New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal...

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New method for the conformal bootstrap with OPE truncations Wenliang LI Kyung Hee University, Seoul, South Korea Symposium: "Bootstrap Approach to Conformal Field Theories and Applications”, OIST, March19-23, 2018 based on 1711.09075

Transcript of New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal...

Page 1: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

New method for the conformal bootstrap with

OPE truncationsWenliang LI

Kyung Hee University, Seoul, South Korea

Symposium: "Bootstrap Approach to Conformal Field Theories and Applications”, OIST, March19-23, 2018

based on 1711.09075

Page 2: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Bootstrap equation viewed geometrically

1

Direction Bound Length ?

1 +X

i

Piv�F�i, li(u, v)� (u $ v)

v� � u�= 0

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non-unitary/boundary/defect/thermal bootstrap problems have no definite sign

Page 3: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Outline

• OPE truncation

• OPE hierarchy

• Effective Field Theory analogue

• Variational method: error minimization

Page 4: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Outline

• OPE truncation

• OPE hierarchy

• Effective Field Theory analogue

• Variational method: error minimization

Page 5: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Conformal bootstrap• Conformal bootstrap: classify & solve CFTs by OPE associativity

OPE: for Wilson’s Operator Product Expansion

• Proposed around 1973 by Ferrara-Grillo-Gatto & Polyakov

• 2d success in 1984 by Belavin-Polyakov-Zamolodchikov

• Revived in 2008 by Rattazzi-Rychkov-Tonni-Vichi

• Carving out the parameter space of unitary CFTs

• 3d Ising CFT at the kink, later in a tiny island, and a lot of progress in other CFTs

• Extract the spectrum from extremal functionals

Page 6: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

OPE truncation• Let us go back to the original question:

solve the bootstrap equation (approximately, in certain regime)

• Why truncate OPEs?Difficult problem (infinitely many unknowns) becomes simpler (finitely) Experiments/Monte Carlo simulations measure low-lying CFT data

• Does it work? Gliozzi’s determinant method Universality for different CFTsOPE hierarchy, error minimization (in this talk)

• Focus on 4-point scalar correlators & Euclidean regimez̄ = z⇤

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Page 7: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Determinant method• Gliozzi 2013

1) Truncate the OPE to a few terms2) Set the truncated bootstrap equation to zero3) Linear dependence of truncated equations (from derivatives) indicates vanishing determinants of minors

• Less numerical demand & not using unitarity/sign-definiteness(Gliozzi-Rago ’14, Gliozzi-Liendo-Meineri-Rago ’15, Nakayama ’16, Gliozzi ’16, Esterlis-Fitzpatrick-Ramirez ’16, Hikami ’17 & ’18, LeClair-Squires ’18)

• Error estimation is subtle

Page 8: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

3d free theory• Truncated OPE

�⇥ � = 1 + [��2 , 0] + [D, 2] + [��2 + 4, 4]<latexit sha1_base64="fXyaju0FTMOj4jmVi17E4/DmS8w=">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</latexit><latexit sha1_base64="fXyaju0FTMOj4jmVi17E4/DmS8w=">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</latexit><latexit sha1_base64="fXyaju0FTMOj4jmVi17E4/DmS8w=">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</latexit><latexit sha1_base64="fXyaju0FTMOj4jmVi17E4/DmS8w=">AAACLnicbZDLSsNAFIYn3q23qks3wSIILSUpBXUhiHbhsoJVIYllMj1pBycXZk6EEvoqvoPv4Fa34kJ06WM4abuwrT8MfPznHM6Z308EV2hZH8bc/MLi0vLKamFtfWNzq7i9c6PiVDJosVjE8s6nCgSPoIUcBdwlEmjoC7j1Hy7y+u0jSMXj6Br7CXgh7UY84IyittrFYzfpcRd5CCqnU7vsuI12lvN9bVCxvLLTqNS8Cbdcr9S9drFkVa2hzFmwx1AiYzXbxW+3E7M0hAiZoEo5tpWgl1GJnAkYFNxUQULZA+2CozGi+iQvG/5wYB5op2MGsdQvQnPo/p3IaKhUP/R1Z0ixp6ZruflfzUkxOPYyHiUpQsRGi4JUmBibeVxmh0tgKPoaKJNc32qyHpWUoQ614HYgcBuZ2wCBdKAjsacDmIVWrXpSta/qpbPzcTYrZI/sk0NikyNyRi5Jk7QII0/khbySN+PZeDc+ja9R65wxntklEzJ+fgFlp6dA</latexit>

Page 9: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Universality for CFTs• Truncated OPE a approximate relations for low-lying CFT data

Universal: consistent with different CFTs in various dimensions(W. Li, JHEP01(2018)077)

• “Inverse” bootstrap: require general bootstrap solutions reproduce truncated OPEs Ansatz (Alday-Zhiboedov ’15):

• Toy example

• In 2d, the degenerate case was found by the determinant method (Esterlis-Fitzpatrick-Ramirez, ’16)

�1 ⇥ �1 = 1 + �2<latexit sha1_base64="wp5DB8c0w4g1UX7KuZSxYmOnVKU=">AAACEXicbZDLSsNAFIYnXmu9RV12EyyCIJSkCOpCKNqFywrGFppSJtOTdujkwsyJUEIXvoPv4Fa3rsStT+DOR3HaZqGtPwx8/OcczpnfTwRXaNtfxtLyyuraemGjuLm1vbNr7u3fqziVDFwWi1i2fKpA8Ahc5CiglUigoS+g6Q+vJ/XmA0jF4+gORwl0QtqPeMAZRW11zZKXDHjX8ZCHoGZ86ZxModo1y3bFnspaBCeHMsnV6JrfXi9maQgRMkGVajt2gp2MSuRMwLjopQoSyoa0D22NEdU7O9n0E2PrSDs9K4ilfhFaU/f3REZDpUahrztDigM1X5uY/9XaKQbnnYxHSYoQsdmiIBUWxtYkEavHJTAUIw2USa5vtdiASspQ51b0ehB49cyrg0A61pE48wEsglutXFSc29Ny7SrPpkBK5JAcE4eckRq5IQ3iEkYeyTN5Ia/Gk/FmvBsfs9YlI585IH9kfP4AHEqc6Q==</latexit><latexit sha1_base64="wp5DB8c0w4g1UX7KuZSxYmOnVKU=">AAACEXicbZDLSsNAFIYnXmu9RV12EyyCIJSkCOpCKNqFywrGFppSJtOTdujkwsyJUEIXvoPv4Fa3rsStT+DOR3HaZqGtPwx8/OcczpnfTwRXaNtfxtLyyuraemGjuLm1vbNr7u3fqziVDFwWi1i2fKpA8Ahc5CiglUigoS+g6Q+vJ/XmA0jF4+gORwl0QtqPeMAZRW11zZKXDHjX8ZCHoGZ86ZxModo1y3bFnspaBCeHMsnV6JrfXi9maQgRMkGVajt2gp2MSuRMwLjopQoSyoa0D22NEdU7O9n0E2PrSDs9K4ilfhFaU/f3REZDpUahrztDigM1X5uY/9XaKQbnnYxHSYoQsdmiIBUWxtYkEavHJTAUIw2USa5vtdiASspQ51b0ehB49cyrg0A61pE48wEsglutXFSc29Ny7SrPpkBK5JAcE4eckRq5IQ3iEkYeyTN5Ia/Gk/FmvBsfs9YlI585IH9kfP4AHEqc6Q==</latexit><latexit sha1_base64="wp5DB8c0w4g1UX7KuZSxYmOnVKU=">AAACEXicbZDLSsNAFIYnXmu9RV12EyyCIJSkCOpCKNqFywrGFppSJtOTdujkwsyJUEIXvoPv4Fa3rsStT+DOR3HaZqGtPwx8/OcczpnfTwRXaNtfxtLyyuraemGjuLm1vbNr7u3fqziVDFwWi1i2fKpA8Ahc5CiglUigoS+g6Q+vJ/XmA0jF4+gORwl0QtqPeMAZRW11zZKXDHjX8ZCHoGZ86ZxModo1y3bFnspaBCeHMsnV6JrfXi9maQgRMkGVajt2gp2MSuRMwLjopQoSyoa0D22NEdU7O9n0E2PrSDs9K4ilfhFaU/f3REZDpUahrztDigM1X5uY/9XaKQbnnYxHSYoQsdmiIBUWxtYkEavHJTAUIw2USa5vtdiASspQ51b0ehB49cyrg0A61pE48wEsglutXFSc29Ny7SrPpkBK5JAcE4eckRq5IQ3iEkYeyTN5Ia/Gk/FmvBsfs9YlI585IH9kfP4AHEqc6Q==</latexit><latexit sha1_base64="wp5DB8c0w4g1UX7KuZSxYmOnVKU=">AAACEXicbZDLSsNAFIYnXmu9RV12EyyCIJSkCOpCKNqFywrGFppSJtOTdujkwsyJUEIXvoPv4Fa3rsStT+DOR3HaZqGtPwx8/OcczpnfTwRXaNtfxtLyyuraemGjuLm1vbNr7u3fqziVDFwWi1i2fKpA8Ahc5CiglUigoS+g6Q+vJ/XmA0jF4+gORwl0QtqPeMAZRW11zZKXDHjX8ZCHoGZ86ZxModo1y3bFnspaBCeHMsnV6JrfXi9maQgRMkGVajt2gp2MSuRMwLjopQoSyoa0D22NEdU7O9n0E2PrSDs9K4ilfhFaU/f3REZDpUahrztDigM1X5uY/9XaKQbnnYxHSYoQsdmiIBUWxtYkEavHJTAUIw2USa5vtdiASspQ51b0ehB49cyrg0A61pE48wEsglutXFSc29Ny7SrPpkBK5JAcE4eckRq5IQ3iEkYeyTN5Ia/Gk/FmvBsfs9YlI585IH9kfP4AHEqc6Q==</latexit>

