Subir Sachdev. Talk online: sachdev.physics.harvard.edu

Transcription

1 HARVARD Gauge theory for the cuprates near optimal doping Developments in Quantum Field Theory and Condensed Matter Physics Simons Center for Geometry and Physics, Stony Brook University November 7, 2018 Subir Sachdev Talk online: sachdev.physics.harvard.edu

2 HARVARD Mathias Scheurer Grigory Tarnopolsky Harley Scammell arxiv:1811.xxxxx

3 Fermi Surface and Pseudogap Evolution in a Cuprate Superconductor Yang He, Yi Yin, M. Zech, A. Soumyanarayanan, I. Zeljkovic, M. M. Yee, M. C. Boyer, K. Chatterjee, W. D. Wise, Takeshi Kondo, T. Takeuchi, H. Ikuta, P. Mistark, R. S. Markiewicz, A. Bansil, S. Sachdev, E. W. Hudson, and J. E. Hoffman, Science 344, 608 (2014) Simultaneous Transitions in Cuprate Momentum-Space Topology and Electronic Symmetry Breaking K. Fujita, Chung Koo Kim, Inhee Lee, Jinho Lee, M. H. Hamidian, I. A. Firmo, S. Mukhopadhyay, H. Eisaki, S. Uchida, M. J. Lawler, E.-A. Kim, J. C. Davis, Science 344, 612 (2014) 0.4 Intensity Smoothed S x S y doping

4 The remarkable underlying ground states of cuprate superconductors Cyril Proust and Louis Taillefer, arxiv:

5 1. SU(2) gauge theory with adjoint scalars Confining and Higgs phases in 2+1 dimensions 2. SU(2) gauge theory of fluctuating antiferromagnetism 3. Role of fermions Pseudogap as a FL* phase

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3, ` =1...N h ), with potential V (H à ) and SU(2) gauge field A a µ.thereis O(N h ) global flavor symmetry. L = 1 4g 2 F a µ F a µ + 1 µh à abc A b µh c` 2 + V (H à ) Fµ a µ A A a µ abc A b µa c V (H à ) = sh à H à + u 0 H à H à HmH b m b +u 1 H à H a mh b` H b m 1 N h H à H à H b mh b m

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sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> N h =1 <latexit sha1_base64="+kg6ggfxejcfzjgwboxcu/bor3u=">aaab7nicdvdjsgnbek1xjxglevtsgarpq49mmrkiqs+ejijziblct6cnadkz0n0jhcef4cwdil79hm/+jz1funehby/3qqiqf6sck43xh7wyura+svnykm7v7o7tlw4owyrjjgvnmohedgkimoaxa2qubeukkpeoekwdjk9mfvuescwt+e5puuzhzbjzkfoijds+6y/qbxl6ptk2xq97nqdhg1fdssu1pikrxs1djo3nkmmsjx7pvtdiabaxwfnbloo6onv+tqtmvlbpszcplhi6jkpwntqmevn+pj93ik6nmkbhik3fgs3v7xm5izsarihpjigeqd/etpzl62y6rpk5j9nms5gufowzqdpbs9/rgetgtzgyqqjk5lzer0qsqk1crrpc16fof9i6tx1so7duux65jkmax3acz+cab3w4hgy0gciyhuajnq3uerrerndf64q1ndmch7depge5ji7x</latexit> <latexit sha1_base64="+kg6ggfxejcfzjgwboxcu/bor3u=">aaab7nicdvdjsgnbek1xjxglevtsgarpq49mmrkiqs+ejijziblct6cnadkz0n0jhcef4cwdil79hm/+jz1funehby/3qqiqf6sck43xh7wyura+svnykm7v7o7tlw4owyrjjgvnmohedgkimoaxa2qubeukkpeoekwdjk9mfvuescwt+e5puuzhzbjzkfoijds+6y/qbxl6ptk2xq97nqdhg1fdssu1pikrxs1djo3nkmmsjx7pvtdiabaxwfnbloo6onv+tqtmvlbpszcplhi6jkpwntqmevn+pj93ik6nmkbhik3fgs3v7xm5izsarihpjigeqd/etpzl62y6rpk5j9nms5gufowzqdpbs9/rgetgtzgyqqjk5lzer0qsqk1crrpc16fof9i6tx1so7duux65jkmax3acz+cab3w4hgy0gciyhuajnq3uerrerndf64q1ndmch7depge5ji7x</latexit> <latexit sha1_base64="+kg6ggfxejcfzjgwboxcu/bor3u=">aaab7nicdvdjsgnbek1xjxglevtsgarpq49mmrkiqs+ejijziblct6cnadkz0n0jhcef4cwdil79hm/+jz1funehby/3qqiqf6sck43xh7wyura+svnykm7v7o7tlw4owyrjjgvnmohedgkimoaxa2qubeukkpeoekwdjk9mfvuescwt+e5puuzhzbjzkfoijds+6y/qbxl6ptk2xq97nqdhg1fdssu1pikrxs1djo3nkmmsjx7pvtdiabaxwfnbloo6onv+tqtmvlbpszcplhi6jkpwntqmevn+pj93ik6nmkbhik3fgs3v7xm5izsarihpjigeqd/etpzl62y6rpk5j9nms5gufowzqdpbs9/rgetgtzgyqqjk5lzer0qsqk1crrpc16fof9i6tx1so7duux65jkmax3acz+cab3w4hgy0gciyhuajnq3uerrerndf64q1ndmch7depge5ji7x</latexit> <latexit sha1_base64="+kg6ggfxejcfzjgwboxcu/bor3u=">aaab7nicdvdjsgnbek1xjxglevtsgarpq49mmrkiqs+ejijziblct6cnadkz0n0jhcef4cwdil79hm/+jz1funehby/3qqiqf6sck43xh7wyura+svnykm7v7o7tlw4owyrjjgvnmohedgkimoaxa2qubeukkpeoekwdjk9mfvuescwt+e5puuzhzbjzkfoijds+6y/qbxl6ptk2xq97nqdhg1fdssu1pikrxs1djo3nkmmsjx7pvtdiabaxwfnbloo6onv+tqtmvlbpszcplhi6jkpwntqmevn+pj93ik6nmkbhik3fgs3v7xm5izsarihpjigeqd/etpzl62y6rpk5j9nms5gufowzqdpbs9/rgetgtzgyqqjk5lzer0qsqk1crrpc16fof9i6tx1so7duux65jkmax3acz+cab3w4hgy0gciyhuajnq3uerrerndf64q1ndmch7depge5ji7x</latexit> Phase diagrams of SU(2) gauge theory V (H à ) hh à i6=0 V (H à ) hh à i =0 Higgs H à Condensation of H a breaks SU(2) to U(1) U(1) confines because of proliferation of thooft-polyakov monopoles Monopole action p s, leading to an exponentially large confinement scale Crossover H à Confinement s A. C. Davis, A. Hart, T. W. B. Kibble, and A. Rajantie, Phys. Rev. D 65, (2002)

