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News on tensor network algorithms Román Orús

Singh, M. Rizzi, RO, arxiv:1809.08180 A.

1 News on tensor network algorithms Román Orús Donostia International Physics Center (DIPC) December 6th 2018 S. S. Jahromi, RO, M. Kargarian, A. Langari, PRB 97, (2018) S. S. Jahromi, RO, PRB 98, 1558 (2018) S. S. Jahromi, RO, arxiv: P. Schmoll, S. Singh, M. Rizzi, RO, arxiv: A. Kshetrimayum, M. Rizzi, J. Eisert, RO, arxiv: P. Schmoll, A. Haller, M. Rizzi, RO, arxiv:

2 Applications welcomed

3 Tensor networks everywhere HEP and lattice gauge theories Quantum information and computation Quantum gravity, string theory and AdS/CFT Strongly correlated systems Classical statistical mechanics Quantum simulations Entanglement and Tensor Networks Numerical tensor calculus Nuclear physics Quantum chemistry Materials science AI, Deep learning & Linguistics

4 Outline 1) ipeps for the ruby model 2) ipeps & VBS phases on the star lattice 3) gpeps and the structure matrix 4) idmrg for SU(2)-invariant chiral ladder Overview, omit many technical details

5 Outline 1) ipeps for the ruby model 2) ipeps & VBS phases on the star lattice 3) gpeps and the structure matrix 4) idmrg for SU(2)-invariant chiral ladder S. S. Jahromi, RO, M. Kargarian, A. Langari, PRB 97, (2018)

6 The ruby model H= X α=x,y,z Jα X σiα σjα α links <latexit sha1_base64="4zpoby6jmn0eeuamn8sqowsfki4=">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</latexit> Reproduces the low-energy sector of the topological color code model in its gapped phase (Z2 x Z2 topological order) E. Bombin, M. A. Martin-Delgado, PRL 97, (2006) Coarse-graining triangles leads to brickwall lattice Study with ipeps

Energies and phase diagram Ground state energy 1.05 1.06 0.55 Jc=(0.8, 0.8, 0.4) J=(1, 1, 1) 0.56 ipeps FU ED (N=24) 1.07 Multicritical point ipeps FU ED (N=24) Comparison to exact diag. for 24 sites.

QPTs meet at a multicritical point. A1 phase: gapped, Z2 x Z2 TO. A2 & A3 phases: gapless. Gapped by a perturbation, colored Ising anyons. S. S. Jahromi, M. Kargarian, S. F. Masoudi, A.

7 Energies and phase diagram Ground state energy Jc=(0.8, 0.8, 0.4) J=(1, 1, 1) 0.56 ipeps FU ED (N=24) 1.07 Multicritical point ipeps FU ED (N=24) Comparison to exact diag. for 24 sites ε0 ε Good convergence. Multicritical point challenging /D /D 3 phases separated by 2nd order QPTs. QPTs meet at a multicritical point. A1 phase: gapped, Z2 x Z2 TO. A2 & A3 phases: gapless. Gapped by a perturbation, colored Ising anyons. S. S. Jahromi, M. Kargarian, S. F. Masoudi, A. Langari, PRB 94, (2016) Phase diagram J x + Jy + J z = 2 <latexit sha1_base64="vafdml2snczwpqlikq/ymgnv+eg=">aaab/hicbzdlsgmxfibp1futt9eu3qsliahlpgi6eypupksk9gltmgtstbuauzbkxlhuv3hjqhg3pog738z0ogttpzdw8f/nkjpfizmtyrk+jclk6tr6rngztlw9s7tn7h+0zzqiqlsk4phoelhszklaukxx2o0fxyhhaccbx8/8zj0vkkxhnupj6gr4gdkfeay05jrlhvuatlhdtbp7ev2immtwrkqvfvogo4ck5nv0za/+icjjqenfojayz1uxcizykey4nzb6iaqxjmm8pd2niq6odcbz8ln0rjub8ioht6hqpv6emobayjtwdgea1uguejpxp6+xkp/cmbawthqnyfwhp+firwiwbbowqyniqqzmbno7ijlcahol8yrpeozfly9du1a1nd+evepxerxfoiqjoaebzqeon9cefhbi4rle4c14ml6md+nj3low8pky/cnj8wflfzju</latexit>

