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Flavio S. Nogueira Institute for Theoretical Solid State Physics, IFW Dresden Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018

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Page 1: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Flavio S. Nogueira Institute for Theoretical Solid State Physics, IFW Dresden

Conformal phase transition in topological insulators and superconductors

UKRATOP Workshop, Dec 2018

Page 2: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

CollaboratorsJeroen van den Brink

Inst. for Theor. Solid State Physics IFW Dresden

Asle Sudbø Dept. of Physics

Norwegian University of Science and Technology (NTNU)

Page 3: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Conformal phase transitionVladimir Miransky

The University of Western Ontario

• V. A. Miransky and Koichi Yamawaki, “Conformal phase transition in gauge theories,” Phys. Rev. D 55, 5051–5066 (1997).

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�c<latexit sha1_base64="eHXK8cm0xPxdYmqwWud/a0lgSP0=">AAAB9HicdVDLSgMxFL1TX7W+qi7dBFvBVZkpWNtd0Y3LCvYB7VAymUwbmsmMSaZQhn6HGxeKuPVj3Pk3pu0IKnogcDjnXO7N8WLOlLbtDyu3tr6xuZXfLuzs7u0fFA+POipKJKFtEvFI9jysKGeCtjXTnPZiSXHocdr1JtcLvzulUrFI3OlZTN0QjwQLGMHaSG55wE3Wx8OUzMvDYsmuNGoNu26jFblwMlKrI6diL1GCDK1h8X3gRyQJqdCEY6X6jh1rN8VSM8LpvDBIFI0xmeAR7RsqcEiVmy6PnqMzo/goiKR5QqOl+n0ixaFSs9AzyRDrsfrtLcS/vH6ig7qbMhEnmgqyWhQkHOkILRpAPpOUaD4zBBPJzK2IjLHERJueCqaEr5+i/0mnWnEMv62WmldZHXk4gVM4BwcuoQk30II2ELiHB3iCZ2tqPVov1usqmrOymWP4AevtE8l3kh0=</latexit><latexit sha1_base64="eHXK8cm0xPxdYmqwWud/a0lgSP0=">AAAB9HicdVDLSgMxFL1TX7W+qi7dBFvBVZkpWNtd0Y3LCvYB7VAymUwbmsmMSaZQhn6HGxeKuPVj3Pk3pu0IKnogcDjnXO7N8WLOlLbtDyu3tr6xuZXfLuzs7u0fFA+POipKJKFtEvFI9jysKGeCtjXTnPZiSXHocdr1JtcLvzulUrFI3OlZTN0QjwQLGMHaSG55wE3Wx8OUzMvDYsmuNGoNu26jFblwMlKrI6diL1GCDK1h8X3gRyQJqdCEY6X6jh1rN8VSM8LpvDBIFI0xmeAR7RsqcEiVmy6PnqMzo/goiKR5QqOl+n0ixaFSs9AzyRDrsfrtLcS/vH6ig7qbMhEnmgqyWhQkHOkILRpAPpOUaD4zBBPJzK2IjLHERJueCqaEr5+i/0mnWnEMv62WmldZHXk4gVM4BwcuoQk30II2ELiHB3iCZ2tqPVov1usqmrOymWP4AevtE8l3kh0=</latexit><latexit sha1_base64="eHXK8cm0xPxdYmqwWud/a0lgSP0=">AAAB9HicdVDLSgMxFL1TX7W+qi7dBFvBVZkpWNtd0Y3LCvYB7VAymUwbmsmMSaZQhn6HGxeKuPVj3Pk3pu0IKnogcDjnXO7N8WLOlLbtDyu3tr6xuZXfLuzs7u0fFA+POipKJKFtEvFI9jysKGeCtjXTnPZiSXHocdr1JtcLvzulUrFI3OlZTN0QjwQLGMHaSG55wE3Wx8OUzMvDYsmuNGoNu26jFblwMlKrI6diL1GCDK1h8X3gRyQJqdCEY6X6jh1rN8VSM8LpvDBIFI0xmeAR7RsqcEiVmy6PnqMzo/goiKR5QqOl+n0ixaFSs9AzyRDrsfrtLcS/vH6ig7qbMhEnmgqyWhQkHOkILRpAPpOUaD4zBBPJzK2IjLHERJueCqaEr5+i/0mnWnEMv62WmldZHXk4gVM4BwcuoQk30II2ELiHB3iCZ2tqPVov1usqmrOymWP4AevtE8l3kh0=</latexit><latexit sha1_base64="eHXK8cm0xPxdYmqwWud/a0lgSP0=">AAAB9HicdVDLSgMxFL1TX7W+qi7dBFvBVZkpWNtd0Y3LCvYB7VAymUwbmsmMSaZQhn6HGxeKuPVj3Pk3pu0IKnogcDjnXO7N8WLOlLbtDyu3tr6xuZXfLuzs7u0fFA+POipKJKFtEvFI9jysKGeCtjXTnPZiSXHocdr1JtcLvzulUrFI3OlZTN0QjwQLGMHaSG55wE3Wx8OUzMvDYsmuNGoNu26jFblwMlKrI6diL1GCDK1h8X3gRyQJqdCEY6X6jh1rN8VSM8LpvDBIFI0xmeAR7RsqcEiVmy6PnqMzo/goiKR5QqOl+n0ixaFSs9AzyRDrsfrtLcS/vH6ig7qbMhEnmgqyWhQkHOkILRpAPpOUaD4zBBPJzK2IjLHERJueCqaEr5+i/0mnWnEMv62WmldZHXk4gVM4BwcuoQk30II2ELiHB3iCZ2tqPVov1usqmrOymWP4AevtE8l3kh0=</latexit>