P2 =4�1

3�2 � 4�1<latexit sha1_base64="8pxD3drArazo58thqkbav60tU6c=">AAACE3icbZDLSsNAFIYnXmu9RV2KECyCG0tSC+pCKNqFywrGFpoQJtOTdujkwsxEKKEr38F3cKtbV+LWB3Dnozhps9DWH4b55z/nMDOfnzAqpGl+aQuLS8srq6W18vrG5ta2vrN7L+KUE7BJzGLe8bEARiOwJZUMOgkHHPoM2v7wOq+3H4ALGkd3cpSAG+J+RANKsFSRpx+0vNqlE3BMsrrT9Kxxdqq22sn04OkVs2pOZMwbqzAVVKjl6d9OLyZpCJEkDAvRtcxEuhnmkhIG47KTCkgwGeI+dJWNcAjCzSbfGBtHKukZQczViqQxSX9PZDgUYhT6qjPEciBma3n4X62byuDczWiUpBIiMr0oSJkhYyNnYvQoByLZSBlMOFVvNcgAKyZSkSs7PQicZuY0gUmcI7FmAcwbu1a9qFq39UrjqmBTQvvoEB0jC52hBrpBLWQjgh7RM3pBr9qT9qa9ax/T1gWtmNlDf6R9/gAGDJy9</latexit><latexit sha1_base64="8pxD3drArazo58thqkbav60tU6c=">AAACE3icbZDLSsNAFIYnXmu9RV2KECyCG0tSC+pCKNqFywrGFpoQJtOTdujkwsxEKKEr38F3cKtbV+LWB3Dnozhps9DWH4b55z/nMDOfnzAqpGl+aQuLS8srq6W18vrG5ta2vrN7L+KUE7BJzGLe8bEARiOwJZUMOgkHHPoM2v7wOq+3H4ALGkd3cpSAG+J+RANKsFSRpx+0vNqlE3BMsrrT9Kxxdqq22sn04OkVs2pOZMwbqzAVVKjl6d9OLyZpCJEkDAvRtcxEuhnmkhIG47KTCkgwGeI+dJWNcAjCzSbfGBtHKukZQczViqQxSX9PZDgUYhT6qjPEciBma3n4X62byuDczWiUpBIiMr0oSJkhYyNnYvQoByLZSBlMOFVvNcgAKyZSkSs7PQicZuY0gUmcI7FmAcwbu1a9qFq39UrjqmBTQvvoEB0jC52hBrpBLWQjgh7RM3pBr9qT9qa9ax/T1gWtmNlDf6R9/gAGDJy9</latexit><latexit sha1_base64="8pxD3drArazo58thqkbav60tU6c=">AAACE3icbZDLSsNAFIYnXmu9RV2KECyCG0tSC+pCKNqFywrGFpoQJtOTdujkwsxEKKEr38F3cKtbV+LWB3Dnozhps9DWH4b55z/nMDOfnzAqpGl+aQuLS8srq6W18vrG5ta2vrN7L+KUE7BJzGLe8bEARiOwJZUMOgkHHPoM2v7wOq+3H4ALGkd3cpSAG+J+RANKsFSRpx+0vNqlE3BMsrrT9Kxxdqq22sn04OkVs2pOZMwbqzAVVKjl6d9OLyZpCJEkDAvRtcxEuhnmkhIG47KTCkgwGeI+dJWNcAjCzSbfGBtHKukZQczViqQxSX9PZDgUYhT6qjPEciBma3n4X62byuDczWiUpBIiMr0oSJkhYyNnYvQoByLZSBlMOFVvNcgAKyZSkSs7PQicZuY0gUmcI7FmAcwbu1a9qFq39UrjqmBTQvvoEB0jC52hBrpBLWQjgh7RM3pBr9qT9qa9ax/T1gWtmNlDf6R9/gAGDJy9</latexit><latexit sha1_base64="8pxD3drArazo58thqkbav60tU6c=">AAACE3icbZDLSsNAFIYnXmu9RV2KECyCG0tSC+pCKNqFywrGFpoQJtOTdujkwsxEKKEr38F3cKtbV+LWB3Dnozhps9DWH4b55z/nMDOfnzAqpGl+aQuLS8srq6W18vrG5ta2vrN7L+KUE7BJzGLe8bEARiOwJZUMOgkHHPoM2v7wOq+3H4ALGkd3cpSAG+J+RANKsFSRpx+0vNqlE3BMsrrT9Kxxdqq22sn04OkVs2pOZMwbqzAVVKjl6d9OLyZpCJEkDAvRtcxEuhnmkhIG47KTCkgwGeI+dJWNcAjCzSbfGBtHKukZQczViqQxSX9PZDgUYhT6qjPEciBma3n4X62byuDczWiUpBIiMr0oSJkhYyNnYvQoByLZSBlMOFVvNcgAKyZSkSs7PQicZuY0gUmcI7FmAcwbu1a9qFq39UrjqmBTQvvoEB0jC52hBrpBLWQjgh7RM3pBr9qT9qa9ax/T1gWtmNlDf6R9/gAGDJy9</latexit>

3�2 = 4�1<latexit sha1_base64="Gtul+DRdUGk6V53QezeTUZK/z6g=">AAAB/3icbVBNS8NAEN3Ur1q/qh69BIvgqSS1oB6Eoj14rGBsIQlls5m0SzfZsLsRSujF/+BVr57Eqz/Fmz/FbZuDtj6Y4fHeDDv7gpRRqSzryyitrK6tb5Q3K1vbO7t71f2DB8kzQcAhnHHRC7AERhNwFFUMeqkAHAcMusHoZup3H0FIypN7NU7Bj/EgoRElWGnJPfPa/cZVU3e7X61ZdWsGc5nYBamhAp1+9dsLOcliSBRhWErXtlLl51goShhMKl4mIcVkhAfgaprgGKSfz06emCdaCc2IC12JMmfq740cx1KO40BPxlgN5aI3Ff/z3ExFF35OkzRTkJD5Q1HGTMXN6f/NkAogio01wURQfatJhlhgonRKFS+EyGvnXhuYwhMdib0YwDJxGvXLun3XrLWui2zK6Agdo1Nko3PUQreogxxEEEfP6AW9Gk/Gm/FufMxHS0axc4j+wPj8AYR/lX4=</latexit><latexit sha1_base64="Gtul+DRdUGk6V53QezeTUZK/z6g=">AAAB/3icbVBNS8NAEN3Ur1q/qh69BIvgqSS1oB6Eoj14rGBsIQlls5m0SzfZsLsRSujF/+BVr57Eqz/Fmz/FbZuDtj6Y4fHeDDv7gpRRqSzryyitrK6tb5Q3K1vbO7t71f2DB8kzQcAhnHHRC7AERhNwFFUMeqkAHAcMusHoZup3H0FIypN7NU7Bj/EgoRElWGnJPfPa/cZVU3e7X61ZdWsGc5nYBamhAp1+9dsLOcliSBRhWErXtlLl51goShhMKl4mIcVkhAfgaprgGKSfz06emCdaCc2IC12JMmfq740cx1KO40BPxlgN5aI3Ff/z3ExFF35OkzRTkJD5Q1HGTMXN6f/NkAogio01wURQfatJhlhgonRKFS+EyGvnXhuYwhMdib0YwDJxGvXLun3XrLWui2zK6Agdo1Nko3PUQreogxxEEEfP6AW9Gk/Gm/FufMxHS0axc4j+wPj8AYR/lX4=</latexit><latexit sha1_base64="Gtul+DRdUGk6V53QezeTUZK/z6g=">AAAB/3icbVBNS8NAEN3Ur1q/qh69BIvgqSS1oB6Eoj14rGBsIQlls5m0SzfZsLsRSujF/+BVr57Eqz/Fmz/FbZuDtj6Y4fHeDDv7gpRRqSzryyitrK6tb5Q3K1vbO7t71f2DB8kzQcAhnHHRC7AERhNwFFUMeqkAHAcMusHoZup3H0FIypN7NU7Bj/EgoRElWGnJPfPa/cZVU3e7X61ZdWsGc5nYBamhAp1+9dsLOcliSBRhWErXtlLl51goShhMKl4mIcVkhAfgaprgGKSfz06emCdaCc2IC12JMmfq740cx1KO40BPxlgN5aI3Ff/z3ExFF35OkzRTkJD5Q1HGTMXN6f/NkAogio01wURQfatJhlhgonRKFS+EyGvnXhuYwhMdib0YwDJxGvXLun3XrLWui2zK6Agdo1Nko3PUQreogxxEEEfP6AW9Gk/Gm/FufMxHS0axc4j+wPj8AYR/lX4=</latexit><latexit sha1_base64="Gtul+DRdUGk6V53QezeTUZK/z6g=">AAAB/3icbVBNS8NAEN3Ur1q/qh69BIvgqSS1oB6Eoj14rGBsIQlls5m0SzfZsLsRSujF/+BVr57Eqz/Fmz/FbZuDtj6Y4fHeDDv7gpRRqSzryyitrK6tb5Q3K1vbO7t71f2DB8kzQcAhnHHRC7AERhNwFFUMeqkAHAcMusHoZup3H0FIypN7NU7Bj/EgoRElWGnJPfPa/cZVU3e7X61ZdWsGc5nYBamhAp1+9dsLOcliSBRhWErXtlLl51goShhMKl4mIcVkhAfgaprgGKSfz06emCdaCc2IC12JMmfq740cx1KO40BPxlgN5aI3Ff/z3ExFF35OkzRTkJD5Q1HGTMXN6f/NkAogio01wURQfatJhlhgonRKFS+EyGvnXhuYwhMdib0YwDJxGvXLun3XrLWui2zK6Agdo1Nko3PUQreogxxEEEfP6AW9Gk/Gm/FufMxHS0axc4j+wPj8AYR/lX4=</latexit>

Page 10: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Add stress tensor

• Slightly larger errors (2-3%) for 2d minimal models with m = 4, 5, 6, 7.

• Universal because effective OPEs coincide

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Page 11: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Outline

• OPE truncation

• OPE hierarchy

• Effective Field Theory analogue

• Variational method: error minimization

Page 12: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

OPE convergence• In unitary CFTs, the OPE convergence is exponentially rapid

(Pappadopulo-Rychkov-Espin-Rattazzi, ’12)“The prefactor is not optimal”, concrete CFTs work even better

• For the bootstrap equation, we want to knowuse will overestimate the truncation error (smaller due to the minus sign)

⇢(z = 1/2) = 3� 2p2 ⇠ 0.17

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Page 13: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Radial coordinate• Using the radial coordinates , conformal blocks are

rapidly convergent power series (Hogervorst-Rychkov, ’13)