8 Fidelity from CTMs With CTMs Ground state fidelity per site C1 H.-Q- Zhou, RO, G. Vidal, PRL 0, (2008) Tr C4 Td C3 C1 Tu C2 N <latexit sha1_base64="xo9iuse/jf0lb+viestxhxz0sjq=">aaacvnicbvhlsgmxfm1mra31nerstbailuizkyjuhkigrqscfucnlkwmbuotzjbkhnl2j3wjn+jgtb9cxxccj+ece5ocbdgjsrvut2wndtk7mexebv/g8ojyotmtqyirmnrwxclzdjaijaps01qz0owlqtxgpbemhqz6451irspxqocxaxpue7rlmdkg6jj8seazyw9rx7uc/7bchhdwbhaixwj0q4owlqtxkumvos/ntqmoymg4febbc8fjuyv3vnateauqb4uqdpwpp4xwwonqmcglwp4b6/yisu0xi5ocnygsizxapdiyucbovhs0i2uclw0twm4kzriaztjljhhisg15yjwc6b5a16bknq2v6o5te0rfnggi8pygbskgjua0yxhssbbmqwmqlttcfei+kghr8xm5e4k3/urnuc+xpinfrvov+0ucwxaolkabeoagvmatqiiawoat/fi2lbk+rf87bwfmvtta9jyblbkdp045sb8=</latexit> ln F (λ1, λ2 ) ln d(λ1, λ2 ) lim N N C2 a Tl F (λ1, λ2 ) = hψ(λ2 ) Ψ(λ1 )i d(λ1, λ2 ) Tu d(λ1, λ2 ) = <latexit sha1_base64="zjjusofcsz+na81gnowvb+6pcm4=">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</latexit> C4 A2 - A3 QPT Td C1 C2 C4 C3 C1 C2 Tl Tr C4 C3 C3 Compatible with other observables: - Entanglement entropy - 2-point correlators Many open questions: - Top. nature of A2 and A3? - Multicritical point?

9 Outline 1) ipeps for the ruby model 2) ipeps & VBS phases on the star lattice 3) gpeps and the structure matrix 4) idmrg for SU(2)-invariant chiral ladder

10 Outline 1) ipeps for the ruby model 2) ipeps & VBS phases on the star lattice 3) gpeps and the structure matrix 4) idmrg for SU(2)-invariant chiral ladder S. S. Jahromi, RO, PRB 98, 1558 (2018)

11 Star Heisenberg Antiferromagnet X H = Je S i S j + Jt hiji2e X Si Sj hiji2t <latexit sha1_base64="pwrwknsaiyfqrybw6/+aw+po7cc=">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</latexit> (spin-1/2) J. Richter et al, PRB 70, 1 (2004) H. Yao, S. A. Kivelson, PRL 99, (2007) G. Kells et al, PRB 81, 4429 (20) S. Dusuel et al, PRB 78, 1252 (2008) G. Misguich, P. Sindzingre, JPCM 19, (2007) B. J. Yang, A. Paramekanti, Y. B. Kim, PRB (20) T. P. Choy, Y. B. Kim, PRB 80, (2009) S. J. Ran et al, PRB 97, (2018) G. Y. Huang, S. D. Liang, D. X. Yao, EPJB 86, 379 (2013) (a) σyσy σxσx σzσz (b) Frustrated triangles. Ground state not fully clear yet. Chiral QSL, VBS, spin-1 trimerized... Coarse-graining leads to brick-wall lattice (different setups) (c) (d) Same ipeps strategy than the ruby lattice T=0 phase diagram as a function of Jt/Je?

12 ipeps phase diagram Agrees with previous results Continuous QPT New phase, breaks C3 symmetry (4x4 unit cell, D=9)

13 Coarse-graining setups Setup A Setup B Jt/Je << 1 Jt/Je >> Je=1, Jt= Je=0.05, Jt= Setup A Setup B Setup A Setup B ε0 ε Fast convergence! Better for large-d /D /D

14 Bond energies Ebond 0.2 Jt/Je << 1 Je=0.05, Jt= Jt-bond1 Jt-bond2 Jt-bond3 Je-bond Jt/Je >> /D Jt/Je >> 1 (a) (b) (c) Degeneracy of the three configurations. RVB possible. ipeps favors one of them because it has less entanglement.