8 N h =2 <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> There is a global O(N h ) symmetry, and we can define a gauge-invariant order parameter Q`m = H à H a m `m N h H a nh a n For u 1 < 0, the Higgs condensate is of the form H a 1 =(H 0, 0, 0), H a 2 =(0, 0, 0) This breaks gauge SU(2) to U(1), and the U(1) confines, as for N h = 1. But the global O(2) is broken because hq`m i6=0

9 <latexit sha1_base64="i1bublbespmz3su6ck4qikr7tka=">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</latexit> <latexit 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sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> Phase diagrams of SU(2) gauge theory V (H à ) hh à i6=0 V (H à ) hh à i =0 Higgs H à H à u 1 < 0 Confinement Condensation of H a breaks SU(2) to U(1), which confines Broken O(2) symmetry, hq`m i6= 0. 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sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> Phase diagrams of SU(2) gauge theory V (H à ) hh à i6=0 V (H à ) hh à i =0 Higgs H à H à u 1 < 0 Condensation of H a breaks SU(2) to U(1), which confines Broken O(2) symmetry, hq`m i6= 0. Possible Deconfined critical SU(2) gauge theory <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">aaacphicbva9s8raen34bfw6tbrzpartjisfp52ohewjngqx49hsjrnfzw7y3qhhub9m44+ws7kxumtw2r0pxa8ha48382ayf+wcaen5d87e5nt0zozcvluwuls8ulldu9cyubsavhkprikigtmbtcmmh6tcackidpfr9dggf3kdsjmpzk0vh3zgusesromxuqdyfkaqmlfseazu321irzk1h6f7dfskxo6nwxbtxyw1cuvpmtvbdk5jkdopblogvc8nqcsfszqvqlfzhsbelfdru8e+/ll8mamimrqdyn0ys1pk1k450brle7lpl0tzkxz6blhoyam9jim0lbuka90uh8/38zzvypxizusypfs/o0qsad3lijuzedpvv3sd8b9eqzdjxrtkii8mcdo6lbqcg4khsekykacg9ywho3gw7rjfqm1auzye//flf8lfupo9mn8ava8ox3hmoq20ibarj+roaj2gbmoiim7ri3pgl86d8+s8om+j0qln7flhp+c8fwa2rq6w</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> Confinement Multi-universality of a Landau-allowed transition? s

11 <latexit 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<latexit 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<latexit 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N h =2 <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> There is a global O(N h ) symmetry, and we can define a gauge-invariant order parameter Q`m = H à H a m `m N h H a nh a n For u 1 < 0, the Higgs condensate is of the form H a 1 =(H 0, 0, 0), H a 2 =(0, 0, 0) This breaks gauge SU(2) to U(1), and the U(1) confines, as for N h = 1. But the global O(2) is broken because hq`m i6=0 For u 1 > 0, the Higgs condensate is of the form H a 1 =(H 0, 0, 0), H a 2 =(0,H 0, 0) This breaks gauge SU(2) to Z 2, leading to Z 2 topological order (as in the toric code ). But now the global O(2) is unbroken because hq`m i =0

12 <latexit sha1_base64="anz1or50rs1l8che7q+fjb8gpgo=">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</latexit> <latexit 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sha1_base64="caqojbdcu2upy7msykoucurlfai=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> N h =2 <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> <latexit sha1_base64="kdvsngxhs1hnkr5ufwov+in2mcs=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnpckslfchrxjbhmaskqejo9szoehe4eiqz5cc8efphq93jzb+wsgoo+khi8v0vvpt8rxgmmp6zc2vrg5lz+u7czu7d/udw8aqs4lzs1acxi2fwjyojhrkw5fqybsezcx7cop7ma+517jhwpozs9tzgxklhea06jnllnzjbgf6g8kjawxxgxw3cqtngtuq1wdknimlt3kwpjbuqwqnnqfo8py5qglnjuekv6dk60lxgporvsvuiniiwetsii9qynsmiuly3onaezowxreettkuyl9fterkklpqfvokoix+q3nxf/8nqpdupexqmk1syiy0vbkpco0fx3nossus2mhhaqubkv0tgrhgqtumge8pup+p+0y7adbee2umpcruliwwmcwjk44eidrqejlaawgqd4gmcrsr6tf+t12zqzvjph8apw2yc6qi7y</latexit> Phase diagrams of SU(2) gauge theory V (H à ) hh à i6=0 V (H à ) hh à i =0 Higgs H à H à u 1 > 0 Confinement Condensation of H a breaks SU(2) to Z 2, leading to Z 2 topological order. Possible Deconfined critical SU(2) gauge theory <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> Unbroken O(2) symmetry, hq`m i = 0. s

13 N h =3 <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> For u 1 < 0, the Higgs condensate is of the form H a 1 =(H 0, 0, 0), H a 2 =(0, 0, 0), H a 3 =(0, 0, 0) This breaks gauge SU(2) to U(1), and the U(1) confines, as for N h =1, 2. The global O(3) is broken to O(2) and hq`m i6=0