• An estimate at z=1/2exponential decay for large exponents

• Sign flipping due to Gegenbauer polynomials

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⇢ = r ei✓<latexit sha1_base64="B9JqtUSlEaSzfV7urEXY4yEhSJw=">AAACCXicbZDLSsNAFIYn9VbrLdqlm2ARXEhJRFAXQtEuXFYwttDEMpmcNEMnF2ZOhBL6Br6DW926Erc+hTsfxfSy0NYfBj7+/xzm8Hup4ApN80srLS2vrK6V1ysbm1vbO/ru3r1KMsnAZolIZMejCgSPwUaOAjqpBBp5Atre4Hqctx9BKp7EdzhMwY1oP+YBZxQLq6dXHRkml9I5hoecOxgC0lFPr5l1cyJjEawZ1MhMrZ7+7fgJyyKIkQmqVNcyU3RzKpEzAaOKkylIKRvQPnQLjGkEys0nx4+Mw8LxjSCRxYvRmLi/N3IaKTWMvGIyohiq+Wxs/pd1MwzO3ZzHaYYQs+lHQSYMTIxxE4bPJTAUwwIok7y41WAhlZRh0VfF8SFwmrnTBDGtxJovYBHsk/pF3bo9rTWuZt2UyT45IEfEImekQW5Ii9iEkSF5Ji/kVXvS3rR37WM6WtJmO1XyR9rnD0emmmw=</latexit><latexit sha1_base64="B9JqtUSlEaSzfV7urEXY4yEhSJw=">AAACCXicbZDLSsNAFIYn9VbrLdqlm2ARXEhJRFAXQtEuXFYwttDEMpmcNEMnF2ZOhBL6Br6DW926Erc+hTsfxfSy0NYfBj7+/xzm8Hup4ApN80srLS2vrK6V1ysbm1vbO/ru3r1KMsnAZolIZMejCgSPwUaOAjqpBBp5Atre4Hqctx9BKp7EdzhMwY1oP+YBZxQLq6dXHRkml9I5hoecOxgC0lFPr5l1cyJjEawZ1MhMrZ7+7fgJyyKIkQmqVNcyU3RzKpEzAaOKkylIKRvQPnQLjGkEys0nx4+Mw8LxjSCRxYvRmLi/N3IaKTWMvGIyohiq+Wxs/pd1MwzO3ZzHaYYQs+lHQSYMTIxxE4bPJTAUwwIok7y41WAhlZRh0VfF8SFwmrnTBDGtxJovYBHsk/pF3bo9rTWuZt2UyT45IEfEImekQW5Ii9iEkSF5Ji/kVXvS3rR37WM6WtJmO1XyR9rnD0emmmw=</latexit><latexit sha1_base64="B9JqtUSlEaSzfV7urEXY4yEhSJw=">AAACCXicbZDLSsNAFIYn9VbrLdqlm2ARXEhJRFAXQtEuXFYwttDEMpmcNEMnF2ZOhBL6Br6DW926Erc+hTsfxfSy0NYfBj7+/xzm8Hup4ApN80srLS2vrK6V1ysbm1vbO/ru3r1KMsnAZolIZMejCgSPwUaOAjqpBBp5Atre4Hqctx9BKp7EdzhMwY1oP+YBZxQLq6dXHRkml9I5hoecOxgC0lFPr5l1cyJjEawZ1MhMrZ7+7fgJyyKIkQmqVNcyU3RzKpEzAaOKkylIKRvQPnQLjGkEys0nx4+Mw8LxjSCRxYvRmLi/N3IaKTWMvGIyohiq+Wxs/pd1MwzO3ZzHaYYQs+lHQSYMTIxxE4bPJTAUwwIok7y41WAhlZRh0VfF8SFwmrnTBDGtxJovYBHsk/pF3bo9rTWuZt2UyT45IEfEImekQW5Ii9iEkSF5Ji/kVXvS3rR37WM6WtJmO1XyR9rnD0emmmw=</latexit><latexit sha1_base64="B9JqtUSlEaSzfV7urEXY4yEhSJw=">AAACCXicbZDLSsNAFIYn9VbrLdqlm2ARXEhJRFAXQtEuXFYwttDEMpmcNEMnF2ZOhBL6Br6DW926Erc+hTsfxfSy0NYfBj7+/xzm8Hup4ApN80srLS2vrK6V1ysbm1vbO/ru3r1KMsnAZolIZMejCgSPwUaOAjqpBBp5Atre4Hqctx9BKp7EdzhMwY1oP+YBZxQLq6dXHRkml9I5hoecOxgC0lFPr5l1cyJjEawZ1MhMrZ7+7fgJyyKIkQmqVNcyU3RzKpEzAaOKkylIKRvQPnQLjGkEys0nx4+Mw8LxjSCRxYvRmLi/N3IaKTWMvGIyohiq+Wxs/pd1MwzO3ZzHaYYQs+lHQSYMTIxxE4bPJTAUwwIok7y41WAhlZRh0VfF8SFwmrnTBDGtxJovYBHsk/pF3bo9rTWuZt2UyT45IEfEImekQW5Ii9iEkSF5Ji/kVXvS3rR37WM6WtJmO1XyR9rnD0emmmw=</latexit>

Page 14: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Example: 2d Ising• Spin-spin OPE

• Bootstrap equation at z = 1/2

• Only when close to z=1/2 ?

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+[4, 4] + [4, 0] + [5, 4] + [6, 6] + [6, 2] + [7, 6] + . . .<latexit sha1_base64="OwMhF+N8jWpQANOSz5TgZvNv838=">AAACIHicbZDLSsNAFIYnXmu9RV26CRZBVEoi9bYr2oXLClYLTSiT6Uk7OLkwcyKU0HfwHXwHt7p1JS515aM4iV1o9cD8fPznHGbm9xPBFdr2uzE1PTM7N19aKC8uLa+smmvr1ypOJYMWi0Us2z5VIHgELeQooJ1IoKEv4Ma/Pc/7N3cgFY+jKxwm4IW0H/GAM4ra6pq7e53afs3L1dZ6WPDR/lGhB1qPc3Z7MaquWbGrdlHWX3DGUCHjanbNT73H0hAiZIIq1XHsBL2MSuRMwKjspgoSym5pHzoaIxqC8rLiTyNrWzs9K4ilPhFahftzI6OhUsPQ15MhxYGa7OXmf71OisGJl/EoSREi9n1RkAoLYysPyOpxCQzFUANlkuu3WmxAJWWoYyy7PQjcRuY2QCAd6UicyQD+Quugelp1LmuV+tk4mxLZJFtkhzjkmNTJBWmSFmHknjySJ/JsPBgvxqvx9j06ZYx3NsivMj6+AN79nvE=</latexit><latexit sha1_base64="OwMhF+N8jWpQANOSz5TgZvNv838=">AAACIHicbZDLSsNAFIYnXmu9RV26CRZBVEoi9bYr2oXLClYLTSiT6Uk7OLkwcyKU0HfwHXwHt7p1JS515aM4iV1o9cD8fPznHGbm9xPBFdr2uzE1PTM7N19aKC8uLa+smmvr1ypOJYMWi0Us2z5VIHgELeQooJ1IoKEv4Ma/Pc/7N3cgFY+jKxwm4IW0H/GAM4ra6pq7e53afs3L1dZ6WPDR/lGhB1qPc3Z7MaquWbGrdlHWX3DGUCHjanbNT73H0hAiZIIq1XHsBL2MSuRMwKjspgoSym5pHzoaIxqC8rLiTyNrWzs9K4ilPhFahftzI6OhUsPQ15MhxYGa7OXmf71OisGJl/EoSREi9n1RkAoLYysPyOpxCQzFUANlkuu3WmxAJWWoYyy7PQjcRuY2QCAd6UicyQD+Quugelp1LmuV+tk4mxLZJFtkhzjkmNTJBWmSFmHknjySJ/JsPBgvxqvx9j06ZYx3NsivMj6+AN79nvE=</latexit><latexit sha1_base64="OwMhF+N8jWpQANOSz5TgZvNv838=">AAACIHicbZDLSsNAFIYnXmu9RV26CRZBVEoi9bYr2oXLClYLTSiT6Uk7OLkwcyKU0HfwHXwHt7p1JS515aM4iV1o9cD8fPznHGbm9xPBFdr2uzE1PTM7N19aKC8uLa+smmvr1ypOJYMWi0Us2z5VIHgELeQooJ1IoKEv4Ma/Pc/7N3cgFY+jKxwm4IW0H/GAM4ra6pq7e53afs3L1dZ6WPDR/lGhB1qPc3Z7MaquWbGrdlHWX3DGUCHjanbNT73H0hAiZIIq1XHsBL2MSuRMwKjspgoSym5pHzoaIxqC8rLiTyNrWzs9K4ilPhFahftzI6OhUsPQ15MhxYGa7OXmf71OisGJl/EoSREi9n1RkAoLYysPyOpxCQzFUANlkuu3WmxAJWWoYyy7PQjcRuY2QCAd6UicyQD+Quugelp1LmuV+tk4mxLZJFtkhzjkmNTJBWmSFmHknjySJ/JsPBgvxqvx9j06ZYx3NsivMj6+AN79nvE=</latexit><latexit sha1_base64="OwMhF+N8jWpQANOSz5TgZvNv838=">AAACIHicbZDLSsNAFIYnXmu9RV26CRZBVEoi9bYr2oXLClYLTSiT6Uk7OLkwcyKU0HfwHXwHt7p1JS515aM4iV1o9cD8fPznHGbm9xPBFdr2uzE1PTM7N19aKC8uLa+smmvr1ypOJYMWi0Us2z5VIHgELeQooJ1IoKEv4Ma/Pc/7N3cgFY+jKxwm4IW0H/GAM4ra6pq7e53afs3L1dZ6WPDR/lGhB1qPc3Z7MaquWbGrdlHWX3DGUCHjanbNT73H0hAiZIIq1XHsBL2MSuRMwKjspgoSym5pHzoaIxqC8rLiTyNrWzs9K4ilPhFahftzI6OhUsPQ15MhxYGa7OXmf71OisGJl/EoSREi9n1RkAoLYysPyOpxCQzFUANlkuu3WmxAJWWoYyy7PQjcRuY2QCAd6UicyQD+Quugelp1LmuV+tk4mxLZJFtkhzjkmNTJBWmSFmHknjySJ/JsPBgvxqvx9j06ZYx3NsivMj6+AN79nvE=</latexit>