15 Outline 1) ipeps for the ruby model 2) ipeps & VBS phases on the star lattice 3) gpeps and the structure matrix 4) idmrg for SU(2)-invariant chiral ladder

16 Outline 1) ipeps for the ruby model 2) ipeps & VBS phases on the star lattice 3) gpeps and the structure matrix 4) idmrg for SU(2)-invariant chiral ladder S. S. Jahromi, RO, arxiv:

17 Graph-based PEPS (gpeps) Store information about the TN on a structure matrix Example: star lattice Links in unit cell Unit cell Label of edge for tensor Looping over the columns of the structure matrix we can authomatize tensor updates in imaginary-time evolution for essentially any lattice. Simple update (no environment): automatic! Full update (with environment): work in progress

18 Graph-based PEPS (gpeps) Store information about the TN on a structure matrix Example: star lattice Links in unit cell Unit cell Label of edge for tensor Looping over the columns of the structure matrix we can authomatize tensor updates in imaginary-time evolution for essentially any lattice. Simple update (no environment): automatic! Full update (with environment): work in progress

19 All you need is the matrix 1d chain

20 All you need is the matrix 2d square

21 All you need is the matrix 2d triangular

22 All you need is the matrix 2d kagome

23 All you need is the matrix 3d pyrochlore

24 Some ground state energies Simple update, and mean field environment (only lambdas) for observables. Not variational. 1d 2d 3d 2d 3d 2d 2d Very simple, efficient, and fully-atomatized way of getting a first idea about the ground state in many cases

25 An example Antiferromagnetic Heisenberg model on the cubic lattice H= X ij <latexit sha1_base64="mnx9gy58opujhing40lzu/oqauk=">aaacexicbzbns8mwgmft+tbnw9wjl+aqdhqtchorhl52noheyc0ltdmtw5qwjbvg6vfw4lfx4kerr968+w3mtik6+yfal//neuiev58wkpvlfrmlldw19y3yzmvre2d3z9w/6mg4fzi0ccxi0forjixy0lzumdjlbegrz0jxh19p6917iisn+z2ajmsn0idtkgkktowztsa8hi5miy+joxxmjh/c29yj0mfbrh7ui+izvatuzqsxws6gcgq1ppptcwkcroqrzjcufdtkljshoshmjk84qsqjwmm0ih2nheveutlsoxyeaceaysz04qro3n8tgyqknes+7oyqgsrf2tt8r9zpvxjhzpqnqsiczx8kuwzvdkfxwiakghwbaebyup1xiidiikx0ibudgr248jj0tuu25puzauoqikmmjsaxqaebnimgaiiwaammhsateagvxqpxblwz7/pwklhmhii/mj6+avplnba=</latexit> sha1_base64="ck8pdc+ekzh4numsp+zg7r8leyk=">aaab2xicbzdnsgmxfixv1l86vq1rn8eiucozbnqpuhfzwbzco5rm5k4bmskmyr2hdh0bf25efc93vo3pz0jbdwq+zknivscullqubn9ebwd3b/+gfugfnfzjk9nmo2fz0gjsilzl5jnmfpxu2cvjcp8lgzylffbj6f0i77+gstlxtzqrmmr4wmtuck7o6oyaraadlmw2ivxdc9yanb+gss7kddujxa0dhefbucunsafw7g9liwuxuz7ggupnm7rrtrxzzi6dk7a0n+5oykv394ukz9bostjdzdhn7ga2mp/lbiwlt1eldvesarh6kc0vo5wtdmajnchizrxwyasblykjn1yqa8z3hysbg29d77odon4moa7ncafxemin3meddkalahj4hxdv4r15h6uuat66tdp4i+/zbzjgijg=</latexit> sha1_base64="rmyfn5xfpi2uqijpx3omrxghb1s=">aaacbnicbzbns8mwhmb/nw9ztq1evqshsnnovehfelzsong9wdpkmqzbtqqtssqm0q/gxa/ixymifgrvfhuzf0q3hwj88jwjyf8jus6udpwvq7sxubw9u96t7fx3dw7to2phjzkkte0snshegbxllkztztsnvvrslajou8hkzpz3h6hulinv9tsla4ghmysywdpyvl1voivkquz4orsxkpeccn0vpkmecrp9sx8j3645dwcuta7uemqwvmu3p70wizmgssyck9v3nvqpciw1i5wwfs9tnmvkgoe0bzdggqpbpp+oqgfgcvguslnijebu7xs5fkpnrwbocqxhajwbmf9l/uxhl4ocxwmmauwwd0uzrzpbs3pqycqlmk8nyckz+ssiiywx0abeiinbxr15htrnddfwrqnloiftqimlf3antwhbgwg8wjo8wpv1zl1y74u6stayt2p4i+vjg6cympq=</latexit> sha1_base64="gztutzs0z28mekjkccn5wb1pfri=">aaacexicbzc7tsmwfiydrqxcaowsfhvspyphgqwpgqvjefqinvxkoe7r1nyi20gqorwcc6/cwgbcrgxsva1ugyfo+svln/9zjuzzbwmjsjvol7wyura+svnakm/v7o7t2wehbrwnepmwjlksuwfshffbwppqrrqjjighjhsc8fw03rknutfy3oljqvocdqsnkebawl5dbcbl6kmu+xkd5tdzggje5j6fhg5j/xmfqd+uodvnjrgmbgevukjp259egooue6exq0r1xcfr/qxjttejedllfukqhqmb6rkuibpvz2yb5fduocgmymmo0hdm/p7iefdqwgptyzeeqsxa1pyv1kt1dnhpqehstqsepxsldooytuobizueazyxglck5q8qd5fewjsqyyyed3hlzwif1vzdn06lflxeuqlh4arugqvoqr00qbo0aayp4am8gffr0xq23qz3eeukvcwcgt+ypr4buqwcda==</latexit> (spin-1/2) Si Sj

