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 Chaos and Order: from Strongly Correlated Systems to Black Holes, Kavli Institute for Theoretical Physics University of California, Santa Barbara, December 7, 2018

2 HARVARD Wenbo Fu Yingfei Gu Grigory Tarnopolsky

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

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

5 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 S. Sachdev and J. Ye, PRL 70, 3339 (1993) A. Kitaev, unpublished; S. Sachdev, PRX 5, (2015)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> 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)

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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 )

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sha1_base64="p3n7qerjsijwfnhlldvq9sjupeq=">aaagfxicdvtdc9teefdcyor4aaqpvowqndid48qemhqywpsvgqcegpq0nylcoj/w8k1oj+xufcnu8ufyxn/c8aj7klmrpdwmrqf9/o1vd2+ws2fsgp61sfnw1nbv7tvv+o++9/4hd3fuffjuzixmemozmennm2zqcownvlijz3onlj1jfda7/nbpn12jnijtj7bm8txlirjzwzkl0cw9rctizulfqkz/jjofk0i2b7tag4bxrwnngkkyeky1mjxtsvaj4dwtrxzld3on5kjqjywuamvswnemzbz0lousdgutqhekaq30q0hyecmcrq9qyhktyrqhurieryvrpmuuzu1b+ikpbgoylcdibov7ocquesgsbwoirgkyo04dpetfm/jcj2aycfuji6n4hws/cifwv7ygmwpirypm2ftklaqnr2ewhochfmrzmydiibreupolx6r9cutfobssdmap7akc9heoypyamved6gfcyznw2tia4ndfucoecnpfqcyvzktotbvcohlab1hiekryqfadimvpmv1wjqvjmlc1uqsxbczhbp2lkdmurs4hrlcwzuplefacljpzj8r3dfcsj1iyxllsqwr/vrmrxlmnkrixphjywvec89rfd3aoirur0rd2/tgdj4wviiqo8phuqezlajqja+g3vomujwvfl8nxj1dfq2x+pb2ti4iirxjyfoca0xrdat/q55+syvkp3k8/fdbsqbalgmxti09hhnajqcdyj4sa25ksd4c6v0ycndphrtlmttotsgt/7ozhwq1vekvw+imf13cwd0t1zjxjftgt+o2+hahpiicfwjao/6faf6kc2lpvpdmqh4juglghhtfvphy+bzrgk2lmony5xgv7ncadl5bxqrff5bvqq4oa7y6atinr4tv/m4272nknr+hh9odzmysjcrg+ne7pmd2ytqctgi/c5tn1vs/ji51/ojjjruq3e5fmmlnxmnvzarux1pdcym74juvwji6kuq2or0vumun51vyqndwnzuwxkazxwwikxeekpyb2z0awjkrzqq4zr9fozgo7f3heczuxfhvve80lsymc7rafwghat0llkrjxdefxznoyp3wksabpwlajxvb78de7fdhlqq5gu6ecwtemwjsptyfe42h8pnl99m2ktjvex94nxt8be4fei+9h77f36vgtp7f+3t7y3tz+t3e/n+ynwtpnjzxpr96tp3f4hxkwksg=</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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<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 ) 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 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)

11 <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)

12 <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> <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)

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sha1_base64="vexfeqgzbsxvb+tcdrwyixhvcxg=">aaaebxicdznlb9naemfdhecjrxaoxebulckq4krr00peklrvovwt+hr1inbrjbpqem121wngssqnwxfhhrjyofgc3dgxdozailjj8uzmzvxm/l57sedaom6pmvr9ytvr12dvng7eun3n7tz8vsmdjyqyqxqjsj14rdpbjts03ah2eitgqk+wy+9ss4gfj5nspjihjo1zpysb5enoiuhxyh7mp+uxgmummmmyyhshi6yyehzs9psdg0iiu/aipwxbxkxomhazaiazclkwt+wm7i+iiswzibawjijhgqjheyrhpica5jiqreffopfrxdr/j4cdxqunbraj9gg4igcjq5skjudyoa0xdezea5himyqkebg/ylz393powq7ogwmvnyzlwzzu4dm4qklbbq7bz4gkurufvbq4dhconfkgjwpgn+tykdqkzpq33keinpolkjm+3mjlicj28glvbjfmco7dysfdbjzcjv3lzv4ysc/k9lrp2thhbwrvjq5h0rxr9lryw4qpdtvpshwot6s9zrdufpjnnqltndurnbxomiv8yghwzsst0gnqcj8vwamwxc6+sqwrrd7wqcojokptnvt95mhcgtn0uk5nvysap1xork6sr3s70ko8c1a1dgdzv1w/okmi6vqqru9btmz6gvggu8fqvkszmnazerbtncujmv72xzzwpdnpyouawymm+uwlgebsqom9mjyruos09pbkoal+o1aq/i/yawkga/2myzgxtnipajgimbeutx58rhg1ikwdumwxbaajgmqjhfhjm/xxzgbgmwvyozphpnkydrpfzshwh0xg/8yrhnaarb32wsblsrzz64h10fqywlbx2rbewbvwouvrr2sfap9qn+sf61/qx+vfpkdrm1xofevsqn//dttpsp8=</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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15 1. Quantum matter without quasiparticles: the complex SYK model 2. Einstein-Maxwell theory of charged black holes in AdS space 3. Fluctuations, and the Schwarzian 4. Supersymmetric models