0 = 1� e�0.17 � e�1.86<latexit sha1_base64="Y1uBIaRh/ofGIp22Oi+pxhOaMaY=">AAACDXicbZDLSsNAGIUn9VbrLerChZvBIrhpSERsXQhFu3BZwdpCU8tk8qcdOrkwMxFKyDv4Dm5160rc+gzufBTTNgttPTDwcc7/MzPHiTiTyjS/tMLS8srqWnG9tLG5tb2j7+7dyzAWFFo05KHoOEQCZwG0FFMcOpEA4jsc2s7oepK3H0FIFgZ3ahxBzyeDgHmMEpVZff3AvLQq8JBUTMOqplOyjNp52tfLpmFOhRfByqGMcjX7+rfthjT2IVCUEym7lhmpXkKEYpRDWrJjCRGhIzKAboYB8UH2kukHUnycOS72QpGdQOGp+3sjIb6UY9/JJn2ihnI+m5j/Zd1YebVewoIoVhDQ2UVezLEK8aQN7DIBVPFxBoQKlr0V0yERhKqss5Ltgmc3ErsBXJFJJdZ8AYvQOjUuDOv2rFy/yrspokN0hE6Qhaqojm5QE7UQRSl6Ri/oVXvS3rR37WM2WtDynX30R9rnDyoQmX0=</latexit><latexit sha1_base64="Y1uBIaRh/ofGIp22Oi+pxhOaMaY=">AAACDXicbZDLSsNAGIUn9VbrLerChZvBIrhpSERsXQhFu3BZwdpCU8tk8qcdOrkwMxFKyDv4Dm5160rc+gzufBTTNgttPTDwcc7/MzPHiTiTyjS/tMLS8srqWnG9tLG5tb2j7+7dyzAWFFo05KHoOEQCZwG0FFMcOpEA4jsc2s7oepK3H0FIFgZ3ahxBzyeDgHmMEpVZff3AvLQq8JBUTMOqplOyjNp52tfLpmFOhRfByqGMcjX7+rfthjT2IVCUEym7lhmpXkKEYpRDWrJjCRGhIzKAboYB8UH2kukHUnycOS72QpGdQOGp+3sjIb6UY9/JJn2ihnI+m5j/Zd1YebVewoIoVhDQ2UVezLEK8aQN7DIBVPFxBoQKlr0V0yERhKqss5Ltgmc3ErsBXJFJJdZ8AYvQOjUuDOv2rFy/yrspokN0hE6Qhaqojm5QE7UQRSl6Ri/oVXvS3rR37WM2WtDynX30R9rnDyoQmX0=</latexit><latexit sha1_base64="Y1uBIaRh/ofGIp22Oi+pxhOaMaY=">AAACDXicbZDLSsNAGIUn9VbrLerChZvBIrhpSERsXQhFu3BZwdpCU8tk8qcdOrkwMxFKyDv4Dm5160rc+gzufBTTNgttPTDwcc7/MzPHiTiTyjS/tMLS8srqWnG9tLG5tb2j7+7dyzAWFFo05KHoOEQCZwG0FFMcOpEA4jsc2s7oepK3H0FIFgZ3ahxBzyeDgHmMEpVZff3AvLQq8JBUTMOqplOyjNp52tfLpmFOhRfByqGMcjX7+rfthjT2IVCUEym7lhmpXkKEYpRDWrJjCRGhIzKAboYB8UH2kukHUnycOS72QpGdQOGp+3sjIb6UY9/JJn2ihnI+m5j/Zd1YebVewoIoVhDQ2UVezLEK8aQN7DIBVPFxBoQKlr0V0yERhKqss5Ltgmc3ErsBXJFJJdZ8AYvQOjUuDOv2rFy/yrspokN0hE6Qhaqojm5QE7UQRSl6Ri/oVXvS3rR37WM2WtDynX30R9rnDyoQmX0=</latexit><latexit sha1_base64="Y1uBIaRh/ofGIp22Oi+pxhOaMaY=">AAACDXicbZDLSsNAGIUn9VbrLerChZvBIrhpSERsXQhFu3BZwdpCU8tk8qcdOrkwMxFKyDv4Dm5160rc+gzufBTTNgttPTDwcc7/MzPHiTiTyjS/tMLS8srqWnG9tLG5tb2j7+7dyzAWFFo05KHoOEQCZwG0FFMcOpEA4jsc2s7oepK3H0FIFgZ3ahxBzyeDgHmMEpVZff3AvLQq8JBUTMOqplOyjNp52tfLpmFOhRfByqGMcjX7+rfthjT2IVCUEym7lhmpXkKEYpRDWrJjCRGhIzKAboYB8UH2kukHUnycOS72QpGdQOGp+3sjIb6UY9/JJn2ihnI+m5j/Zd1YebVewoIoVhDQ2UVezLEK8aQN7DIBVPFxBoQKlr0V0yERhKqss5Ltgmc3ErsBXJFJJdZ8AYvQOjUuDOv2rFy/yrspokN0hE6Qhaqojm5QE7UQRSl6Ri/oVXvS3rR37WM2WtDynX30R9rnDyoQmX0=</latexit>

�e�6.14 � e�6.71 � e�8.94 � e�9.95 � e�10.6 � e�12.3 � . . .<latexit sha1_base64="Vl+JnENR4fxfDFrWA57D5W5US+4=">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</latexit><latexit sha1_base64="Vl+JnENR4fxfDFrWA57D5W5US+4=">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</latexit><latexit sha1_base64="Vl+JnENR4fxfDFrWA57D5W5US+4=">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</latexit><latexit sha1_base64="Vl+JnENR4fxfDFrWA57D5W5US+4=">AAACOHicbZDLSsNAFIYn9V5vVZdugkVw05B4bXdFu3BZwWqhqWUyPamDkwszJ0IJeR/fwXdwq+50JW59AqdtFmr9YeDj/89hZn4vFlyhbb8ahZnZufmFxaXi8srq2nppY/NKRYlk0GKRiGTbowoED6GFHAW0Ywk08ARce3dno/z6HqTiUXiJwxi6AR2E3OeMorZ6pdMK3KSVY8s5zHI6cSZUtWq5V7NqRxNybOs4p33rIKu4/QhVr1S2LXsscxqcHMokV7NXetN7LAkgRCaoUh3HjrGbUomcCciKbqIgpuyODqCjMaQBqG46/mtm7mqnb/qR1CdEc+z+3EhpoNQw8PRkQPFW/c1G5n9ZJ0G/2k15GCcIIZtc5CfCxMgcFWf2uQSGYqiBMsn1W012SyVlqOstun3w3UbqNkAgzXQlzt8CpqG1b9Us5+KwXD/Nu1kk22SH7BGHnJA6OSdN0iKMPJAn8kxejEfj3fgwPiejBSPf2SK/ZHx9Az2TqFc=</latexit>

e�2 ⇠ 0.14<latexit sha1_base64="S7WM1O0xOjQVtlukzmdOA8N3p9Y=">AAACBXicbZBLS8NAFIUnPmt9NOrSzWAR3FiSUlB3RbtwWcHaQhPLZHLTDp1MwsxEKKF7/4Nb3boSt/4Od/4Up4+Fth4Y+DjnXuZygpQzpR3ny1pZXVvf2CxsFbd3dvdK9v7BvUoySaFFE57ITkAUcCagpZnm0EklkDjg0A6G15O8/QhSsUTc6VEKfkz6gkWMEm2snl2Ch/ysOvYUi7FTcWs9u+xUnKnwMrhzKKO5mj372wsTmsUgNOVEqa7rpNrPidSMchgXvUxBSuiQ9KFrUJAYlJ9PDx/jE+OEOEqkeULjqft7IyexUqM4MJMx0QO1mE3M/7JupqMLP2cizTQIOvsoyjjWCZ60gEMmgWo+MkCoZOZWTAdEEqpNV0UvhMhr5F4DuCZjU4m7WMAytKqVy4p7WyvXr+bdFNAROkanyEXnqI5uUBO1EEUZekYv6NV6st6sd+tjNrpizXcO0R9Znz8uhpd0</latexit><latexit sha1_base64="S7WM1O0xOjQVtlukzmdOA8N3p9Y=">AAACBXicbZBLS8NAFIUnPmt9NOrSzWAR3FiSUlB3RbtwWcHaQhPLZHLTDp1MwsxEKKF7/4Nb3boSt/4Od/4Up4+Fth4Y+DjnXuZygpQzpR3ny1pZXVvf2CxsFbd3dvdK9v7BvUoySaFFE57ITkAUcCagpZnm0EklkDjg0A6G15O8/QhSsUTc6VEKfkz6gkWMEm2snl2Ch/ysOvYUi7FTcWs9u+xUnKnwMrhzKKO5mj372wsTmsUgNOVEqa7rpNrPidSMchgXvUxBSuiQ9KFrUJAYlJ9PDx/jE+OEOEqkeULjqft7IyexUqM4MJMx0QO1mE3M/7JupqMLP2cizTQIOvsoyjjWCZ60gEMmgWo+MkCoZOZWTAdEEqpNV0UvhMhr5F4DuCZjU4m7WMAytKqVy4p7WyvXr+bdFNAROkanyEXnqI5uUBO1EEUZekYv6NV6st6sd+tjNrpizXcO0R9Znz8uhpd0</latexit><latexit sha1_base64="S7WM1O0xOjQVtlukzmdOA8N3p9Y=">AAACBXicbZBLS8NAFIUnPmt9NOrSzWAR3FiSUlB3RbtwWcHaQhPLZHLTDp1MwsxEKKF7/4Nb3boSt/4Od/4Up4+Fth4Y+DjnXuZygpQzpR3ny1pZXVvf2CxsFbd3dvdK9v7BvUoySaFFE57ITkAUcCagpZnm0EklkDjg0A6G15O8/QhSsUTc6VEKfkz6gkWMEm2snl2Ch/ysOvYUi7FTcWs9u+xUnKnwMrhzKKO5mj372wsTmsUgNOVEqa7rpNrPidSMchgXvUxBSuiQ9KFrUJAYlJ9PDx/jE+OEOEqkeULjqft7IyexUqM4MJMx0QO1mE3M/7JupqMLP2cizTQIOvsoyjjWCZ60gEMmgWo+MkCoZOZWTAdEEqpNV0UvhMhr5F4DuCZjU4m7WMAytKqVy4p7WyvXr+bdFNAROkanyEXnqI5uUBO1EEUZekYv6NV6st6sd+tjNrpizXcO0R9Znz8uhpd0</latexit><latexit sha1_base64="S7WM1O0xOjQVtlukzmdOA8N3p9Y=">AAACBXicbZBLS8NAFIUnPmt9NOrSzWAR3FiSUlB3RbtwWcHaQhPLZHLTDp1MwsxEKKF7/4Nb3boSt/4Od/4Up4+Fth4Y+DjnXuZygpQzpR3ny1pZXVvf2CxsFbd3dvdK9v7BvUoySaFFE57ITkAUcCagpZnm0EklkDjg0A6G15O8/QhSsUTc6VEKfkz6gkWMEm2snl2Ch/ysOvYUi7FTcWs9u+xUnKnwMrhzKKO5mj372wsTmsUgNOVEqa7rpNrPidSMchgXvUxBSuiQ9KFrUJAYlJ9PDx/jE+OEOEqkeULjqft7IyexUqM4MJMx0QO1mE3M/7JupqMLP2cizTQIOvsoyjjWCZ60gEMmgWo+MkCoZOZWTAdEEqpNV0UvhMhr5F4DuCZjU4m7WMAytKqVy4p7WyvXr+bdFNAROkanyEXnqI5uUBO1EEUZekYv6NV6st6sd+tjNrpizXcO0R9Znz8uhpd0</latexit>

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Page 15: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Real axis

0.2 0.4 0.6 0.8 1.0z

-12

-10

-8

-6

-4

-2ϵT

[4,4][4,0][5,4][6,6][6,2][7,6]

large gap below T due to decoupling of a spin-2 primary (sparse spectrum)

