The disordered Hubbard model: from Si:P to the high temperature superconductors

Transcription

1 The disordered Hubbard model: from Si:P to the high temperature superconductors Subir Sachdev April 25, 2018 Workshop on 2D Quantum Metamaterials NIST, Gaithersburg, MD HARVARD

2 1. Disordered Hubbard model for Si:P 2. Solvable disordered models: (A) Random matrix model of a `quantum dot Metal with quasiparticles (B) SYK model of a `quantum dot Metal without quasiparticles 3. High temperature superconductivity and strange metals.

3 1. Disordered Hubbard model for Si:P 2. Solvable disordered models: (A) Random matrix model of a `quantum dot Metal with quasiparticles (B) SYK model of a `quantum dot Metal without quasiparticles 3. High temperature superconductivity and strange metals.

4 <latexit sha1_base64="cx0klfibznm9k5vrm8mrowgc8nq=">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</latexit> <latexit sha1_base64="cx0klfibznm9k5vrm8mrowgc8nq=">aaadfhicdvjnbxmxepuuxyv8pxdkwiiiqogm7laghansbzcei0tastmw8nq9ivn717k9ldf2e+qn8cu4wokb4sqda/8fb7kbtgphsub5ztypz+xicqan5/123ctxr12/sxkzcev2nbv3mqv393swk0l7joozooiwppyltg+y4frakopfxol+dpi2iu8fuavzlr43u0mhao9sljccjaxcvefr2g68hjyeohdhwtymjrjrwxps+qbzocblhydgoxfvftlzkbaejbxtzbpnihx+frtuz5b0skx/bv2z2lxipblj8+ofxw/j7di9mo9/wfy6rs5btwy7czqtencpetbbj1aeuqkqtfxb//aetp7j0hvacjstr/nyq9d91qwv483mat/b7g11wa+zfqptn2z+sxvjlmhqcmdad3xpmmgblwge07ir5jpkta7xia4stlggeim+yllp4lcypwojt2wwhirtdqugzuyyumbc66mibkbazqzpxyryotggn0lvwlbu5oamzf4oytmydkofajftlbg+tqatxey1gyyxwstyf1tny9e0xa72nu2eov67f63tn/vkvtbd9bitix910tbaqbuoj4jzyfnifhw+uz/d7+4p9+c81xvqzqn0xtxffwhapvut</latexit> <latexit sha1_base64="cx0klfibznm9k5vrm8mrowgc8nq=">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</latexit> <latexit sha1_base64="cx0klfibznm9k5vrm8mrowgc8nq=">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</latexit> Disordered Hubbard model for Si:P H = X i,j t ij c i c j µ X i c i c i + U X i c i" c i"c i# c i# t ij exp( r i r j /a 0 )

5 Electrons in doped silicon appear to separate into two components: localized spin moments and itinerant electrons 10' P 2 [ " T(K) F1G. 1. Temperature dependence of normalized susceptibility g/gp, ~; of three Si:P,B samples with dia'erent normalized electron densities, n/n, =0. 58, I.I, and 1.8. Solid lines through data are a guide to the eye. M. J. Hirsch, D.F. Holcomb, R.N. Bhatt, and M.A. Paalanen PRL 68, 1418 (1992) M. Milovanovic, S. Sachdev and R.N. Bhatt, PRL 63, 82 (1989) A.C. Potter, M. Barkeshli, J. McGreevy, T. Senthil, PRL 109, (2012)

6 Theory of local moment formation in Si:P Choose an e ective Hamiltonian of the form H e = t ij c i c X j X i,j X ~ hi c i ~ c i i i ic i c i where ~ h i and i are variational parameters. Taking the density matrix defined by H e as the trial density matrix, and minimizing the free energy, we obtain an expansion in power of ~ h i where F e [ ~ h i ]=F X i,j,k ij( jk U jk )( ~ h i ~h k )+... ij X, (i) (i) (j) (j) f( ) f( ) <latexit sha1_base64="ordtzt4fenvvf3ocibhpu6opsry=">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</latexit> where and are the eigenfunctions and eigenvalues of H e at ~ h i = 0. M. Milovanovic, S. Sachdev and R.N. Bhatt, PRL 63, 82 (1989)

