Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 7-Day Trial for You or Your Team.

Learn More →

Benchmarking global $SU(2)$ symmetry in 2d tensor network algorithms

Benchmarking global $SU(2)$ symmetry in 2d tensor network algorithms 1 2, 3, 4 Philipp Schmoll and Rom an Orus Institute of Physics, Johannes Gutenberg University, 55099 Mainz, Germany Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebasti an, Spain Ikerbasque Foundation for Science, Maria Diaz de Haro 3, E-48013 Bilbao, Spain Multiverse Computing, Paseo de Miram on 170, E-20014 San Sebastian,  Spain We implement and benchmark tensor network algorithms with SU (2) symmetry for systems in two spatial dimensions and in the thermodynamic limit. Speci cally, we implement SU (2)-invariant versions of the in nite Projected Entangled Pair States (iPEPS) and in nite Projected Entangled Simplex States (iPESS) methods. Our implementation of SU (2) symmetry follows the formalism based on fusion trees from [P. Schmoll, S. Singh, M. Rizzi, R. Orus,  Annals of Physics 419, 168232]. In order to assess the utility of implementing SU (2) symmetry, the algorithms are benchmarked for three models with di erent local spin: the spin-1 bilinear-biquadratic model on the square lattice, and the Kagome Heisenberg antiferromagnets (KHAF) for spin-1=2 and spin-2. We observe that the implementation of SU (2) symmetry provides better energies in general than non-symmetric simulations, with smooth scalings with respect to the number of parameters in the ansatz, and with the actual improvement depending on the speci cs of the model. In particular, for the spin-2 KHAF model, our SU (2) simulations are compatible with a quantum spin liquid ground state. Introduction.- Tensor networks [1] (TN) are mathemat- However, since SU (2)-invariant tensors are highly con- ical objects tailored to describe highly-correlated struc- strained, we nd that the actual improvement depends a tures in an ecient way. In condensed matter physics lot on the speci cs of the model. In particular, for the they can describe low-energy states of quantum matter. spin-2 KHAF model, the SU (2) simulations produce a The success of TN methods has been particularly im- ground state structure compatible with that of a quan- pressive for one-dimensional (1d) systems [2], with the tum spin liquid to the best of our computational power. Matrix Product State (MPS) as the driving force. On Methods.- We implemented SU (2)-invariant versions top of that, many applications of TN methods have also of iPEPS and iPESS algorithms. We refer the interested been developed to tackle strongly correlated systems in reader to Ref. [4] for details about iPEPS, and to Ref. [5] two spatial dimensions (2d). Here, Projected Entangled about iPESS. Let us just mention that, in this paper, Pair States (PEPS) [3] are widely used, and the in nite- we employ the so-called simple update [13], which pro- PEPS (iPEPS) algorithm [4] is nowadays a standard tool vides an ecient tensor update for an imaginary-time for simulations in the thermodynamic limit. Alternative evolution algorithm, also when combined with SU (2) methods such as the in nite Projected Entangled Sim- symmetry. The accuracy of our calculations could al- plex States (iPESS) [5] have been applied with success ways be improved by more precise tensor optimization to the Kagome lattice [6]. schemes [4, 14, 15], but at the cost of extra computational An important problem in tensor networks, especially expense. Expectation values in all cases are approxi- in 2d algorithms like iPEPS and iPESS, is how to deal mated using well-known Corner Transfer Matrix (CTM) with global non-abelian symmetries, SU (2) being a com- techniques [16], which can also be easily adapted to deal mon example. For instance, numerical simulations of the with SU (2). spin-1=2 Kagome Heisenberg Antiferromagnet (KHAF) seem to indicate [8] that its ground state is a quantum Concerning SU (2) itself, we work here with the imple- spin liquid and therefore an SU (2) singlet. One would mentation from Ref. [11]. Under the action of the group, therefore expect, a priori, that the study of such a ground both the physical and the virtual vector spaces of the ten- state with a TN algorithm would bene t from the explicit sors can be described in terms of basis states jj; t ; m i. j j preservation of SU (2) symmetry. While this has been Here j are the irreducible representations of SU (2), i.e. done already using an SU (2)-invariant implementation of the spin quantum numbers j = 0; 1=2; 1; : : :, t labels DMRG [9], the generalization to true 2d TN algorithms the multiple copies of irrep j, and m labels the states has been achieved only in a very few cases [10]. within the vector space of spin j (m = [j; : : : ; +j]). In this paper we implement SU (2) symmetry in iPEPS Due to the action of the group, a symmetric tensor gen- and iPESS algorithms using the formalism from Ref. [11], erally consists of various symmetric blocks, each of which which is based on fusion trees. We benchmark our imple- factorizes into a degeneracy part and a structural part ac- mentation by computing ground state properties of three cording to the Wigner-Eckard