Quantum Monte Carlo study of a Z 2 gauge theory containing phases with and without a Luttinger volume Fermi surface

Quantum Monte Carlo study of a Z 2 gauge theory containing phases with and without a Luttinger volume Fermi surface V44.00011 APS March Meeting, Los Angeles Fakher Assaad, Snir Gazit, Subir Sachdev, Ashvin Vishwanath March 8, 2018 Talk online: sachdev.physics.harvard.edu

Z2 lattice gauge theory (Wegner, 1971) z H = X z z z z g X i x z z z G i = x x x x Gauss s Law: [H, G i ]=0, G i =1

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sha1_base64="fy9nkk0ijhrx9wayzx00b1bhh3y=">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</latexit> <latexit sha1_base64="fy9nkk0ijhrx9wayzx00b1bhh3y=">aaaca3icdvbnb9qwehvspkr46fi48hwwwfx0afestqvuhv44follk21wi+nmdq06tmu7ofwue/+qgz+bc7+bsbqvamfitp7ezjt5m4wrwvkk+r6ew1euxru+fso6eev2nz3r3d2ptjeww4rrqe1zwrxiowdihzdwziywupawlc5p+vz0m1gntprg1wbmnvsquqnopfkl0de8haq1q6fwnnziwagzvmvtsujajd5in9mkzob/ko2ivicluc0h5cf20ylwvb6kzou24jyppr1ba1syl89opw2e0yksfdgxuygdvlrqbvyugyoveyxgy2x9djhlgc5ihkcqyhwe0htdjmkmthejb3mpevojnevm3cxnjj+36fpwcwixcwayp2dlmcfuray3b4cvo7qhtnk7w6c8hdjffs2rnvvxbvbwzk/c37me/fdu1vjq1bwvyjqefl8yvdwsek37y9nswoberhewbgv6pxzflon4axfhes43pf8hkyx+hafvs/hx2801tslj8ptsk5s8jmfkhtkle8ljj2aneba8dh6g98nh4zol0jdyao6rpylc+wuwhbmo</latexit> Z2 lattice gauge theory (Wegner, 1971) W C = Y C z C Deconfined phase. Perimeter law for Wegner-Wilson loops Confined phase. Area law for Wegner-Wilson loops g H = X z z z z g X i x

Z2 lattice gauge theory Deconfined phase. Z 2 flux expelled. Z 2 (toric code) topological order. Topological phase transition Confined phase. Z 2 flux proliferates. No topological order. g H = X z z z z g X i x E. Fradkin and S. H. Shenker, PRD 19, 3682 (1979); N. Read and S. Sachdev, PRL 66, 1773 (1991); X.-G. Wen, PRB 44, 2664 (1991); A.Y. Kitaev, Annals of Physics 303, 2 (2003)

<latexit sha1_base64="uxxxnog5ohq+bis/fyz/wrzsor4=">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</latexit> <latexit sha1_base64="uxxxnog5ohq+bis/fyz/wrzsor4=">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</latexit> <latexit sha1_base64="uxxxnog5ohq+bis/fyz/wrzsor4=">aaadexicjvjni9raeo3erzv+zerrs+osmikgjoyueltuxipico67obmgtqesabbthbo7yw4hv8glf8wlbxwv3rz5b6zmzsk6kljqoxhv9arei1kthxvr9npzz52/cphs2uxgytvr128m1m++tboxhmzcs232m2zbcgvjj5ye/doaqzije9nb076+dwjgcq3eueun04qvshscm4fqbn0bptkuolykauvg1blmwizrwlnmxrufm/f9goxj8rsdd0gaqsluy0e5mf3wdedvz5dteo6xhgkatomfddrtva6mp7tdscvm5lnwvutmyqytzhne4aggksepuw4zznsmndacu65zuidvp2vkl1gfpnog6/8sfaxvg5enc0kvyoihlk0fuop0y3mwlac2mqxj0kfuupmj0nlgipq3t5ikqsoizwasy2xt0nifdmkqdmedh2muos5rjktm7ssoajdt8szbjsb5y6fm/icvmmfusqrstf1q6ohdrhjaainpobpet0+0rlj2uwxy2rtlz9z68g+1seokr9nwqlpxopjxoqkr1gna/x80fwa4kwtmgdfc9a7pmwecpbabmncilp47gsfh4zb+nqx3nqzcwco3yr0yijf5shbic7jlxor7772p3mfvi//b/+r/9b8dt/reauyw+s38778ap2b9uq==</latexit> <latexit sha1_base64="uxxxnog5ohq+bis/fyz/wrzsor4=">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</latexit> Z2 lattice gauge theory + matter Deconfined phase. Z 2 flux expelled. Z 2 (toric code) topological order. Non-Luttinger volume Fermi surface Topological phase transition Confined phase. Z 2 flux proliferates. No topological order. Luttinger volume Fermi surface g H = X z z z z g X i x +...+... T. Senthil, M. Vojta, and S. Sachdev, PRB 69, 035111 (2004)