14 <latexit sha1_base64="97ipecln4tbkprpzzaleb2t/uje=">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</latexit> <latexit 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sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> N h =3 <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> Phase diagrams of SU(2) gauge theory V (H à ) hh à i6=0 V (H à ) hh à i =0 Higgs H à H à u 1 < 0 Condensation of H a breaks SU(2) to U(1), which confines O(3) symmetry broken to O(2), hq`m i6= 0. Possible Deconfined critical SU(2) gauge theory <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">aaacphicbva9s8raen34bfw6tbrzpartjisfp52ohewjngqx49hsjrnfzw7y3qhhub9m44+ws7kxumtw2r0pxa8ha48382ayf+wcaen5d87e5nt0zozcvluwuls8ulldu9cyubsavhkprikigtmbtcmmh6tcackidpfr9dggf3kdsjmpzk0vh3zgusesromxuqdyfkaqmlfseazu321irzk1h6f7dfskxo6nwxbtxyw1cuvpmtvbdk5jkdopblogvc8nqcsfszqvqlfzhsbelfdru8e+/ll8mamimrqdyn0ys1pk1k450brle7lpl0tzkxz6blhoyam9jim0lbuka90uh8/38zzvypxizusypfs/o0qsad3lijuzedpvv3sd8b9eqzdjxrtkii8mcdo6lbqcg4khsekykacg9ywho3gw7rjfqm1auzye//flf8lfupo9mn8ava8ox3hmoq20ibarj+roaj2gbmoiim7ri3pgl86d8+s8om+j0qln7flhp+c8fwa2rq6w</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> Confinement Multi-universality of a Landau-allowed transition? s

15 <latexit sha1_base64="a6fm79rp880/tomjfi/qmjp+s80=">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</latexit> <latexit sha1_base64="a6fm79rp880/tomjfi/qmjp+s80=">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</latexit> <latexit sha1_base64="a6fm79rp880/tomjfi/qmjp+s80=">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</latexit> <latexit sha1_base64="a6fm79rp880/tomjfi/qmjp+s80=">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</latexit> N h =3 <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> For u 1 < 0, the Higgs condensate is of the form H a 1 =(H 0, 0, 0), H a 2 =(0, 0, 0), H a 3 =(0, 0, 0) This breaks gauge SU(2) to U(1), and the U(1) confines, as for N h =1, 2. The global O(3) is broken to O(2) and hq`m i6=0 For u 1 > 0, the Higgs condensate is of the form H a 1 =(H 0, 0, 0), H a 2 =(0,H 0, 0), H a 3 =(0, 0,H 0 ) This breaks gauge SU(2) to Z 2, leading to Z 2 topological order (as in the toric code ). But now the global O(3) is unbroken because hq`m i =0

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sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> <latexit sha1_base64="zeeoj1q1lpu+nudmf3hte4r4xjg=">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</latexit> N h =3 <latexit 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sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> <latexit sha1_base64="5cy0laabpolrhvljwdzrp11g1r4=">aaab7nicdvdjsgnbek2jw4xb1koxxib4gnp0kslfchrxjbhmaskqejqdpenpqnepeiz8hbcpinj1e7z5n3ywquuffdzeq6kqxpairjtgh1zuzxvtfso/wdja3tndk+4fnfwcssoanbaxbademcej1tbcc9zojcnhifgrgf/n/ny9k4rh0z2ejmwpytdia06jnllrpjdcf+i8vyxh2/wwv3uqtnhflzddq8q44lu95nh4jhisue8v37v9mkyhizqvrkmogxptz0rqtgwbfrqpygmhyzjkhumjejllz/nzp+jekh00ikwpsko5+n0ii6fskzawnshri/xbm4l/ez1ud6p+xqmk1syii0wdvcado9nvqm8lo1pmdcfucnmroimicdumoyij4ett9d9pntkotp1bt1s7xmarhym4hlnwwimaxemdgkbhda/wbm9wyj1al9brojvnlwco4qest088li7z</latexit> Phase diagrams of SU(2) gauge theory V (H à ) hh à i6=0 V (H à ) hh à i =0 Higgs H à H à u 1 > 0 Confinement Condensation of H a breaks SU(2) to Z 2, leading to Z 2 topological order. Possible Deconfined critical SU(2) gauge theory <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> Unbroken O(3) symmetry, hq`m i = 0. s

17 N h 4 <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> N h 4 <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> For u 1 < 0, the Higgs condensate is of the form H a 1 =(H 0, 0, 0), H à >1 =(0, 0, 0) This breaks gauge SU(2) to U(1), and the U(1) confines, as for N h =1, 2. The global O(N h ) is broken to O(N h 1) and hq`m i6=0

18 <latexit sha1_base64="zmr6i3dhdfdcrtkno3mukofzwbw=">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</latexit> <latexit 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sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> Phase diagrams of SU(2) gauge theory V (H à ) hh à i6=0 V (H à ) hh à i =0 Higgs H à H à u 1 < 0 Condensation of H a breaks SU(2) to U(1), which confines O(N h )symmetry broken to O(N h 1), hq`m i6= 0. Possible Deconfined critical SU(2) gauge theory <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">aaacphicbva9s8raen34bfw6tbrzpartjisfp52ohewjngqx49hsjrnfzw7y3qhhub9m44+ws7kxumtw2r0pxa8ha48382ayf+wcaen5d87e5nt0zozcvluwuls8ulldu9cyubsavhkprikigtmbtcmmh6tcackidpfr9dggf3kdsjmpzk0vh3zgusesromxuqdyfkaqmlfseazu321irzk1h6f7dfskxo6nwxbtxyw1cuvpmtvbdk5jkdopblogvc8nqcsfszqvqlfzhsbelfdru8e+/ll8mamimrqdyn0ys1pk1k450brle7lpl0tzkxz6blhoyam9jim0lbuka90uh8/38zzvypxizusypfs/o0qsad3lijuzedpvv3sd8b9eqzdjxrtkii8mcdo6lbqcg4khsekykacg9ywho3gw7rjfqm1auzye//flf8lfupo9mn8ava8ox3hmoq20ibarj+roaj2gbmoiim7ri3pgl86d8+s8om+j0qln7flhp+c8fwa2rq6w</latexit> Confinement Multi-universality of a Landau-allowed transition? s