theorem. The degeneracy 2d models: the spin-1 bilinear-biquadratic model on the part contains the remaining variational parameters, the square lattice, and the spin-1=2 and spin-2 KHAF. We structural part is completely determined by the under- observe that the implementation of SU (2) in the 2d sim- lying symmetry and describes the coupling of the spins ulations in general allows to produce lower energies than on the tensor indices in terms of Clebsch-Gordan coe- the ones obtained using non-symmetric TN algorithms. cients. In our implementation the structural part of the arXiv:2005.02748v2 [cond-mat.str-el] 22 Dec 2020 2 (a) 0.38 no symmetry 0.37 SU(2) CTM SU(2) MF 0.36 extrapolation SU(2) (b) U(1) (Corboz) 0.35 extrapolation U(1) 0.34 0.33 0.32 FIG. 1: (Color online) (a) Decomposition of an SU (2)- 0.31 symmetric 2 2 iPEPS unit cell on a square lattice, in terms 0.3 of degeneracy tensors and a network of fusion trees. The fu- 0 0.05 0.1 0.15 0.2 0.25 sion tree structure of every tensor is shown on the right-hand side, where arrows correspond to incoming/outgoing indices. Every three-index node in the fusion trees is an intertwiner FIG. 2: Ground state energy of the spin-1 BLBQ model on of SU (2), i.e., a tensor of Clebsch-Gordan coecients. (b) the square lattice at  = 0:21, as a function of 1=D . U (1) Decomposition of an SU (2)-symmetric iPESS unit cell on the results are from Ref. [17] and replotted with permission. honeycomb lattice, which is used to simulate its dual Kagome lattice. 0.38 0.38 no symmetry no symmetry symmetric tensors is codi ed in the form of fusion trees SU(2) CTM SU(2) CTM 0.36 0.36 SU(2) MF SU(2) MF decorated by quantum numbers. They are a memory- extrapolation SU(2) extrapolation 0.34 0.34 ecient analytic representation of the group structure, so that no actual Clebsch-Gordan coecients for the struc- 0.32 0.32 tural tensors have to be stored. In Fig. 1 we show the 0.3 0.3 decomposition of a network of iPEPS and iPESS tensors -5 -4 -3 -2 -12 -11 -10 10 10 10 10 10 10 10 in terms of degeneracy parts and structural parts. This implementation allows for clean, accurate calculations, which is of particular importance when dealing with 2d FIG. 3: (Color online) Ground state energy of the spin-1 BLBQ model on the square lattice at  = 0:21, as a func- TN algorithms. Concerning notation, in the following tion of 1=N (left) and as a function of the discarded weight we call D the e ective bond dimension of the PEPS (right). or PESS, i.e., D = D for non-symmetric TNs (with D the usual bond dimension) and D = t jm j e j j i i for symmetric ones (with t the degeneracy of symmetry sector j and jm j = 2j + 1 for an index i  (j ; t ; m ) This phase is reminiscent of coupled spin-1 chains [17]. i j i i j j i i [11]). The symmetric bond dimension is D = t Thus, the point  = 0:21 is a non-trivial benchmark for sym j and N is the number of variational parameters in the a 2d SU (2)-invariant gapped phase. In our simulation, ansatz. we choose to work with the simple update and a 2  2 unit cell. The ground state energy of the system is Results.- Let us now discuss the performance of SU (2)- shown in Fig. 2 as a function 1=D , in Fig. 3 as a func- invariant iPEPS and iPESS. We focus rst on the spin-1 tion of 1=N and as a function of the discarded weight in bilinear-biquadratic (BLBQ) model on the square lattice. the truncations  [18]. The plots show the performance Its Hamiltonian is given by for SU (2)-iPEPS as well as for iPEPS with no symmetry, and we also compare with the results from Ref. [17] using H = cos() (S  S ) + sin() (S  S ) ; (1) i j i j U (1)-iPEPS in Fig. 2 (results replotted with permission). hi;ji In the gures, for the SU (2) simulations we include re- where hi; ji are nearest-neighbour interactions, S is the sults obtained by using a CTM environment to compute vector of spin-1 matrices, and  tunes the relative cou- expectation values, as well as using a mean- eld (MF) pling strength of the bilinear and biquadratic terms. The environment estimation. This last option does not pro- phase diagram of this model has already been computed vide variational energies, but allows us to see the overall previously with iPEPS, both without symmetries but tendency for large bond dimension (for which the calcula- also including U (1) symmetry [17]. Here, we tune the tions using SU (2)-CTM algorithms are computationally coupling parameter to  = 0:21. At this angle the costly). We see that the extrapolation 1=N ! 0 is better magnetization of the ground state vanishes, so that the behaved than the one for 1=D ! 0, and is actually com- system is believed to be in the middle of an SU (2)- parable for SU (2) to the extrapolation in the discarded symmetric Haldane phase existing for  2 [0:189; 0:217]. weight. In this last extrapolation one can also clearly 3 0.06 0.3287 0.3287 0.2977 0.2975 0.2977 0.04 0.3229 0.3229 0.02 0.2975 0.2972 0.2975 0 1 2 3 10 10 10 10 0.3287 0.3287 FIG. 5: (Color online) Ratio of ansatz variational param- FIG. 4: (Color online) Ground state energy per link in the eters between SU (2)-symmetric and non-symmetric simula- iPEPS unit cell. The structure is compatible with vertical tions, for iPEPS and iMPS, as a function of their respective Haldane chains coupled in the horizontal direction. The dif- bond dimensions D and  . e e ferences in the fourth relevant digit between the upper and lower horizontal link energies is due to truncation e ects. state with \simple" tensor updates. In order to get an idea of the e ect of the symmetry on the size of the see that the non-symmetric simulation is far from being variational space we evaluated the ratio between remain- converged. Our extrapolated data for the ground state ing variational parameters in the SU (2)-iPEPS and the energy e is e (1=D ! 