7 <latexit sha1_base64="d+nvezigw+znbxq2zvtszp4xycg=">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</latexit> <latexit sha1_base64="d+nvezigw+znbxq2zvtszp4xycg=">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</latexit> <latexit sha1_base64="d+nvezigw+znbxq2zvtszp4xycg=">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</latexit> <latexit sha1_base64="d+nvezigw+znbxq2zvtszp4xycg=">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</latexit> Theory of local moment formation in Si:P F e [ ~ h i ]=F X i,j,k ij( jk U jk )( ~ h i ~h k )+... We numerically diagonalize ij, and choose eigenmodes with eigenvalues i > 1/U: these are the eigenmodes corresponding to local moment instabilities. M. Milovanovic, S. Sachdev and R.N. Bhatt, PRL 63, 82 (1989)

8 Theory of local moment formation in Si:P Magnetic excitations are localized (eigenmodes of the spin susceptibility) Charged fermionic excitations are extended (eigenmodes of the electron Hamiltonian) M. Milovanovic, S. Sachdev and R.N. Bhatt, PRL 63, 82 (1989)

9 1. Disordered Hubbard model for Si:P 2. Solvable disordered models: (A) Random matrix model of a `quantum dot Metal with quasiparticles (B) SYK model of a `quantum dot Metal without quasiparticles 3. High temperature superconductivity and strange metals.

10 A simple model of a metal with quasiparticles Pick a set of random positions

11 A simple model of a metal with quasiparticles Place electrons randomly on some sites

12 A simple model of a metal with quasiparticles Electrons move one-by-one randomly

13 A simple model of a metal with quasiparticles Electrons move one-by-one randomly

14 A simple model of a metal with quasiparticles Electrons move one-by-one randomly

15 A simple model of a metal with quasiparticles Electrons move one-by-one randomly

16 A simple model of a metal with quasiparticles 1 NX H = (N) 1/2 i,j=1 t ij c i c j +... c i c j + c j c i =0, c i c j + c j c i = ij 1 N X i c i c i = Q t ij are independent random variables with t ij = 0 and t ij 2 = t 2 Fermions occupying the eigenstates of a N x N random matrix

17 Infinite-range model with quasiparticles Feynman graph expansion in t ij.., and graph-by-graph average, yields exact equations in the large N limit: G(i!) = 1 i! + µ (i!) G( =0 )=Q., ( ) =t 2 G( ) G(!) can be determined by solving a quadratic equation. Im G(!) µ!

