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While tensor networks can now be seen as becoming standard tools in the description of such complex many-body systems, close to optimal variational principles based on such states are less obvious to come by. In this work, we generalize a recently proposed variational uniform matrix product state algorithm for capturing one-dimensional quantum lattices in the thermodynamic limit, to the study of regular two-dimensional tensor networks with a non-trivial unit cell. A key property of the algorithm is a computational effort that scales linearly rather than exponentially in the size of the unit cell. We demonstrate the performance of our approach on the computation of the classical partition functions of the antiferromagnetic Ising model and interacting dimers on the square lattice, as well as of a quantum doped resonating valence bond state.\n\nTensor network states provide an efficient class of states that faithfully capture strongly correlated quantum models and systems in classical statistical mechanics. While tensor networks can now be seen as becoming standard tools in the description of such complex many-body systems, close to optimal variational principles based on such states are less obvious to come by. In this work, we generalize a recently proposed variational uniform matrix product state algorithm for capturing one-dimensional quantum lattices in the thermodynamic limit, to the study of regular two-dimensional tensor networks with a non-trivial unit cell. A key property of the algorithm is a computational effort that scales linearly rather than exponentially in the size of the unit cell. We demonstrate the performance of our approach on the computation of the classical partition functions of the antiferromagnetic Ising model and interacting dimers on the square lattice, as well as of a quantum doped resonating valence bond state.<\/jats:p>","DOI":"10.22331\/q-2020-09-21-328","type":"journal-article","created":{"date-parts":[[2020,9,21]],"date-time":"2020-09-21T17:38:16Z","timestamp":1600709896000},"page":"328","update-policy":"http:\/\/dx.doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["Efficient variational contraction of two-dimensional tensor networks with a non-trivial unit cell"],"prefix":"10.22331","volume":"4","author":[{"given":"A.","family":"Nietner","sequence":"first","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, D-14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin f\u00fcr Materialien und Energie, 14109 Berlin, Germany"}]},{"given":"B.","family":"Vanhecke","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium"}]},{"given":"F.","family":"Verstraete","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium"}]},{"given":"J.","family":"Eisert","sequence":"additional","affiliation":[{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, D-14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin f\u00fcr Materialien und Energie, 14109 Berlin, Germany"}]},{"given":"L.","family":"Vanderstraeten","sequence":"additional","affiliation":[{"name":"Department of Physics and Astronomy, University of Ghent, Krijgslaan 281, 9000 Gent, Belgium"},{"name":"Dahlem Center for Complex Quantum Systems, Freie Universit\u00e4t Berlin, D-14195 Berlin, Germany"},{"name":"Helmholtz-Zentrum Berlin f\u00fcr Materialien und Energie, 14109 Berlin, Germany"}]}],"member":"9598","published-online":{"date-parts":[[2020,9,21]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"S. 