26 Outline 1) ipeps for the ruby model 2) ipeps & VBS phases on the star lattice 3) gpeps and the structure matrix 4) idmrg for SU(2)-invariant chiral ladder

27 Outline 1) ipeps for the ruby model 2) ipeps & VBS phases on the star lattice 3) gpeps and the structure matrix 4) idmrg for SU(2)-invariant chiral ladder P. Schmoll, S. Singh, M. Rizzi, RO, arxiv: P. Schmoll, A. Haller, M. Rizzi, RO, arxiv:

28 Chiral interactions on a ladder H= X i <latexit sha1_base64="totcl0mr/kvfmpyycsqtsfvihji=">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</latexit> Ji Si (Si+1 Si+2 ) (spin-1/2) Multiple motivations: wire constructionism, simpler case than 2d, etc

29 Chiral interactions on a ladder H= X i <latexit sha1_base64="totcl0mr/kvfmpyycsqtsfvihji=">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</latexit> 1 Ji Si (Si+1 Si+2 ) (spin-1/2) 3 ~2 S ~3 + S ~3 S ~4 ~1 S ~2 S H1 = S H Singlet ground state S2 = 0 ~3 S ~4 ~2 S ~3 S ~2 S ~1 S H2 = + S H Study with SU(2) idmrg (omit tech. details) ~2 S ~1 S ~3 S ~4 ~2 S ~3 S H3 = S H Non-singlet ground state ~ ~ ~ ~ ~ ~ H4 = + S1 S2 S3 + S2 S3 S4 H4 2 4

30 First intuition S <latexit sha1_base64="dvnxep6x03kjflmge9mhpnn0c4c=">aaab73icbzbns8naeiyn9avwr6phl4tf8fqsefry9okxov2anptndtiu3wzs3y1qqv+efw+kepxveppfug1z0nyxfh7emwfn3iarxbvx/xyka+sbm1vf7dlo7t7+qfnwqknjvdfssfjeqh1qjyjlbbhublythtqkblac0e2s3npcpxksh80kqt+ia8ldzqixvjvrbif5mjjeuejw3bnikng5vcbxvvf+6vzjlkyodrnu647njsbpqdkcczywuqnghlirhwdhoqqraj+b7zslz9bpkzbw9kld5u7viyxgwk+iwhzg1az1cm1m/lfrpca89jmuk9sgziupwlqqe5pz8atpftijjhyou9zustiqksqmjahkq/cwt16f5kxvs3x/wand5heu4qro4rw8uiia3eedgsbawdo8wpszdl6cd+dj0vpw8plj+cpn8wdo0o95</latexit> <latexit S 1) Kadanoff blocking on triangles H= X Ji Si (Si+1 Si+2 ) <latexit sha1_base64="gz+dlajuekmmb8cinuvxtftsnp8=">aaab+3icbzdlssnafiyn9vbrldalm8eiucqjclosunfz0v6gcwuyowmhtizhzikwkfdx40irt76io9/gazuftv4w8pgfczhn/idltgnh+byqa+sbm1vv7dro7t7+gx1y76okkxq6nogj7adeawccopppdv1uaokddr1gcjor9x5bkpaibz1nwy/jslciuaknnbtrnmy8hdz3ggjn90wbi6hdcjroxhgv3biaqfr7ah95yukzgismncg1cj1u+zmrmleorc3lfksetsgibgyfiuh5+fz2ap8aj8rris0tgs/d3xm5izwaxohpjikeq+xazpyvnsh0doxntkszbkexi6kmy53gwra4zbko5lmdhepmbsv0tcsh2srvmyg4y19ehe550zv8d9foxzdxvnexokfnyewxqivuurt1eevp6bm9ojersf6sd+tj0vqxypkj9efw5w8kk5rs</latexit> Hef f i <latexit sha1_base64="totcl0mr/kvfmpyycsqtsfvihji=">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</latexit> <latexit sha1_base64="qcye88vtfubvcaovrnprdiatxpw=">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</latexit> J1 + J2 X S n S n+1 = sgn(j1 J2 ) 3 3 n (1,2) = nn triangles sgn(j1j2) = -1 sgn(j1j2) = +1 Spin-1/2 AFH chain: gapless, c=1 CFT Spin-1/2 FH chain: non-critical 2) Currents by exact diagonalization Jijz <latexit sha1_base64="ybr3kkthpknmtpwszufju2b3qgk=">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</latexit> H1 pbc 1st excited state (triplet) i + = (Si Sj Si Sj+ ) 2 H3 obc g.s. <S2> = 12, Sz = 3 Compatible with original intuition

