Universal theory of complex SYK models and extremal charged black holes

Transcription

1 HARVARD Universal theory of complex SYK models and extremal charged black holes Subir Sachdev Jerusalem Winter School, December 31, 2018

2 HARVARD Wenbo Fu Yingfei Gu Grigory Tarnopolsky

3 1. Quantum matter with quasiparticles: random matrix model 2. Quantum matter without quasiparticles: the complex SYK model 3. Einstein-Maxwell theory of charged black holes in AdS space 4. Fluctuations, and the Schwarzian

4 1. Quantum matter with quasiparticles: random matrix model 2. Quantum matter without quasiparticles: the complex SYK model 3. Einstein-Maxwell theory of charged black holes in AdS space 4. Fluctuations, and the Schwarzian

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sha1_base64="qtfrwbsoxqf63lpegxch09tgfe4=">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</latexit> A simple model of a metal with quasiparticles Feynman graph expansion in t ij.., and graph-by-graph average, yields exact equations in the large N limit: D E G( ) T c i ( )c i (0) G(i!) = 1 i! + µ (i!) G( =0 )=Q., ( ) =t 2 G( ) G(!) can be determined by solving a quadratic equation. (!) = 1 Im G(!) 2t µ 2t µ <latexit sha1_base64="z7bwwciuqjemghi4f4oww/cnaoi=">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</latexit>

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sha1_base64="qtfrwbsoxqf63lpegxch09tgfe4=">aaacixicdzdpthsxema9grby/kvpsrelukmhkvjg2cirluupvcijunyveb2txirxxtmzqltkq/tsa7wkn9rb1rfpesceiaj2jeufvm/gy/+yukmhjp0kghubz55vbe+el16+ev2m+xa370xlbfseucaezdybkhp6kfhbwwmbf5mcqty7xuadc7bogn2k8xlsgk+0hevb0vuj5m6ijdq5aaz3e1paho+pmi3wzgdx93newtti7dcovyi7crx3anrmq2qrdz2mmn+s3iiq8jcixz0brqzetoywpvcwcjpkqcnfje9g6kxmbbhp+bks3uqm9eobc/rbhzkdg+uprrpyhw/xvhbuxms+s+a4du+zpfmvbfjh+dctps4rbc3uf40rrdhqjroaswsc1dwllqz0z6ziyi0x6mmfiqopvu9wwicil3ghc7+n7kq98kgeend/i37hu2xh3zqtoy9ranvkpdkjh0ledsgr+uposi8ickm+kytyhfwibolb4od9aynyz7wjf1xw+w5bm6td</latexit> <latexit sha1_base64="qtfrwbsoxqf63lpegxch09tgfe4=">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</latexit> <latexit sha1_base64="qtfrwbsoxqf63lpegxch09tgfe4=">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</latexit> A simple model of a metal with quasiparticles p Let " be the eigenvalues of the matrix t ij / N. The fermions will occupy the lowest NQ eigenvalues, upto the Fermi energy E F. The singleparticle density of states is (!) =(1/N ) P (! " ), and 0 (! = 0). (!) = 1 Im G(!) " level spacing 1/N 0 2t µ 2t µ <latexit sha1_base64="z7bwwciuqjemghi4f4oww/cnaoi=">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</latexit>

8 <latexit sha1_base64="tljbkts021mdkxljrfxvfeutno8=">aaab73icbvdlsgnbeoynrxhfuy9ebopgkeykomegf08swtwgwclszdyzmjo7zvqkieqnvhhqxku/482/cfjanlggoajqprsrsqww6ptfxm5ldw19i79z2nre2d0r7h/ubzizxmsskylprtryktsvoudjm6nhvewsn6lb9crvphjjralvczjyungefrfgfj3ubfuhyc3btrhkl/0pya8jfkkj5qh2ip/tbsiyxtuysa1tbx6k4ygafezycagdwz5snqa93njuu8vtojreoyyntumsodgunjkp+ntirjw1qxw5tkwxbxe9ifif18owvgxhqqczcs1mi+jmekzi5hnsfyyzlenhkdpc3upynxrk0evuccesvbxm6mflwc8hd+elytu8jjwcwtgcqgaxuiebqeingeh4ghd49r68z+/ne5+15rz5zch8gffxduonj3i=</latexit> <latexit sha1_base64="tljbkts021mdkxljrfxvfeutno8=">aaab73icbvdlsgnbeoynrxhfuy9ebopgkeykomegf08swtwgwclszdyzmjo7zvqkieqnvhhqxku/482/cfjanlggoajqprsrsqww6ptfxm5ldw19i79z2nre2d0r7h/ubzizxmsskylprtryktsvoudjm6nhvewsn6lb9crvphjjralvczjyungefrfgfj3ubfuhyc3btrhkl/0pya8jfkkj5qh2ip/tbsiyxtuysa1tbx6k4ygafezycagdwz5snqa93njuu8vtojreoyyntumsodgunjkp+ntirjw1qxw5tkwxbxe9ifif18owvgxhqqczcs1mi+jmekzi5hnsfyyzlenhkdpc3upynxrk0evuccesvbxm6mflwc8hd+elytu8jjwcwtgcqgaxuiebqeingeh4ghd49r68z+/ne5+15rz5zch8gffxduonj3i=</latexit> <latexit sha1_base64="tljbkts021mdkxljrfxvfeutno8=">aaab73icbvdlsgnbeoynrxhfuy9ebopgkeykomegf08swtwgwclszdyzmjo7zvqkieqnvhhqxku/482/cfjanlggoajqprsrsqww6ptfxm5ldw19i79z2nre2d0r7h/ubzizxmsskylprtryktsvoudjm6nhvewsn6lb9crvphjjralvczjyungefrfgfj3ubfuhyc3btrhkl/0pya8jfkkj5qh2ip/tbsiyxtuysa1tbx6k4ygafezycagdwz5snqa93njuu8vtojreoyyntumsodgunjkp+ntirjw1qxw5tkwxbxe9ifif18owvgxhqqczcs1mi+jmekzi5hnsfyyzlenhkdpc3upynxrk0evuccesvbxm6mflwc8hd+elytu8jjwcwtgcqgaxuiebqeingeh4ghd49r68z+/ne5+15rz5zch8gffxduonj3i=</latexit> <latexit sha1_base64="tljbkts021mdkxljrfxvfeutno8=">aaab73icbvdlsgnbeoynrxhfuy9ebopgkeykomegf08swtwgwclszdyzmjo7zvqkieqnvhhqxku/482/cfjanlggoajqprsrsqww6ptfxm5ldw19i79z2nre2d0r7h/ubzizxmsskylprtryktsvoudjm6nhvewsn6lb9crvphjjralvczjyungefrfgfj3ubfuhyc3btrhkl/0pya8jfkkj5qh2ip/tbsiyxtuysa1tbx6k4ygafezycagdwz5snqa93njuu8vtojreoyyntumsodgunjkp+ntirjw1qxw5tkwxbxe9ifif18owvgxhqqczcs1mi+jmekzi5hnsfyyzlenhkdpc3upynxrk0evuccesvbxm6mflwc8hd+elytu8jjwcwtgcqgaxuiebqeingeh4ghd49r68z+/ne5+15rz5zch8gffxduonj3i=</latexit> A simple model of a metal with quasiparticles There are 2 N many body levels with energy Nt Many-body level spacing 2 N E = NX =1 n ", Quasiparticle excitations with spacing 1/N where n =0, 1. Shown are all values of E for a single cluster of size N = 12. The " have a level spacing 1/N.

