Corner Transfer Matrix Renormalization Group Method

Nishino, Tomotoshi; Okunishi, Kouichi

doi:10.1143/jpsj.65.891

Cited by 416 publications

(501 citation statements)

References 25 publications

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“…In practice, approximate contraction schemes based on corner transfer matrices (CTM) [33,34], MPS [35,36], tensorentanglement renormalization group (TERG) [37,38], and higher-order (HOTRG) variants [39,40] are used for the contraction of tensor networks both in the finite and infinite cases. Here, we have relied on the directional CTM scheme of Ref.…”

Section: B Contractionmentioning

confidence: 99%

Probing the stability of the spin-liquid phases in the Kitaev-Heisenberg model using tensor network algorithms

2014

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Probing the stability of the spin liquid phases in the Kitaev-Heisenberg model using tensor network algorithms Iregui, J.O.; Corboz, P.R.; Troyer, M. General rightsIt is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulationsIf you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: http://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. We study the extent of the spin liquid phases in the Kitaev-Heisenberg model using infinite projected entangledpair states tensor network ansatz wave functions directly in the thermodynamic limit. To assess the accuracy of the ansatz wave functions, we perform benchmarks against exact results for the Kitaev model and find very good agreement for various observables. In the case of the Kitaev-Heisenberg model, we confirm the existence of six different phases: Néel, stripy, ferromagnetic, zigzag, and two spin liquid phases. We find finite extents for both spin liquid phases and discontinuous phase transitions connecting them to symmetry-broken phases.

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Section: B Contractionmentioning

confidence: 99%

Probing the stability of the spin-liquid phases in the Kitaev-Heisenberg model using tensor network algorithms

2014

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“…33 for details). The contraction of the infinite tensor network is done with the corner-transfer matrix method [37,40,41]. For the plateau phases we used tensors with U(1) symmetry to increase the efficiency [42,43].…”

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Crystals of Bound States in the Magnetization Plateaus of the Shastry-Sutherland Model

2014

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Using infinite projected entangled-pair states (iPEPS), we show that the Shastry-Sutherland model in an external magnetic field has low-magnetization plateaus which, in contrast to previous predictions, correspond to crystals of bound states of triplets, and not to crystals of triplets. The first sizable plateaus appear at magnetization 1/8, 2/15 and 1/6, in agreement with experiments on the orthogonal-dimer antiferromagnet SrCu2(BO3)2, and they can be naturally understood as regular patterns of bound states, including the intriguing 2/15 one. We also show that, even in a confined geometry, two triplets bind into a localized bound state with Sz = 2. Finally, we discuss the role of competing domain-wall and supersolid phases as well as that of additional anisotropic interactions. PACS numbers: 75.10.Jm, 75.40.Mg, 75.10.Kt, Predicting the phases of frustrated spin systems is one of the major challenges in theoretical condensed matter physics [1]. A famous example is the Shastry-Sutherland model (SSM) [2], which is believed to accurately capture the physics of the orthogonal-dimer antiferromagnet SrCu 2 (BO 3 ) 2 . A big effort has been invested in understanding the appearance of various magnetization plateaus observed in experiments [3][4][5][6][7][8][9][10][11][12][13]. Early on it was found that the SSM has almost localized triplet excitations [5,14] which suggests that each plateau corresponds to a particular crystal of localized triplets. This viewpoint has been supported by many analytical and (approximate) numerical studies over the last 15 years [14][15][16][17][18][19][20][21][22][23][24][25].The SSM is given by the Hamiltonianwhere the i, j bonds with coupling strength J build an array of orthogonal dimers while the bonds with coupling J denote inter-dimer couplings, and h the strength of the external magnetic field. The main result of this Letter is that -in contrast to the standard belief -the plateaus are not crystals of S z = 1 triplets (see Fig. 1(a)), but of S z = 2 bound states of triplets, which form a pinwheel pattern as shown in Fig. 1(b), and which are shown to be stable even if they are localized. Bound states have previously been predicted to be relevant in the dilute limit of excitations [16,[26][27][28][29], but not for the formation of crystals. The only hint so far that bound states can form crystals was found in a one-dimensional analog -a SSM spin tube [30]. It is also shown that the crystals formed by the bound states naturally explain the sequence of magnetization plateaus observed in SrCu 2 (BO 3 ) 2 . In particular, the 2/15 plateau is made of a simple and regular pattern of bound states, in contrast to the more complicated patterns of triplets which were previously suggested [12,21,24].Method -Our results have been obtained with infinite projected entangled-pair states (iPEPS) -a variational tensornetwork ansatz to represent a two-dimensional wave function in the thermodynamic limit [31][32][33]. It consists of a cell of tensors which is periodically repeated on the lattice, where ...