Conformal symmetry

Broken conformal symmetry

Page 4: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Conformal phase transitionWell known example: BKT transition

⇠�1 ⇠ exp⇣� constp

Tc�T

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• Essential singularity at T = Tc

• No power law behavior in the correlation length

Compare typical 2nd order phase transition:

⇠�1 ⇠ |Tc � T |⌫<latexit sha1_base64="JTL02sBdF/j1Kd8NQmFKG6GKHdU=">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</latexit><latexit sha1_base64="JTL02sBdF/j1Kd8NQmFKG6GKHdU=">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</latexit><latexit sha1_base64="JTL02sBdF/j1Kd8NQmFKG6GKHdU=">AAACsHicdVFNb9NAEF2br2K+Ahy5jEiRuCRyIrWkSEgVFRInVETSRorTaLyexEvXu9buuiW4+X3cufFvWCcpAgRzenozs/Pe27SUwro4/hGEN27eun1n52507/6Dh49aj5+cWF0ZTiOupTbjFC1JoWjkhJM0Lg1hkUo6Tc+Pmv7pBRkrtBq6ZUnTAhdKzAVH56lZ61uS0kKoWjgqxFdaRVHSQHhnLSknUIIValFJNMItAR3sDt8MZ3z3eu6DhlJfkgGJl5BSjhdCGxAKXE7AtTEk15dAklq4PIqOdFGiIfBivAgJfZWBNpl/ocy9D3AGlRXNyuso2k2+iLO601slVhRX/m5neHVWJ6paNQJIZb90z1rtuHuwfxAPYtiAvd4W7A+g143X1WbbOp61vieZ5lXhXXKJ1k56cemmNRonuPQ5JJWlEvk5LmjiocKC7LReB76CF57JYO6dzrVysGZ/36ixsHZZpH6yQJfbv3sN+a/epHLzwbQWqqwcKb45NK8kON0ERpAJQ9zJpQfI/ZcIDjxHg9z5P458CNdO4f/gpN/tefyx3z58u41jhz1jz9lL1mOv2CF7z47ZiPGgE3wKkmAa9sNxOAtxMxoG252n7I8KP/8EsPXTng==</latexit><latexit sha1_base64="JTL02sBdF/j1Kd8NQmFKG6GKHdU=">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</latexit>

2−π κ

y

PlasmaDielectric

0

Page 5: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Conformal phase transitionExample in 2+1 dimensions:

graphene with Coulomb interaction• E. V. Gorbar, V. P. Gusynin, V. A. Miransky, and I. A. Shovkovy, “Magnetic field driven metal-insulator phase transition in planar systems,” Phys.