By exact 2d blocks, examine each contribution, not their sums/truncated sum rules

Page 16: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Imaginary axis

-3 -2 -1 1 2 3θ

-15

-10

-5

5

10

ϵT

[4,4][4,0][5,4]

[�i, li] ! log

����Piv�F�i,li(u, v)� (u $ v)

v� � u�

����Re[z]=1/2

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Page 17: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Some comments• OPE hierarchy most clear at z=1/2

but persists in a large region

• Delicate balance of different contributions Remarkable consequence: OPE truncation error is dominated by 1 conformal multiplet (come back to this later)

• 2d Lee-Yang CFT is similar (non-unitary)

0.2 0.4 0.6 0.8 1.0z

-12

-10

-8

-6

-4

-2

2

[-0.4,0]T

[3.6,4][4,0][5.6,6][7,6]

-3 -2 -1 1 2 3

-15

-10

-5

5

10

[-0.4,0]T

[3.6,4][4,0]

Page 18: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Outline

• OPE truncation

• OPE hierarchy

• Effective Field Theory analogue

• Variational method: error minimization

Page 19: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Effective Field Theory• Another great legacy of Ken Wilson

• EFT is useful when a large hierarchy

• Large dimension operators suppressed by small coupling constants

• A few terms in effective Lagrangians capture main physics

• Effective large N CFTs are related to AdS locality (Heemskerk-Penedones-Polchinski-Sully ’09, Fitzpatrick-Katz-Poland-Simmons-Duffin ’10)

Page 20: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Effective OPE

• Truncated/effective OPE 34 effective Lagrangian

• OPE coefficient 34 coupling constant

• OPE hierarchy: high-lying operators suppressed by small OPE coefficients

• Effective OPEs simplify the strong coupling problemShifman–Vainshtein–Zakharov sum rules in QCDGliozzi’s determinant method in conformal bootstrap

Page 21: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Energy scale

• What are the analogue of energy scale?

• “Low energy” limit: crossing symmetric point z=1/2OPE hierarchies are largest

• Effective OPEs remains good approximations in a large region around z=1/2

• EFT needs to know when effective description breaks down

Page 22: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

UV cutoff• “High energy” limit: branch cuts (z>1, z<0, infinity)

• Approaching branch cuts, high-lying operators become more and more important

• Gradually, an effective OPE does not make sense: one of the omitted operators becomes comparableExample: 2d Ising

• Cutoff also depends on precision0.00000 0.00002 0.00004 0.00006 0.00008 0.00010

z

-4.0

-3.5

-3.0ϵT

[4,4][4,0][5,4][6,6][6,2][7,6]

Page 23: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Example: 2d Ising0.2 0.4 0.6 0.8 1.0

z

-12

-10

-8

-6

-4

-2ϵT

[4,4][4,0][5,4][6,6][6,2][7,6]

: “UV cutoff” for the effective OPE � ⇥ � = 1 + ✏+ T<latexit sha1_base64="jtZL5cxjNxvlmRmHCLRB8YsGDgY=">AAACFXicbZA9SwNBEIb34leMX1FLCw+DIATCnQhqIQRNYRkhMYFcCHubuWTJ3t6xOyeEI6X/wf9gq62V2Frb+VPcfBSa+MLCwzszzOzrx4JrdJwvK7O0vLK6ll3PbWxube/kd/fudZQoBnUWiUg1fapBcAl15CigGSugoS+g4Q9uxvXGAyjNI1nDYQztkPYkDzijaKxO/tDTvBdSD3kIespXbtGDWHMRyWKtky84JWciexHcGRTITNVO/tvrRiwJQSITVOuW68TYTqlCzgSMcl6iIaZsQHvQMiip2dtOJx8Z2cfG6dpBpMyTaE/c3xMpDbUehr7pDCn29XxtbP5XayUYXLRTLuMEQbLpoiARNkb2OBW7yxUwFEMDlClubrVZnyrK0GSX87oQeJXUq4BAOjKRuPMBLEL9tHRZcu/OCuXrWTZZckCOyAlxyTkpk1tSJXXCyCN5Ji/k1Xqy3qx362PamrFmM/vkj6zPH0Rxnzg=</latexit><latexit sha1_base64="jtZL5cxjNxvlmRmHCLRB8YsGDgY=">AAACFXicbZA9SwNBEIb34leMX1FLCw+DIATCnQhqIQRNYRkhMYFcCHubuWTJ3t6xOyeEI6X/wf9gq62V2Frb+VPcfBSa+MLCwzszzOzrx4JrdJwvK7O0vLK6ll3PbWxube/kd/fudZQoBnUWiUg1fapBcAl15CigGSugoS+g4Q9uxvXGAyjNI1nDYQztkPYkDzijaKxO/tDTvBdSD3kIespXbtGDWHMRyWKtky84JWciexHcGRTITNVO/tvrRiwJQSITVOuW68TYTqlCzgSMcl6iIaZsQHvQMiip2dtOJx8Z2cfG6dpBpMyTaE/c3xMpDbUehr7pDCn29XxtbP5XayUYXLRTLuMEQbLpoiARNkb2OBW7yxUwFEMDlClubrVZnyrK0GSX87oQeJXUq4BAOjKRuPMBLEL9tHRZcu/OCuXrWTZZckCOyAlxyTkpk1tSJXXCyCN5Ji/k1Xqy3qx362PamrFmM/vkj6zPH0Rxnzg=</latexit><latexit sha1_base64="jtZL5cxjNxvlmRmHCLRB8YsGDgY=">AAACFXicbZA9SwNBEIb34leMX1FLCw+DIATCnQhqIQRNYRkhMYFcCHubuWTJ3t6xOyeEI6X/wf9gq62V2Frb+VPcfBSa+MLCwzszzOzrx4JrdJwvK7O0vLK6ll3PbWxube/kd/fudZQoBnUWiUg1fapBcAl15CigGSugoS+g4Q9uxvXGAyjNI1nDYQztkPYkDzijaKxO/tDTvBdSD3kIespXbtGDWHMRyWKtky84JWciexHcGRTITNVO/tvrRiwJQSITVOuW68TYTqlCzgSMcl6iIaZsQHvQMiip2dtOJx8Z2cfG6dpBpMyTaE/c3xMpDbUehr7pDCn29XxtbP5XayUYXLRTLuMEQbLpoiARNkb2OBW7yxUwFEMDlClubrVZnyrK0GSX87oQeJXUq4BAOjKRuPMBLEL9tHRZcu/OCuXrWTZZckCOyAlxyTkpk1tSJXXCyCN5Ji/k1Xqy3qx362PamrFmM/vkj6zPH0Rxnzg=</latexit><latexit sha1_base64="jtZL5cxjNxvlmRmHCLRB8YsGDgY=">AAACFXicbZA9SwNBEIb34leMX1FLCw+DIATCnQhqIQRNYRkhMYFcCHubuWTJ3t6xOyeEI6X/wf9gq62V2Frb+VPcfBSa+MLCwzszzOzrx4JrdJwvK7O0vLK6ll3PbWxube/kd/fudZQoBnUWiUg1fapBcAl15CigGSugoS+g4Q9uxvXGAyjNI1nDYQztkPYkDzijaKxO/tDTvBdSD3kIespXbtGDWHMRyWKtky84JWciexHcGRTITNVO/tvrRiwJQSITVOuW68TYTqlCzgSMcl6iIaZsQHvQMiip2dtOJx8Z2cfG6dpBpMyTaE/c3xMpDbUehr7pDCn29XxtbP5XayUYXLRTLuMEQbLpoiARNkb2OBW7yxUwFEMDlClubrVZnyrK0GSX87oQeJXUq4BAOjKRuPMBLEL9tHRZcu/OCuXrWTZZckCOyAlxyTkpk1tSJXXCyCN5Ji/k1Xqy3qx362PamrFmM/vkj6zPH0Rxnzg=</latexit>

Page 24: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Outline

• OPE truncation

• OPE hierarchy

• Effective Field Theory analogue

• Variational method: error minimization

Page 25: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

How to tame the truncation error?

From a review talk by Slava Rychkov in 2015

Page 26: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Error minimization• OPE truncation induces truncation error

• Errors suppressed by small OPE coefficients of omitted operators

• OPE hierarchy is delicate: arbitrary CFT data will not show hierarchy.

• RHS is complicated, but equal to LHS when bootstrap equation is satisfied

• Assume CFT data of effective OPEs lead to smaller truncation errors

• New bootstrap method extract low-lying CFT data by minimizing OPE truncation error (= LHS)

1 +X

in

Piu�F�i,li(u, v)� (u $ v)

u� � v�= �

X

i>n

Piu�F�i,li(u, v)� (u $ v)

u� � v�<latexit sha1_base64="syJSVnPCt2pmBS7zIdMBPFuefrk=">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</latexit><latexit sha1_base64="syJSVnPCt2pmBS7zIdMBPFuefrk=">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</latexit><latexit sha1_base64="syJSVnPCt2pmBS7zIdMBPFuefrk=">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</latexit><latexit sha1_base64="syJSVnPCt2pmBS7zIdMBPFuefrk=">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</latexit>

Page 27: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Error function• Error function

• Geometric interpretation (p-norm with p=2)

• Reduce to Gliozzi’s determinant method when numbers of equations and parameters match , all truncated equations =0, linear dependence, vanishing determinants