18 Now add weak interactions Infinite-range model with quasiparticles H = 1 (N) 1/2 NX i,j=1 t ij c i c j + 1 (2N) 3/2 NX i,j,k,`=1 U ij;k` c i c j c k c` J 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 E. By Fermi s Golden rule, for E at the Fermi energy 1 = U Z de de de f(e )(1 f(e ))(1 f(e )) (E + E E E ) = 3 U T 2 where 0 is the density of states at the Fermi energy. <latexit sha1_base64="cmzytw7hgbfdnmmx0ij330jbm9e=">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</latexit> <latexit sha1_base64="cmzytw7hgbfdnmmx0ij330jbm9e=">aaafo3icdvrtb9s2efzbe2u1t3rdp/xlove7b3m9yu2afkwbolu3yaocboibapft0njjziyrkkklnvttj+3/7mp+y46ynspfxi863j18epfcibmi49r4/j/xrt/odd/7/oyt94svv/r6m63b377rslqrjikzsfv2xjrmxodicjph20ihy2czhs8wp9n48rkqzau4msscxzllbu94xay5prdv/b0kyuwmwrih8hxyhmm5sgvwyvcxykk0g84w5ajigu9f7r7acwgtclzbxfuodyzv8oowrihuzt6tep/0evbpdgembu5riohtt8kypskqsfty/tbigillehszo7/oh5hlk6qrzxq2sa6c9imdxxbg09owvzhu4vkxz8lihnzum4i9emwiyhdlpkplhijityneb60bpwakwypuykmukcziijljijnswsm5n3pwqkls205u7yrrkssndhg3er/aka+touwnjknvamcikcyl0icyoulgezq8r5ajmhhxms0lpgypmuwtowhlngqz1ex1kue6bva99q2qpyi0yutjhyxm02ofq6hhdzzlztkthqgbpjhgotdsccjpkkkbkl2c92pzxqbeluexvdn/xsovmim1qju2k0sqfg6yaegb6jrmkyl68hecwkvygckcwykgvhm59a1bwwgxkc/qsmziwz6vrbgzw9ykt47uqc+ah9dsg9bo29oaybmyept0aro35a/hgr0xvyc+tgmeofvgl7sufgjkd/lousj1tvvdkxvcovznc1s2kq3aa64bquhkzc3snqbpmr5kw8f0a9sf+e/3d4n98afby3/38zcmvd29vb0hbao/wdvoer2ebv0bxjiqcxrbkgnanwr+ycbvuuo1g5yacxytwionzaqwo+7hz7zqjtmumjemhvsujjsuj5k60xjbhyuwa73mz4tmmznrt2pwevxspdtjk3hfhr14ea0ussomjat7yehmfuymw5lbisupbyjmzd5n9ky5pmemaph/481wejbafwy3x7xck3ptuevcc3po4ow7l5wd57uzcqlonc6zzs+dv9373d+7f3apvtdr19znvnmure74p5ge2ec=</latexit> <latexit sha1_base64="cmzytw7hgbfdnmmx0ij330jbm9e=">aaafo3icdvrtb9s2efzbe2u1t3rdp/xlove7b3m9yu2afkwbolu3yaocboibapft0njjziyrkkklnvttj+3/7mp+y46ynspfxi863j18epfcibmi49r4/j/xrt/odd/7/oyt94svv/r6m63b377rslqrjikzsfv2xjrmxodicjph20ihy2czhs8wp9n48rkqzau4msscxzllbu94xay5prdv/b0kyuwmwrih8hxyhmm5sgvwyvcxykk0g84w5ajigu9f7r7acwgtclzbxfuodyzv8oowrihuzt6tep/0evbpdgembu5riohtt8kypskqsfty/tbigillehszo7/oh5hlk6qrzxq2sa6c9imdxxbg09owvzhu4vkxz8lihnzum4i9emwiyhdlpkplhijityneb60bpwakwypuykmukcziijljijnswsm5n3pwqkls205u7yrrkssndhg3er/aka+touwnjknvamcikcyl0icyoulgezq8r5ajmhhxms0lpgypmuwtowhlngqz1ex1kue6bva99q2qpyi0yutjhyxm02ofq6hhdzzlztkthqgbpjhgotdsccjpkkkbkl2c92pzxqbeluexvdn/xsovmim1qju2k0sqfg6yaegb6jrmkyl68hecwkvygckcwykgvhm59a1bwwgxkc/qsmziwz6vrbgzw9ykt47uqc+ah9dsg9bo29oaybmyept0aro35a/hgr0xvyc+tgmeofvgl7sufgjkd/lousj1tvvdkxvcovznc1s2kq3aa64bquhkzc3snqbpmr5kw8f0a9sf+e/3d4n98afby3/38zcmvd29vb0hbao/wdvoer2ebv0bxjiqcxrbkgnanwr+ycbvuuo1g5yacxytwionzaqwo+7hz7zqjtmumjemhvsujjsuj5k60xjbhyuwa73mz4tmmznrt2pwevxspdtjk3hfhr14ea0ussomjat7yehmfuymw5lbisupbyjmzd5n9ky5pmemaph/481wejbafwy3x7xck3ptuevcc3po4ow7l5wd57uzcqlonc6zzs+dv9373d+7f3apvtdr19znvnmure74p5ge2ec=</latexit> <latexit sha1_base64="cmzytw7hgbfdnmmx0ij330jbm9e=">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</latexit> 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.