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Levin and Cody P. Nave. Tensor renormalization group approach to two-dimensional classical lattice models. Phys. Rev. Lett., 99: 120601, Sep 2007. 10.1103\/PhysRevLett.99.120601.","DOI":"10.1103\/PhysRevLett.99.120601"},{"key":"89","doi-asserted-by":"publisher","unstructured":"Z. Y. Xie, J. Chen, M. P. Qin, J. W. Zhu, L. P. Yang, and T. Xiang. Coarse-graining renormalization by higher-order singular value decomposition. Phys. Rev. B, 86: 045139, Jul 2012. 10.1103\/PhysRevB.86.045139.","DOI":"10.1103\/PhysRevB.86.045139"},{"key":"90","doi-asserted-by":"publisher","unstructured":"G. Evenbly and G. Vidal. Tensor network renormalization. Phys. Rev. Lett., 115: 180405, Oct 2015. 10.1103\/PhysRevLett.115.180405.","DOI":"10.1103\/PhysRevLett.115.180405"},{"key":"91","doi-asserted-by":"publisher","unstructured":"S. Yang, Z.-G. Gu, and X.-G. Wen. Loop optimization for tensor network renormalization. Phys. Rev. Lett., 118: 110504, Mar 2017. 10.1103\/PhysRevLett.118.110504.","DOI":"10.1103\/PhysRevLett.118.110504"},{"key":"92","doi-asserted-by":"publisher","unstructured":"M. Bal, M. Mari\u00ebn, J. Haegeman, and F. Verstraete. Renormalization group flows of hamiltonians using tensor networks. Phys. Rev. Lett., 118: 250602, Jun 2017. 10.1103\/PhysRevLett.118.250602.","DOI":"10.1103\/PhysRevLett.118.250602"},{"key":"93","doi-asserted-by":"publisher","unstructured":"M. Hauru, C. Delcamp, and S. Mizera. Renormalization of tensor networks using graph-independent local truncations. Phys. Rev. B, 97: 045111, Jan 2018. 10.1103\/PhysRevB.97.045111.","DOI":"10.1103\/PhysRevB.97.045111"},{"key":"94","doi-asserted-by":"publisher","unstructured":"R. J. Baxter. Dimers on a rectangular lattice. J. Math. Phys., 9 (4): 650\u2013654, 1968. 10.1063\/1.1664623.","DOI":"10.1063\/1.1664623"},{"key":"95","doi-asserted-by":"publisher","unstructured":"R. J. Baxter. Variational approximations for square lattice models in statistical mechanics. J. Stat. Phys., 19: 461\u2013478, Nov 1978. ISSN 1572-9613. 10.1007\/BF01011693.","DOI":"10.1007\/BF01011693"},{"key":"96","doi-asserted-by":"publisher","unstructured":"T. Nishino and K. Okunishi. Corner transfer matrix renormalization group method. J. Phys. Soc. Jap., 65 (4): 891\u2013894, 1996. 10.1143\/JPSJ.65.891.","DOI":"10.1143\/JPSJ.65.891"},{"key":"97","doi-asserted-by":"publisher","unstructured":"T. Nishino and K. Okunishi. Corner transfer matrix algorithm for classical renormalization group. J. Phys. Soc. Jap., 66 (10): 3040\u20133047, 1997. 10.1143\/JPSJ.66.3040.","DOI":"10.1143\/JPSJ.66.3040"},{"key":"98","doi-asserted-by":"publisher","unstructured":"R. Or\u00fas and G. Vidal. Simulation of two-dimensional quantum systems on an infinite lattice revisited: Corner transfer matrix for tensor contraction. Phys. Rev. B, 80: 094403, Sep 2009. 10.1103\/PhysRevB.80.094403.","DOI":"10.1103\/PhysRevB.80.094403"},{"key":"99","doi-asserted-by":"publisher","unstructured":"P. Corboz, J. Jordan, and G. Vidal. Simulation of fermionic lattice models in two dimensions with projected entangled-pair states: Next-nearest neighbor hamiltonians. Phys. Rev. B, 82: 245119, Dec 2010. 10.1103\/PhysRevB.82.245119.","DOI":"10.1103\/PhysRevB.82.245119"},{"key":"100","doi-asserted-by":"publisher","unstructured":"Guifr\u00e9 Vidal. Efficient classical simulation of slightly entangled quantum computations. Phys. Rev. Lett., 91: 147902, Oct 2003. 