• Generalization: more equations than unknown parameters

⌘(k) =

vuuuut1

k

kX

j=1

0

@1 +X

in

Piu�F�i,li(u, v)� (u $ v)

u� � v�

1

A2

(j)<latexit sha1_base64="rLOuSCsPzeSjrotKyJ/fN7PWgaY=">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</latexit><latexit sha1_base64="rLOuSCsPzeSjrotKyJ/fN7PWgaY=">AAAClHicbVFdb9MwFHXCgFG+CjzwwItFhZRAVyVjEvBQaWID8YSKRNmkuo0c12m9Oh+zr4uqyH+F/8UbPwU3jRBsHMnSuefeY1vnppUUGqLop+ff2Lt56/b+nc7de/cfPOw+evxNl0YxPmalLNV5SjWXouBjECD5eaU4zVPJz9LVybZ/tuZKi7L4CpuKT3O6KEQmGAUnJd0fhAMNVuGQ6EsFNckUZTjGK0y0yZP6YhjbmSskzyCIX+1E4cpLXFjSHyUCN5bazMgpl0Dxx6TesUT0ZSJsYPrr8CAwzRVKLJZAlSq/43Vo/5gO1i2xpJkIkzq4CO3s0CbdXjSIGuDrJG5JD7UYJd1fZF4yk/MCmKRaT+KogmlNFQgmue0Qo3lF2You+MTRguZcT+smR4tfOGWOs1K5UwBu1L8dNc213uSpm8wpLPXV3lb8X29iIHs7rUVRGeAF2z2UGYmhxNul4LlQnIHcOEKZEu6vmC2pyxXc6jpkzjNy2sa6jSS+GsB1Mj4cvBvEX456x+/bbPbRM/QcBShGb9Ax+oRGaIyYt+e99F57R/5Tf+if+B92o77Xep6gf+B//g3pHsjV</latexit><latexit sha1_base64="rLOuSCsPzeSjrotKyJ/fN7PWgaY=">AAAClHicbVFdb9MwFHXCgFG+CjzwwItFhZRAVyVjEvBQaWID8YSKRNmkuo0c12m9Oh+zr4uqyH+F/8UbPwU3jRBsHMnSuefeY1vnppUUGqLop+ff2Lt56/b+nc7de/cfPOw+evxNl0YxPmalLNV5SjWXouBjECD5eaU4zVPJz9LVybZ/tuZKi7L4CpuKT3O6KEQmGAUnJd0fhAMNVuGQ6EsFNckUZTjGK0y0yZP6YhjbmSskzyCIX+1E4cpLXFjSHyUCN5bazMgpl0Dxx6TesUT0ZSJsYPrr8CAwzRVKLJZAlSq/43Vo/5gO1i2xpJkIkzq4CO3s0CbdXjSIGuDrJG5JD7UYJd1fZF4yk/MCmKRaT+KogmlNFQgmue0Qo3lF2You+MTRguZcT+smR4tfOGWOs1K5UwBu1L8dNc213uSpm8wpLPXV3lb8X29iIHs7rUVRGeAF2z2UGYmhxNul4LlQnIHcOEKZEu6vmC2pyxXc6jpkzjNy2sa6jSS+GsB1Mj4cvBvEX456x+/bbPbRM/QcBShGb9Ax+oRGaIyYt+e99F57R/5Tf+if+B92o77Xep6gf+B//g3pHsjV</latexit><latexit sha1_base64="rLOuSCsPzeSjrotKyJ/fN7PWgaY=">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</latexit>

⌘ =��1 + ~F1 + · · ·+ ~Fn

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Page 28: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

2d Ising• Test error-minimization by 2d Ising

• Effective OPE with

• More truncated equations than free parameters, so minima of error function > 0

• Not Taylor expansion around z=1/2 because higher derivatives converge much slower. Instead, Taylor expansion around u=v=1, converge more uniformly

• Assume to rule out the unphysical minimum .

� ⇥ � = 1 + ✏+ T<latexit sha1_base64="jtZL5cxjNxvlmRmHCLRB8YsGDgY=">AAACFXicbZA9SwNBEIb34leMX1FLCw+DIATCnQhqIQRNYRkhMYFcCHubuWTJ3t6xOyeEI6X/wf9gq62V2Frb+VPcfBSa+MLCwzszzOzrx4JrdJwvK7O0vLK6ll3PbWxube/kd/fudZQoBnUWiUg1fapBcAl15CigGSugoS+g4Q9uxvXGAyjNI1nDYQztkPYkDzijaKxO/tDTvBdSD3kIespXbtGDWHMRyWKtky84JWciexHcGRTITNVO/tvrRiwJQSITVOuW68TYTqlCzgSMcl6iIaZsQHvQMiip2dtOJx8Z2cfG6dpBpMyTaE/c3xMpDbUehr7pDCn29XxtbP5XayUYXLRTLuMEQbLpoiARNkb2OBW7yxUwFEMDlClubrVZnyrK0GSX87oQeJXUq4BAOjKRuPMBLEL9tHRZcu/OCuXrWTZZckCOyAlxyTkpk1tSJXXCyCN5Ji/k1Xqy3qx362PamrFmM/vkj6zPH0Rxnzg=</latexit><latexit sha1_base64="jtZL5cxjNxvlmRmHCLRB8YsGDgY=">AAACFXicbZA9SwNBEIb34leMX1FLCw+DIATCnQhqIQRNYRkhMYFcCHubuWTJ3t6xOyeEI6X/wf9gq62V2Frb+VPcfBSa+MLCwzszzOzrx4JrdJwvK7O0vLK6ll3PbWxube/kd/fudZQoBnUWiUg1fapBcAl15CigGSugoS+g4Q9uxvXGAyjNI1nDYQztkPYkDzijaKxO/tDTvBdSD3kIespXbtGDWHMRyWKtky84JWciexHcGRTITNVO/tvrRiwJQSITVOuW68TYTqlCzgSMcl6iIaZsQHvQMiip2dtOJx8Z2cfG6dpBpMyTaE/c3xMpDbUehr7pDCn29XxtbP5XayUYXLRTLuMEQbLpoiARNkb2OBW7yxUwFEMDlClubrVZnyrK0GSX87oQeJXUq4BAOjKRuPMBLEL9tHRZcu/OCuXrWTZZckCOyAlxyTkpk1tSJXXCyCN5Ji/k1Xqy3qx362PamrFmM/vkj6zPH0Rxnzg=</latexit><latexit sha1_base64="jtZL5cxjNxvlmRmHCLRB8YsGDgY=">AAACFXicbZA9SwNBEIb34leMX1FLCw+DIATCnQhqIQRNYRkhMYFcCHubuWTJ3t6xOyeEI6X/wf9gq62V2Frb+VPcfBSa+MLCwzszzOzrx4JrdJwvK7O0vLK6ll3PbWxube/kd/fudZQoBnUWiUg1fapBcAl15CigGSugoS+g4Q9uxvXGAyjNI1nDYQztkPYkDzijaKxO/tDTvBdSD3kIespXbtGDWHMRyWKtky84JWciexHcGRTITNVO/tvrRiwJQSITVOuW68TYTqlCzgSMcl6iIaZsQHvQMiip2dtOJx8Z2cfG6dpBpMyTaE/c3xMpDbUehr7pDCn29XxtbP5XayUYXLRTLuMEQbLpoiARNkb2OBW7yxUwFEMDlClubrVZnyrK0GSX87oQeJXUq4BAOjKRuPMBLEL9tHRZcu/OCuXrWTZZckCOyAlxyTkpk1tSJXXCyCN5Ji/k1Xqy3qx362PamrFmM/vkj6zPH0Rxnzg=</latexit><latexit sha1_base64="jtZL5cxjNxvlmRmHCLRB8YsGDgY=">AAACFXicbZA9SwNBEIb34leMX1FLCw+DIATCnQhqIQRNYRkhMYFcCHubuWTJ3t6xOyeEI6X/wf9gq62V2Frb+VPcfBSa+MLCwzszzOzrx4JrdJwvK7O0vLK6ll3PbWxube/kd/fudZQoBnUWiUg1fapBcAl15CigGSugoS+g4Q9uxvXGAyjNI1nDYQztkPYkDzijaKxO/tDTvBdSD3kIespXbtGDWHMRyWKtky84JWciexHcGRTITNVO/tvrRiwJQSITVOuW68TYTqlCzgSMcl6iIaZsQHvQMiip2dtOJx8Z2cfG6dpBpMyTaE/c3xMpDbUehr7pDCn29XxtbP5XayUYXLRTLuMEQbLpoiARNkb2OBW7yxUwFEMDlClubrVZnyrK0GSX87oQeJXUq4BAOjKRuPMBLEL9tHRZcu/OCuXrWTZZckCOyAlxyTkpk1tSJXXCyCN5Ji/k1Xqy3qx362PamrFmM/vkj6zPH0Rxnzg=</latexit>

�� = 1/8<latexit sha1_base64="jUSHA3aq+LZF8NI4WAaX4/OcSeQ=">AAACAXicbZDNSsNAFIUn9a/Wv6pLN4NFcFUTEawLoWgXLitYW2hCmUxu2qEzSZyZCCV05zu41a0rceuTuPNRnLZZaOuBgY9z7mUux084U9q2v6zC0vLK6lpxvbSxubW9U97du1dxKim0aMxj2fGJAs4iaGmmOXQSCUT4HNr+8HqStx9BKhZHd3qUgCdIP2Iho0Qby3MbPVexviCXzkmtV67YVXsqvAhODhWUq9krf7tBTFMBkaacKNV17ER7GZGaUQ7jkpsqSAgdkj50DUZEgPKy6dFjfGScAIexNC/SeOr+3siIUGokfDMpiB6o+Wxi/pd1Ux3WvIxFSaohorOPwpRjHeNJAzhgEqjmIwOESmZuxXRAJKHa9FRyAwjdRuY2gGsyNpU48wUsQuu0elF1bs8q9au8myI6QIfoGDnoHNXRDWqiFqLoAT2jF/RqPVlv1rv1MRstWPnOPvoj6/MHyoaWzg==</latexit><latexit sha1_base64="jUSHA3aq+LZF8NI4WAaX4/OcSeQ=">AAACAXicbZDNSsNAFIUn9a/Wv6pLN4NFcFUTEawLoWgXLitYW2hCmUxu2qEzSZyZCCV05zu41a0rceuTuPNRnLZZaOuBgY9z7mUux084U9q2v6zC0vLK6lpxvbSxubW9U97du1dxKim0aMxj2fGJAs4iaGmmOXQSCUT4HNr+8HqStx9BKhZHd3qUgCdIP2Iho0Qby3MbPVexviCXzkmtV67YVXsqvAhODhWUq9krf7tBTFMBkaacKNV17ER7GZGaUQ7jkpsqSAgdkj50DUZEgPKy6dFjfGScAIexNC/SeOr+3siIUGokfDMpiB6o+Wxi/pd1Ux3WvIxFSaohorOPwpRjHeNJAzhgEqjmIwOESmZuxXRAJKHa9FRyAwjdRuY2gGsyNpU48wUsQuu0elF1bs8q9au8myI6QIfoGDnoHNXRDWqiFqLoAT2jF/RqPVlv1rv1MRstWPnOPvoj6/MHyoaWzg==</latexit><latexit sha1_base64="jUSHA3aq+LZF8NI4WAaX4/OcSeQ=">AAACAXicbZDNSsNAFIUn9a/Wv6pLN4NFcFUTEawLoWgXLitYW2hCmUxu2qEzSZyZCCV05zu41a0rceuTuPNRnLZZaOuBgY9z7mUux084U9q2v6zC0vLK6lpxvbSxubW9U97du1dxKim0aMxj2fGJAs4iaGmmOXQSCUT4HNr+8HqStx9BKhZHd3qUgCdIP2Iho0Qby3MbPVexviCXzkmtV67YVXsqvAhODhWUq9krf7tBTFMBkaacKNV17ER7GZGaUQ7jkpsqSAgdkj50DUZEgPKy6dFjfGScAIexNC/SeOr+3siIUGokfDMpiB6o+Wxi/pd1Ux3WvIxFSaohorOPwpRjHeNJAzhgEqjmIwOESmZuxXRAJKHa9FRyAwjdRuY2gGsyNpU48wUsQuu0elF1bs8q9au8myI6QIfoGDnoHNXRDWqiFqLoAT2jF/RqPVlv1rv1MRstWPnOPvoj6/MHyoaWzg==</latexit><latexit sha1_base64="jUSHA3aq+LZF8NI4WAaX4/OcSeQ=">AAACAXicbZDNSsNAFIUn9a/Wv6pLN4NFcFUTEawLoWgXLitYW2hCmUxu2qEzSZyZCCV05zu41a0rceuTuPNRnLZZaOuBgY9z7mUux084U9q2v6zC0vLK6lpxvbSxubW9U97du1dxKim0aMxj2fGJAs4iaGmmOXQSCUT4HNr+8HqStx9BKhZHd3qUgCdIP2Iho0Qby3MbPVexviCXzkmtV67YVXsqvAhODhWUq9krf7tBTFMBkaacKNV17ER7GZGaUQ7jkpsqSAgdkj50DUZEgPKy6dFjfGScAIexNC/SeOr+3siIUGokfDMpiB6o+Wxi/pd1Ux3WvIxFSaohorOPwpRjHeNJAzhgEqjmIwOESmZuxXRAJKHa9FRyAwjdRuY2gGsyNpU48wUsQuu0elF1bs8q9au8myI6QIfoGDnoHNXRDWqiFqLoAT2jF/RqPVlv1rv1MRstWPnOPvoj6/MHyoaWzg==</latexit>