0) = 0:309 0:003, e (1=N ! 0 0 e 0 number of variational parameters in the corresponding 0) = 0:311 0:004, and e ( ! 0) = 0:310 0:002. unconstrained TN for di erent bond dimensions. This We notice from our plots that the simulations without is shown in Fig. 5, alongside with the same information symmetry yield the lowest ground state energy for for an SU (2)-symmetric in nite MPS simulation of a small bond dimensions and the data points with SU (2) critical spin-1=2 ladder system [12]. The comparison symmetry are considerably higher than those with lower between both cases allows us to understand better the or no symmetry. We take this as a rst indication that e ect of dimensionality in the reduction of variational the SU (2)-symmetric ansatz in 2d may sometimes be parameters in a SU (2)-invariant TN ansatz. What we too restrictive, which is especially true for small bond conclude from the plot is that the SU (2)-invariant ansatz dimensions. However, for large bond dimension the becomes very restrictive with the bond dimension, as situation is the opposite, and the SU (2) simulation expected, but at a much faster rate in 2d than in 1d. In produces lower energies. It is interesting, though, that other words, SU (2) in 2d restricts the variational space the SU (2) numbers computed by CTM (which are faster than in 1d. A priori, this could be good news, variational, since the CTM bond dimension is converged since the number of parameters to optimize is much [16]) tend to be always slightly above those obtained more drastically reduced in 2d than in 1d. However, this with an U (1)-symmetric ansatz. We will comment on needs to be taken into account with care when assessing the restrictiveness and expressiveness of our simulations symmetric TN simulations since the optimization space in the next paragraph. For the record, the obtained may actually be too constrained in some cases for extrapolated energy with U (1) symmetry in Fig. 2 is nding low variational energies. The systematically e (1=D ! 0) = 0:307 0:001, and therefore very close 0 e higher SU (2) energies could also hint at the fact, that to the SU (2) number. Finally, in order to understand the ground state weakly breaks the symmetry, albeit better the nature of the SU (2)-invariant ground state predicted to be symmetric. In this scenario, a manifestly that we obtain, we also plot its energy on each link of the SU (2)-invariant TN ansatz is expected to yield higher iPEPS unit cell in Fig. 4. The ground state cultivates energies than an ansatz with a lower symmetry. di erent energies in x- and y-directions, thus breaking lattice rotation symmetry. This is however compatible The next model that we considered was the spin-1=2 with vertical coupled Haldane 1d chains, in accordance KHAF. The Hamiltonian is given by with the results from Ref. [17]. This di erence in bond energies is caused by converging to half-integer spin H = S  S ; (2) i j representations on the vertical bonds, and integer ones hi;ji on the horizontal bonds, leading to di erent e ective bond dimensions. where hi; ji denotes nearest-neighbour interactions be- These ndings point towards an interesting fact: SU (2) tween sites of the Kagome lattice, and S is the spin-1=2 symmetry in 2d, even if generically useful, can be highly (vector) operator at site i. The Kagome lattice exhibits restrictive in some cases. The variational space is highly corner-sharing triangles resulting in huge quantum uc- constrained, and in some situations this could be too tuations around the ground state due to strong geomet- limited to nd a good approximation to the ground ric frustration, with many states very close in energy -0.1017 -0.0114 -0.4394 -0.1496 -0.0196 -0.4092 -0.0196 -0.4092 -0.2169 -0.2169 -0.38 -0.4091 -0.4347 1 2 1 2 3-PESS no symmetry 6-PESS no symmetry -0.39 6-PESS SU(2) CTM -0.2169 3 3 1 2 extrapolation SU(2) -0.4 -0.0197 -0.4091 -0.1180 -0.2519 3 5 4 5 5 4 5 -0.41 -0.2169 2 1 6 6 -0.42 -0.43 -0.0197 -0.2189 2 1 2 1 iDMRG comparison -0.44 0 0.1 0.2 0.3 0.4 0.5 FIG. 7: (Color online) Spin-spin correlation hS S i on each i j link of the unit cell for the non-symmetric 3-PESS, the non- -0.38 symmetric 6-PESS and the SU (2)-invariant 6-PESS (from left 3-PESS no symmetry to right). 6-PESS no symmetry -0.39 6-PESS SU(2) CTM extrapolation SU(2) (a) (b) -0.4 -0.41 -0.42 -0.43 iDMRG comparison -0.44 -4 -3 -2 10 10 10 FIG. 8: (Color online) Di erence between the strongest and the weakest bond in the unit cell (correlator skewness) of a 6- FIG. 6: (Color online) Ground state energy of the spin-1=2 PESS, for the spin-1=2 KHAF simulations, with no symmetry KHAF, as a function of 1=D and 1=N , with the yellow line and with SU (2) symmetry, as a function of (a) 1=D and (b) denoting the extrapolation to in nite bond dimension. 