19 <latexit sha1_base64="8hosc9qgjzxc2cffieedlrrjsua=">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</latexit> <latexit sha1_base64="8hosc9qgjzxc2cffieedlrrjsua=">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</latexit> <latexit sha1_base64="8hosc9qgjzxc2cffieedlrrjsua=">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</latexit> <latexit sha1_base64="8hosc9qgjzxc2cffieedlrrjsua=">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</latexit> N h 4 <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> N h 4 <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> For u 1 < 0, the Higgs condensate is of the form H a 1 =(H 0, 0, 0), H à >1 =(0, 0, 0) This breaks gauge SU(2) to U(1), and the U(1) confines, as for N h =1, 2. The global O(N h ) is broken to O(N h 1) and hq`m i6=0 For u 1 > 0, the Higgs condensate is of the form H a 1 =(H 0, 0, 0), H a 2 =(0,H 0, 0), H a 3 =(0, 0,H 0 ), H à >3 =(0, 0, 0) This breaks gauge SU(2) to Z 2, leading to Z 2 topological order (as in the toric code ). But now the global O(N h ) is broken to O(N h 3) O(3), and hq`m i6=0

20 <latexit sha1_base64="9rravhyj/0hvwtot4narafpgxwy=">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</latexit> <latexit 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sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> <latexit sha1_base64="ykt0tqbb0okec5mpuqcvk9dgs7s=">aaab8xicdvdlsgnbejynrxhfuy9ebopgazmvttbhobdpese8malhdtjjhszorjozqgj5cy8efphq33jzb5w8bbutaciquunuchpbtshkw8msrk6tb2q3c1vbo7t7+f2duo5txadgyhgrzkg1cc6hzrgr0ewu0cgu0ahhfzo/cq9k81jemhecnygojo9zro2vbq+6q9wewb32u/kccf2abgupe5eu/glrt6riske5wj5l5iigjard/hu7f7m0ammyofq3pjkyzoqqw5maaa6dakgog9ebtcyvnaldmcwvnuitq/rwp1a2pmfz9fvehezaj6pqdkbudpvvbyb+5bvs0y93jlwmqqhjfov6qcamxrp3cy8ryeamlafmcxsrzkoqkdm2pjwn4ett/d+pn7kecb1rv1a5x8arrufogj0idwwogi5rfduqqxi9ocf07gjn0xlxxhetgwc5c4h+whn7bmsykfq=</latexit> Phase diagrams of SU(2) gauge theory V (H à ) hh à i6=0 V (H à ) hh à i =0 Higgs H à H à u 1 > 0 Confinement Condensation of H a breaks SU(2) to Z 2, leading to Z 2 topological order. Possible Deconfined critical SU(2) gauge theory <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">aaacphicbva9s8raen34bfw6tbrzpartjisfp52ohewjngqx49hsjrnfzw7y3qhhub9m44+ws7kxumtw2r0pxa8ha48382ayf+wcaen5d87e5nt0zozcvluwuls8ulldu9cyubsavhkprikigtmbtcmmh6tcackidpfr9dggf3kdsjmpzk0vh3zgusesromxuqdyfkaqmlfseazu321irzk1h6f7dfskxo6nwxbtxyw1cuvpmtvbdk5jkdopblogvc8nqcsfszqvqlfzhsbelfdru8e+/ll8mamimrqdyn0ys1pk1k450brle7lpl0tzkxz6blhoyam9jim0lbuka90uh8/38zzvypxizusypfs/o0qsad3lijuzedpvv3sd8b9eqzdjxrtkii8mcdo6lbqcg4khsekykacg9ywho3gw7rjfqm1auzye//flf8lfupo9mn8ava8ox3hmoq20ibarj+roaj2gbmoiim7ri3pgl86d8+s8om+j0qln7flhp+c8fwa2rq6w</latexit> <latexit sha1_base64="39cwgtj3dzqamqnto81eorewojm=">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</latexit> O(N h )symmetry broken to O(N h 3) O(3), hq`m i6= 0. s

21 1. SU(2) gauge theory with adjoint scalars Confining and Higgs phases in 2+1 dimensions 2. SU(2) gauge theory of fluctuating antiferromagnetism 3. Role of fermions Pseudogap as a FL* phase

22 <latexit sha1_base64="ydzkz6d8fyef9krt77uxnx2rpti=">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</latexit> <latexit sha1_base64="ydzkz6d8fyef9krt77uxnx2rpti=">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</latexit> <latexit sha1_base64="ydzkz6d8fyef9krt77uxnx2rpti=">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</latexit> <latexit sha1_base64="ydzkz6d8fyef9krt77uxnx2rpti=">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</latexit> Gauge theory of fluctuating antiferromagnetism We can (exactly) transform the Hubbard model to the spin-fermion model: electrons c i on the square lattice with dispersion X H c = t c i, c i+v, + c i+v, c i, i, µ X i c i, c i, + H int are coupled to a magnetic moment order parameter p (i), p = x, y, z H int = X i p (i)c i, p c i, + V

23 <latexit sha1_base64="a3asiy3wafzborgf9n7bi3nsxeq=">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</latexit> <latexit sha1_base64="a3asiy3wafzborgf9n7bi3nsxeq=">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</latexit> <latexit sha1_base64="a3asiy3wafzborgf9n7bi3nsxeq=">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</latexit> <latexit sha1_base64="a3asiy3wafzborgf9n7bi3nsxeq=">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</latexit> Gauge theory of fluctuating antiferromagnetism For fluctuating antiferromagnetism (spin density waves (SDW)), we transform to a rotating reference frame using the SU(2) rotation R i ci" c i# = R i i,+ i,, in terms of fermionic chargons s and a Higgs field H a (i) p p (i) =R i a H a (i) R i The Higgs field is the SDW order in the rotating reference frame.