9 A simple model of a metal with quasiparticles The grand potential (T ) at low T is (from the Sommerfeld expansion) (T ) E 0 = N 2 6 0T 2 + O(T 4 ) +... where 0 (0) is the single particle density of states at the Fermi level. We can also define the many body density of states, D(E), via Z = e (T )/T = Z 1 1 ded(e)e E/T The inversion from (T )tod(e) has to performed with care (it need not commute with the 1/N expansion), and we obtain r! 2N 0 (E E 0 ) 1 D(E) exp, E > E 0, 3 N 0(E E 0 ) N and D(E) = 0 for E<E 0. This is related to the asymptotic growth of the partitions of an integer, p(n) exp( p 2n/3). Near the lower bound, there are large sampleto-sample fluctuations due to variations in the lowest quasiparticle energies. <latexit 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<latexit 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<latexit 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<latexit 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10 <latexit 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A simple model of a metal with quasiparticles Now add weak interactions H = 1 (N) 1/2 NX i,j=1 t ij c i c j µ X i c i c i + 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` c i c j c k c` U ij;k` are independent random variables with U ij;k` = 0 and U ij;k` 2 = U 2.We compute the lifetime of a quasiparticle,, in an exact eigenstate (i) of the free particle Hamitonian with energy ". By Fermi s Golden rule, for " at the Fermi energy 1 = U Z d" d" d" f(" )(1 f(" ))(1 f(" )) (" + " " " ) = 3 U T 2 where 0 is the density of states at the Fermi energy, and f( ) =1/(e /T + 1) is the Fermi function. Fermi liquid state: Two-body interactions lead to a scattering time of quasiparticle excitations from in (random) single-particle eigenstates which diverges as T 2 at the Fermi level.

11 1. Quantum matter with quasiparticles: random matrix model 2. Quantum matter without quasiparticles: the complex SYK model 3. Einstein-Maxwell theory of charged black holes in AdS space 4. Fluctuations, and the Schwarzian

12 H = The complex SYK model (See also: the 2-Body Random Ensemble in nuclear physics; did not obtain the large N limit; T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, and S.S.M. Wong, Rev. Mod. Phys. 53, 385 (1981)) 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` c i c j c k c` µ X i c i c j + c j c i =0, c i c j + c j c i = ij Q = 1 X c i N c i i c i c i U ij;k` are independent random variables with U ij;k` = 0 and U ij;k` 2 = U 2 N!1yields critical strange metal. S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Kitaev, unpublished; S. Sachdev, PRX 5, (2015)

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sha1_base64="h54fe11doi0rewnp9nuj3ouhzuu=">aaab6hicbvdlsgnbeoynrxhfuy9ebopgkewgqb0fpegxafoazamzk95kzozsmjmrhcvf4mwdil79jg/+jzphqrmlgoqqbrq7gkrwbvz328mtrw9sbuw3czu7e/shxcojpo5txbdbyhgrdka1ci6xybgr2e4u0igq2apgt1o/9yrk81g+mhgcfkqhkoecuwol+l2vwhllbuxaq1sijtmqb5muyifar/jv7ccsjvaajqjwhc9njj9rztgtocl0u40jzsm6wi6lkkao/wx26iscwavpwljzkobm1n8tgy20hkeb7yyogeplbyr+53vse175gzdjalcy+aiwfctezpo16xofziixjzqpbm8lbegvzczmu7ahrly8spqvsuewvfpfqxqzicmpj3ak5+dbjvthhmrqaayiz/akb86j8+k8ox/z1pyzmdmgp3a+fwc1gozc</latexit> <latexit sha1_base64="h54fe11doi0rewnp9nuj3ouhzuu=">aaab6hicbvdlsgnbeoynrxhfuy9ebopgkewgqb0fpegxafoazamzk95kzozsmjmrhcvf4mwdil79jg/+jzphqrmlgoqqbrq7gkrwbvz328mtrw9sbuw3czu7e/shxcojpo5txbdbyhgrdka1ci6xybgr2e4u0igq2apgt1o/9yrk81g+mhgcfkqhkoecuwol+l2vwhllbuxaq1sijtmqb5muyifar/jv7ccsjvaajqjwhc9njj9rztgtocl0u40jzsm6wi6lkkao/wx26iscwavpwljzkobm1n8tgy20hkeb7yyogeplbyr+53vse175gzdjalcy+aiwfctezpo16xofziixjzqpbm8lbegvzczmu7ahrly8spqvsuewvfpfqxqzicmpj3ak5+dbjvthhmrqaayiz/akb86j8+k8ox/z1pyzmdmgp3a+fwc1gozc</latexit> <latexit sha1_base64="h54fe11doi0rewnp9nuj3ouhzuu=">aaab6hicbvdlsgnbeoynrxhfuy9ebopgkewgqb0fpegxafoazamzk95kzozsmjmrhcvf4mwdil79jg/+jzphqrmlgoqqbrq7gkrwbvz328mtrw9sbuw3czu7e/shxcojpo5txbdbyhgrdka1ci6xybgr2e4u0igq2apgt1o/9yrk81g+mhgcfkqhkoecuwol+l2vwhllbuxaq1sijtmqb5muyifar/jv7ccsjvaajqjwhc9njj9rztgtocl0u40jzsm6wi6lkkao/wx26iscwavpwljzkobm1n8tgy20hkeb7yyogeplbyr+53vse175gzdjalcy+aiwfctezpo16xofziixjzqpbm8lbegvzczmu7ahrly8spqvsuewvfpfqxqzicmpj3ak5+dbjvthhmrqaayiz/akb86j8+k8ox/z1pyzmdmgp3a+fwc1gozc</latexit> <latexit sha1_base64="h54fe11doi0rewnp9nuj3ouhzuu=">aaab6hicbvdlsgnbeoynrxhfuy9ebopgkewgqb0fpegxafoazamzk95kzozsmjmrhcvf4mwdil79jg/+jzphqrmlgoqqbrq7gkrwbvz328mtrw9sbuw3czu7e/shxcojpo5txbdbyhgrdka1ci6xybgr2e4u0igq2apgt1o/9yrk81g+mhgcfkqhkoecuwol+l2vwhllbuxaq1sijtmqb5muyifar/jv7ccsjvaajqjwhc9njj9rztgtocl0u40jzsm6wi6lkkao/wx26iscwavpwljzkobm1n8tgy20hkeb7yyogeplbyr+53vse175gzdjalcy+aiwfctezpo16xofziixjzqpbm8lbegvzczmu7ahrly8spqvsuewvfpfqxqzicmpj3ak5+dbjvthhmrqaayiz/akb86j8+k8ox/z1pyzmdmgp3a+fwc1gozc</latexit> <latexit sha1_base64="h54fe11doi0rewnp9nuj3ouhzuu=">aaab6hicbvdlsgnbeoynrxhfuy9ebopgkewgqb0fpegxafoazamzk95kzozsmjmrhcvf4mwdil79jg/+jzphqrmlgoqqbrq7gkrwbvz328mtrw9sbuw3czu7e/shxcojpo5txbdbyhgrdka1ci6xybgr2e4u0igq2apgt1o/9yrk81g+mhgcfkqhkoecuwol+l2vwhllbuxaq1sijtmqb5muyifar/jv7ccsjvaajqjwhc9njj9rztgtocl0u40jzsm6wi6lkkao/wx26iscwavpwljzkobm1n8tgy20hkeb7yyogeplbyr+53vse175gzdjalcy+aiwfctezpo16xofziixjzqpbm8lbegvzczmu7ahrly8spqvsuewvfpfqxqzicmpj3ak5+dbjvthhmrqaayiz/akb86j8+k8ox/z1pyzmdmgp3a+fwc1gozc</latexit> The complex SYK model Feynman graph expansion in U ijk`, and graph-by-graph average, yields exact equations in the large N limit: G(i!) = 1 i! + µ (i!) G( =0 )=Q., ( ) = U 2 G 2 ( )G( ) = U G<latexit sha1_base64="h54fe11doi0rewnp9nuj3ouhzuu=">aaab6hicbvdlsgnbeoynrxhfuy9ebopgkewgqb0fpegxafoazamzk95kzozsmjmrhcvf4mwdil79jg/+jzphqrmlgoqqbrq7gkrwbvz328mtrw9sbuw3czu7e/shxcojpo5txbdbyhgrdka1ci6xybgr2e4u0igq2apgt1o/9yrk81g+mhgcfkqhkoecuwol+l2vwhllbuxaq1sijtmqb5muyifar/jv7ccsjvaajqjwhc9njj9rztgtocl0u40jzsm6wi6lkkao/wx26iscwavpwljzkobm1n8tgy20hkeb7yyogeplbyr+53vse175gzdjalcy+aiwfctezpo16xofziixjzqpbm8lbegvzczmu7ahrly8spqvsuewvfpfqxqzicmpj3ak5+dbjvthhmrqaayiz/akb86j8+k8ox/z1pyzmdmgp3a+fwc1gozc</latexit> U S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