19 1. Disordered Hubbard model for Si:P 2. Solvable disordered models: (A) Random matrix model of a `quantum dot Metal with quasiparticles (B) SYK model of a `quantum dot Metal without quasiparticles 3. High temperature superconductivity and strange metals.

20 The Sachdev-Ye-Kitaev (SYK) model Pick a set of random positions

21 The SYK model Place electrons randomly on some sites

22 The SYK model Entangle electrons pairwise randomly

23 The SYK model Entangle electrons pairwise randomly

24 The SYK model Entangle electrons pairwise randomly

25 The SYK model Entangle electrons pairwise randomly

26 The SYK model Entangle electrons pairwise randomly

27 The SYK model Entangle electrons pairwise randomly

28 The SYK model This describes both a strange metal and a black hole!

29 H = The 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)

30 The 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( ) S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

31 The 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!) X G( =0 )=Q., ( ) = U 2 G 2 ( )G( ) Low frequency analysis shows that the solutions must be gapless and obey (z) =µ 1 Ap z +..., G(z) = A pz where A = e i /4 ( /U 2 ) 1/4 at half-filling. The ground state is a non-fermi liquid, with a continuously variable density Q. <latexit sha1_base64="0uo4waqgszzn7hs7cocap/zkzug=">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</latexit> <latexit sha1_base64="0uo4waqgszzn7hs7cocap/zkzug=">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</latexit> S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993)

32 <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> The SYK model T = 0 fermion Green s function is incoherent: G( ) 1/2 at large. (Fermi liquids with quasiparticles have the coherent: G( ) 1/ ) S. Sachdev and J. Ye, PRL 70, 3339 (1993) T>0 Green s function has conformal invariance G (T/sin( k B T /~)) 1/2 The last property indicates eq ~/(k B T ), and this has been found in a recent numerical study.

33 <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">aaaem3icdvpnbtnaedyjv+gvwjhlibqrinz10kicegibsjkgbckfst02wtutenx12t1dtxgrt8gzcoakt4g4ia6iv2dsbih/k9mand2zb75vdojmcmn9/+orrvposemntp5qnt5z9tz5uqsxx5g01yguh6lm9ubaduqhcn0kk3ej08itqollyodbdf5yd7urqrraisothe+ugiuqw3knljqwwiatouphmrgvcdpildv6no4dktstygesrorfii0kifazyqtix2ast2gya5fpog2vdoazkqbul0/ajpxptvvxm5ldbcsetfcxv0d+b54am/ak4vo6v5vzakh+q49c4a743srnlchkwpm8fxg7qaxqysjiuhoba6tcre6zatznffpdwfwfmwfigyaj1brx+5gj5q7v+c6musdeg9y0orpmyjyrsnibhaghwjrgtblva4fhwmssf8ywemen2szyfmaxmrflxe5sb+octyc5nidpqg7xa7vlzhiujbc6o2ugc5zcwwozpnjnrwrgx9gyilgrgddwhbinxhwpwcyih8pyowzbxaorpp1fwrjdu8tqlzyxn5rfjvnnhrcxpa4fvyfw3vo7sabt4vlviskwcdgz3sfzqvrlla647ns2y5pei6gkfr2n0dy87/nl/f7ap14t+4pbrcrw+yu3+n3oen695p3zejqa+8ainmwtohnkbsxm18/svjkjty8tn5jxcidpcjnmxrm0w2ut3bsukiccykgfslb7d0eupdgmsak6mxabmz/pkue/zjzzox5slujluuuvhgcncwk2hwpgibiaqyvpvqseakg1qhhzzunlk9ribmnm1mtgjbp4yu6lihdkze+gufns6icm8h/jrc/r+l73ww/+3v2zviedy84vp+10nb5zz3nkphxwnbdxpvgu8b7xofm2+an5ufnl4grjyczmkvpban79dicevsy=</latexit> <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> The SYK model T = 0 fermion Green s function is incoherent: G( ) 1/2 at large. (Fermi liquids with quasiparticles have the coherent: G( ) 1/ ) S. Sachdev and J. Ye, PRL 70, 3339 (1993) T>0 Green s function has conformal invariance G (T/sin( k B T /~)) 1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999) The last property indicates eq ~/(k B T ), and this has been found in a recent numerical study.