24 Field Symbol Statistics SU(2) gauge SU(2) spin U(1) e.m.charge Electron c fermion AF order boson Chargon fermion Spinon R or z boson Higgs H boson Note that this representation is ambiguous up to a SU(2) gauge transformation, V i! V i i,+ i, i,+ R i! R i V i a H a (i)! V i b H b (i) V i. S. Sachdev, M. A. Metlitski, Y. Qi, and C. Xu, PRB 80, (2009) i,

25 Field Symbol Statistics SU(2) gauge SU(2) spin U(1) e.m.charge Electron c fermion AF order boson Chargon fermion Spinon R or z boson Higgs H boson Begin with an effective theory for the Higgs field alone

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sha1_base64="xwwxn/g6zumr8ermj4enjfvlira=">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</latexit> Gauge theory of fluctuating antiferromagnetism For the hole-doped cuprates, N h = 4, we define complex Higgs fields Hx a = H1 a + ih2 a, Hy a = H3 a + ih4 a. The SU(2) gauge theory is L = 1 µhx a abc A b µhx c µhy a abc A b µhy c + 1 4g 2 F µ F a µ a + V (Hx,y) a 2 F a µ µ A A a µ abc A b µa c V (H a x,y) = s H a x H a x + H a u 1 4 H a x H a x +u 3 H a xh a y y Hy a + u 0 Hx a Hx a + H a H a y H a y 2 + H a x H a y 2 u y H a y h H a xh a x 2 + H a yh a y 2 2 i

28 <latexit sha1_base64="nrcn2w7kazopmgqfcngdqzesyye=">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</latexit> <latexit sha1_base64="nrcn2w7kazopmgqfcngdqzesyye=">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</latexit> <latexit sha1_base64="nrcn2w7kazopmgqfcngdqzesyye=">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</latexit> <latexit sha1_base64="nrcn2w7kazopmgqfcngdqzesyye=">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</latexit> Gauge theory of fluctuating antiferromagnetism There are multiple order parameters for di erent broken symmetries (Note: spin rotations are preserved and there is no SDW order) Ising nematic order = H a x H a x H a y H a y Charge density wave (CDW) order at wavevectors 2K x,y x = H a xh a x, y = H a yh a y Charge density wave (CDW) order at wavevectors K x ± K y (Modulated) scalar spin chirality + = H a xh a y, = H a xh a y ijk = abc H a (r i )H b (r j )H c (r k )

29 Broken symmetries and topological order in the Higgs phase

30 Broken symmetries and topological order in the Higgs phase

31 Broken symmetries and topological order in the Higgs phase Phenomenologically attractive state with uni-directional CDW, Ising-nematic, and scalar spin chirality order: H a x =sin (0, 0, 1), H a y = cos (1,i,0) i.e. short-range collinear spin correlations along x, and short-range spiral spin correlations along y. <latexit sha1_base64="zfrjdiuoaaycz63malmhen6/lvm=">aaada3icdvlbbtnaelublixcwvogpixokrwsy+xs0gojqai8llci9sz1q7ret+xv17vw7rpnzoxl+bieeqxxdbboqtuiotjkoznnlntmkljwy6pom7/qunp33v3fb+2hjx4/ebq0/ozyqeozpgjkkh2auiocszyy3ao8ltxsihf4kpzvnfgtc9sgk3loxyx2c5pjpusmwgcnlv1dkmdgzc1qwtst9kgouhxuczu5lhbjonzqyiy/qdcwworlbnoojo+mxcnrclebex9pavhkumy6egtxngvazqrgfaeatmklsjxrkrgdg9ip6neetdvecxphqfcng9excu+bucjabi6wrkdbfmqbnqbu8myzhlnmzop1oaaeqltriqrqggxweepyegimv9p2nzuzgtoreepqhku1iqksbqhqmopoqdmf+1qai7uxb88zd8i2qzleithywo3crv5vk+5bfeztc85w9hy73of4jqx6czsylp0mqwkv090yqy05i6ps9muqnywcj21sgswpo6cznjlx0gjnv56ufwjrdklhqlr70siuvz5r08kycze4zqoh+tfwgp+lnvv2unovuswri5lngg0ravzbc0sww7+7kjrtprmb1w2ynrfilu5gl6rwfzdozlnmnq+jxzg95knrw78jyy4ntrg/ssdtzvfmgedh/hlzdffdxltf74x3ylv3ym/b2/x2vqpvygp+v/+7/8p/ufcrtdj63no5oy7485wv74a11v4agqkxjg==</latexit> <latexit sha1_base64="zfrjdiuoaaycz63malmhen6/lvm=">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</latexit> <latexit sha1_base64="zfrjdiuoaaycz63malmhen6/lvm=">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</latexit> <latexit sha1_base64="zfrjdiuoaaycz63malmhen6/lvm=">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</latexit>

32 1. SU(2) gauge theory with adjoint scalars Confining and Higgs phases in 2+1 dimensions 2. SU(2) gauge theory of fluctuating antiferromagnetism 3. Role of fermions Pseudogap as a FL* phase

33 Gauge theory of fluctuating antiferromagnetism Field Symbol Statistics SU(2) gauge SU(2) spin U(1) e.m.charge Electron c fermion AF order boson Chargon fermion Spinon R or z boson Higgs H boson 3 1 0

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sha1_base64="dr87cfgsximpswdewvrs7dxm6fq=">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</latexit> Gauge theory of fluctuating antiferromagnetism Field Symbol Statistics SU(2) gauge SU(2) spin U(1) e.m.charge Electron c fermion AF order boson Chargon fermion Spinon R or z boson Higgs H boson SU(2) gauge theory: fractionalize the SDW order parameter into the Higgs field (H) and the spinons (R); fractionalize the electron (c) into chargons ( ) and spinons (R). When the Higgs field is condensed, the fermions and R bosons are deconfined particles in an algebraic charge liquid (ACL) state, but with a strong residual attractive interaction from the t ij.