14 <latexit 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<latexit sha1_base64="p3n7qerjsijwfnhlldvq9sjupeq=">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</latexit> <latexit 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<latexit sha1_base64="p3n7qerjsijwfnhlldvq9sjupeq=">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</latexit> The complex SYK model Solution of these equations and corresponding evaluation of the free energy yields the following universal results (i.e. all results are quantitatively unchanged by adding additional higher q fermion terms): At long times, and at T = 0, G( ) 2 with = 1/q () indication there are no quasiparticles) At general charge Q, there is a spectral symmetry determined by a parameter E: G( ) 2 > 0 e 2 E ( ) 2, T =0 < 0 There is a universal Luttinger relation between the total charge 0 < Q < 1 1 < E < 1 and e 2 E sin( + ) = sin( ) Q = sin(2 ) sin(2 )

15 <latexit 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sha1_base64="p3n7qerjsijwfnhlldvq9sjupeq=">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</latexit> The complex SYK model Solution of these equations and corresponding evaluation of the free energy yields the following universal results (i.e. all results are quantitatively unchanged by adding additional higher q fermion terms): At long times, and at T = 0, G( ) 2 with = 1/q () indication there are no quasiparticles) At general charge Q, there is a spectral symmetry determined by a parameter E: G( ) 2 > 0 e 2 E ( ) 2, T =0 < 0 There is a universal Luttinger relation between the total charge 0 < Q < 1 1 < E < 1 and e 2 E sin( + ) = sin( ) Q = sin(2 ) sin(2 ) S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

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The complex SYK model Solution of these equations and corresponding evaluation of the free energy yields the following universal results (i.e. all results are quantitatively unchanged by adding additional higher q fermion terms): At long times, and at T = 0, G( ) 2 with = 1/q () indication there are no quasiparticles) At general charge Q, there is a spectral symmetry determined by a parameter E: G( ) 2 > 0 e 2 E ( ) 2, T =0 < 0 There is a universal Luttinger relation between the total charge 0 < Q < 1 1 < E < 1 and e 2 E sin( + ) = sin( ) Q = sin(2 ) sin(2 ) S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

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<latexit sha1_base64="p3n7qerjsijwfnhlldvq9sjupeq=">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</latexit> The complex SYK model Solution of these equations and corresponding evaluation of the free energy yields the following universal results (i.e. all results are quantitatively unchanged by adding additional higher q fermion terms): At long times, and at T = 0, G( ) 2 with = 1/q () indication there are no quasiparticles) At general charge Q, there is a spectral symmetry determined by a parameter E: G( ) 2 > 0 e 2 E ( ) 2, T =0 < 0 There is a universal Luttinger relation between the total charge 0 < Q < 1 e 2 E sin( + ) = sin( ) Q = sin(2 ) sin(2 ) 1 < E < 1 and A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001) R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, PRB 95, (2017)

18 <latexit sha1_base64="uaurchyj8zbbewvt2t0qygobcka=">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</latexit> <latexit sha1_base64="uaurchyj8zbbewvt2t0qygobcka=">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</latexit> <latexit sha1_base64="uaurchyj8zbbewvt2t0qygobcka=">aaadixicdvlfb9mwee7baanjy4nhxk6sie60xvi1bekaniyej0pqfkhlnxzn2lhznmy+rjsofygv/ce84bsdte1wuutpvs+fl99dmelhyhv/vaq1pqcphy0/rq88wv17ur7x7nikuez4xfoz6toqgzrc4rejkniaawrjkpekvdwo8yfxqi1ivz8mgq4snljikdgje3sxurkhkhuqqkx1imsruiugtmrpnnadekf9n6dr/+g2wjbgseniqgedk8q8licxonjueaqgqu6ybem655rrvbgmhmkkmysm4mn9szmglm/bb2jdpgrdzxib50w7c0emichjnmni8xt6udkn0yiwv5okd95asezdj9qaqokqmgu5fnlfqgbjm4tswyywo5i2zgtvu2t3vzmlolvyxxg/o9nf2+4hrvy9qbxdrxsco0aggof7xjqjlredpxkbmumsionjaxagiituiw1gboeegrfkbmxfb9y9wn90o+6o33vrgdvxxhfx9y3we77vd8hrulpydbzxelh+j4hsnie2u1wyy848n6nbsajfdis3mdf+yuz4zqficzpwdc0ym4odyjysu3hlkxhyxtlpecxob18uwglsl4wwwrzv7udmdvwjd5btchdqcjxlhirphxrmeiifcvigeho5yykfjgthywyem9sw65hvmmihv40olglchzqwkx2n6hv8ouqzbhyb/4pjrvwx433rbu59wti27lxwxjpnx3n2nd3nq3pohdm88ru6vf2trtvwal5tt/zutq1wfneeo3eidvaxmkefew==</latexit> <latexit sha1_base64="uaurchyj8zbbewvt2t0qygobcka=">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</latexit> The complex SYK model Solution of these equations and corresponding evaluation of the free energy yields the following universal results (i.e. all results are quantitatively unchanged by adding additional higher q fermion terms): At T>0, we obtain a solution with a conformal structure G( ) = A e 2 ET p 1+e 4 E 1/2 T, 0 < < 1/T, sin( T ) where the particle-hole asymmetry is determined by E A. Georges and O. Parcollet PRB 59, 5341 (1999) S. Sachdev, PRX 5, (2015)

19 The complex SYK model We now examine the behavior of the chemical potential, µ, as T! 0 at fixed Q. For this we relate the long-time conformal Greens function, (valid for 1/U) to its short-time behavior. In particular at! n U we have G(i! n )= 1 i! n µ (i! n ) which implies for the spectral density of the Green s function, ( ) µ = Z 1 1 d ( ), which makes it evident that µ depends only upon the particle-hole asymmetric part of the spectral density. Next, we can relate the integrals to the derivative of the imaginary time correlator µ G( =0 + G( =(1/T ) ).

20 The complex SYK model We pull out an explicitly particle-hole asymmetric part of G( ) by defining G( ) e 2 ET G c ( ), 0 < < 1 T. where G c will be given by a particle-hole symmetric conformal form at low T and low!. Then we obtain µ = 2 ET G( =0 + )+G( =(1/T ) g( =0 + ) + terms dependent on G c = 2 ET + terms dependent on G c It can be shown that all the terms dependent upon G c have a T dependence that is weaker than linear in T provided Q is held fixed. Hence we have µ = µ 0 2 ET + terms vanishing as T p with p>1 with µ 0 a non-universal constant. From this relation we obtain

21 ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> ="(null)">(null)</latexit> The complex SYK model E with µ 0 a non-universal constant. From this relation we = 2 E, T! Using a Maxwell relation we then have =2 E 6= 0 as T! 0. Q T

22 <latexit sha1_base64="yi2wwv614t/akulca8+x7kfn7mg=">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</latexit> <latexit sha1_base64="yi2wwv614t/akulca8+x7kfn7mg=">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</latexit> <latexit sha1_base64="yi2wwv614t/akulca8+x7kfn7mg=">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</latexit> <latexit sha1_base64="yi2wwv614t/akulca8+x7kfn7mg=">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</latexit> The complex SYK model Solution of these equations and corresponding evaluation of the free energy yields the following universal results (i.e. all results are quantitatively unchanged by adding additional higher q fermion terms): There is a non-vanishing entropy in the zero temperature limit S(T! 0) = Ns The saddle point equations imply the relation ds 0 dq =2 E Integrating this relation from s 0 = 0, Q = 0, allows us to compute s 0 as a function of Q. A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001)