31 H1 : energy and entropy Algebraic energy convergence: gapless Entanglement entropy: c=1 CFT

32 H1 : correlation functions Critical correlation functions Spin-spin 0-1 Matches continuum limit Z 1 C(r) r -2 H g <latexit sha1_base64="mmshpbspzffgt/r8rrfh3k1g+tq=">aaacanicbzdlsgmxfiyz9vbrbdsvuakwow7kjai6lhbjsok9qduutjppqznjsdjigqy3voobf4q49snc+tam7sy09yfax3/o4et8owrug8/7dgorq2vrg8xn0tb2zu6eu3/q0ijrmdsxyej1qqqjo5w0dtwmdkqika4zayfj+rtevidku8hvzessiezdtiokkbfw3z2qv9qz7ceplxiavughnppzqjlyd8te1zsjloofqxnkavtdr95a4cqm3gcgto76njrbipshmjgs1es0kqip0zb0lxiuex2ksxmyegqdayyeso8bohn/t6qo1nosh7yzrmakf2tt879anzhrvzbslhndoj4vihigjydtpocakoinm1hawfh7v4hhymzgbgolg4k/epiytm6rvuxbi3ltoo+jci7bcagah1ycgrgbddaegdycz/ak3pwn58v5dz7mrqunnzkef+r8/gaeo5am</latexit> dx (JL,1 JR,2 JR,1 JL,2 ) <latexit sha1_base64="fy46rwicdfpahmwssggr5gigm/g=">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</latexit> -3 E.g., P.-Huang et al, PRB 95, (2017) Vertical dimer-dimer C(r) -4 <latexit sha1_base64="32oadrjo3tnzecis5mjetwsqehc=">aaacd3icbzc7tsmwfiydrqxcaowsfhwolfwcimcs6mjyjhqrmla5rtnadrlldhcvltdg4vvygecilzwnt8ftm0dll1n69j9zdhz+gdmqlen8w0vlk6tr64wn4ubw9s6uvbffkkkqmgnihcwieybjgi1ju1hfsiclgqkakxywqk/q7xsije3iwzxmxi/qikyhxugzq2ef1mvifhqic5e8qc8ucgs30+joz/g809usy3p2yak4u8ffchmogvynnv3l9rocrirwmcepu67dla+rubqzkhw9vbko8agnsndgjciift29j4phxundmbhmxqpo3d8tgkvsjqpadezidev8bwl+v+umkrz0ny15qkimz4vclegvwek4se8fwyqndsasqpkrxenkclamwqijwz0/erfazxxx8e21vlvk4yiaq3aeysaff6agrkednaegj+azvii368l6sd6tj1nrkpxphia/sj5/an76nju=</latexit> r 5 4 Similar algebraic decays for other correlators

33 H1 : entanglement spectrum Entanglement spectrum of half an infinite chain comparison with spin-1/2 AFH chain 30 Same entang. spectrum Same boundary CFT 15 Same (1+1)d-CFT 5 SU(2)1 WZW theory Compatible with bosonization, Kadanoff blocking, etc Many open questions: H3? More legs? WZW limit?

34 OUTLOOK 1) Phase diagram and phase transitions of the topological ruby model with ipeps 2) AFH model on the star lattice has a competition between VBS phases. ipeps shows that one of them is a new phase with a 6-site unit cell 3) Structure matrix can be used to authomatize (at least some) PEPS methods to any lattice 4) SU(2) idmrg simulations of a chiral ladder suggest that it corresponds to a SU(2)1 WZW theory

Non-magnetic states. The Néel states are product states; φ N a. , E ij = 3J ij /4 2 The Néel states have higher energy (expectations; not eigenstates)

Non-magnetic states. The Néel states are product states; φ N a. , E ij = 3J ij /4 2 The Néel states have higher energy (expectations; not eigenstates) Non-magnetic states Two spins, i and j, in isolation, H ij = J ijsi S j = J ij [Si z Sj z + 1 2 (S+ i S j + S i S+ j )] For Jij>0 the ground state is the singlet; φ s ij = i j i j, E ij = 3J ij /4 2 The