<latexit sha1_base64="i3tpemnawz7lgweiliiohswjdhg=">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</latexit> <latexit sha1_base64="i3tpemnawz7lgweiliiohswjdhg=">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</latexit> <latexit sha1_base64="i3tpemnawz7lgweiliiohswjdhg=">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</latexit> <latexit sha1_base64="i3tpemnawz7lgweiliiohswjdhg=">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</latexit> Orthogonal metal c ) z f Electron fractionalizes into an orthogonal fermion f, carrying both the spin and charge of the electron, and an Ising matter field, z, carrying only energy ( dark matter ). Theory should be invariant under the Z 2 gauge transformation f i! i f i, z i! z i i Also need a Z 2 gauge field ij z ij z! i ij z j transforming as Rahul Nandkishore, Max A. Metlitski, and T. Senthil. PRB 86, 045128 (2012)

<latexit sha1_base64="zzl2xh4fu4ximzflgfx5oue/wmk=">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</latexit> <latexit sha1_base64="zzl2xh4fu4ximzflgfx5oue/wmk=">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</latexit> <latexit sha1_base64="zzl2xh4fu4ximzflgfx5oue/wmk=">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</latexit> <latexit sha1_base64="zzl2xh4fu4ximzflgfx5oue/wmk=">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</latexit> Lattice model with no sign problem H = H Z2 + H + H f + H c, where Y H Z2 = K X x b b2 z b g X b H = J X r, z r, z r z r+ h X r x r H f = w X r,, z r, f r, f r+, + h.c µ X r, f r, f r, H c = t X r,, z r f r, z r+ f r+, + h.c

<latexit sha1_base64="a19yqjmg86cgevytwuetsdgxf/0=">aaab9xicbzdnssnafiun9a/gqnxtjlgevyfrrn0jblxwmlbqhdkz3lrdj5mwc6ow0kubtz6gk/fpfapfwenpqlspdbzomehe+ejcci2e92xv1ty3nrfq2/zow97d2282hnrekgyby0wuujhvilieadkk6bykabyl6msjm2nfeqsles7vcvxalngb5clnfe3u7jdbnuvn5kwaf2fazkf+8ztmclzmijejqnxp9wqmkqqqmwetoyw1fjsn6ab6xkqagy6q2zot59gkizpmyhyjziz9/akimdbjldy3m4pdvdxnw/+6xonpzvrxwzqiks0hpavwmhemf3ysrochgbtdmejmv4cnqaimdrk71gcwyqeoqxdhgz94yuzuz+45lxmdyf/gswqcu/fk9e88uieh5iicej9ckgtys9okiiwk5jw8ws/wu/ux51izfkapyb9znz+whpzm</latexit> <latexit sha1_base64="dbjypyfcwlhl7feg+mjegb5n//m=">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</latexit> <latexit sha1_base64="tx/6lrhambdoazq558ley03dvos=">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</latexit> <latexit sha1_base64="px/1pmhyvpzbf+vevdmv5eonw7o=">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</latexit> Orthogonal semi-metal At large h and small g, the Ising matter field is gapped, and the Ising gauge flux does not fluctuate. Then the orthogonal fermions f are deconfined, and they optimize their energy by inducing flux in each plaquette: we obtain an orthogonal semi-metal at half-filling. π-flux No Luttinger volume Fermi surface Snir Gazit, Mohit Randeria, and Ashvin Vishwanath, Nature Physics 13, 484 (2017)

<latexit sha1_base64="bh9nj0ynturulhhp7i2st2vg1ea=">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</latexit> <latexit sha1_base64="bh9nj0ynturulhhp7i2st2vg1ea=">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</latexit> <latexit sha1_base64="bh9nj0ynturulhhp7i2st2vg1ea=">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</latexit> <latexit sha1_base64="bh9nj0ynturulhhp7i2st2vg1ea=">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</latexit> As we decrease h (which reduces the gap of the z matter fields), with t 6= 0, a Fermi h i surface of c f z appears, with a Luttinger volume. (a) (b) (c) (d) (e) (f) D E Figure 3: Momentum occupation n k = c k c k for t = 0.09, h =0.05, 0.44, 0.82 and K = 1. (top row) Density plot in the first Brillouin zone and (bottom row) along diagonal momentum cut, ~ k = {q, q}.

As we increase g (which induces confinement via Z 2 flux fluctuations), we obtain a phase with AF order. The large Fermi surface of c appears unstable to AF order, likely due to Fermi surface nesting. <latexit sha1_base64="qjpadlq4tsro9aejoppojpuhzdm=">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</latexit> <latexit sha1_base64="qjpadlq4tsro9aejoppojpuhzdm=">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</latexit> <latexit sha1_base64="qjpadlq4tsro9aejoppojpuhzdm=">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</latexit> <latexit sha1_base64="qjpadlq4tsro9aejoppojpuhzdm=">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</latexit> 1.2 1.1 L=8, W=1.0, J=0.4, h = 0.5, K = -1.0 AFM Confinement 1 AFM Gauge field confined 0.9 g 0.8 0.7 0.6 FL or Z 2 D. Gauge field deconfined 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 t