23 <latexit sha1_base64="yi2wwv614t/akulca8+x7kfn7mg=">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</latexit> <latexit sha1_base64="yi2wwv614t/akulca8+x7kfn7mg=">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</latexit> <latexit sha1_base64="yi2wwv614t/akulca8+x7kfn7mg=">aaady3icdvlbbtrade13uzrwa+grf4suuhgwyq4a2pdkfqged0arepoa1wp24irwjznhztkydxnko/gufounjtmwtlasrbhsofbxssefigod4odcp3vt+o2b87f823fu3ru/sphg16hsc9zhsii9p2egbuncswqf7hcawt4rudc5fn3k945qg1jy204lhouslzqqz9afxotzpykpsmyorr9nmcvzkcwctrd2o8ydfztdjuaggeglxxwxssyjmyldafvmgstydoeetqkhkfazwzqiojz+vi3jfijnc2az2v+8va2rpjsztgt1dmftwiepymybpinixmoap0izx0ad8whd4lggfi66bfxsttmyolwq05amrthgrmopfproxqu4avi7x9seorpvvl07ikoicjopvqmv4pjewkzdni0onnpindwerksces0o6xd0nkpvqvx1jsceumcgsgnwavd5uvpsat1gjchjkxlbsswxwl3+jmfzfsyls0e/wa1xxg4g6a+cyc0mnrouhge4hee/ag3jo7xn8ckvkfa8zn3eugdghaycwo4qpi1x0ey7nfgwfshsphcuzdma5/erfaz1r1v7bju8cckyeqxd5+rvoxfbfcunmeyt97jhb/7ozzbwb+6gtmnaqclzkcl5rffsikan5nyhjo3cug3hxlgmrxt4xtwqrbtwpzlozl+mnqsii1/tmcwut7xsd0k2yp0pav93dodox/5ga7i08eputnnvkffyw/yg3qq34b3znr0dj3eg87bzqbpz/da13zput9ntztwp5qf3ybrffwpq+jtf</latexit> <latexit sha1_base64="yi2wwv614t/akulca8+x7kfn7mg=">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</latexit> The complex SYK model Solution of these equations and corresponding evaluation of the free energy yields the following universal results (i.e. all results are quantitatively unchanged by adding additional higher q fermion terms): There is a non-vanishing entropy in the zero temperature limit S(T! 0) = Ns The saddle point equations imply the relation ds 0 dq =2 E Integrating this relation from s 0 = 0, Q = 0, allows us to compute s 0 as a function of Q. A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001)

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sha1_base64="vexfeqgzbsxvb+tcdrwyixhvcxg=">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</latexit> The complex SYK model NU Many-body level spacing 2 N = e N ln 2 Non-quasiparticle excitations with spacing e Ns 0 There are 2 N many body levels with energy E. Shown are all values of E for a single cluster of size N = 12. The T! 0 state has an entropy S GP S = Ns 0,where s 0 < ln 2 is determined by integrating ds 0 dq =2 E. At Q =1/2, s 0 = G + ln(2) 4 = where G is Catalan s constant. GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001) W. Fu and S. Sachdev, PRB 94, (2016)

25 1. Quantum matter with quasiparticles: random matrix model 2. Quantum matter without quasiparticles: the complex SYK model 3. Einstein-Maxwell theory of charged black holes in AdS space 4. Fluctuations, and the Schwarzian

26 1. Quantum matter with quasiparticles: random matrix model 2. Quantum matter without quasiparticles: the complex SYK model 3. Einstein-Maxwell theory of charged black holes in AdS space S. Sachdev, Phys. Rev. Lett. 105, (2010) S. Sachdev, PRX 5, (2015) 4. Fluctuations, and the Schwarzian

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sha1_base64="j/odcfzzeni2gnxhuj2gqux4/go=">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</latexit> S 2 of radius r 0 r Solutions of S EM have metric and gauge field (F = da) ds 2 = V (r)dt 2 + r 2 d 2 d + dr2 V (r) V (r) =1+ r2 L M r 2d 2 r d 1., A = µ 1! r0 d 1 r d 1 dt ~x where d 2 d is the metric of the d-sphere. All parameters of the solution are determined in terms of the chemical potential µ, and the Hawking temperature of horizon, T. <latexit sha1_base64="8v9kmhcq0oktcpsbqyojm8vxbqw=">aaaecxicdvplbtqwfe1nejtwaoedmysaqqmygswjoralsi0i1auvrx1kzxtksz3eaukkttnqimip8bvsycuoseurwpav3kqzbufwptjh93gofw8yzgkhjev+ngo0r12/cxp6ln37zt1792dmh+zpnfeu79i0ttxbkggec8l3jtaxp8guj8kw5vvd45dvfp+uky1suwngge8njjqiejqyda1mg498mqrjudt2dhrnlvddgoczpshebzyorosuq8knehsizbcspoqqcb4zadmvyrxyurng+0meclnwe0muiqpsnp8hpo96go/axkstado4ewykvwb+24shzmbqlx8oqgugkbkoukvwt3lcod1+ryojn+tgxzwwlfcq8bxeddyjgnw8sizugicvrbizsg7wajjphlyb71fn+taxuz7qkjjnxgvtn7hcepydibtsuzg6xzo95j7obl7qds9vbj9ldqudzxfxhjyrl3vaaddrruox0dxoyu5hz1v6f9bjgdkicgbg3cyzejwbiejjqlcce2cgjft4io9gpmhhikvqckyckyotc9r17kqmdxj2lgsizungfte5emjxlcrxppvtchac7mbmzu263vliogciakow9hzfxembn/bmwwpbgsz88llk8wqlauy0pvtczpqlooygms9tp9c8i/syhpwqoctb6ty7fzmuyb+ov+qsnmkqqzaqfksb2nu1uccj1qnkijkjmzh+o1y5/xu7ze2w3c+ezhjscz0xcviytarvbwfmke5npejaqbj4bgwltofwq7cxh5nlw//bxq/ruv3vxw9u7cw4m9pwy+uj1bi8a8laszaslwvxoo2i8anxufgl+bh5tfmt+f08tte1rnlo/whnh78bfxphpq==</latexit> A. Chamblin, R. Emparan, C.V. Johnson, and R.C. Myers, PRD 60, (1999)