35 Antiferromagnet with p holes per square

36 =( "#i #"i) / p 2 ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers),

37 =( "#i #"i) / p 2 ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers),

38 =( "#i #"i) / p 2 ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers),

39 =( "#i #"i) / p 2 ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers),

40 =( "#i #"i) / p 2 ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers),

41 =( "#i #"i) / p 2 ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers),

42 =( "#i #"i) / p 2 ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers),

43 =( "#i #"i) / p 2 ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers),

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45 <latexit sha1_base64="jx/ceeyyvtqfzi6u8uvsipnhl3i=">aaab93icbzdnsgmxfiuz9a+ovevatbairoazguhscooygv2btimzzj02njmzkjtqgbp14dbxccc+jg/hi5j+llt1qobwtsk9+ajccoo+/+vutrz3dveq++5bzt08oq7xoiyrnic2z2smexeziiwcngqu0ms1sdss0i0mt/o++wjaiew94dshqcpgsiscm7rra1hv+j6/en00wco0yerd+ncyz7xiqsgxzjh+4oc4kjlgwsxm3lawkdm+yspow6tycmzqltac0xobxdtjtd0k6sl9/ajkqthtnli3u4zjs97nw/+6fohj9aauki8qff8osgpjmapzp9nyaoaop9ywroxdlfix04yjjeogbiw2ncjxgsi845oi7zwy8jqxqs0sowcdx6bpnl3a94j7n1tjktkjfyqgv+sg3jewarnoyvjk3pwx5935wjksocukj+spnm8f6h2wxg==</latexit> <latexit sha1_base64="h/iu7f317d4b0bqmd1umbpr1m8a=">aaacnxicbvfdb9mwfhxcx0yzuhjl5yofaadrjzvgvccb9sidd+wj61btvy5zk1hz7mjxyvrrfww/i3/bt8dnkgqbv7j0dk7pufa5wamkutj+gyq3bt66vbn7z3b37979b8ohe6dkwitwkowy9izjhepqndrpfj41fnmdkzxl5yeb/uwbwpjgf3grbhc1l7uspodou8vhjztduupoohzo14n3jx/gqrokctqk3qqijgjtadvskyscam3tlvkaqlgtjsai0ozkdagdrngw3ghk3pyihusvexy4ciftluiu/qv6dpimwhxee0l0+u1ynpbqzfbv9ikapkjzpy9advfjudwxxafjfuyzbu2ww19pbkrbe71qngiexi1bdnw6krsub2ll2hbxzkuce6h5jbto+ijx8nqzortg+um/0rn/kzpee63qzn+suavoam9d/q83b13xetfj3bqotbgcvlqknihnxnw2fovtkw+4snjta+bch+cdcp1qdemqlnx43v3i3m/pkth4pdrrh1fynzdr4hq8sujr8jfmu+wxe8iowmko2vv2nk3ylilgj3grvaqow+fhjjxdhhkg21qfsx8q/pobh+hm5q==</latexit> <latexit sha1_base64="lsyfxgd0zdnesltzmzl4rbq7s8g=">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</latexit> <latexit sha1_base64="xtxypdqsvjzqs2gvpeccsr5d290=">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</latexit> ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers), and spin S =1/2, bosons R =( "#i #"i) / p 2

46 <latexit sha1_base64="jx/ceeyyvtqfzi6u8uvsipnhl3i=">aaab93icbzdnsgmxfiuz9a+ovevatbairoazguhscooygv2btimzzj02njmzkjtqgbp14dbxccc+jg/hi5j+llt1qobwtsk9+ajccoo+/+vutrz3dveq++5bzt08oq7xoiyrnic2z2smexeziiwcngqu0ms1sdss0i0mt/o++wjaiew94dshqcpgsiscm7rra1hv+j6/en00wco0yerd+ncyz7xiqsgxzjh+4oc4kjlgwsxm3lawkdm+yspow6tycmzqltac0xobxdtjtd0k6sl9/ajkqthtnli3u4zjs97nw/+6fohj9aauki8qff8osgpjmapzp9nyaoaop9ywroxdlfix04yjjeogbiw2ncjxgsi845oi7zwy8jqxqs0sowcdx6bpnl3a94j7n1tjktkjfyqgv+sg3jewarnoyvjk3pwx5935wjksocukj+spnm8f6h2wxg==</latexit> <latexit sha1_base64="h/iu7f317d4b0bqmd1umbpr1m8a=">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</latexit> <latexit sha1_base64="lsyfxgd0zdnesltzmzl4rbq7s8g=">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</latexit> <latexit sha1_base64="xtxypdqsvjzqs2gvpeccsr5d290=">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</latexit> ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers), and spin S =1/2, bosons R =( "#i #"i) / p 2

47 <latexit sha1_base64="jx/ceeyyvtqfzi6u8uvsipnhl3i=">aaab93icbzdnsgmxfiuz9a+ovevatbairoazguhscooygv2btimzzj02njmzkjtqgbp14dbxccc+jg/hi5j+llt1qobwtsk9+ajccoo+/+vutrz3dveq++5bzt08oq7xoiyrnic2z2smexeziiwcngqu0ms1sdss0i0mt/o++wjaiew94dshqcpgsiscm7rra1hv+j6/en00wco0yerd+ncyz7xiqsgxzjh+4oc4kjlgwsxm3lawkdm+yspow6tycmzqltac0xobxdtjtd0k6sl9/ajkqthtnli3u4zjs97nw/+6fohj9aauki8qff8osgpjmapzp9nyaoaop9ywroxdlfix04yjjeogbiw2ncjxgsi845oi7zwy8jqxqs0sowcdx6bpnl3a94j7n1tjktkjfyqgv+sg3jewarnoyvjk3pwx5935wjksocukj+spnm8f6h2wxg==</latexit> <latexit sha1_base64="h/iu7f317d4b0bqmd1umbpr1m8a=">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</latexit> <latexit sha1_base64="lsyfxgd0zdnesltzmzl4rbq7s8g=">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</latexit> <latexit sha1_base64="xtxypdqsvjzqs2gvpeccsr5d290=">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</latexit> ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers), and spin S =1/2, bosons R =( "#i #"i) / p 2

48 <latexit sha1_base64="jx/ceeyyvtqfzi6u8uvsipnhl3i=">aaab93icbzdnsgmxfiuz9a+ovevatbairoazguhscooygv2btimzzj02njmzkjtqgbp14dbxccc+jg/hi5j+llt1qobwtsk9+ajccoo+/+vutrz3dveq++5bzt08oq7xoiyrnic2z2smexeziiwcngqu0ms1sdss0i0mt/o++wjaiew94dshqcpgsiscm7rra1hv+j6/en00wco0yerd+ncyz7xiqsgxzjh+4oc4kjlgwsxm3lawkdm+yspow6tycmzqltac0xobxdtjtd0k6sl9/ajkqthtnli3u4zjs97nw/+6fohj9aauki8qff8osgpjmapzp9nyaoaop9ywroxdlfix04yjjeogbiw2ncjxgsi845oi7zwy8jqxqs0sowcdx6bpnl3a94j7n1tjktkjfyqgv+sg3jewarnoyvjk3pwx5935wjksocukj+spnm8f6h2wxg==</latexit> <latexit sha1_base64="h/iu7f317d4b0bqmd1umbpr1m8a=">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</latexit> <latexit sha1_base64="lsyfxgd0zdnesltzmzl4rbq7s8g=">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</latexit> <latexit sha1_base64="xtxypdqsvjzqs2gvpeccsr5d290=">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</latexit> ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers), and spin S =1/2, bosons R =( "#i #"i) / p 2