Rev. B 66, 045108 (2002)

• Excitonic gap generation by spontaneous chiral symmetry breaking:

m ⇠ exp⇣� constp

↵c�↵

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Graphene “fine-structure constant”:↵ = e2

✏~vF<latexit sha1_base64="rDOyrq8muLuzDpnC60ZtiLoNKbE=">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</latexit><latexit sha1_base64="rDOyrq8muLuzDpnC60ZtiLoNKbE=">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</latexit><latexit sha1_base64="rDOyrq8muLuzDpnC60ZtiLoNKbE=">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</latexit><latexit sha1_base64="rDOyrq8muLuzDpnC60ZtiLoNKbE=">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</latexit>

Page 6: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Conformal phase transitionAlso in d=2+1: Deconfined quantum critical points in

SU(N) antiferromagnets

<n> 0 <ψ >VBS 0

Higgs Confinement

gcg

• Conventional 2nd order phase transition at large N

• Conformal phase transition at low N (e.g., N = 2)

[Nogueira and Sudbø, EPL 104, 56004 (2013)]<latexit sha1_base64="l1RAox0suQUqJ4r65DDOdVrs5do=">AAACw3icdVHfaxNBEN47f9X4o1EffRlMhBRK2DvbmD4IwVLwQUpE0xZyIeztzV2W3O0eu3uVeOSf9E3/GveSKFXqwMK3M9/HzHwTl7kwltIfnn/n7r37D/Yeth49fvJ0v/3s+YVRleY44SpX+ipmBnMhcWKFzfGq1MiKOMfLeHna1C+vURuh5Be7KnFWsEyKVHBmXWre/hnFmAlZC4uF+IbrVitqIJwqeY2y4bAcQpmA0glqKBeuF1jNpBFNEZiFnOkMoXvevaFNlS6c8Fa6+tqQoYf9rH/o4Luwe+CkqVJWKovGjVFPz1VWodAMmOv9uUriOlLrQzgbf4Q6ilMI6JH7Hg8oPYJeSIM3B7NmdpTJn1Xm7Q7tnwxO6JDCFhwHOzAYQtCnm+iQXYzn7e9RonhVuL15zoyZBrS0s5ppK3jurIkqgyXjS5bh1EHJCjSzenODNbx2mQTc3u5JC5vsTUXNCmNWReyYBbML82+tSd5Wm1Y2Hc5qIcvKouTbRmmVg1XQHBQSoZHbfOUA49rZzIEvmGbcurO3nAm/N4X/g4uwHzj8KeyM3u/s2CMvySvSIwF5S0bkAxmTCeHeyEs95ZX+mb/0tW+3VN/baV6Qv8Jf/wKDoNXr</latexit><latexit sha1_base64="l1RAox0suQUqJ4r65DDOdVrs5do=">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</latexit><latexit sha1_base64="l1RAox0suQUqJ4r65DDOdVrs5do=">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</latexit><latexit sha1_base64="l1RAox0suQUqJ4r65DDOdVrs5do=">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</latexit>

Page 7: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

L = 2 ✏µ⌫�A

µ@⌫A� � jµAµ + . . .<latexit sha1_base64="sQGJfd/uQRDMK8phhXsPvGlnhbk=">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</latexit><latexit sha1_base64="sQGJfd/uQRDMK8phhXsPvGlnhbk=">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</latexit><latexit sha1_base64="sQGJfd/uQRDMK8phhXsPvGlnhbk=">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</latexit><latexit sha1_base64="sQGJfd/uQRDMK8phhXsPvGlnhbk=">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</latexit>

1. jµ = ̄�µ (Mott TI)