28 <latexit sha1_base64="gfxjj3bkhiedoeaohn4ssxh2tvy=">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</latexit> <latexit 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sha1_base64="gfxjj3bkhiedoeaohn4ssxh2tvy=">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</latexit> <latexit 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sha1_base64="c5pejxneqs6nhyn9lpno1mfmatw=">aaad9nicdvndb9mwfm1apkb52ucrlytwpfasu1inbxtamoae9gdsbvus6lk5tpnydzzmdravkl/cg+kvv8mf4dfwye3wcjyns5gpz7f97z25hmdkwuf7pxcazvu379xdvne6/+dho8dly0+objobjg5zqljzmqzwkknfoznoizpmcjqmltget95u+vgzmfam+sbnmzfmakrlkbl1si2wf34rnurnhxytmhar1iv0ipgfrdkifylwqszapy4qyde1kqcpwse3vprnie4vmnz+meubktpnqivlfpbqzlqv+yw8bhiayoopo/htwxtbccseglf0oralqjeadn8zdcu5d/asv72g+67cxmgodsoylmiezhlflhq9xfye5lcd1hnxce+iiddwcal7u/r8inuewkjjsykykvcplyrs43agftohsxjumpqc/dz02nbe21hxxslfryena9zm7r6xcv6o/qennjchzaksker4i9gsfkz0b3tky+daqc76ykc+1ksut50fcpptdquzuzlmxvwbzbiz/tch9havksf4twzuzpxnjpzw/du/2fxfdwbbprbsvnryt/oqzjgvbzb2rku/cu9zjou6pqi1g8dp3lcgxkmmqubirciwawgkbgg1tyrd5wcyszucfnvvlvacrq5havdtdmr238mftaydjmpcwavtr2t1ltdog9yfm8nc6ix3qrplqggusfwhantg0gjm1bqbzuzi2nx/uubwobsiffhwdotigjhx4c4lxzjfutsvvxm/4p/gql8w+gvbfn9le2dm2ql3zhvudbza2/c2vv1vzzv0won9wza+nmrmrfnr81vz++xwxslszfpvymj++amkikhc</latexit> <latexit sha1_base64="c5pejxneqs6nhyn9lpno1mfmatw=">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</latexit> <latexit sha1_base64="c5pejxneqs6nhyn9lpno1mfmatw=">aaad9nicdvndb9mwfm1apkb52ucrlytwpfasu1inbxtamoae9gdsbvus6lk5tpnydzzmdravkl/cg+kvv8mf4dfwye3wcjyns5gpz7f97z25hmdkwuf7pxcazvu379xdvne6/+dho8dly0+objobjg5zqljzmqzwkknfoznoizpmcjqmltget95u+vgzmfam+sbnmzfmakrlkbl1si2wf34rnurnhxytmhar1iv0ipgfrdkifylwqszapy4qyde1kqcpwse3vprnie4vmnz+meubktpnqivlfpbqzlqv+yw8bhiayoopo/htwxtbccseglf0oralqjeadn8zdcu5d/asv72g+67cxmgodsoylmiezhlflhq9xfye5lcd1hnxce+iiddwcal7u/r8inuewkjjsykykvcplyrs43agftohsxjumpqc/dz02nbe21hxxslfryena9zm7r6xcv6o/qennjchzaksker4i9gsfkz0b3tky+daqc76ykc+1ksut50fcpptdquzuzlmxvwbzbiz/tch9havksf4twzuzpxnjpzw/du/2fxfdwbbprbsvnryt/oqzjgvbzb2rku/cu9zjou6pqi1g8dp3lcgxkmmqubirciwawgkbgg1tyrd5wcyszucfnvvlvacrq5havdtdmr238mftaydjmpcwavtr2t1ltdog9yfm8nc6ix3qrplqggusfwhantg0gjm1bqbzuzi2nx/uubwobsiffhwdotigjhx4c4lxzjfutsvvxm/4p/gql8w+gvbfn9le2dm2ql3zhvudbza2/c2vv1vzzv0won9wza+nmrmrfnr81vz++xwxslszfpvymj++amkikhc</latexit> Charged black holes In the T! 0 limit, at fixed µ, we obtain a charged black hole solution with radius r 0 (T! 0,µ)=R h. All properties of this black hole can be expressed in terms of R h The total charge in the black hole is p Q = Rd 1 h 2d [(d + 1)R 2 h +(d 1)L 2 ] apple 2 g F The Bekenstein-Hawking entropy remains finite as T! 0(s d is the area of the d-dimensional surface of a unit sphere) S(T! 0) = s , s 0 = 2 s d apple 2 R d h A. Chamblin, R. Emparan, C.V. Johnson, and R.C. Myers, PRD 60, (1999)

29 <latexit sha1_base64="gfxjj3bkhiedoeaohn4ssxh2tvy=">aaac5nicdvjnb9naelxnvwlfay5crsriryoiowpqe6hu4ak3gpq2uhxf6/ukxnw9a+2om0rw/gi3xjv/hfgnhfm7qsovjlts08zsvldvni2lsbrfp/3g1u07d+/t3o88epjo8zpu02envleg45hrqc15yixkoxbmgiselwzzkuo8sy/en/wzszrwahvc6xknbvsomreckuvnuj8spyxkufhnowlkecitsixy5msm0uuiqpcienqhrjaxk8wgtioq7mmsqafehaigpgdm4uqpzpwcci0rrjzvwwflqtkylonkqmhmeezeyoidg/gaduhzla8h8fzkki0u0zbac3ruzal7ftrncliexlmnwutonqzcu7tnytolm+v2okeu749gmtjqhgn7bw6igyhe20zp28bxrpsrytsvcucel8zasryvnk2ze8elbjpjzbf0ctgcjw4qvqdtz5eitc2c1u0unvcqagtntxfhebtz65drvli7lllxwtdk7c1ak/xxbvlrfh9ac1vwhipfec0rcashwsxkwianuxaacsoc7hyrjdtn3csl7m+obev1qriipcgctz0sauos+umh/b+cdgdxnig/dxth77am7xgvvjferhd7e96r98e79sye9w997ku/cplgs/a1+hbvgvjbo8+9vyl4/htomuxv</latexit> <latexit 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sha1_base64="c5pejxneqs6nhyn9lpno1mfmatw=">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</latexit> <latexit sha1_base64="c5pejxneqs6nhyn9lpno1mfmatw=">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</latexit> <latexit sha1_base64="c5pejxneqs6nhyn9lpno1mfmatw=">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</latexit> Charged black holes In the T! 0 limit, at fixed µ, we obtain a charged black hole solution with radius r 0 (T! 0,µ)=R h. All properties of this black hole can be expressed in terms of R h The total charge in the black hole is p Q = Rd 1 h 2d [(d + 1)R 2 h +(d 1)L 2 ] apple 2 g F The Bekenstein-Hawking entropy remains finite as T! 0(s d is the area of the d-dimensional surface of a unit sphere) S(T! 0) = s , s 0 = 2 s d apple 2 R d h A. Chamblin, R. Emparan, C.V. Johnson, and R.C. Myers, PRD 60, (1999)

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<latexit 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<latexit 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Charged black holes In the T! 0 limit, at fixed µ, we obtain a charged black hole solution with radius r 0 (T! 0,µ)=R h. All properties of this black hole can be expressed in terms of R h In the near-horizon region, we change co-ordinates from r to so that r R h = R2 2, R 2 = LR h p d(d + 1)R 2 h +(d 1) 2 L 2. Then the near-horizon metric becomes AdS 2 S d,with ds 2 = R 2 2 apple dt 2 + d R 2 hd 2 d, A = E dt. where the dimensionless electric field E is E = g F R h p 2d [(d + 1)R 2 h +(d 1)L 2 ] 2[d(d + 1)R 2 h +(d 1)2 L 2 ].

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sha1_base64="rc+dbuas+iis7q2cx3mkmpgqwfi=">aaab93icdvdlssnafj3uv62vqks3g63gqsiknltlohuxfu1bagoztcft0mmdmukhlh6dw125e7d+jgv/xwlaquupxdiccy/33umnnemf0ltrwfvf2nwqbpd2dvf2d8qhr20zp4jql8q8fl0fs8pzrf3ffkfdrfac+px2/mnvwu9mqzasju5ullavxkoibyxgpsw3entvvwflcjkdmum4nkqmatsdrk0tczwqxyowixjuwaqtqfmjp4xjgtjiey6l7fkoud4mc8uip/nsp5u0wwscr7snayrdki+gu5binhqz/o45pnpmeaax0bupmkvfh2c4ldilfd0zyjwwv72f+jfxs1xq8gyssljfi7jcfkqcqhguqobdjihrpnmee8h02zcmscbe6ahkoo+vp+h/pg3rhezrxq40l1fjfmejoaxnwaj10atxoavcqaadd+arpbmz8wy8gk/l1okxmjkgp2c8fqlo7plg</latexit> Charged black holes Black hole horizon of radius R h and entropy s 0 AdS 2 S 2 ds 2 =(d 2 dt 2 )/ 2 + d~x 2 Gauge field: A =(E/ )dt total charge Q <latexit 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sha1_base64="j/odcfzzeni2gnxhuj2gqux4/go=">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</latexit> <latexit sha1_base64="j/odcfzzeni2gnxhuj2gqux4/go=">aaaclhicdvdlsgmxfm34dnxvxbojvsgflezp4cxfny4t2cp0sslkbmswkxmsjfcg/oqf4fe41zubebd+h+m0bru9ediccy6590sz4noq8ubmzs8sli2vrlpr6xubw5xtnbzoc8wgxvkrqpuiahbcqstwi+amu0ctsmb1dhc+9q/vqwmeyiszzkcb0ihkfc6osvkveoqrdlgsgegdausa1farhpjdujuafbam1nwykorm6manqcazzk9sjtw/4ft+hzmacu78oggjrwjsb2cvrkpu0rsxvcphgkcst+w4e1trjkcy0y2ompwjgllhrigj7i4oogoppano4/iez7qk3ai8doqprrnjfqrskwax6vfhgizad5pijse769/ewpzl6+smh3qllrpcggstj/q5wcbf4+zwzbuwi4awuka4xbtsitjbh3zth7oj8f+kxbcn1bxmvxp6nm1mbe2hfxsephscttefukqtxnadeklp6mv5df6dd+djep1zpjo76aeczy9wt6kr</latexit> S 2 = 1 ~x The entropy s 0, the charge Q, and the dimensionless electric field E obey ds 0 dq =2 E A. Sen, JHEP 0509, 038 (2005)