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

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

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sha1_base64="j/odcfzzeni2gnxhuj2gqux4/go=">aaaclhicdvdlsgmxfm34dnxvxbojvsgflezp4cxfny4t2cp0sslkbmswkxmsjfcg/oqf4fe41zubebd+h+m0bru9ediccy6590sz4noq8ubmzs8sli2vrlpr6xubw5xtnbzoc8wgxvkrqpuiahbcqstwi+amu0ctsmb1dhc+9q/vqwmeyiszzkcb0ihkfc6osvkveoqrdlgsgegdausa1farhpjdujuafbam1nwykorm6manqcazzk9sjtw/4ft+hzmacu78oggjrwjsb2cvrkpu0rsxvcphgkcst+w4e1trjkcy0y2ompwjgllhrigj7i4oogoppano4/iez7qk3ai8doqprrnjfqrskwax6vfhgizad5pijse769/ewpzl6+smh3qllrpcggstj/q5wcbf4+zwzbuwi4awuka4xbtsitjbh3zth7oj8f+kxbcn1bxmvxp6nm1mbe2hfxsephscttefukqtxnadeklp6mv5df6dd+djep1zpjo76aeczy9wt6kr</latexit> <latexit 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=">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</latexit> A. Chamblin, R. Emparan, C.V. Johnson, and R.C. Myers, PRD 60, (1999)

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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)

20 <latexit sha1_base64="gfxjj3bkhiedoeaohn4ssxh2tvy=">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</latexit> <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 sha1_base64="gfxjj3bkhiedoeaohn4ssxh2tvy=">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</latexit> <latexit sha1_base64="gfxjj3bkhiedoeaohn4ssxh2tvy=">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</latexit> <latexit 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> <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)

21 <latexit sha1_base64="gfxjj3bkhiedoeaohn4ssxh2tvy=">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</latexit> <latexit sha1_base64="gfxjj3bkhiedoeaohn4ssxh2tvy=">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</latexit> <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> <latexit 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 sha1_base64="j/odcfzzeni2gnxhuj2gqux4/go=">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</latexit> <latexit sha1_base64="j/odcfzzeni2gnxhuj2gqux4/go=">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</latexit> <latexit sha1_base64="j/odcfzzeni2gnxhuj2gqux4/go=">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</latexit> <latexit sha1_base64="j/odcfzzeni2gnxhuj2gqux4/go=">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</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)

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sha1_base64="geqm+fl4qxcp8+2byzrif1lxcgy=">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</latexit> <latexit sha1_base64="geqm+fl4qxcp8+2byzrif1lxcgy=">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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 A probe fermion has a near-horizon Green s function 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. This is identical to the complex SYK model. T. Faulkner, Hong Liu, J. McGreevy, D. Vegh, PRD 83, (2011)

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

25 <latexit sha1_base64="+gg5vvgebgxdmf1x9q6o58vzagm=">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</latexit> <latexit sha1_base64="+gg5vvgebgxdmf1x9q6o58vzagm=">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</latexit> <latexit sha1_base64="+gg5vvgebgxdmf1x9q6o58vzagm=">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</latexit> <latexit sha1_base64="+gg5vvgebgxdmf1x9q6o58vzagm=">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</latexit> Fluctuations The fluctuations can be expressed as a path integral over a Green s function G( 1, 2 ) which is bilocal in time. At frequencies U, the action is invariant under reparameterizations f( ) and gauge transformations ( )! f( ) G( 1, 2 )! [f 0 ( 1 )f 0 ( 2 )] e i( ( 1) ( 2 )) G( 1, 2) A. Kitaev, 2015 R. Davison, Wenbo Fu, A. Georges, Yingfei Gu, K. Jensen, S. Sachdev, PRB 95, (2017)