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“…The number of exact results for such models is limited, and numerical studies are hard. A clear illustration of the numerical difficulty is the disagreement which existed over the numerical determination of critical temperature and exponents for the standard θ-point model, see for example references [3].To date the numerical methods available for the study of interacting self-avoiding walks in two dimensions are series expansions of walks of lengths of a few tens of steps [4], transfer matrices for lattice widths up to about 12[5] and increasingly complicated Monte-Carlo simulation methods [6], limited in practice to only a portion of the phase diagram.Motivated by these numerical difficulties, we decided to extend the Corner Transfer Matrix Renormalisation Group (CTMRG) method [7]. The CTMRG method is based on White's Density Matrix Renormalisation Group method (DMRG)[8] and Baxter's corner matrix formalism [9].…”

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confidence: 99%

“…The CTMRG method is based on White's Density Matrix Renormalisation Group method (DMRG)[8] and Baxter's corner matrix formalism [9]. To date the CTMRG method has only been applied to discrete spin models, where it is shown to be computationally efficient [7].Our extension to interacting self-avoiding walk models exploits the connection betweeen these models and the O(n) invariant spin models [10], which contains as special cases the Ising model (n = 1), the XY model (n = 2) and the Heisenberg model (n = 3). The method therefore has applications well beyond the self-avoiding walk type models (n = 0).…”

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confidence: 99%

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Corner-transfer-matrix renormalization-group method for two-dimensional self-avoiding walks and otherO(n)models

Foster

Pinettes

2003

Phys. Rev. E

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We present an extension of the corner transfer matrix renormalisation group (CTMRG) method to O(n) invariant models, with particular interest in the self-avoiding walk class of models (O(n = 0)). The method is illustrated using an interacting self-avoiding walk model. Based on the efficiency and versatility when compared to other available numerical methods, we present CTMRG as the method of choice for two-dimensional self-avoiding walk problems.The self-avoiding walk class of models on the twodimensional square lattice, with a variety of possible interactions, has mobilised the scientific community for about half a century [1,2]. The number of exact results for such models is limited, and numerical studies are hard. A clear illustration of the numerical difficulty is the disagreement which existed over the numerical determination of critical temperature and exponents for the standard θ-point model, see for example references [3].To date the numerical methods available for the study of interacting self-avoiding walks in two dimensions are series expansions of walks of lengths of a few tens of steps [4], transfer matrices for lattice widths up to about 12[5] and increasingly complicated Monte-Carlo simulation methods [6], limited in practice to only a portion of the phase diagram.Motivated by these numerical difficulties, we decided to extend the Corner Transfer Matrix Renormalisation Group (CTMRG) method [7]. The CTMRG method is based on White's Density Matrix Renormalisation Group method (DMRG)[8] and Baxter's corner matrix formalism [9]. To date the CTMRG method has only been applied to discrete spin models, where it is shown to be computationally efficient [7].Our extension to interacting self-avoiding walk models exploits the connection betweeen these models and the O(n) invariant spin models [10], which contains as special cases the Ising model (n = 1), the XY model (n = 2) and the Heisenberg model (n = 3). The method therefore has applications well beyond the self-avoiding walk type models (n = 0).The O(n) spin model is defined through the partition function [11]

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Corner Transfer Matrix Renormalization Group Method

Cited by 416 publications

References 25 publications

Probing the stability of the spin-liquid phases in the Kitaev-Heisenberg model using tensor network algorithms

Probing the stability of the spin-liquid phases in the Kitaev-Heisenberg model using tensor network algorithms

Crystals of Bound States in the Magnetization Plateaus of the Shastry-Sutherland Model

Corner-transfer-matrix renormalization-group method for two-dimensional self-avoiding walks and otherO(n)models

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