32 Charged black holes Dimensional reduction to 2D gravity We neglect all dependence of the metric on angular co-ordinates, introduce a scalar field (r, ), and write the 4D metric as ds 2 = 1 p h ab dx a dx b + d 2 2 where a, b = x, and h ab is a 2D metric. Then the 4D action S EM reduces to an action for 2D gravity S JT = 2 Z apple 2 d 2 x p Z( )L 2 h R 2D + V ( ) F 2 g 2 F with V ( )= 2 p + 6p L 2, Z( )= 3/2 This describes Jackiw-Teitelbaum 2D gravity, along with a 2D electromagnetic field. <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit> <latexit sha1_base64="(null)">(null)</latexit>

33 <latexit sha1_base64="geqm+fl4qxcp8+2byzrif1lxcgy=">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</latexit> <latexit sha1_base64="geqm+fl4qxcp8+2byzrif1lxcgy=">aaadyxicdvjbt9rafc67xnc9aproy4lgwokytjs1qkijabqthss4xaxdcdo93u6ytsvmffiapvmr/cn+ex893v0med1pm1/p+c6l35kgk8jy1/01vavfuxvv/vsdxsnhj5/mzm493tnprjnu8lsm+ibgbqvquguflxiqawrjihe/ohlfxffpuburqp4dzthp2ecjshbmyxu8n/xtv6lqisrb8amccfuii4m4xllhv6ixczloa4qiduipedmddbqyvrynwlysa0sjqkudua54vbzxlmxmjj6qknujk2cszrnnnv41cyxjjbsmzmzly6vpw5yvwvtyhk3wi814guffcgf8tibfktitxycselaxy7lwzam2hqevocj2bxpjwjcy2eakxk9keapz8czfloihxsc2s0eft9khr9ochdcanc68gdho8fz6fqvv8fgfn+c+j1ej2bjhe8a0fvwiasirmbkmcvo9xarhierbaacwhgaic9fgxghdlyzg9vqrolvisomoje+ttoiffp22dukaomypj/i7nupzebftemvdrgcerkzg9fw6u94bb+kzdya2czz72w9tnifuietmzkhnzrzftcanfecgm8zp2aapcsqwogmfzyizi9gvrtethjcudie2s6+ympjety5yykiagjjvr5rbszf4/4gd5jza6xdczblfxcenonyscn1dcivgbuwqaona0njay0brjx2pkkg6/mpg48k3eghpruh9iq5qjul1pqf8h+x12p7b9r505jfetusbdp47l5ym4zmrzobzydlxdh1eg9rh2ufatn27flq/qf+oqbwpsc4z54bvf/wb4bc41q==</latexit> <latexit sha1_base64="geqm+fl4qxcp8+2byzrif1lxcgy=">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</latexit> <latexit sha1_base64="geqm+fl4qxcp8+2byzrif1lxcgy=">aaadyxicdvjbt9rafc67xnc9aproy4lgwokytjs1qkijabqthss4xaxdcdo93u6ytsvmffiapvmr/cn+ex893v0med1pm1/p+c6l35kgk8jy1/01vavfuxvv/vsdxsnhj5/mzm493tnprjnu8lsm+ibgbqvquguflxiqawrjihe/ohlfxffpuburqp4dzthp2ecjshbmyxu8n/xtv6lqisrb8amccfuii4m4xllhv6ixczloa4qiduipedmddbqyvrynwlysa0sjqkudua54vbzxlmxmjj6qknujk2cszrnnnv41cyxjjbsmzmzly6vpw5yvwvtyhk3wi814guffcgf8tibfktitxycselaxy7lwzam2hqevocj2bxpjwjcy2eakxk9keapz8czfloihxsc2s0eft9khr9ochdcanc68gdho8fz6fqvv8fgfn+c+j1ej2bjhe8a0fvwiasirmbkmcvo9xarhierbaacwhgaic9fgxghdlyzg9vqrolvisomoje+ttoiffp22dukaomypj/i7nupzebftemvdrgcerkzg9fw6u94bb+kzdya2czz72w9tnifuietmzkhnzrzftcanfecgm8zp2aapcsqwogmfzyizi9gvrtethjcudie2s6+ympjety5yykiagjjvr5rbszf4/4gd5jza6xdczblfxcenonyscn1dcivgbuwqaona0njay0brjx2pkkg6/mpg48k3eghpruh9iq5qjul1pqf8h+x12p7b9r505jfetusbdp47l5ym4zmrzobzydlxdh1eg9rh2ufatn27flq/qf+oqbwpsc4z54bvf/wb4bc41q==</latexit> Charged black holes Probe fermion in the AdS2 near horizon A probe fermion has a near-horizon Green s function with a conformal structure G( ) = A e 2 ET p 1+e 4 E T sin( T ) 1/2, 0 < < 1/T, where the particle-hole asymmetry is determined by E. This is identical to the complex SYK model. T. Faulkner, Hong Liu, J. McGreevy, D. Vegh, PRD 83, (2011)

34 1. Quantum matter with quasiparticles: random matrix model 2. Quantum matter without quasiparticles: the complex SYK model 3. Einstein-Maxwell theory of charged black holes in AdS space 4. Fluctuations, and the Schwarzian

35 The SYK model 1 G(i!) =, ( ) = U 2 G 2 ( )G( ) i! + µ (i!) 1 A (z) =µ z +..., G(z) = pz Ap S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

36 The SYK model 1 G(i!) =, ( ) = U 2 G 2 ( )G( ) Xi! + Xµ (i!) 1 A (z) =µ Ap X z +..., G(z) = pz At frequencies U, thei! + µ can be dropped, and without it equations are invariant under the reparametrization and gauge transformations. The singular part of the self-energy and the Green s function obey Z 0 d 2 sing ( 1, 2 )G( 2, 3 )= ( 1 3 ) sing ( 1, 2 )= U 2 G 2 ( 1, 2 )G( 2, 1 ) A. Kitaev, 2015 S. Sachdev, PRX 5, (2015)

37 Z 0 The SYK model d 2 ( 1, 2 )G( 2, 3 )= ( 1 3 ) ( 1, 2 )= U 2 G 2 ( 1, 2 )G( 2, 1 ) These equations are invariant under = f( ) G( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] ( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] 1/4 g( 1 ) g( 2 ) e G( 1, 2) 3/4 g( 1 ) g( 2 ) e ( 1, 2) where f( ) and g( ) are arbitrary functions. By using f( ) = tan( T )/( T ) we can now obtain the T>0 solution from the T = 0 solution. A. Kitaev, 2015 S. Sachdev, PRX 5, (2015)

38 The SYK model Let us write the large N saddle point solutions of S as G s ( 1 2 ) ( 1 2 ) 1/2 s ( 1 2 ) ( 1 2 ) 3/2. The saddle point will be invariant under a reperamaterization f( ) when choosing G( 1, 2 )=G s ( 1 2 ) leads to a transformed G( e 1, 2) =G s ( 1 2 ) (and similarly for ). It turns out this is true only for the SL(2, R) transformations under which f( ) = a + b c + d, ad bc =1. So the (approximate) reparametrization symmetry is spontaneously broken down to SL(2, R) by the saddle point. A. Kitaev