2. jµ = i('⇤@µ'� '@µ'⇤) + 2|'|2Aµ (Top. SC)<latexit sha1_base64="MK2tVjsZocNCSy9zqe6LkJMZirw=">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</latexit><latexit sha1_base64="MK2tVjsZocNCSy9zqe6LkJMZirw=">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</latexit><latexit sha1_base64="MK2tVjsZocNCSy9zqe6LkJMZirw=">AAACpHicdVHLbhMxFPUMrxJeAZZsDCkobUU0iURJF0iFLgAJpFRN2kpxEt1xnMTU9lj2nUrRtF/GX7Djb/Akg1SqciXLx+f46L5Sq6THJPkdxbdu37l7b+N+7cHDR4+f1J8+O/ZZ7rgY8Exl7jQFL5Q0YoASlTi1ToBOlThJzw5K/eRcOC8z08elFSMNcyNnkgMGalL/yVIxl6YQJtfCAYrLWo1JFJpu/pgwnX9gKTjKrJdsDlpDyZWvTdr8niHS/tetawbZZOfg7EKOt5kFhxLUyrQm31b3DdJ4e2unc1E9Lsadj6UW8vQz26JHB2UeYaZXKp3UG0lrb3cv6SZ0Dd61K7Dbpe1WsooGqaI3qf9i04wHu0GuwPthO7E4KspKuAqds9wLC/wM5mIYoAEt/KhYDfmSvg7MlM4yF45BumKvOgrQ3i91Gn5qwIW/rpXkTdowx1l3VEhjcxSGrxPNckUxo+XG6FQ6wVEtAwDuZKiV8gU44Bj2WgtD+Nsp/T847rTaAR92GvufqnFskBfkFWmSNnlP9skX0iMDwqOX0eeoFx3Gb+Jv8VE8WH+No8rznPwT8fgPRjDO2Q==</latexit><latexit sha1_base64="MK2tVjsZocNCSy9zqe6LkJMZirw=">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</latexit>

We mainly consider two cases:

Page 8: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

Current correlation function in conformal invariant theory in 2+1 dimensions (Euclidean):

Cµ⌫(p) = hjµ(p)j⌫(�p)i

=1

2⇡

v|p|

✓�µ⌫ � pµp⌫

p2

◆+ w✏µ⌫�p�

<latexit sha1_base64="raRyODRKVbbizBWvuJR0HIRNMPM=">AAAC2HicdVJdixMxFM2MX+v4VfXRl2BxaZFdZgquXUFYLIKPK9jdxaaWTOZOGzeTySaZlZId8EERX/1pvvkr/AtmOtVV0Qshh3POvbm5SaoENzaOvwXhhYuXLl/ZuBpdu37j5q3O7TsHpqw0gzErRamPUmpAcAljy62AI6WBFqmAw/R41OiHp6ANL+Uru1QwLehc8pwzaj0163wnKcy5dHAiqdZ0WUdE0BSEe37yZDSqo9HMkaIisqp7qr/5dNOrci4Av5152lN+l1VvS/WJXgkRkaWsihQ0IVHjzzVlLqndgCheEwG5nZyeqbMV6pEMhKW/jthq3aqpjVVTuXbqzaAmms8Xth89fEdAGS584+sU306RZrT25ha11ul5GxEBmZ1fb9bpxtu7O7vxMMYteJSswc4QJ9vxKrpoHfuzzleSlawqQFomqDGTJFZ26qi2nAnw46oMKMqO6RwmHkpagJm61cPU+IFnMpyX2i9p8Yr9PcPRwphlkXpnQe3C/K015L+0SWXz4dRxqSoLkrUH5ZXAtsTNK+OMa2BWLD2gTHPfK2YL6odr/V+I/BB+3hT/HxwMthOPXw66e8/W49hA99B91EMJeoz20Au0j8aIBePABR+Cj+Hr8H34KfzcWsNgnXMX/RHhlx94nOfW</latexit><latexit sha1_base64="raRyODRKVbbizBWvuJR0HIRNMPM=">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</latexit><latexit sha1_base64="raRyODRKVbbizBWvuJR0HIRNMPM=">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</latexit><latexit sha1_base64="raRyODRKVbbizBWvuJR0HIRNMPM=">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</latexit>