1=N . and competing to be the true ground state. This makes ever, one can see again that the limit of in nite bond the simulation of the model very challenging. For the dimension is better achieved by the SU (2)-invariant sim- sake of this study, our goal here is not to provide bet- ulations as a function of 1=N (see Fig. 6(b)), with ex- ter ground-state numbers than those obtained by other trapolated values e (1=D ! 0) = 0:435  0:004 and 0 e simulations [8], but rather to benchmark the utility of e (1=N ! 0) = 0:435  0:002. The iDMRG compari- SU (2) symmetry in 2d TN algorithms, and in partic- son shows the current energetically lowest ground state ular in iPESS. Previous results have shown that using energy [7]. Let us mention that for all the iPESS simula- three-site iPESS without symmetries produces reason- tions that we performed, the extrapolation in the dis- ably good numbers for the ground state energy [5]. For carded weight was not possible because the discarded the SU (2)-symmetric simulations, however, we need to weight was always too small. The spin-spin correlators resort to the six-site unit cell in order to accommodate for each link of the unit cell are shown in Fig. 7, for consistently the SU (2) quantum numbers on all the in- the non-symmetric 3-PESS, the non-symmetric 6-PESS dices of the symmetric TN ansatz. Since the physical and the SU (2)-invariant 6-PESS respectively. While the sites carry spin-1=2, the geometry and the unit cell force 3-site unit cell produces a state that seems compatible us to use mixed spins (integer and half-integer) on the with a quantum spin liquid, the 6-site unit cells seem bond indices of the iPESS. to produce valence-bond crystal structures with strong We computed the ground state energy of the model and weak links, thus breaking invariance under trans- for a 3-site and a 6-site unit cell without symmetry, and lations and lattice rotations. We observe, in any case, for a 6-site unit cell with SU (2) symmetry. The results that the valence bond crystal tends to melt when we in- are shown in Fig. 6. The symmetric results are com- crease the bond dimension of the iPESS ansatz, both patible with those obtained without symmetries, with for the non-symmetric and the SU (2)-invariant simula- an algebraic convergence of the ground state energy as tions, thus slowly recovering translation invariance (see a function of 1=D (see Fig. 6(a)), in turn reinforc- Fig. 8 for plots of the correlator skewness as D in- e e ing the observation that the ground state is a quan- creases). Interestingly, we also observe in the gure that tum spin liquid and therefore SU (2)-invariant. How- the non-symmetric simulations tend to melt faster than -0.2169 -0.2169 -0.2207 -0.2207 -0.2205 -0.2208 -0.5577 -0.0907 -0.2430 +0.0124 5 lower variational energy than the non-symmetric simula- tions. Moreover, both in the non-symmetric and symmet- ric cases we observe in Figs. 9(c,d) a structure of spin- spin correlators in the unit cell that seems compatible with that of a quantum spin liquid, which is also com- patible with the algebraic behaviour of the ground state energy with D in Fig. 9(a). The energies are, however, dicult to extrapolate to in nite bond dimension, and hence we cannot be sure whether this is the true nature of the ground state. But we can claim that, to the best of our calculations, here the SU (2)-invariant iPESS with a 3-site unit cell produces the best variational energy for the ground state, which seems compatible with a quan- tum spin liquid. Moreover, we computed the expectation value of the chiral correlators S  (S  S ) on all tri- i j k angles, and obtained exactly 0 everywhere, in turn also compatible with the structure of a non-chiral quantum spin liquid. FIG. 9: (Color online) Ground state energy of the spin-2 KHAF, as a function of (a) 1=D and (b) 1=N , with the Model No symmetry SU (2) red line denoting the extrapolation to in nite bond dimen- sion. Notice that in this case, this extrapolation is just for s = 1 BLBQ (7; 0:3188) (6; 19:5; 0:3108) completeness since the numbers still do not show convergence s = 1=2 KHAF (10;0:4348) (7; 17:75;0:4349) for the achievable bond dimensions due to the large local spin s = 2 KHAF (10;4:7975) (5; 19;4:8227) at every site. (c,d) Spin-spin correlation hS S i on each link i j of the unit cell for the non-symmetric and the SU (2)-invariant TABLE I: Ground state energies obtained for the maxi- 3-PESS respectively. mum achievable bond dimension for the bilinear-biquadratic (BLBQ) and KHAF models that we considered. We show (D; e ) for non-symmetric simulations and (D ; D ; e ) for 0 sym e 0 the SU (2)-symmetric ones. The observed pattern of local SU (2)-invariant ones, with D the non-symmetric bond di- correlations for the non-symmetric 6-site iPESS in Fig. 7 mension, D the symmetric bond dimension and D the sym e is expected, since TN simulations tend to trade symme- e ective bond dimension when using SU (2) averaged for all try for injectivity of the target state. For the symmetric bonds, which can be integer or fractional. 