26 <latexit sha1_base64="dmnlhgvdbndar4oweq3v3wjmvqw=">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</latexit> <latexit sha1_base64="dmnlhgvdbndar4oweq3v3wjmvqw=">aaae4xiclvrlb9naehabacw8wjhygvgbejwj7fc3vvqql8gbq4h0idvptf6v7vxxa9e7bgmwuxndxplrnpkrnbg7tktlobdyyuoz+wb3+2bxbik40qb5c26+0bx2/cbczdat23fu3ltcur+r4iylbifgik73xaky4jltak4f209srijxsd336euz3zthqekxhopjwkyrcst3osuaxeol+r+ojln0mnqtx2ublznxlokfwnfysqs1dbko4nmc9rlm1ddl87hkbjleridf63xb0sqbw9cdyhh0ybptz+d4kae5o8x7a3asdk6zt4nixxuwhl9rrze76jjvuqurumjwl8nwgcoyr9t18wej4bjd5l1rsoppkq9c3tnmcq8vmdaexux6fylunjpyxagxjytlbcnhukwgq5vs2p7wtj3owvso6jri5cdpvemp4dtknisijkbrmpwyeub5sgxdcjogrrfrfhei9hacusqmngjmqucygf2qonozcb5i3f7fftljqjn6fvxkl1ffbccttag1iedu5jse4prn6bpcqc+qk+bckdp/bxmaqh5nxotox4bpd1wkmxz6zcyzyexnhkf1+ejndzuahhzare657x5mxwmt6155vkrvmw9g+7zithvtncaly2bfxldx1yww+5zpbtg2gvaqbdsdsppmnzanemypf385xkyzcg8afdisa8tm9cgnqebyonq9uywh9ige7abnsskmut4jt1rljvlqehbwgimeypdw4pgpvh+ccxipnylczcx5qcux6hbeetvitl8xyrlmms0kns7kzwlwqju3gjyemqrxghqc0jtjtoggbdun8d63hmxwpyadhwij2ud9yj1cj3/ax+evwdnf4n/g7gb17fvvbstbz2vzfoyhxiojbvjgurflvdg2jr2dntyatom1/cztfml+bx6bps7p1zghxoxr/p4b6aywnw==</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)

27 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=">aaafxhicdvrrb9s2efbreovcdutxyc97oswa4qcuj7utswwougwymkdd0cfow8b0djqijkiiqzguxu/qr9qv6csetj+xxx0lo0ucllcfe+873t33nbgscmfdfl2/dbuz1/3o4zuf9o5+eu+zz/fvf/hk6tiwfsz0rs2bjbu8f4qfoefy/qywnmplzl8v3/7o/a9x3fih1drtcj6xnfuieyw63frc7/xclbyq5sr1tjdscmc2qe1astyxa6aqbv4ok2hcadoykf0tis6b6tw4jdp8cobjwpktkw6ueqai2ynlngpvxciwod1i6rk6d7qoijeexsmsgqapmjghz0asq1n164u6gtdahhkl6lwaftutiqbiaal9uldjbm0xzaupg4cgod9gwwdxxzoavz/vmd2cq+dk6hwmj2cxhkruslpxt3zk+qfyeayt8qbyqqvmiel7x9hroagqetajxmvx21tn3hait9aqwqakuivttbimjc0gmvpcoisggonmith96w3qkml0izsr1el4arfctucwlso81goidotbcja8p47h/ja8wegdbxsvmdhmrqyof8y20wsoguwvsuvrmouuzwtq/hbufx+jykrkdw0d4b+xyrwlmt08pkzxwsoz7c7jvrycj67/p6jfecpszhkdhjfqdlpw1kxlnm98lxcjifqo6wwvcum4pmgowumzyjpcc8mrk9xumnspdjfetjx2iszriu3suiz2doechw3l9ymzdfbw6hnuxizy4k+juyjqytuch4v9g2gyhz9mto4bjzonk2icxutx6gqyhtewatzbsf0vf/v/klizplowu2tno6hw88rplss5ko8yfzs9psmfoymscjuiv6kwjtmvmhughm8aornt8n+qhruxgysqrd3ijsi94fa6r1hhbt+sdml380oo/o65ym2ipmybocprbgjhui18gwzlrmdzwdkkxdu8dhrecryrvoqyijj+zq1fjhmqp0lvsnwod/iw8wo8hexd0w/jg+c/bem7e3wvfb30g1fwhdwpfg5ebmcb6/zzed/5u/pp3l/dve7d7r0wevvwnuzbcgv1v/wpgia92q==</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=">aaafxhicdvrrb9s2efbreovcdutxyc97oswa4qcuj7utswwougwymkdd0cfow8b0djqijkiiqzguxu/qr9qv6csetj+xxx0lo0ucllcfe+873t33nbgscmfdfl2/dbuz1/3o4zuf9o5+eu+zz/fvf/hk6tiwfsz0rs2bjbu8f4qfoefy/qywnmplzl8v3/7o/a9x3fih1drtcj6xnfuieyw63frc7/xclbyq5sr1tjdscmc2qe1astyxa6aqbv4ok2hcadoykf0tis6b6tw4jdp8cobjwpktkw6ueqai2ynlngpvxciwod1i6rk6d7qoijeexsmsgqapmjghz0asq1n164u6gtdahhkl6lwaftutiqbiaal9uldjbm0xzaupg4cgod9gwwdxxzoavz/vmd2cq+dk6hwmj2cxhkruslpxt3zk+qfyeayt8qbyqqvmiel7x9hroagqetajxmvx21tn3hait9aqwqakuivttbimjc0gmvpcoisggonmith96w3qkml0izsr1el4arfctucwlso81goidotbcja8p47h/ja8wegdbxsvmdhmrqyof8y20wsoguwvsuvrmouuzwtq/hbufx+jykrkdw0d4b+xyrwlmt08pkzxwsoz7c7jvrycj67/p6jfecpszhkdhjfqdlpw1kxlnm98lxcjifqo6wwvcum4pmgowumzyjpcc8mrk9xumnspdjfetjx2iszriu3suiz2doechw3l9ymzdfbw6hnuxizy4k+juyjqytuch4v9g2gyhz9mto4bjzonk2icxutx6gqyhtewatzbsf0vf/v/klizplowu2tno6hw88rplss5ko8yfzs9psmfoymscjuiv6kwjtmvmhughm8aornt8n+qhruxgysqrd3ijsi94fa6r1hhbt+sdml380oo/o65ym2ipmybocprbgjhui18gwzlrmdzwdkkxdu8dhrecryrvoqyijj+zq1fjhmqp0lvsnwod/iw8wo8hexd0w/jg+c/bem7e3wvfb30g1fwhdwpfg5ebmcb6/zzed/5u/pp3l/dve7d7r0wevvwnuzbcgv1v/wpgia92q==</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:

28 <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)

29 <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=">aaae1xicdvrnb9tgegus5uttejs95jkowubcy5lsrdgxa04kawucpakqxw5mrvgth9tayyw7u7stelofufy/9v/kt+tsiskhdpmuigf2z2fem6e3muwkrauczzfwwu2bt27fudv54cd79x+sbzx8a7pcsdyumcrm8uxyvktx0jftejwbfolm4dhs9lcqf3sgxlkmx26r4yqviaayphb8nn1y+yfugekites811cgmajeconc7sxkws2btmpecauz9we3r8a4runodihvcrdqkcillquzcazudk46gxy+saykqgba4l8faom2e84wiv2sw5q+4litvlfnzjwjyqodgxxcmyhgkyzgnjhcjcatchsrzpid/wrclklx6/lnen6f3ieuz181vtq9obog0il4iharzzvyvfxlzse0hz04mazcgszgydj/p1ywq3gfl5kp6uj6/fgclfaleqvhbf2xfsorgldmkqtulc1ikvtwknthv/ht86bwqtjwx3ub328ghnpifgp4k6mrrjbvm2ow1vsrnixqsnxr1zdcctfdyft9op4vdg5v8luz/tkdq2gomtthdkmfj69q12ouloylstsjbfrirjokgfehijjn66yz8h7ijxq6h9vj3ir3rvbqibvhzh5kpzp2tm+uxnpi85akngqyud6rncvmfuktf+ydzvfw9oq58srdsrr4+rs9bvjsqtp1zaaf7i52ng4g6a+c4nloxmfozzqadwhqd+q16a3w6+n6lzdkzjeyc6metsedihetuhhhulwwlczmqp6kbe841cjf+zg6o9zw4ass390sfufkxiy3/ogb6torxavirv2km75zsbf/ztwsfzs7kvz8bfkszgvhj6ybigtve4ifmuru2uqtobdsenmgorfszmdpvrna5p8bnbh5gtq8cocumu75pl87il2pdskj/h/wdshc9gdvhpv7l1a63feeet97xw/g7xr73u/ea+/qk61e64/wcetd+6i9bh9sf2qurt1y1fzkxvvtv/8fanwvvg==</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:

30 <latexit sha1_base64="cy6nebszhgugt2hkw1zm4wevmbq=">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</latexit> <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=">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</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.

37 <latexit sha1_base64="ukkutu1lj91h/j0rlgqhz/bojqg=">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</latexit> <latexit sha1_base64="ukkutu1lj91h/j0rlgqhz/bojqg=">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</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 ).

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