39 <latexit sha1_base64="dmnlhgvdbndar4oweq3v3wjmvqw=">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</latexit> <latexit sha1_base64="dmnlhgvdbndar4oweq3v3wjmvqw=">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</latexit> <latexit sha1_base64="dmnlhgvdbndar4oweq3v3wjmvqw=">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</latexit> <latexit sha1_base64="dmnlhgvdbndar4oweq3v3wjmvqw=">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</latexit> Fluctuations The saddle-point G( 1 2 )= A e 2 ET ( 1 2 ) p 1+e 4 E 2 T sin( T ( 1 2 )) is invariant only under PSL(2, R) transformations which map the thermal circle onto itself, and an associated gauge transformation tan( Tf( )) T = tan( T ) a T tan( T ) c T + b + d, ad bc =1, i ( ) = i 0 +2 ET ( f( )) A. Kitaev, 2015 R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, PRB 95, (2017)

40 Infinite-range (SYK) model without quasiparticles After introducing replicas a = 1...n, and integrating out the disorder, the partition function can be written as Z " X Z Z = Dc ia ( )exp d ia µ c U 2 X 4N 3 ab Z 0 ia 0 d d 0 X i c ia ( )c ib( 0 ) For simplicity, we neglect the replica indices, and introduce the identity Z " Z 1= D ( 1, 2 )exp N d 1 d 2 ( 1, 2 ) G( 2, 1 ) N X i c i ( 2 )c i ( 1)!#.

41 Infinite-range (SYK) model without quasiparticles Then the partition function can be written as a path integral with an action S analogous to a Luttinger-Ward functional Z Z = DG( 1, 2 )D ( 1, 2 )exp( NS) S =lndet[ ( 1 2 )(@ 1 + µ) ( 1, 2 )] Z + d 1 d 2 ( 1, 2 ) G( 2, 1 )+(U 2 /2)G 2 ( 2, 1 )G 2 ( 1, 2 ) At frequencies U, the time derivative in the determinant is less important, and without it the path integral is invariant under the reparametrization and gauge transformations = f( ) G( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] ( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] where f( ) and g( ) are arbitrary functions. 1/4 g( 1 ) g( 2 ) G( 1, 2) 3/4 g( 1 ) g( 2 ) ( 1, 2) A. Georges and O. Parcollet PRB 59, 5341 (1999) A. Kitaev, 2015 S. Sachdev, PRX 5, (2015)

42 J. Maldacena and D. Stanford, arxiv: ; R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, arxiv ; S. Sachdev, PRX 5, (2015); J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv: ; K. Jensen, arxiv: ; J. Engelsoy, T.G. Mertens, and H. Verlinde, arxiv: The SYK model Reparametrization and phase zero modes We can write the path integral for the SYK model as Z Z = DG( 1, 1 )D ( 1, 2 )e NS[G, ] for a known action S[G, ]. We find the saddle point, G s, s, and only focus on the Nambu-Goldstone modes associated with breaking reparameterization and U(1) gauge symmetries by writing G( 1, 2 )=[f 0 ( 1 )f 0 ( 2 )] 1/4 G s (f( 1 ) f( 2 ))e i ( 1) i ( 2 ) (and similarly for ). Then the path integral is approximated by Z Z = Df( )D ( )e NS eff [f, ].

43 Fluctuations Symmetry arguments, and explicit computations, show that the e ective action is S e [f, ]= NK 2 Z 1/T 0 d (@ + i(2 ET )@ f) 2 N 4 2 Z 1/T 0 d {tan( Tf( )), }, where f( ) is a monotonic map from [0, 1/T ]to[0, 1/T ], the couplings K,, and E can be related to thermodynamic derivatives and we have used the Schwarzian: {g, } g000 3 g 0 2 g 00 g 0 2. Specifically, an argument constraining the e ective at T = 0is apple S e f( ) = a + b, ( ) =0 =0, c + d and this is origin of the Schwarzian. <latexit sha1_base64="zgavwmithffkqayyyzh/niezp2c=">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</latexit> <latexit sha1_base64="zgavwmithffkqayyyzh/niezp2c=">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</latexit> <latexit sha1_base64="zgavwmithffkqayyyzh/niezp2c=">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</latexit> <latexit sha1_base64="zgavwmithffkqayyyzh/niezp2c=">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</latexit> J. Maldacena and D. Stanford, arxiv: ; R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, PRB 95, (2017); A. Gaikwad, L.K. Joshi, G. Mandal, and S.R. Wadia, arxiv:

44 <latexit sha1_base64="qfgjwabjdgrryfho9vd8uorr4z4=">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</latexit> <latexit sha1_base64="qfgjwabjdgrryfho9vd8uorr4z4=">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</latexit> <latexit sha1_base64="qfgjwabjdgrryfho9vd8uorr4z4=">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</latexit> <latexit sha1_base64="qfgjwabjdgrryfho9vd8uorr4z4=">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</latexit> Fluctuations An exact path integral over the e ective action leads to the following physical consequences The ground state energy with fermion number N Q + p (p integer) varies as E p = E 0 + p2 2NK This identifies K with the compressibility K = dq/dµ at T = 0. The low temperature corrections to the entropy at fixed Q are h i S(T! 0, Q) =N s 0 + T ln(U/T )... This defines as the co-e cient of the linear-in-t specific heat (at fixed Q)

45 <latexit sha1_base64="qfgjwabjdgrryfho9vd8uorr4z4=">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</latexit> <latexit sha1_base64="qfgjwabjdgrryfho9vd8uorr4z4=">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</latexit> <latexit sha1_base64="qfgjwabjdgrryfho9vd8uorr4z4=">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</latexit> <latexit sha1_base64="qfgjwabjdgrryfho9vd8uorr4z4=">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</latexit> Fluctuations An exact path integral over the e ective action leads to the following physical consequences The ground state energy with fermion number N Q + p (p integer) varies as E p = E 0 + p2 2NK This identifies K with the compressibility K = dq/dµ at T = 0. The low temperature corrections to the entropy at fixed Q are h i S(T! 0, Q) =N s 0 + T ln(U/T )... This defines as the co-e cient of the linear-in-t specific heat (at fixed Q) D. Stanford and E. Witten, arxiv: ; A. Kitaev and S. Josephine Suh, arxiv:

46 <latexit 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Fluctuations An exact path integral over the e ective action leads to the following physical consequences The ground state energy with fermion number N Q + p The many-body density of states, D(E), is related to the grand potential, (p integer) varies as (T )by Z 1 Z = e (T )/T = E p = E NK We obtain This identifies K with 1X the compressibility K = dq/dµ at T = 0. D(E) = e 2 pe d (E E p ) p= 1 ded(e)e p2 E/T where NQ The+ low p istemperature the integer fermion corrections number, to the entropy at fixed Q are p d(e) exp (Ns 0 )sinh 2N h E, E i> 0, e cn E N S(T! 0, Q) =N s 0 + T ln(U/T )... There are exponentially more low energy states than for the quasiparticle case, and This D(E) defines self-averages as the co-e down cient to energies of theexponentially linear-in-t specific small in N. heat (at fixed Q) J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, arxiv: ; A.M. Garcia-Garcia and J.J.M. Verbaarschot, arxiv: ; D. Bagrets, A. Altland, and A. Kamenev, arxiv: ; D. Stanford and E. Witten, arxiv: ; A. Kitaev and S.J. Suh, arxiv:

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sha1_base64="vexfeqgzbsxvb+tcdrwyixhvcxg=">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</latexit> The complex SYK model NU Many-body level spacing 2 N = e N ln 2 Non-quasiparticle excitations with spacing e Ns 0 There are 2 N many body levels with energy E. Shown are all values of E for a single cluster of size N = 12. The T! 0 state has an entropy S GP S = Ns 0,where s 0 < ln 2 is determined by integrating ds 0 dq =2 E. At Q =1/2, s 0 = G + ln(2) 4 = where G is Catalan s constant. GPS: A. Georges, O. Parcollet, and S. Sachdev, PRB 63, (2001) W. Fu and S. Sachdev, PRB 94, (2016)