6-site iPESS the correlators are even more skewed pre- sumably due to non-uniform e ective bond dimensions Conclusions.- In Table I we make a comparison of the caused by the mixture of integer and half-integer repre- computed ground state energies for the maximum achiev- sentations. This e ect is expected to vanish in the limit able bond dimensions, for the three models considered of large bond dimensions, and also explains the slower here, and for non-symmetric and SU (2)-symmetric sim- melting in Fig. 8. ulations. We conclude that implementing SU (2) symme- Finally, we computed the ground state energy of the try in 2d TN algorithms usually produces better energies spin-2 KHAF for a 3-site unit cell without and with than non-symmetric simulations, but the performance SU (2) symmetry. Unlike in the spin-1=2 case, the fact depends on the speci cs of the model and in particular that we have spin-2 in the physical indices allows us to use on the gap of the phase being targeted. For the spin-2 the 3-site unit cell (this, in fact, is true for all integer-spin model, the SU (2) simulations point towards a quantum Heisenberg models on the Kagome lattice). The results spin liquid as a plausible ground state. are shown in Fig. 9. This time, due to the large dimension of the physical spin at every site, we cannot reach values Acknowledgments.- We acknowledge Andreas of D as large as for the spin-1=2 case. However, the ef- Haller, Saeed Jahromi, Matteo Rizzi, and Sukhbinder sym fective bond dimension is larger for the spin-2 KHAF, as Singh for insightful discussions. We also acknowledge shown in Table I. We see in Figs. 9(a,b) that the SU (2)- DFG funding through project GZ OR 381/3-1, as well invariant simulations can handle larger D and produce as the MAINZ Graduate School of Excellence. [1] R. Orus,  Ann. Phys. 349, 117 (2014); R. Orus,  Nat. Rev. [4] J. Jordan, R. Orus,  G. Vidal, F. Verstraete, J. I. Cirac, Phys. 1, 538 (2019). Phys. Rev. Lett. 101, 250602 (2008); J. Jordan, R. Orus, [2] S. R. White, Phys. Rev. Lett.69, 28632866 (1992); S. R. G. Vidal, Phys. Rev. B 79, 174515 (2009). White, Phys. Rev. B 48, 10345 (1992). [5] Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, T. [3] F. Verstraete, J. I. Cirac, cond-mat/0407066. Xiang, Phys. Rev. X 4, 011025 (2014). 6 [6] A. Kshetrimayum, T. Picot, R. Orus,  D. Poilblanc, Phys. Capponi, D. Poilblanc, Phys. Rev. B 98, 184409 (2018). Rev. B 94, 235146 (2016); A. Kshetrimayum, C. Balz, B. [11] P. Schmoll, S. Singh, M. Rizzi, R. Orus,  Annals of Physics Lake, J. Eisert, arXiv:1904.00028; S. S. Jahromi, R. Orus,  419, 168232 (2020) D. Poilblanc, F. Mila, arXiv:1912.10756. [12] P. Schmoll, A. Haller, M. Rizzi, R. Orus,  Phys. Rev. B [7] S. Depenbrock, I. P. McCulloch, U. Schollw ock, Phys. 99, 205121 (2019). Rev. Lett. 109, 067201 (2012). [13] H. C. Jiang, Z. Y. Weng, T. Xiang, Phys. Rev. Lett. 101, [8] H. C. Jiang, Z. Y. Weng, D. N. Sheng, Phys. Rev. Lett. 090603 (2008). 101, 117203 (2008); S. Yan, D. A. Huse, S. White, Sci- [14] M. Lubasch, I. Cirac, M.C. Banuls, ~ Phys. Rev. B 90, ence 332, 1173 (2011); S. Depenbrock, I. P. McCulloch, 064425 (2014); H. N. Phien, J. A. Bengua, H. D. Tuan, U. Schollw ock, Phys. Rev. Lett. 109, 067201 (2012); G. P. Corboz, R. Orus,  Phys. Rev. B 92, 035142 (2015). Evenbly, G. Vidal, Phys. Rev. Lett. 104, 187203 (2010); [15] L. Vanderstraeten, J. Haegeman, P. Corboz, F. Ver- T. Picot, M. Ziegler, R. Orus,  D. Poilblanc, Phys. Rev. straete, Phys. Rev. B 94, 155123 (2016); P. Corboz, B 93, 060407 (2016). Phys. Rev. B 94, 035133 (2016). [9] I. McCulloch, M. Gulacsi, Eur. Phys. Lett. 57, 852 [16] R. J. Baxter, Physica A 106, pp18-27 (1981); R. J. Bax- (2002); S. Singh, H.-Q. Zhou, G. Vidal, New J. Phys. ter, Exactly Solved Models in Statistical Mechanics, Aca- 033029 (2010). demic Press, London, (1982); R. J. Baxter, J. Math. [10] T. Liu, W. Li, A. Weichselbaum, J. von Delft, G. Su, Phys. 9, 650 (1968); R. J. Baxter, J. Stat. Phys. 19, 461 Phys. Rev. B 91, 060403 (2015); W. Li, A. Weichsel- (1978); T. Nishino, K. Okunishi, J. Phys. Soc. Jpn. 65, baum, J. von Delft, H.-H. Tu, Phys. Rev. B 91, 224414 pp. 891-894 (1996); T. Nishino, K. Okunishi, J. Phys. (2015); J.-W. Mei, J.-Y. Chen, H. He, X.-G. Wen, Phys. Soc. Jp. 66, 3040 (1997); R. Orus,  G. Vidal, Phys. Rev. Rev. B 95, 235107 (2017); S.-J. Ran, W. Li, S.-S. Gong, B 80, 094403 (2009); R. Orus,  Phys. Rev. B 85, 205117 A. Weichselbaum, J. von Delft, G. Su, Phys. Rev. B (2012); L. Vanderstraeten, M. Mari'en, F. Verstraete, J. 97, 075146 (2018); M. Mambrini, R. Orus,  D. Poilblanc, Haegeman, Phys. Rev. B 92, 201111 (2015). Phys. Rev. B 94, 205124 (2016); D. Poilblanc, M. Mam- [17] I. Niesen, P. Corboz, SciPost Phys. 3, 030 (2017). brini, Phys. Rev. B 96, 014414 (2017); C. Hubig, SciPost [18] P. Corboz, Phys. Rev. B 93, 045116 (2016). Phys. 5, 047 (2018) ; J.-Y. Chen, L. Vanderstraeten, S. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Condensed Matter arXiv (Cornell University)

Benchmarking global $SU(2)$ symmetry in 2d tensor network algorithms

Schmoll, Philipp; Orus, Roman
Condensed Matter , Volume 2020 (2005) – May 6, 2020
Download PDF
6 pages

Loading next page...