34 <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">aaaem3icdvpnbtnaedyjv+gvwjhlibqrinz10kicegibsjkgbckfst02wtutenx12t1dtxgrt8gzcoakt4g4ia6iv2dsbih/k9mand2zb75vdojmcmn9/+orrvposemntp5qnt5z9tz5uqsxx5g01yguh6lm9ubaduqhcn0kk3ej08itqollyodbdf5yd7urqrraisothe+ugiuqw3knljqwwiatouphmrgvcdpildv6no4dktstygesrorfii0kifazyqtix2ast2gya5fpog2vdoazkqbul0/ajpxptvvxm5ldbcsetfcxv0d+b54am/ak4vo6v5vzakh+q49c4a743srnlchkwpm8fxg7qaxqysjiuhoba6tcre6zatznffpdwfwfmwfigyaj1brx+5gj5q7v+c6musdeg9y0orpmyjyrsnibhaghwjrgtblva4fhwmssf8ywemen2szyfmaxmrflxe5sb+octyc5nidpqg7xa7vlzhiujbc6o2ugc5zcwwozpnjnrwrgx9gyilgrgddwhbinxhwpwcyih8pyowzbxaorpp1fwrjdu8tqlzyxn5rfjvnnhrcxpa4fvyfw3vo7sabt4vlviskwcdgz3sfzqvrlla647ns2y5pei6gkfr2n0dy87/nl/f7ap14t+4pbrcrw+yu3+n3oen695p3zejqa+8ainmwtohnkbsxm18/svjkjty8tn5jxcidpcjnmxrm0w2ut3bsukiccykgfslb7d0eupdgmsak6mxabmz/pkue/zjzzox5slujluuuvhgcncwk2hwpgibiaqyvpvqseakg1qhhzzunlk9ribmnm1mtgjbp4yu6lihdkze+gufns6icm8h/jrc/r+l73ww/+3v2zviedy84vp+10nb5zz3nkphxwnbdxpvgu8b7xofm2+an5ufnl4grjyczmkvpban79dicevsy=</latexit> <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> <latexit sha1_base64="4qtgybnvf4bwaohansesw2ahoq8=">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</latexit> The SYK model T = 0 fermion Green s function is incoherent: G( ) 1/2 at large. (Fermi liquids with quasiparticles have the coherent: G( ) 1/ ) S. Sachdev and J. Ye, PRL 70, 3339 (1993) T>0 Green s function has conformal invariance G (T/sin( k B T /~)) 1/2 A. Georges and O. Parcollet PRB 59, 5341 (1999) The last property indicates eq ~/(k B T ), and this has been found in a recent numerical study. A. Eberlein, V. Kasper, S. Sachdev, and J. Steinberg, PRB 96, (2017)

35 Quantum matter without quasiparticles: If there are no quasiparticles, then E 6= X n " + X, F n n +... If there are no quasiparticles, then eq =# ~ k B T Systems without quasiparticles are the fastest possible in reaching local equilibrium, and all many-body quantum systems obey, as T! 0 eq >C ~ k B T. S. Sachdev, Quantum Phase Transitions, Cambridge (1999) In Fermi liquids eq 1/T 2, and so the bound is obeyed as T! 0. This bound rules out quantum systems with e.g. eq ~/(Jk B T ) 1/2. There is no bound in classical mechanics (~! 0). By cranking up frequencies, we can attain equilibrium as quickly as we desire.