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Fluctuations An exact path integral over the e ective action leads to the following physical consequences The ground state energy with fermion number N Q + p The many-body density of states, D(E), is related to the grand potential, (p integer) varies as (T )by Z 1 Z = e (T )/T = E p = E NK We obtain This identifies K with 1X the compressibility K = dq/dµ at T = 0. D(E) = e 2 pe d (E E p ) p= 1 ded(e)e p2 E/T where NQ The+ low p istemperature the integer fermion corrections number, to the entropy at fixed Q are p d(e) exp (Ns 0 )sinh 2N h E, E i> 0, e cn E N S(T! 0, Q) =N s 0 + T ln(U/T )... There are exponentially more low energy states than for the quasiparticle case, and This D(E) defines self-averages as the co-e down cient to energies of theexponentially linear-in-t specific small in N. heat (at fixed Q) J. S. Cotler, G. Gur-Ari, M. Hanada, J. Polchinski, P. Saad, S. H. Shenker, D. Stanford, A. Streicher, and M. Tezuka, arxiv: ; A.M. Garcia-Garcia and J.J.M. Verbaarschot, arxiv: ; D. Bagrets, A. Altland, and A. Kamenev, arxiv: ; D. Stanford and E. Witten, arxiv: ; A. Kitaev and S.J. Suh, arxiv:

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Fluctuations An exact path integral over the e ective action leads to the following physical consequences The ground state energy with fermion number N Q + p At charge (p integer) NQ + p, varies the prefactor as of the sinh( p 2N (E E p )) term is exp [Ns 0 (Q)+2 pe] E p = E 0 + exp [Ns 0 (Q + p/n)] 2NK using This identifies K with the ds compressibility K = dq/dµ 0 at T = 0. dq =2 E The low temperature corrections to the entropy at fixed Q are h i S(T! 0, Q) =N s 0 + T ln(U/T )... p2 This defines as the co-e cient of the linear-in-t specific heat (at fixed Q)

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sha1_base64="czr51u4y8wilwuuyvow46w/24j8=">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</latexit> <latexit sha1_base64="czr51u4y8wilwuuyvow46w/24j8=">aaadohicdvjnb9naehxcvwlflry5jgibipxisdwuccogjcpxkiv+se0urddjz1r719odp6rrfwk/hhuch8kng+lkkrpjnewaycrlt/n2zt48t1rk5dkmp9fq585fuhhp7nljytvr12/ml9zc9bz0gne0zazbj5thjazumhgg+4vdlucz7kwhzyp+b4tokzvvefxgp1epoys0ykknfmr3esasidfwoxdhsmzcjdkd40mjvyhyxki5lsojo+npfzazkufkaatyocdghayzfapnc3q8bpvaczkekcx9dzxe66bj1kkr8xjwelnair4yslimhffgsfx6adbeqvpkfqxtblm0aespq5hvipurbl8ojy+bj+zdwmpmtz7kwjcueaiapk1mv4bbycqirwi6dcpfpcmmblihwchjj5lztg1dkt44ep1yub2y/adrqxp/mmmwvxg2w/vup7soarprnbajolpa6nba0gqg01gmzre1mp/ri60urrzrthl/0aol7k+uy9jzzw7pxvh9qfi8egjeyb8sj6jwu9iftp/tcdwvmobeovle4jt7e/fe5d6p80he5oqh/m+usv6loyg5edsfkclkrqnpbyvlbmyhohsiyahmbcxaaucig/rqlkhliugnj3jrjuxhpmf4lo8oljmt1eyamcqsm0fg/2c33wyfzdar9ulg05ltc8ht4e6whlsc9wajebfsbtubrr2rva99rh2qf6h/qx+tfzt9wq/nam4ff0t9+0/fpwxr</latexit> The Schwarzian theory and black holes Reparameterization invariance is a defining property of Einstein s theory of gravity In imaginary time, AdS 2 is the homogeneous hyperbolic space: two-dimensional surface of constant negative curvature. Its metric is invariant under SL(2,R) ds 2 =(d 2 + d 2 )/ 2 is invariant under 0 + i 0 = a( + i )+b c( + i )+d with ad bc = 1. A. Kitaev, 2015 J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv:

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sha1_base64="czr51u4y8wilwuuyvow46w/24j8=">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</latexit> P. Nayak, A. Shukla, R.M. Soni, S.P. Trivedi, and V. Vishal, arxiv: U. Moitra, S. P. Trivedi, and V. Vishal, arxiv: P. Chaturvedi, Yingfei Gu, Wei Song, Boyang Yu, arxiv: A. Gaikwad, L.K. Joshi, G. Mandal, and S.R. Wadia, arxiv: The Schwarzian theory and black holes Reparameterization invariance is a defining property of Einstein s theory of gravity In imaginary time, AdS 2 is the homogeneous hyperbolic space: two-dimensional surface of constant negative curvature. Its metric is invariant under SL(2,R) Semiclassical fluctuations about the saddle-point of Einstein-Maxwell theory of a charged black holes in d 2 spatial dimensions lead to the same Schwarzian+phase theory of fluctuations.

53 <latexit sha1_base64="ukkutu1lj91h/j0rlgqhz/bojqg=">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</latexit> <latexit sha1_base64="ukkutu1lj91h/j0rlgqhz/bojqg=">aaaecxicdvplbtnafhuthsw8wliyuaicpuoboslnyajsbqihavjl0xypk1itmyk96nhszyybpsy7dgj+hr1iy1fwjuisgdujacu4qzp3ee718tarxbvp+7lqqv65eu364g335q3bd+4uld871hgqcdsgsyjv+yhwthdjdgw3gr1pfmprulcj4fhzin50wptmseyascl6eq4kh3gcjxx5ywu/kiy5pewafw1zwgxgdyv4gctdvcbwuygdf1xqw7hcf4tpx0wimcgl1qqew1sdiys3jgih4jgxasry4ygzo9c6vrncj+eyqzoozt0jlefzh/lqlhggc/cxvabybityydqajrqmgquuyrmsllk9pc+faq5i8sa0h/ysu+3lrumz26rr9eyqvbm0qpsu3vnhikprg/m0yw9ojdztvpc33dqu2ta3yj827ofzncxwxxboy5wkencydwqs02jifcbkae7xurmkrfq8bm/tpmtvs0j7htqbbtikmhqlkghs9kvgs/ic/xnacbdzaum/odvdod60m8/jwnrn1oala6zmak3xd4almktz45z4/nx9prwv4w112p0tskcz2fbafrq3mp12c5onr7qvz2a7/tjvrgosrlyprgcte00vmf0mk8ojknssapzgcowd1rnqwghonxrce13cflaknydhnkhlwyxiaad0ni/ockt1jbrazokg+nkscp4r1kvn6gk/4zjjdznkomiuilkdvvlauwleiikfmchuaqmjrvbjovmxawz/hhmymeognzoxp3zotthy5li41vwi8h9w2go0vuzzr7wy82x2tkxngfpqqtlnz8vzcv45u86bqyqdysfk58qx6qfq1+q36vdpamvhvnpfuwdvh38ah+pigq==</latexit> <latexit sha1_base64="ukkutu1lj91h/j0rlgqhz/bojqg=">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</latexit> <latexit sha1_base64="ukkutu1lj91h/j0rlgqhz/bojqg=">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</latexit> The Schwarzian theory and black holes The Einstein-Maxwell theory leads to the following parameters for the Schwarzian+phase theory K = dq = 2(d 1)L2 s d R d 3 d(d + 1)R 2 h h +(d dµ (d + 1)g 2 T =0 F apple2 1) 2 L 2 S(T! 0, Q) =s 0 + T +... = 4 2 ds d L 2 R d+1 h apple 2 (d(d + 1)R 2 h +(d 1)2 L 2 ).