 
/lp/arxiv-cornell-university/benchmarking-global-su-2-symmetry-in-2d-tensor-network-algorithms-N9BFtqtlZ3

References

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

ISSN
2469-9950
eISSN
ARCH-3331
DOI
10.1103/PhysRevB.102.241101
Publisher site
See Article on Publisher Site

Abstract

1 2, 3, 4 Philipp Schmoll and Rom an Orus Institute of Physics, Johannes Gutenberg University, 55099 Mainz, Germany Donostia International Physics Center, Paseo Manuel de Lardizabal 4, E-20018 San Sebasti an, Spain Ikerbasque Foundation for Science, Maria Diaz de Haro 3, E-48013 Bilbao, Spain Multiverse Computing, Paseo de Miram on 170, E-20014 San Sebastian,  Spain We implement and benchmark tensor network algorithms with SU (2) symmetry for systems in two spatial dimensions and in the thermodynamic limit. Speci cally, we implement SU (2)-invariant versions of the in nite Projected Entangled Pair States (iPEPS) and in nite Projected Entangled Simplex States (iPESS) methods. Our implementation of SU (2) symmetry follows the formalism based on fusion trees from [P. Schmoll, S. Singh, M. Rizzi, R. Orus,  Annals of Physics 419, 168232]. In order to assess the utility of implementing SU (2) symmetry, the algorithms are benchmarked for three models with di erent local spin: the spin-1 bilinear-biquadratic model on the square lattice, and the Kagome Heisenberg antiferromagnets (KHAF) for spin-1=2 and spin-2. We observe that the implementation of SU (2) symmetry provides better energies in general than non-symmetric simulations, with smooth scalings with respect to the number of parameters in the ansatz, and with the actual improvement depending on the speci cs of the model. In particular, for the spin-2 KHAF model, our SU (2) simulations are compatible with a quantum spin liquid ground state. Introduction.- Tensor networks [1] (TN) are mathemat- However, since SU (2)-invariant tensors are highly con- ical objects tailored to describe highly-correlated struc- strained, we nd that the actual improvement depends a tures in an ecient way. In condensed matter physics lot on the speci cs of the model. In particular, for the they can describe low-energy states of quantum matter. spin-2 KHAF model, the SU (2) simulations produce a The success of TN methods has been particularly im- ground state structure compatible with that of a quan- pressive for one-dimensional (1d) systems [2], with the tum spin liquid to the best of our computational power. Matrix Product State (MPS) as the driving force. On Methods.- We implemented SU (2)-invariant versions top of that, many applications of TN methods have also of iPEPS and iPESS algorithms. We refer the interested been developed to tackle strongly correlated systems in reader to Ref. [4] for details about iPEPS, and to Ref. [5] two spatial dimensions (2d). Here, Projected Entangled about iPESS. Let us just mention that, in this paper, Pair States (PEPS) [3] are widely used, and the in nite- we employ the so-called simple update [13], which pro- PEPS (iPEPS) algorithm [4] is nowadays a standard tool vides an ecient tensor update for an imaginary-time for simulations in the thermodynamic limit. Alternative evolution algorithm, also when combined with SU (2) methods such as the in nite Projected Entangled Sim- symmetry. The accuracy of our calculations could al- plex States (iPESS) [5] have been applied with success ways be improved by more precise tensor optimization to the Kagome lattice [6]. schemes [4, 14, 15], but at the cost of extra computational An important problem in tensor networks, especially expense. Expectation values in all cases are approxi- in 2d algorithms like iPEPS and iPESS, is how to deal mated using well-known Corner Transfer Matrix (CTM) with global non-abelian symmetries, SU (2) being a com- techniques [16], which can also be easily adapted to deal mon example. For instance, numerical simulations of the with SU (2). spin-1=2 Kagome Heisenberg Antiferromagnet (KHAF) seem to indicate [8] that its ground state is a quantum Concerning SU (2) itself, we work here with the imple- spin liquid and therefore an SU (2) singlet. One would mentation from Ref. [11]. Under the action of the group, therefore expect, a priori, that the study of such a ground both the physical and the virtual vector spaces of the ten- state with a TN algorithm would bene t from the explicit sors can be described in terms of basis states jj; t ; m i. j j preservation of SU (2) symmetry. While this has been Here j are the irreducible representations of SU (2), i.e. done already using an SU (2)-invariant implementation of the spin quantum numbers j = 0; 1=2; 1; : : :, t labels DMRG [9], the generalization to true 2d TN algorithms the multiple copies of irrep j, and m labels the states has been achieved only in a very few cases [10]. within the vector space of spin j (m = [j; : : : ; +j]). In this paper we implement SU (2) symmetry in iPEPS Due to the action of the group, a symmetric tensor gen- and iPESS algorithms using the formalism from Ref. [11], erally consists of various symmetric blocks, each of which which is based on fusion trees. We benchmark our imple- factorizes into a degeneracy part and a structural part ac- mentation by computing ground state properties of three cording to the Wigner-Eckard theorem. The degeneracy 2d models: the spin-1 bilinear-biquadratic model on the part contains the remaining variational parameters, the square lattice, and the spin-1=2 and spin-2 KHAF. We structural part is completely determined