49 <latexit sha1_base64="jx/ceeyyvtqfzi6u8uvsipnhl3i=">aaab93icbzdnsgmxfiuz9a+ovevatbairoazguhscooygv2btimzzj02njmzkjtqgbp14dbxccc+jg/hi5j+llt1qobwtsk9+ajccoo+/+vutrz3dveq++5bzt08oq7xoiyrnic2z2smexeziiwcngqu0ms1sdss0i0mt/o++wjaiew94dshqcpgsiscm7rra1hv+j6/en00wco0yerd+ncyz7xiqsgxzjh+4oc4kjlgwsxm3lawkdm+yspow6tycmzqltac0xobxdtjtd0k6sl9/ajkqthtnli3u4zjs97nw/+6fohj9aauki8qff8osgpjmapzp9nyaoaop9ywroxdlfix04yjjeogbiw2ncjxgsi845oi7zwy8jqxqs0sowcdx6bpnl3a94j7n1tjktkjfyqgv+sg3jewarnoyvjk3pwx5935wjksocukj+spnm8f6h2wxg==</latexit> <latexit sha1_base64="h/iu7f317d4b0bqmd1umbpr1m8a=">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</latexit> <latexit sha1_base64="lsyfxgd0zdnesltzmzl4rbq7s8g=">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</latexit> <latexit sha1_base64="xtxypdqsvjzqs2gvpeccsr5d290=">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</latexit> ACL with density p of spinless fermionic chargons, emergent gauge fields (the blue dimers), and spin S =1/2, bosons R =( "#i #"i) / p 2

50 FL* Metal with electron-like quasiparticles on a Fermi surface of size p, and emergent gauge fields =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

51 FL* Metal with electron-like quasiparticles on a Fermi surface of size p, and emergent gauge fields =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

52 FL* Metal with electron-like quasiparticles on a Fermi surface of size p, and emergent gauge fields =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

53 FL* Metal with electron-like quasiparticles on a Fermi surface of size p, and emergent gauge fields =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

54 FL* Metal with electron-like quasiparticles on a Fermi surface of size p, and emergent gauge fields =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

55 FL* Metal with electron-like quasiparticles on a Fermi surface of size p, and emergent gauge fields =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

56 FL* Metal with electron-like quasiparticles on a Fermi surface of size p, and emergent gauge fields =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

57 FL* Metal with electron-like quasiparticles on a Fermi surface of size p, and emergent gauge fields =( "#i #"i) / p 2 =( " i + "i) / p 2 M. Punk, A. Allais, and S. Sachdev, PNAS 112, 9552 (2015)

58 Constraints on volume enclosed by the Fermi surface In a conventional Fermi liquid state, Fermi volume must equal (1-p) (mod 2). When the unit cell is doubled by SDW order, total Fermi volume must equal (1-p) (mod 1). A state with Fermi volume (-p) (mod 2), but no translational symmetry breaking, must have nonquasiparticle excitations with vanishing energy on a torus i.e. emergent gauge fields (bulk topological order) M. Oshikawa, Phys. Rev. Lett. 84, 3370 (2000) T. Senthil, M. Vojta, and S. Sachdev, Phys. Rev. B 69, (2004)

59 Gauge theory of fluctuating antiferromagnetism Field Symbol Statistics SU(2) gauge SU(2) spin U(1) e.m.charge Electron c fermion AF order boson Chargon fermion Spinon R or z boson Higgs H boson SU(2) gauge theory: The ACL was used to model the photoemission and the cluster DMFT results in the intermediate phase. <latexit sha1_base64="l8n7x68qq8zeloz25axstj4fniu=">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</latexit> <latexit sha1_base64="l8n7x68qq8zeloz25axstj4fniu=">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</latexit> <latexit sha1_base64="l8n7x68qq8zeloz25axstj4fniu=">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</latexit> <latexit sha1_base64="l8n7x68qq8zeloz25axstj4fniu=">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</latexit> M. S. Scheurer, S. Chatterjee, Wei Wu, M. Ferrero, A. Georges, and S. Sachdev, Proceedings of the National Academy of Sciences 115, E3665 (2018)

60 Theory of optimal doping criticality Field Symbol Statistics SU(2) gauge SU(2) spin U(1) e.m.charge Electron c fermion AF order boson Chargon fermion Spinon R or z boson Higgs H boson SU(2) gauge theory for quantum criticality: fractionalize the SDW order parameter into the Higgs field (H) and the spinons (R); but do not fractionalize the electron (c) into chargons ( ) and spinons. In the FL* state, the low energy fermionic excitations have the quantum numbers of an electron <latexit sha1_base64="0e0ppdrci/zbtzrxubupfr9gsig=">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</latexit> <latexit sha1_base64="0e0ppdrci/zbtzrxubupfr9gsig=">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</latexit> <latexit sha1_base64="0e0ppdrci/zbtzrxubupfr9gsig=">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</latexit> <latexit sha1_base64="0e0ppdrci/zbtzrxubupfr9gsig=">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</latexit>

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sha1_base64="o71pbgzg0bpv5mld1tvk59w2xge=">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</latexit> Theory of optimal doping criticality SU(2) gauge theory SU(2) gauge theory with adjoint complex Higgs fields H a x,y (a =1, 2, 3), and gauge-invariant, electron-like fermions c with a large Fermi surface. L = 1 µhx a abc A b µhx c µhy a abc A b µhy c g 2 F µ F a µ a + V (Hx,y) a X t c j, c j+v, + c j+v, c j, µ X c j, c j, j, j + X j c j, c j, H a (j)h a (j) The fermions do not have Yukawa coupling to the Higgs fields, or a minimal coupling to the gauge fields: both are prohibited by gauge invariance. We treat the quartic coupling, perturbatively.