10.1103\/PhysRevLett.91.147902.","DOI":"10.1103\/PhysRevLett.91.147902"},{"key":"101","doi-asserted-by":"publisher","unstructured":"R. Or\u00fas and G. Vidal. Infinite time-evolving block decimation algorithm beyond unitary evolution. Phys. Rev. B, 78: 155117, Oct 2008. 10.1103\/PhysRevB.78.155117.","DOI":"10.1103\/PhysRevB.78.155117"},{"key":"102","doi-asserted-by":"publisher","unstructured":"Shi-Ju Ran. Ab initio optimization principle for the ground states of translationally invariant strongly correlated quantum lattice models. Phys. Rev. E, 93: 053310, May 2016. 10.1103\/PhysRevE.93.053310.","DOI":"10.1103\/PhysRevE.93.053310"},{"key":"103","doi-asserted-by":"publisher","unstructured":"Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, Gang Su, and Maciej Lewenstein. Tensor network contractions. Lecture Notes in Physics, 2020. 10.1007\/978-3-030-34489-4.","DOI":"10.1007\/978-3-030-34489-4"},{"key":"104","doi-asserted-by":"publisher","unstructured":"V. Zauner-Stauber, L. Vanderstraeten, M. T. Fishman, F. Verstraete, and J. Haegeman. Variational optimization algorithms for uniform matrix product states. Phys. Rev. B, 97: 045145, Jan 2018. 10.1103\/PhysRevB.97.045145.","DOI":"10.1103\/PhysRevB.97.045145"},{"key":"105","doi-asserted-by":"publisher","unstructured":"J. Haegeman, C. Lubich, I. Oseledets, B. Vandereycken, and F. Verstraete. Unifying time evolution and optimization with matrix product states. Phys. Rev. B, 94: 165116, Oct 2016. 10.1103\/PhysRevB.94.165116.","DOI":"10.1103\/PhysRevB.94.165116"},{"key":"106","doi-asserted-by":"publisher","unstructured":"L. Vanderstraeten, J. Haegeman, and F. Verstraete. Tangent-space methods for uniform matrix product states. SciPost Phys. Lect. Notes, page 7, 2019b. 10.21468\/SciPostPhysLectNotes.7.","DOI":"10.21468\/SciPostPhysLectNotes.7"},{"key":"107","doi-asserted-by":"publisher","unstructured":"M. T. Fishman, L. Vanderstraeten, V. Zauner-Stauber, J. Haegeman, and F. Verstraete. Faster methods for contracting infinite two-dimensional tensor networks. Phys. Rev. B, 98: 235148, Dec 2018. 10.1103\/PhysRevB.98.235148.","DOI":"10.1103\/PhysRevB.98.235148"},{"key":"108","doi-asserted-by":"publisher","unstructured":"P. Corboz, S. R. White, G. Vidal, and M. Troyer. Stripes in the two-dimensional $t$-$j$ model with infinite projected entangled-pair states. Phys. Rev. B, 84: 041108, Jul 2011. 10.1103\/PhysRevB.84.041108.","DOI":"10.1103\/PhysRevB.84.041108"},{"key":"109","unstructured":"A. Bauer, J. Eisert, and C. Wille. Towards a mathematical formalism for classifying phases of matter. 2019."},{"key":"110","doi-asserted-by":"publisher","unstructured":"M. B. Hastings and T. Koma. Spectral gap and exponential decay of correlations. Comm. Math. Phys., 265 (3): 781\u2013804, 2006. 10.1007\/s00220-006-0030-4.","DOI":"10.1007\/s00220-006-0030-4"},{"key":"111","doi-asserted-by":"publisher","unstructured":"F. G. S. L. Brand\u00e3o and M. Horodecki. Exponential decay of correlations implies area law. Comm. Math. Phys., 333: 761\u2013798, Jan 2015. ISSN 1432-0916. 10.1007\/s00220-014-2213-8.","DOI":"10.1007\/s00220-014-2213-8"},{"key":"112","unstructured":"N. Schuch and F. Verstraete. Matrix product state approximations for infinite systems. 2017."