by the under- observe that the implementation of SU (2) in the 2d sim- lying symmetry and describes the coupling of the spins ulations in general allows to produce lower energies than on the tensor indices in terms of Clebsch-Gordan coe- the ones obtained using non-symmetric TN algorithms. cients. In our implementation the structural part of the arXiv:2005.02748v2 [cond-mat.str-el] 22 Dec 2020 2 (a) 0.38 no symmetry 0.37 SU(2) CTM SU(2) MF 0.36 extrapolation SU(2) (b) U(1) (Corboz) 0.35 extrapolation U(1) 0.34 0.33 0.32 FIG. 1: (Color online) (a) Decomposition of an SU (2)- 0.31 symmetric 2 2 iPEPS unit cell on a square lattice, in terms 0.3 of degeneracy tensors and a network of fusion trees. The fu- 0 0.05 0.1 0.15 0.2 0.25 sion tree structure of every tensor is shown on the right-hand side, where arrows correspond to incoming/outgoing indices. Every three-index node in the fusion trees is an intertwiner FIG. 2: Ground state energy of the spin-1 BLBQ model on of SU (2), i.e., a tensor of Clebsch-Gordan coecients. (b) the square lattice at  = 0:21, as a function of 1=D . U (1) Decomposition of an SU (2)-symmetric iPESS unit cell on the results are from Ref. [17] and replotted with permission. honeycomb lattice, which is used to simulate its dual Kagome lattice. 0.38 0.38 no symmetry no symmetry symmetric tensors is codi ed in the form of fusion trees SU(2) CTM SU(2) CTM 0.36 0.36 SU(2) MF SU(2) MF decorated by quantum numbers. They are a memory- extrapolation SU(2) extrapolation 0.34 0.34 ecient analytic representation of the group structure, so that no actual Clebsch-Gordan coecients for the struc- 0.32 0.32 tural tensors have to be stored. In Fig. 1 we show the 0.3 0.3 decomposition of a network of iPEPS and iPESS tensors -5 -4 -3 -2 -12 -11 -10 10 10 10 10 10 10 10 in terms of degeneracy parts and structural parts. This implementation allows for clean, accurate calculations, which is of particular importance when dealing with 2d FIG. 3: (Color online) Ground state energy of the spin-1 BLBQ model on the square lattice at  = 0:21, as a func- TN algorithms. Concerning notation, in the following tion of 1=N (left) and as a function of the discarded weight we call D the e ective bond dimension of the PEPS (right). or PESS, i.e., D = D for non-symmetric TNs (with D the usual bond dimension) and D = t jm j e j j i i for symmetric ones (with t the degeneracy of symmetry sector j and jm j = 2j + 1 for an index i  (j ; t ; m ) This phase is reminiscent of coupled spin-1 chains [17]. i j i i j j i i [11]). The symmetric bond dimension is D = t Thus, the point  = 0:21 is a non-trivial benchmark for sym j and N is the number of variational parameters in the a 2d SU (2)-invariant gapped phase. In our simulation, ansatz. we choose to work with the simple update and a 2  2 unit cell. The ground state energy of the system is Results.- Let us now discuss the performance of SU (2)- shown in Fig. 2 as a function 1=D , in Fig. 3 as a func- invariant iPEPS and iPESS. We focus rst on the spin-1 tion of 1=N and as a function of the discarded weight in bilinear-biquadratic (BLBQ) model on the square lattice. the truncations  [18]. The plots show the performance Its Hamiltonian is given by for SU (2)-iPEPS as well as for iPEPS with no symmetry, and we also compare with the results from Ref. [17] using H = cos() (S  S ) + sin() (S  S ) ; (1) i j i j U (1)-iPEPS in Fig. 2 (results replotted with permission). hi;ji In the gures, for the SU (2) simulations we include re- where hi; ji are nearest-neighbour interactions, S is the sults obtained by using a CTM environment to compute vector of spin-1 matrices, and  tunes the relative cou- expectation values, as well as using a mean- eld (MF) pling strength of the bilinear and biquadratic terms. The environment estimation. This last option does not pro- phase diagram of this model has already been computed vide variational energies, but allows us to see the overall previously with iPEPS, both without symmetries but tendency for large bond dimension (for which the calcula- also including U (1) symmetry [17]. Here, we tune the tions using SU (2)-CTM algorithms are computationally coupling parameter to  = 0:21. At this angle the costly). We see that the extrapolation 1=N ! 0 is better magnetization of the ground state vanishes, so that the behaved than the one for 1=D ! 0, and is actually com- system is believed to be in the middle of an SU (2)- parable for SU (2) to the extrapolation in the discarded symmetric Haldane phase existing for  2 [0:189; 0:217]. weight. In this last extrapolation one can also clearly 3 0.06 0.3287 0.3287 0.2977 0.2975 0.2977 0.04 0.3229 0.3229 0.02 0.2975 0.2972 0.2975 0 1 2 3 10 10 10 10 0.3287 0.3287 FIG. 5: (Color online) Ratio of ansatz variational param- FIG. 4: (Color online) Ground state energy per link in the eters between SU (2)-symmetric and non-symmetric simula- iPEPS unit cell. The structure is compatible with vertical tions, for iPEPS and iMPS, as a function of their respective Haldane chains coupled in the horizontal direction. The dif- bond dimensions D and  . e e ferences in the fourth relevant digit between the upper and lower horizontal link energies is due to truncation e ects. state with \simple" tensor updates. In order to get an idea of the e ect of the symmetry on the size of the see that the non-symmetric simulation is far from being variational space we evaluated the ratio between remain- converged. Our extrapolated data for the ground state ing variational parameters in the SU (2)-iPEPS and the energy e is e (1=D ! 