36 Quantum matter without quasiparticles: If there are no quasiparticles, then E 6= X n " + X, F n n +... If there are no quasiparticles, then eq =# ~ k B T Systems without quasiparticles are the fastest possible in reaching local equilibrium, and all many-body quantum systems obey, as T! 0 eq >C ~ k B T. S. Sachdev, Quantum Phase Transitions, Cambridge S. Sachdev, (1999) Quantum Phase Transitions, Cambridge (1999) In Fermi liquids eq 1/T 2, and so the bound is obeyed as T! 0. This bound rules out quantum systems with e.g. eq ~/(Jk B T ) 1/2. There is no bound in classical mechanics (~! 0). By cranking up frequencies, we can attain equilibrium as quickly as we desire.

37 Quantum matter without quasiparticles: If there are no quasiparticles, then E 6= X n " + X, F n n +... If there are no quasiparticles, then eq =# ~ k B T Systems without quasiparticles are the fastest possible in reaching local equilibrium, and all many-body quantum systems obey, as T! 0 eq >C ~ k B T. S. Sachdev, Quantum Phase Transitions, Cambridge (1999) In Fermi liquids eq 1/T 2, and so the bound is obeyed as T! 0. This bound rules out quantum systems with e.g. eq ~/(Jk B T ) 1/2. There is no bound in classical mechanics (~! 0). By cranking up frequencies, we can attain equilibrium as quickly as we desire.

38 SYK models and black holes Black holes have an entropy and a temperature, T H = ~c 3 /(8 GMk B ). Black holes relax to thermal equilibrium in a time ~/(k B T H )=8 GM/c 3. Black holes

39 <latexit sha1_base64="c/xovl74roynoxdul6vugijr6gk=">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</latexit> <latexit sha1_base64="c/xovl74roynoxdul6vugijr6gk=">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</latexit> <latexit sha1_base64="c/xovl74roynoxdul6vugijr6gk=">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</latexit> <latexit sha1_base64="c/xovl74roynoxdul6vugijr6gk=">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</latexit> SYK models and black holes Black holes have an entropy and a temperature, T H = ~c 3 /(8 GMk B ). Black holes relax to thermal equilibrium in a time ~/(k B T H )=8 GM/c 3. The entropy of black holes is proportional to their surface area: the SYK model (in 0+1 dimensions) maps to a black hole in 1+1 dimensions. Black holes

40 Black hole horizon SYK and black holes ~x Black holes with a near-horizon AdS2 geometry (described by quantum gravity in 1+1 spacetime dimensions) match the properties of the 0+1 dimensional SYK model in the previous slide: Ns0 is the Bekenstein-Hawking entropy S. Sachdev, PRL 105, (2010); A. Kitaev (2015); J. Maldacena, D. Stanford, and Zhenbin Yang, arxiv:

41 1. Disordered Hubbard model for Si:P 2. Solvable disordered models: (A) Random matrix model of a `quantum dot Metal with quasiparticles (B) SYK model of a `quantum dot Metal without quasiparticles 3. High temperature superconductivity and strange metals.