},{"key":"113","doi-asserted-by":"publisher","unstructured":"Alexander M Dalzell and Fernando GSL Brand\u00e3o. Locally accurate mps approximations for ground states of one-dimensional gapped local hamiltonians. Quantum, 3: 187, 2019. https:\/\/doi.org\/10.22331\/q-2019-09-23-187.","DOI":"10.22331\/q-2019-09-23-187"},{"key":"114","unstructured":"Y. Huang. Approximating local properties by tensor network states with constant bond dimension. 2019."},{"key":"115","doi-asserted-by":"publisher","unstructured":"J. Haegeman, T. J. Osborne, and F. Verstraete. Post-matrix product state methods: To tangent space and beyond. Phys. Rev. B, 88: 075133, Aug 2013. 10.1103\/PhysRevB.88.075133.","DOI":"10.1103\/PhysRevB.88.075133"},{"key":"116","doi-asserted-by":"publisher","unstructured":"F. Alet, J. Lykke Jacobsen, G. Misguich, V. Pasquier, F. Mila, and M. Troyer. Interacting classical dimers on the square lattice. Phys. Rev. Lett., 94: 235702, Jun 2005. 10.1103\/PhysRevLett.94.235702.","DOI":"10.1103\/PhysRevLett.94.235702"},{"key":"117","doi-asserted-by":"publisher","unstructured":"F. Alet, Y. Ikhlef, J. L. Jacobsen, G. Misguich, and V. Pasquier. Classical dimers with aligning interactions on the square lattice. Phys. Rev. E, 74: 041124, Oct 2006. 10.1103\/PhysRevE.74.041124.","DOI":"10.1103\/PhysRevE.74.041124"},{"key":"118","doi-asserted-by":"publisher","unstructured":"P. W. Kasteleyn. The statistics of dimers on a lattice: I. the number of dimer arrangements on a quadratic lattice. Physica, 27 (12): 1209 \u2013 1225, 1961. ISSN 0031-8914. https:\/\/doi.org\/10.1016\/0031-8914(61)90063-5.","DOI":"10.1016\/0031-8914(61)90063-5"},{"key":"119","doi-asserted-by":"publisher","unstructured":"H. N. V. Temperley and M. E. Fisher. Dimer problem in statistical mechanics-an exact result. Phil. Mag. A, 6 (68): 1061\u20131063, 1961. 10.1080\/14786436108243366.","DOI":"10.1080\/14786436108243366"},{"key":"120","doi-asserted-by":"publisher","unstructured":"M. E. Fisher. Statistical mechanics of dimers on a plane lattice. Phys. Rev., 124: 1664\u20131672, Dec 1961. 10.1103\/PhysRev.124.1664.","DOI":"10.1103\/PhysRev.124.1664"},{"key":"121","doi-asserted-by":"publisher","unstructured":"M. E. Fisher and John Stephenson. Statistical mechanics of dimers on a plane lattice. ii. dimer correlations and monomers. Phys. Rev., 132: 1411\u20131431, 1963. 10.1103\/PhysRev.132.1411.","DOI":"10.1103\/PhysRev.132.1411"},{"key":"122","doi-asserted-by":"publisher","unstructured":"Y. Li, D. Wu, X. Huang, and C. Ding. Percolation of interacting classical dimers on the square lattice. Physica A, 404: 285 \u2013 290, 2014. ISSN 0378-4371. https:\/\/doi.org\/10.1016\/j.physa.2014.02.076.","DOI":"10.1016\/j.physa.2014.02.076"},{"key":"123","doi-asserted-by":"publisher","unstructured":"N. Schuch, D. Poilblanc, J. I. Cirac, and D. P\u00e9rez-Garc\u00eda. Resonating valence bond states in the PEPS formalism. Phys. Rev. B, 86: 115108, Sep 2012. 10.1103\/PhysRevB.86.115108.","DOI":"10.1103\/PhysRevB.86.115108"},{"key":"124","doi-asserted-by":"publisher","unstructured":"D. Poilblanc. Entanglement Hamiltonian of the quantum N\u00e9el state. J. Stat. Mech., 2014 (10): P10026, oct 2014. 10.1088\/1742-5468\/2014\/10\/p10026.","DOI":"10.1088\/1742-5468\/2014\/10\/p10026"},{"key":"125","doi-asserted-by":"publisher","unstructured":"D. Poilblanc, P. Corboz, N. Schuch, and J. I. Cirac. Resonating-valence-bond superconductors with fermionic projected entangled pair states. Phys. Rev. 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