�✏ > 1/2<latexit sha1_base64="ytllAO6q++x5NJvUsaKTbwInm5I=">AAACBXicbZDNSsNAFIUn9a/Wn0ZdugkWwVVNiqBupGgXLisYW2hCmExv2qGTSZiZCCV07zu41a0rcetzuPNRnLZZaOuBgY9z7mUuJ0wZlcq2v4zSyura+kZ5s7K1vbNbNff2H2SSCQIuSVgiuiGWwCgHV1HFoJsKwHHIoBOObqZ55xGEpAm/V+MU/BgPOI0owUpbgVn1WoEHqaQs4VfOaSMwa3bdnslaBqeAGirUDsxvr5+QLAauCMNS9hw7VX6OhaKEwaTiZRJSTEZ4AD2NHMcg/Xx2+MQ61k7fihKhH1fWzP29keNYynEc6skYq6FczKbmf1kvU9GFn1OeZgo4mX8UZcxSiTVtwepTAUSxsQZMBNW3WmSIBSZKd1Xx+hB5rdxrAVN4oitxFgtYBrdRv6w7d2e15nXRTRkdoiN0ghx0jproFrWRiwjK0DN6Qa/Gk/FmvBsf89GSUewcoD8yPn8A/huX9w==</latexit><latexit sha1_base64="ytllAO6q++x5NJvUsaKTbwInm5I=">AAACBXicbZDNSsNAFIUn9a/Wn0ZdugkWwVVNiqBupGgXLisYW2hCmExv2qGTSZiZCCV07zu41a0rcetzuPNRnLZZaOuBgY9z7mUuJ0wZlcq2v4zSyura+kZ5s7K1vbNbNff2H2SSCQIuSVgiuiGWwCgHV1HFoJsKwHHIoBOObqZ55xGEpAm/V+MU/BgPOI0owUpbgVn1WoEHqaQs4VfOaSMwa3bdnslaBqeAGirUDsxvr5+QLAauCMNS9hw7VX6OhaKEwaTiZRJSTEZ4AD2NHMcg/Xx2+MQ61k7fihKhH1fWzP29keNYynEc6skYq6FczKbmf1kvU9GFn1OeZgo4mX8UZcxSiTVtwepTAUSxsQZMBNW3WmSIBSZKd1Xx+hB5rdxrAVN4oitxFgtYBrdRv6w7d2e15nXRTRkdoiN0ghx0jproFrWRiwjK0DN6Qa/Gk/FmvBsf89GSUewcoD8yPn8A/huX9w==</latexit><latexit sha1_base64="ytllAO6q++x5NJvUsaKTbwInm5I=">AAACBXicbZDNSsNAFIUn9a/Wn0ZdugkWwVVNiqBupGgXLisYW2hCmExv2qGTSZiZCCV07zu41a0rcetzuPNRnLZZaOuBgY9z7mUuJ0wZlcq2v4zSyura+kZ5s7K1vbNbNff2H2SSCQIuSVgiuiGWwCgHV1HFoJsKwHHIoBOObqZ55xGEpAm/V+MU/BgPOI0owUpbgVn1WoEHqaQs4VfOaSMwa3bdnslaBqeAGirUDsxvr5+QLAauCMNS9hw7VX6OhaKEwaTiZRJSTEZ4AD2NHMcg/Xx2+MQ61k7fihKhH1fWzP29keNYynEc6skYq6FczKbmf1kvU9GFn1OeZgo4mX8UZcxSiTVtwepTAUSxsQZMBNW3WmSIBSZKd1Xx+hB5rdxrAVN4oitxFgtYBrdRv6w7d2e15nXRTRkdoiN0ghx0jproFrWRiwjK0DN6Qa/Gk/FmvBsf89GSUewcoD8yPn8A/huX9w==</latexit><latexit sha1_base64="ytllAO6q++x5NJvUsaKTbwInm5I=">AAACBXicbZDNSsNAFIUn9a/Wn0ZdugkWwVVNiqBupGgXLisYW2hCmExv2qGTSZiZCCV07zu41a0rcetzuPNRnLZZaOuBgY9z7mUuJ0wZlcq2v4zSyura+kZ5s7K1vbNbNff2H2SSCQIuSVgiuiGWwCgHV1HFoJsKwHHIoBOObqZ55xGEpAm/V+MU/BgPOI0owUpbgVn1WoEHqaQs4VfOaSMwa3bdnslaBqeAGirUDsxvr5+QLAauCMNS9hw7VX6OhaKEwaTiZRJSTEZ4AD2NHMcg/Xx2+MQ61k7fihKhH1fWzP29keNYynEc6skYq6FczKbmf1kvU9GFn1OeZgo4mX8UZcxSiTVtwepTAUSxsQZMBNW3WmSIBSZKd1Xx+hB5rdxrAVN4oitxFgtYBrdRv6w7d2e15nXRTRkdoiN0ghx0jproFrWRiwjK0DN6Qa/Gk/FmvBsf89GSUewcoD8yPn8A/huX9w==</latexit>

�✏ = 0, P✏ = �1, PT = 0<latexit sha1_base64="aYkpLNuQhBrQGA79ZQwvwWO0xp8=">AAACG3icbVDLSsNAFJ34rPUVdekmWAouaklEUBeFol24rNDaQhPKZHrTDp1MwsxEKKFf4D/4D25160rcunDnpzhtA2rrgQvnnnMvM/f4MaNS2fansbS8srq2ntvIb25t7+yae/t3MkoEgSaJWCTaPpbAKIemoopBOxaAQ59Byx9eT/zWPQhJI95Qoxi8EPc5DSjBSktds+jWui7EkrKIV+ySW6r/tCfOtG9U7K5ZsMv2FNYicTJSQBnqXfPL7UUkCYErwrCUHceOlZdioShhMM67iYQYkyHuQ0dTjkOQXjo9Z2wVtdKzgkjo4sqaqr83UhxKOQp9PRliNZDz3kT8z+skKrjwUsrjRAEns4eChFkqsibZWD0qgCg20gQTQfVfLTLAAhOlE8y7PQjcWurWgCk81pE48wEskuZp+bLs3J4VqldZNjl0iI7QMXLQOaqiG1RHTUTQA3pCz+jFeDRejTfjfTa6ZGQ7B+gPjI9vDM6f6A==</latexit><latexit sha1_base64="aYkpLNuQhBrQGA79ZQwvwWO0xp8=">AAACG3icbVDLSsNAFJ34rPUVdekmWAouaklEUBeFol24rNDaQhPKZHrTDp1MwsxEKKFf4D/4D25160rcunDnpzhtA2rrgQvnnnMvM/f4MaNS2fansbS8srq2ntvIb25t7+yae/t3MkoEgSaJWCTaPpbAKIemoopBOxaAQ59Byx9eT/zWPQhJI95Qoxi8EPc5DSjBSktds+jWui7EkrKIV+ySW6r/tCfOtG9U7K5ZsMv2FNYicTJSQBnqXfPL7UUkCYErwrCUHceOlZdioShhMM67iYQYkyHuQ0dTjkOQXjo9Z2wVtdKzgkjo4sqaqr83UhxKOQp9PRliNZDz3kT8z+skKrjwUsrjRAEns4eChFkqsibZWD0qgCg20gQTQfVfLTLAAhOlE8y7PQjcWurWgCk81pE48wEskuZp+bLs3J4VqldZNjl0iI7QMXLQOaqiG1RHTUTQA3pCz+jFeDRejTfjfTa6ZGQ7B+gPjI9vDM6f6A==</latexit><latexit sha1_base64="aYkpLNuQhBrQGA79ZQwvwWO0xp8=">AAACG3icbVDLSsNAFJ34rPUVdekmWAouaklEUBeFol24rNDaQhPKZHrTDp1MwsxEKKFf4D/4D25160rcunDnpzhtA2rrgQvnnnMvM/f4MaNS2fansbS8srq2ntvIb25t7+yae/t3MkoEgSaJWCTaPpbAKIemoopBOxaAQ59Byx9eT/zWPQhJI95Qoxi8EPc5DSjBSktds+jWui7EkrKIV+ySW6r/tCfOtG9U7K5ZsMv2FNYicTJSQBnqXfPL7UUkCYErwrCUHceOlZdioShhMM67iYQYkyHuQ0dTjkOQXjo9Z2wVtdKzgkjo4sqaqr83UhxKOQp9PRliNZDz3kT8z+skKrjwUsrjRAEns4eChFkqsibZWD0qgCg20gQTQfVfLTLAAhOlE8y7PQjcWurWgCk81pE48wEskuZp+bLs3J4VqldZNjl0iI7QMXLQOaqiG1RHTUTQA3pCz+jFeDRejTfjfTa6ZGQ7B+gPjI9vDM6f6A==</latexit><latexit sha1_base64="aYkpLNuQhBrQGA79ZQwvwWO0xp8=">AAACG3icbVDLSsNAFJ34rPUVdekmWAouaklEUBeFol24rNDaQhPKZHrTDp1MwsxEKKFf4D/4D25160rcunDnpzhtA2rrgQvnnnMvM/f4MaNS2fansbS8srq2ntvIb25t7+yae/t3MkoEgSaJWCTaPpbAKIemoopBOxaAQ59Byx9eT/zWPQhJI95Qoxi8EPc5DSjBSktds+jWui7EkrKIV+ySW6r/tCfOtG9U7K5ZsMv2FNYicTJSQBnqXfPL7UUkCYErwrCUHceOlZdioShhMM67iYQYkyHuQ0dTjkOQXjo9Z2wVtdKzgkjo4sqaqr83UhxKOQp9PRliNZDz3kT8z+skKrjwUsrjRAEns4eChFkqsibZWD0qgCg20gQTQfVfLTLAAhOlE8y7PQjcWurWgCk81pE48wEskuZp+bLs3J4VqldZNjl0iI7QMXLQOaqiG1RHTUTQA3pCz+jFeDRejTfjfTa6ZGQ7B+gPjI9vDM6f6A==</latexit>