62 Fermi surface reconstruction in electron-doped cuprates without antiferromagnetic long-range order Junfeng He, C. R. Rotundu, M. S. Scheurer, Y. He, M. Hashimoto, K. Xu, Y. Wang, E. W. Huang, T. Jia, S.-D. Chen, B. Moritz, D.-H. Lu, Y. S. Lee, T. P. Devereaux and Z.-X. Shen New photoemission measurements at zero magnetic field show Fermi surfaces in quantitative agreement with quantum oscillation measurements. The energy gap between the electron and hole pockets collapses near x = 0.17 like an order parameter. The totality of the data points to a mysterious order between x =0.14 and x =0.17, whose appearance favors the FS reconstruction and disappearance defines the quantum critical doping. A recent topological proposal provides an ansatz for its origin. <latexit sha1_base64="i1a7r3cs4dcjgpmetup+tfiv9+0=">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</latexit> <latexit sha1_base64="i1a7r3cs4dcjgpmetup+tfiv9+0=">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</latexit> <latexit sha1_base64="i1a7r3cs4dcjgpmetup+tfiv9+0=">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</latexit> <latexit sha1_base64="i1a7r3cs4dcjgpmetup+tfiv9+0=">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</latexit>

63 Mathias Scheurer S. Sachdev, Topological order and Fermi surface reconstruction, arxiv: M. S. Scheurer, S. Chatterjee, Wei Wu, M. Ferrero, A. Georges, and S. Sachdev, Proceedings of the National Academy of Sciences 115, E3665 (2018)

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sha1_base64="/ufmky5dun7/ufoso4fnu2rt8+s=">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</latexit> <latexit sha1_base64="/ufmky5dun7/ufoso4fnu2rt8+s=">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</latexit> <latexit 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sha1_base64="/ufmky5dun7/ufoso4fnu2rt8+s=">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</latexit> <latexit sha1_base64="/ufmky5dun7/ufoso4fnu2rt8+s=">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</latexit> Hole-doped cuprates N h =4 <latexit sha1_base64="8cj07kqocmoheyc4svvhq6ov1mg=">aaab7nicdvdjsgnbek1xjxglevtsgarpq49mmrkiqs+ejijziblct6cnadkz0n0jhcef4cwdil79hm/+jz1funehby/3qqiqf6sck43xh7wyura+svnykm7v7o7tlw4owyrjjgvnmohedgkimoaxa2qubeukkpeoekwdjk9mfvuescwt+e5puuzhzbjzkfoijds+6y/qbxl7ptk2xq97nqdhg1fdssu1pikrxs1djo3nkmmsjx7pvtdiabaxwfnbloo6onv+tqtmvlbpszcplhi6jkpwntqmevn+pj93ik6nmkbhik3fgs3v7xm5izsarihpjigeqd/etpzl62y6rpk5j9nms5gufowzqdpbs9/rgetgtzgyqqjk5lzer0qsqk1crrpc16fof9i6tx1so7duux65jkmax3acz+cab3w4hgy0gciyhuajnq3uerrerndf64q1ndmch7depge9si7a</latexit> <latexit sha1_base64="8cj07kqocmoheyc4svvhq6ov1mg=">aaab7nicdvdjsgnbek1xjxglevtsgarpq49mmrkiqs+ejijziblct6cnadkz0n0jhcef4cwdil79hm/+jz1funehby/3qqiqf6sck43xh7wyura+svnykm7v7o7tlw4owyrjjgvnmohedgkimoaxa2qubeukkpeoekwdjk9mfvuescwt+e5puuzhzbjzkfoijds+6y/qbxl7ptk2xq97nqdhg1fdssu1pikrxs1djo3nkmmsjx7pvtdiabaxwfnbloo6onv+tqtmvlbpszcplhi6jkpwntqmevn+pj93ik6nmkbhik3fgs3v7xm5izsarihpjigeqd/etpzl62y6rpk5j9nms5gufowzqdpbs9/rgetgtzgyqqjk5lzer0qsqk1crrpc16fof9i6tx1so7duux65jkmax3acz+cab3w4hgy0gciyhuajnq3uerrerndf64q1ndmch7depge9si7a</latexit> <latexit 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sha1_base64="8cj07kqocmoheyc4svvhq6ov1mg=">aaab7nicdvdjsgnbek1xjxglevtsgarpq49mmrkiqs+ejijziblct6cnadkz0n0jhcef4cwdil79hm/+jz1funehby/3qqiqf6sck43xh7wyura+svnykm7v7o7tlw4owyrjjgvnmohedgkimoaxa2qubeukkpeoekwdjk9mfvuescwt+e5puuzhzbjzkfoijds+6y/qbxl7ptk2xq97nqdhg1fdssu1pikrxs1djo3nkmmsjx7pvtdiabaxwfnbloo6onv+tqtmvlbpszcplhi6jkpwntqmevn+pj93ik6nmkbhik3fgs3v7xm5izsarihpjigeqd/etpzl62y6rpk5j9nms5gufowzqdpbs9/rgetgtzgyqqjk5lzer0qsqk1crrpc16fof9i6tx1so7duux65jkmax3acz+cab3w4hgy0gciyhuajnq3uerrerndf64q1ndmch7depge9si7a</latexit> hh x,y i6=0 hh <latexit 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sha1_base64="jxvlirved9obd1oy/iumxv0r2ay=">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</latexit> <latexit sha1_base64="jxvlirved9obd1oy/iumxv0r2ay=">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</latexit> H x,y One or more of Ising-nematic, CDW, scalar spin chirality, and Z 2 topological orders <latexit 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sha1_base64="o53gxr1fjqvotg5im17xot5fsua=">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</latexit> <latexit sha1_base64="o53gxr1fjqvotg5im17xot5fsua=">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</latexit> <latexit sha1_base64="o53gxr1fjqvotg5im17xot5fsua=">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</latexit> Reconstructed (FL*) Fermi surfaces, with large length scale confinement in the U(1) cases Confinement Fermi liquid with large Fermi surface <latexit 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