42 High temperature superconductors Cu O CuO 2 plane Described by a Hubbard model on the Cu sites YBa 2 Cu 3 O 6+x

43 SM FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

44 SM Strange Metal FL YBa 2 Cu 3 O 6+x Figure: K. Fujita and J. C. Seamus Davis

45 <latexit sha1_base64="k7jx8efbma2lrujxy27hncmbd18=">aaadi3icdvjnb9naelxnvzffkry5jehqilrsjwkl4vsvc9ykagilbejw67g9qr3r7q4brvb4l1z4k1w4qmwfa/+ftekigmboz++9mz1967diudzb8mp1rly9dv3gxk3/1u07d+81nu+/07judidmzlidh1rjxguodtczhhckar5mebsevfzqr2eonjfi0mwlhoc0etzmjbplttbdfytehiukotcofv4w5kcln7luhpgvwz1qkywwmj2+nyqkbnvtbui+whk63afrpa/xgskulz30m5tk9g0z2/j0tktclhm7vbb8jozmkfinfbrqe1f+xs18g1pepbk1dbousxrkiho9nrqsggiew2grnk5ftazcgjy5arnrakitsqlpdr7vtqibleswmwkrz1es09bp7ax/tlwxwmsbca0mragxhvjwazku96ocyccnkkklrcanztajukfvtwswrmqc/mdqf/4mujxtdoo6mdtossrzatcwlknaj7pbycyvvyazdbc+ktuwlj3qbecwcpqjhlerj1jay8teentayykmrnjlhrxntz7noxxm1kt6b21j/ksblsbehvdcfkvbwdyhxwugrslyj7grkwqmm1tamej2v2apvztzdlrvq7i4kfwfdhudqsd402vu7ddpbdgpnufoltn1+s6e88o5ciyocz+6n92v7jfvk/ffo/e+r62ew/c8cp4o7+cvfkr9bq==</latexit> Ubiquitous Strange, Bad, Incoherent, or Ultra-quantum metal has a resistivity,, which obeys T, and in some cases h/e 2 (in two dimensions), where h/e 2 is the quantum unit of resistance.

46 Ultra-quantum metals just got stranger B-linear magnetoresistance!? Ba-122 LSCO I. M. Hayes et. al., Nat. Phys P. Giraldo-Gallo et. al., arxiv:

47 Ultra-quantum metals just got stranger Scaling between B and T!? Ba-122 Ba-122 I. M. Hayes et. al., Nat. Phys. 2016

48 SYK building blocks for a strange metal SYK quantum dots of electrons with random hopping between them. t U H = X x X U ijkl,x c ix c jx c kx c lx i<j,k<l hxx 0 i i,j X X X X + t ij,xx 0c i,x c j,x 0 U ijkl 2 = 2U 2 N 3 Gaussian distrib d t ij,x,x 0 2 = t 2 0 /N. sm, one studies P Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, (2017)

49 Low coherence scale E c t2 0 U (a) Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, (2017)

50 Low coherence scale E c t2 0 U For E c <T <U,the resistivity,, and entropy density, s, are (a) h e 2 T E c, s = s 0 Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, (2017)

51 Low coherence scale E c t2 0 U For T<E c,the resistivity,, and entropy density, s, are = h e 2 " c 1 + c 2 T E c 2 # (a) s s 0 T E c Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, (2017)

52 Low coherence scale E c t2 0 U (a) But this model exhibits negligible magnetoresistivity! Xue-Yang Song, Chao-Ming Jian, and L. Balents, PRL 119, (2017)

53 Infecting a Fermi liquid and making it SYK Mobile electrons (c) interacting with SYK quantum dots (f) with exchange interactions. This yields the first model agreeing with magnetotransport in strange metals! c f Aavishkar A. Patel, John McGreevy, Daniel P. Arovas, Subir Sachdev, arxiv:

54 Infecting a Fermi liquid and making it SYK Mobile electrons (c) interacting with SYK quantum dots (f) with exchange interactions. This yields the first model agreeing with magnetotransport in strange metals! c f Debanjan Chowdhury, Yochai Werman, Erez Berg, T. Senthil, arxiv:

55 Infecting a Fermi liquid and making it SYK Mobile electrons (c) interacting with SYK quantum dots (f) with exchange interactions. This yields the first model agreeing with magnetotransport in strange metals! Large N solution (with or without microscopic disorder) yields a marginal Fermi liquid metal, with conductivities of the form: xx(b,t) = 1 B T L T xy(b,t) = B B T 2 H T where the scaling functions interpolate as L,H(b! 0) constant ; L,H(b!1) 1/b 2 This solution exhibits B/T scaling, but the magnetoresistance xx saturates for B T. <latexit sha1_base64="nvhrmartpt4gwjshxqwwoa/+row=">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</latexit> Aavishkar A. Patel, John McGreevy, Daniel P. Arovas, Subir Sachdev, arxiv:

56 Infecting a Fermi liquid and making it SYK Need mesoscopic disorder to obtain linear-in-b magnetoresistance Current path length increases linearly with B at large B due to local Hall effect, which causes the global resistance to increase linearly with B at large B. Exact numerical solution of charge-transport equations in a random-resistor network. (M. M. Parish and P. Littlewood, Nature 426, 162 (2003))

57 Infecting a Fermi liquid and making it SYK n b /n a =0.8 b/ a =0.8 a =0.1k B T t/100 (B =0.0025) Scaling between B and T (T = t/100) ~ 50 T (a = 3.82 A) Aavishkar A. Patel, John McGreevy, Daniel P. Arovas, Subir Sachdev, arxiv:

58 <latexit sha1_base64="8ddjrrjnv7czdq/lbzn5isozyp8=">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</latexit> <latexit sha1_base64="8ddjrrjnv7czdq/lbzn5isozyp8=">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</latexit> <latexit sha1_base64="8ddjrrjnv7czdq/lbzn5isozyp8=">aaaegnicfvnlb9qwee53eztl0rbeicuicqli3vv2qz7epruvuo9f9cv1v5xjtbkrjh3zttml9j9w4gq/ghviyoufwx9gnn22gablksyznu/75ugokmk6mpw+02rfuhnr9uydzt179x/mzs88plc6nbz3uzbahexmohqk951weo8kgyypjb5gp9s+fnigxgqt9ty4wfhouiuswzkj18lc6/ewwlsowjjmxvu86ay91dnlhaur8kiiuep3uc4lrva5yhwmemk03igiltbqwif1zchobhlmhbqomseziawsiwk3hmgmkje7o5ufocblcctvoujnovwrlckuzrhxr9gqhlhybwi0vyky6gwzbbykddbveue1c/ocengocbfqhowrgaso0ajuobuzx7pjcfekj7j/q+9sx+zuh5dros/coeimzonheorebur175rrxew81vgjl8gsno2hs/f8cm+5qrlpk4vjtay43/qgpykgmmkwj3hm0ar+ffjzjjucbkm5awykklv2e16df1rvtncskponelfljcv+6kh0wjjegjquopo/tmq6sxkutubkfjhshatrg/uv8mzgbxwnji2vsb+g0o+fzvkmpmf3zp7hmna89jrcmmup+2hhrjx1rndpt6q0wdb+yli8jloxho2obvb3ap6rhwql7isaymq8v2bulld2ned0k0rn9s+yd/4tdly6zgnuc1wujsy4iupkcu6dfww0a0mtl2mygk0zaqwemb8o9gq6q4v0nltqsnro8nxviiaeekuo35/ljsc/jynbrx/2+q8hi1svp52adz4et4olob+sb1vbtrab7ae89a71ofwx9an9vv25/ax9dxk1ntpnert8dtrffglipl0u</latexit> <latexit sha1_base64="8ddjrrjnv7czdq/lbzn5isozyp8=">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</latexit> This simple two-component model describes a new state of matter which is realized by electrons in the presence of strong interactions and disorder. Can such a model be realized as a fixed-point of a generic theory of strongly-interacting electrons in the presence of disorder? Can we start from a single-band Hubbard model with (or without) disorder, and end up with such two-band fixed point, with emergent local conservation laws? Experiments on Si:P indicate an a rmative answer.