v and w are universal numbers<latexit sha1_base64="jz0lR3pnnNLZ/Deep4JtGG2osdc=">AAACCnicdZDLSgMxFIYzXmu9jbp0E20FV2WmYG13RTcuK9gLtEPJZNI2NMkMSaZShq7d+CpuXCji1idw59uYaUdQ0QMhH/9/Dsn5/YhRpR3nw1paXlldW89t5De3tnd27b39lgpjiUkThyyUHR8pwqggTU01I51IEsR9Rtr++DL12xMiFQ3FjZ5GxONoKOiAYqSN1LePipMiRCKAxVtzSwJjQdN+xKCIuW+obxecUq1Sc6oOXMCZm0GlCt2SM68CyKrRt997QYhjToTGDCnVdZ1IewmSmmJGZvlerEiE8BgNSdegQJwoL5mvMoMnRgngIJTmCA3n6veJBHGlptw3nRzpkfrtpeJfXjfWg6qXUBHFmgi8eGgQM6hDmOYCAyoJ1mxqAGFJzV8hHiGJsDYZ5E0IX5vC/6FVLrmGr8uF+kUWRw4cgmNwClxwDurgCjRAE2BwBx7AE3i27q1H68V6XbQuWdnMAfhR1tsnuG+ZtA==</latexit><latexit sha1_base64="jz0lR3pnnNLZ/Deep4JtGG2osdc=">AAACCnicdZDLSgMxFIYzXmu9jbp0E20FV2WmYG13RTcuK9gLtEPJZNI2NMkMSaZShq7d+CpuXCji1idw59uYaUdQ0QMhH/9/Dsn5/YhRpR3nw1paXlldW89t5De3tnd27b39lgpjiUkThyyUHR8pwqggTU01I51IEsR9Rtr++DL12xMiFQ3FjZ5GxONoKOiAYqSN1LePipMiRCKAxVtzSwJjQdN+xKCIuW+obxecUq1Sc6oOXMCZm0GlCt2SM68CyKrRt997QYhjToTGDCnVdZ1IewmSmmJGZvlerEiE8BgNSdegQJwoL5mvMoMnRgngIJTmCA3n6veJBHGlptw3nRzpkfrtpeJfXjfWg6qXUBHFmgi8eGgQM6hDmOYCAyoJ1mxqAGFJzV8hHiGJsDYZ5E0IX5vC/6FVLrmGr8uF+kUWRw4cgmNwClxwDurgCjRAE2BwBx7AE3i27q1H68V6XbQuWdnMAfhR1tsnuG+ZtA==</latexit><latexit sha1_base64="jz0lR3pnnNLZ/Deep4JtGG2osdc=">AAACCnicdZDLSgMxFIYzXmu9jbp0E20FV2WmYG13RTcuK9gLtEPJZNI2NMkMSaZShq7d+CpuXCji1idw59uYaUdQ0QMhH/9/Dsn5/YhRpR3nw1paXlldW89t5De3tnd27b39lgpjiUkThyyUHR8pwqggTU01I51IEsR9Rtr++DL12xMiFQ3FjZ5GxONoKOiAYqSN1LePipMiRCKAxVtzSwJjQdN+xKCIuW+obxecUq1Sc6oOXMCZm0GlCt2SM68CyKrRt997QYhjToTGDCnVdZ1IewmSmmJGZvlerEiE8BgNSdegQJwoL5mvMoMnRgngIJTmCA3n6veJBHGlptw3nRzpkfrtpeJfXjfWg6qXUBHFmgi8eGgQM6hDmOYCAyoJ1mxqAGFJzV8hHiGJsDYZ5E0IX5vC/6FVLrmGr8uF+kUWRw4cgmNwClxwDurgCjRAE2BwBx7AE3i27q1H68V6XbQuWdnMAfhR1tsnuG+ZtA==</latexit><latexit sha1_base64="jz0lR3pnnNLZ/Deep4JtGG2osdc=">AAACCnicdZDLSgMxFIYzXmu9jbp0E20FV2WmYG13RTcuK9gLtEPJZNI2NMkMSaZShq7d+CpuXCji1idw59uYaUdQ0QMhH/9/Dsn5/YhRpR3nw1paXlldW89t5De3tnd27b39lgpjiUkThyyUHR8pwqggTU01I51IEsR9Rtr++DL12xMiFQ3FjZ5GxONoKOiAYqSN1LePipMiRCKAxVtzSwJjQdN+xKCIuW+obxecUq1Sc6oOXMCZm0GlCt2SM68CyKrRt997QYhjToTGDCnVdZ1IewmSmmJGZvlerEiE8BgNSdegQJwoL5mvMoMnRgngIJTmCA3n6veJBHGlptw3nRzpkfrtpeJfXjfWg6qXUBHFmgi8eGgQM6hDmOYCAyoJ1mxqAGFJzV8hHiGJsDYZ5E0IX5vC/6FVLrmGr8uF+kUWRw4cgmNwClxwDurgCjRAE2BwBx7AE3i27q1H68V6XbQuWdnMAfhR1tsnuG+ZtA==</latexit>