Page 29: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Error bar• CFT data errors come from OPE truncation errors

• Predictions with error bars

• Exact values

• Solve the truncated bootstrap equations (labeled by j) by relaxing 1 parameterError bar set by the largest deviation

exact : �✏ = 1, P✏ = 0.25, PT = 1/64 ⇠ 0.0156<latexit sha1_base64="eNq7wfl/jE5B/l1IZupq7YjkrbQ=">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</latexit><latexit sha1_base64="eNq7wfl/jE5B/l1IZupq7YjkrbQ=">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</latexit><latexit sha1_base64="eNq7wfl/jE5B/l1IZupq7YjkrbQ=">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</latexit><latexit sha1_base64="eNq7wfl/jE5B/l1IZupq7YjkrbQ=">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</latexit>

k = 10 : �✏ = 0.9955(88), P✏ = 0.2484(22), PT = 0.0142(13)<latexit sha1_base64="fnFLmkKP9AzXub1EHZ+Z6j2HMg8=">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</latexit><latexit sha1_base64="fnFLmkKP9AzXub1EHZ+Z6j2HMg8=">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</latexit><latexit sha1_base64="fnFLmkKP9AzXub1EHZ+Z6j2HMg8=">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</latexit><latexit sha1_base64="fnFLmkKP9AzXub1EHZ+Z6j2HMg8=">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</latexit>

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2d minimal models

0.05 0.10 0.15 0.20 0.25 0.30Δ0.6

0.8

1.0

1.2

1.4

Δ

6789101112131415

exact relation : �1,3 =2

3(1 + 4�1,2)

<latexit sha1_base64="PTdktmj9bgYtIKqnkSB6VfOT4KA=">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</latexit><latexit sha1_base64="PTdktmj9bgYtIKqnkSB6VfOT4KA=">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</latexit><latexit sha1_base64="PTdktmj9bgYtIKqnkSB6VfOT4KA=">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</latexit><latexit sha1_base64="PTdktmj9bgYtIKqnkSB6VfOT4KA=">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</latexit>

Interpolating 2d minimal models ( Liendo-Rastelli-van Rees ’12 )

Page 31: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

More aboutOPE truncation error

• Derivatives of truncated bootstrap equation not well controlled. How to assign proper weights in error function?

• Examine OPE truncation error more closely

• At z=1/2, truncation error is approximated by cutoff OPE coefficient (squared)

• Near z=1/2, error is cutoff OPE coefficient times conformal block combination2d Ising on the imaginary axis

-3 -2 -1 1 2 3θ

-5

5

1+ϵT

1 + ✏ ⇠ �T<latexit sha1_base64="Ly7/NgDcoq3VevRsPGiYLdiJkGA=">AAACB3icbZDNSsNAFIUn/tb6l+rSzWARBLEkIqi7ol24rNDYQlPKZHrTDp1MwsxEKaEv4Du41a0rcetjuPNRnLZZaOuBgY9z7mUuJ0g4U9pxvqyl5ZXVtfXCRnFza3tn1y7t3as4lRQ8GvNYtgKigDMBnmaaQyuRQKKAQzMY3kzy5gNIxWLR0KMEOhHpCxYySrSxunbJPfEhUYzHwlcswqeNrl12Ks5UeBHcHMooV71rf/u9mKYRCE05UartOonuZERqRjmMi36qICF0SPrQNihIBKqTTU8f4yPj9HAYS/OExlP390ZGIqVGUWAmI6IHaj6bmP9l7VSHl52MiSTVIOjsozDlWMd40gPuMQlU85EBQiUzt2I6IJJQbdoq+j0I/Vrm14BrMjaVuPMFLIJ3VrmquHfn5ep13k0BHaBDdIxcdIGq6BbVkYcoekTP6AW9Wk/Wm/VufcxGl6x8Zx/9kfX5A5wxmN4=</latexit><latexit sha1_base64="Ly7/NgDcoq3VevRsPGiYLdiJkGA=">AAACB3icbZDNSsNAFIUn/tb6l+rSzWARBLEkIqi7ol24rNDYQlPKZHrTDp1MwsxEKaEv4Du41a0rcetjuPNRnLZZaOuBgY9z7mUuJ0g4U9pxvqyl5ZXVtfXCRnFza3tn1y7t3as4lRQ8GvNYtgKigDMBnmaaQyuRQKKAQzMY3kzy5gNIxWLR0KMEOhHpCxYySrSxunbJPfEhUYzHwlcswqeNrl12Ks5UeBHcHMooV71rf/u9mKYRCE05UartOonuZERqRjmMi36qICF0SPrQNihIBKqTTU8f4yPj9HAYS/OExlP390ZGIqVGUWAmI6IHaj6bmP9l7VSHl52MiSTVIOjsozDlWMd40gPuMQlU85EBQiUzt2I6IJJQbdoq+j0I/Vrm14BrMjaVuPMFLIJ3VrmquHfn5ep13k0BHaBDdIxcdIGq6BbVkYcoekTP6AW9Wk/Wm/VufcxGl6x8Zx/9kfX5A5wxmN4=</latexit><latexit sha1_base64="Ly7/NgDcoq3VevRsPGiYLdiJkGA=">AAACB3icbZDNSsNAFIUn/tb6l+rSzWARBLEkIqi7ol24rNDYQlPKZHrTDp1MwsxEKaEv4Du41a0rcetjuPNRnLZZaOuBgY9z7mUuJ0g4U9pxvqyl5ZXVtfXCRnFza3tn1y7t3as4lRQ8GvNYtgKigDMBnmaaQyuRQKKAQzMY3kzy5gNIxWLR0KMEOhHpCxYySrSxunbJPfEhUYzHwlcswqeNrl12Ks5UeBHcHMooV71rf/u9mKYRCE05UartOonuZERqRjmMi36qICF0SPrQNihIBKqTTU8f4yPj9HAYS/OExlP390ZGIqVGUWAmI6IHaj6bmP9l7VSHl52MiSTVIOjsozDlWMd40gPuMQlU85EBQiUzt2I6IJJQbdoq+j0I/Vrm14BrMjaVuPMFLIJ3VrmquHfn5ep13k0BHaBDdIxcdIGq6BbVkYcoekTP6AW9Wk/Wm/VufcxGl6x8Zx/9kfX5A5wxmN4=</latexit><latexit sha1_base64="Ly7/NgDcoq3VevRsPGiYLdiJkGA=">AAACB3icbZDNSsNAFIUn/tb6l+rSzWARBLEkIqi7ol24rNDYQlPKZHrTDp1MwsxEKaEv4Du41a0rcetjuPNRnLZZaOuBgY9z7mUuJ0g4U9pxvqyl5ZXVtfXCRnFza3tn1y7t3as4lRQ8GvNYtgKigDMBnmaaQyuRQKKAQzMY3kzy5gNIxWLR0KMEOhHpCxYySrSxunbJPfEhUYzHwlcswqeNrl12Ks5UeBHcHMooV71rf/u9mKYRCE05UartOonuZERqRjmMi36qICF0SPrQNihIBKqTTU8f4yPj9HAYS/OExlP390ZGIqVGUWAmI6IHaj6bmP9l7VSHl52MiSTVIOjsozDlWMd40gPuMQlU85EBQiUzt2I6IJJQbdoq+j0I/Vrm14BrMjaVuPMFLIJ3VrmquHfn5ep13k0BHaBDdIxcdIGq6BbVkYcoekTP6AW9Wk/Wm/VufcxGl6x8Zx/9kfX5A5wxmN4=</latexit>

-3 -2 -1 1 2 3θ

-15

-10

-5

5

10

ϵT

[4,4][4,0][5,4]

Page 32: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Refined error function• Refined version

• Weights/measure set by the cutoff operator, not identity

• Multipoint sampling: bootstrap equation evaluated at many points(Hogervorst-Rychkov ’13, Echeverri-von Harling-Serone ’16)

• Refined error estimation (not error bound) of OPE truncations: replaced by cutoff conformal block (with spin).“The prefactor is not optimal”. Here, prefactor = cutoff OPE coefficient (squared).

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Page 33: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Subtleties

• UV cutoff depends on omitted operators

• Results more accurate when sampling near UV cutoff Equations more independent Delicate balance of different operators requires precise CFT data

• Spurious zeros/singularities due to spinning blocks

• Recursive procedure 1) error minimization a CFT data2) high-lying CFT data a improve sampling choice a 1)

Page 34: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

Summary• Q: Lengths of the bootstrap equation?

• Many physical CFTs exhibit OPE hierarchy ( so ~ EFT )Ex: Ising, Lee-Yang, 2d minimal models…

• Length of truncated vector sum = OPE truncation error (suppressed by small OPE coefficients)

• New bootstrap methodExtract low-lying CFT data by Error Minimization

• Preliminary positive results have been obtained

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Outlook• A more systematic procedure is under development

weight (measure) by cutoff conformal blockmultipoint sampling where effective OPEs are validrecursive procedure

• Test in mind: make predictions about 3d Isingconsistent with rigorous bound methodsmall numerical effort, reasonably good precision and error bars

• Could be useful for d>2 non-unitary CFTs from statistical physicsLee-Yang edge singularity, percolation, self-avoiding walk/polymer, branched polymer, quenched disorderintriguing dimensional reductions by Parisi-Sourlas supersymmetry

• And bootstrap problems without sign-definitenessboundary bootstrap, defect bootstrap, thermal bootstrap

Page 36: New method for the conformal bootstrap with OPE truncations · Conformal bootstrap • Conformal bootstrap: classify & solve CFTs by OPE associativity OPE: for Wilson’s Operator

More outlook• Classification of effective OPEs, instead of CFTs ?

• Extremal functional method + error minimization ?Many operator dimensions of 3d Ising are predicted by EFM (Simmons-Duffin ’16)Require precise CFT data to be compatible with fragile OPE hierarchy

• New type of rigorous bounds from the triangle inequality in the normed vector space?

• Analytic methods in the Euclidean regime?Perturbation theory based on small OPE coefficientsReproducing approximate zeros/singularities of cutoff spinning blocks

x y

z < x + y

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Thank you!