FyThUZ9I9YfNokmplaPp/8+i/IHjx6MWrHs30qLi7FgS+fF9VKvyK2uhInH8eZFeuXrt+Y2t7Z3fv5q3bwzt3j6NvglQT6Y0PpwVGZbRTE9Jk1GkdFNrCqJPi7OU6P3mvQtTevaVrWYWK6dLZGSNR8WogwoW7FQhpC96f4onPukAvjXRV0tSQMwX9gwqCrjBIFBlFHzUSF1uI7zja30KfskPWPshftgah1Nx/u8xHvi/0V+UWxP37+be8jABzNh1/FwsvGKkfSYIzTnNc0azGQlkZ1O6KJqkZ5hpWaJunQqjhrexYde5ScBSt9SMcR691/J1q0Ma5skTot0jJezNbm/7JpQ+WzWatd3ZBycrOobAwjz9Zg2UIHJcmskAZdPork0tMHCjhP7elsGsmlwhcFsdPRjkf5a8TnDFsagsewEN4Dk8hTG8giOYgIRP8B1+wM/BlyzLtrPdTWs2+D1zD85Vdv8XB0S5Zg=</latexit> <latexit sha1_base64="2RYOUgzS3/V3n5i6GtpRZdJn3Eo=">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</latexit> <latexit sha1_base64="2RYOUgzS3/V3n5i6GtpRZdJn3Eo=">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</latexit> <latexit sha1_base64="Fo2EQC2ZXIM150xtQC+nYbynE2Q=">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</latexit> original fermion composite fermion magnetic field density density magnetic field � µ ( @ µ − ia µ ) − 1 4 ⇡ ✏ µ νλ A µ @ ν a λ + 1 L = i ¯ 8 ⇡ ✏ µ νλ A µ @ ν A λ ρ = δ S = B − b δ A 0 4 π δ S ψγ 0 ψ i = B ! h ¯ = 0 � δ a 0 4 π
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