Page 9: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

Cµ⌫(p) = ⇧(p)(p2�µ⌫ � pµp⌫) +2⇡ ✏µ⌫�p�

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Vacuum polarization

Large Nf limit: ⇧(p) = A2⇡|p| A = ⇡Nf

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Dµ⌫(p) =2⇡

A2 + 2

⇢A

|p|

�µ⌫

+(↵� 1)(A2 + 2) + 2

A2

pµp⌫p2

��

✏µ⌫�p�p2

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Gauge field propagator:(2Nf two-component fermions)

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Page 10: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

= +-1

Complete fermion propagator

Gauge field propagator

M± ⇠ ± exp

� 2⇡p

8A/[⇡(A2+2)]�1

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Solution of the gap equation:

Page 11: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

Physical interpretation:

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h(�†a"�a" � �†

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Page 12: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

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Page 13: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

Superconductor Chern-Simons-Higgs electrodynamics

L =1

4e2F 2µ⌫ + |(@µ � iAµ)'|2 +m2|'|2 + g

2|'|4

+ i

2✏µ⌫�Aµ@⌫A�

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Page 14: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

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Topological conformal phase transition

Renormalization group flow

Page 15: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

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Topological conformal phase transition

Renormalization group flow

Strong-coupling (IR) regime(|p| ⌧ e2⇤)

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Page 16: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

RG equation

�(g) = µ dgdµ =

�N+48

� ⇥�⌦+ (g � g⇤)2

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g⇤ = 4N+4

�8

3⇡2N � 1� 8N

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⌦ =�

83⇡2N � 1� 8

N

�2 � 48N

�1 + 4

N

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• Quantum criticality: ⌦ > 0

• Conformality lost: ⌦ < 0

=) g± = g⇤ ± ip

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Page 17: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

Conformality lost phase:

3⇡2

8

hN + 8� 4

p3(N + 4)

i< < 3⇡2

8

hN + 8 + 4

p3(N + 4)

i

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⇤IR⇤UV

⇡ exp

� 8⇡

(N+4)p

|⌦|

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Page 18: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Topological conformal phase transition

Physical implications

RG equation for superfluid stiffness:

µd⇢s

dµ =h1� �(g)

g

i⇢s

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Universal jump

Page 19: Conformal phase transition in topological insulators and ... · Conformal phase transition in topological insulators and superconductors UKRATOP Workshop, Dec 2018. Collaborators

Conclusions• Conformal phase transitions leads to BKT-like scaling in

higher dimensions

• Unconventional quantum critical behavior

• Mott TI: CSB induced by Chern-Simons term

• Top. SC: fluctuation induced Higgs mechanism with BKT scaling