0) = 0:309 0:003, e (1=N ! 0 0 e 0 number of variational parameters in the corresponding 0) = 0:311 0:004, and e ( ! 0) = 0:310 0:002. unconstrained TN for di erent bond dimensions. This We notice from our plots that the simulations without is shown in Fig. 5, alongside with the same information symmetry yield the lowest ground state energy for for an SU (2)-symmetric in nite MPS simulation of a small bond dimensions and the data points with SU (2) critical spin-1=2 ladder system [12]. The comparison symmetry are considerably higher than those with lower between both cases allows us to understand better the or no symmetry. We take this as a rst indication that e ect of dimensionality in the reduction of variational the SU (2)-symmetric ansatz in 2d may sometimes be parameters in a SU (2)-invariant TN ansatz. What we too restrictive, which is especially true for small bond conclude from the plot is that the SU (2)-invariant ansatz dimensions. However, for large bond dimension the becomes very restrictive with the bond dimension, as situation is the opposite, and the SU (2) simulation expected, but at a much faster rate in 2d than in 1d. In produces lower energies. It is interesting, though, that other words, SU (2) in 2d restricts the variational space the SU (2) numbers computed by CTM (which are faster than in 1d. A priori, this could be good news, variational, since the CTM bond dimension is converged since the number of parameters to optimize is much [16]) tend to be always slightly above those obtained more drastically reduced in 2d than in 1d. However, this with an U (1)-symmetric ansatz. We will comment on needs to be taken into account with care when assessing the restrictiveness and expressiveness of our simulations symmetric TN simulations since the optimization space in the next paragraph. For the record, the obtained may actually be too constrained in some cases for extrapolated energy with U (1) symmetry in Fig. 2 is nding low variational energies. The systematically e (1=D ! 0) = 0:307 0:001, and therefore very close 0 e higher SU (2) energies could also hint at the fact, that to the SU (2) number. Finally, in order to understand the ground state weakly breaks the symmetry, albeit better the nature of the SU (2)-invariant ground state predicted to be symmetric. In this scenario, a manifestly that we obtain, we also plot its energy on each link of the SU (2)-invariant TN ansatz is expected to yield higher iPEPS unit cell in Fig. 4. The ground state cultivates energies than an ansatz with a lower symmetry. di erent energies in x- and y-directions, thus breaking lattice rotation symmetry. This is however compatible The next model that we considered was the spin-1=2 with vertical coupled Haldane 1d chains, in accordance KHAF. The Hamiltonian is given by with the results from Ref. [17]. This di erence in bond energies is caused by converging to half-integer spin H = S  S ; (2) i j representations on the vertical bonds, and integer ones hi;ji on the horizontal bonds, leading to di erent e ective bond dimensions. where hi; ji denotes nearest-neighbour interactions be- These ndings point towards an interesting fact: SU (2) tween sites of the Kagome lattice, and S is the spin-1=2 symmetry in 2d, even if generically useful, can be highly (vector) operator at site i. The Kagome lattice exhibits restrictive in some cases. The variational space is highly corner-sharing triangles resulting in huge quantum uc- constrained, and in some situations this could be too tuations around the ground state due to strong geomet- limited to nd a good approximation to the ground ric frustration, with many states very close in energy -0.1017 -0.0114 -0.4394 -0.1496 -0.0196 -0.4092 -0.0196 -0.4092 -0.2169 -0.2169 -0.38 -0.4091 -0.4347 1 2 1 2 3-PESS no symmetry 6-PESS no symmetry -0.39 6-PESS SU(2) CTM -0.2169 3 3 1 2 extrapolation SU(2) -0.4 -0.0197 -0.4091 -0.1180 -0.2519 3 5 4 5 5 4 5 -0.41 -0.2169 2 1 6 6 -0.42 -0.43 -0.0197 -0.2189 2 1 2 1 iDMRG comparison -0.44 0 0.1 0.2 0.3 0.4 0.5 FIG. 7: (Color online) Spin-spin correlation hS S i on each i j link of the unit cell for the non-symmetric 3-PESS, the non- -0.38 symmetric 6-PESS and the SU (2)-invariant 6-PESS (from left 3-PESS no symmetry to right). 6-PESS no symmetry -0.39 6-PESS SU(2) CTM extrapolation SU(2) (a) (b) -0.4 -0.41 -0.42 -0.43 iDMRG comparison -0.44 -4 -3 -2 10 10 10 FIG. 8: (Color online) Di erence between the strongest and the weakest bond in the unit cell (correlator skewness) of a 6- FIG. 6: (Color online) Ground state energy of the spin-1=2 PESS, for the spin-1=2 KHAF simulations, with no symmetry KHAF, as a function of 1=D and 1=N , with the yellow line and with SU (2) symmetry, as a function of (a) 1=D and (b) denoting the extrapolation to in nite bond dimension. 1=N . and competing to be the true ground state. This makes ever, one can see again that the limit of in nite bond the simulation of the model very challenging. For the dimension is better achieved by the SU (2)-invariant sim- sake of this study, our goal here is not to provide bet- ulations as a function of 1=N (see Fig. 6(b)), with ex- ter ground-state numbers than those obtained by other trapolated values e (1=D ! 0) = 0:435  0:004 and 0 e simulations [8], but rather to benchmark the utility of e (1=N ! 0) = 0:435  0:002. The iDMRG compari- SU (2) symmetry in 2d TN algorithms, and in partic- son shows the current energetically lowest ground state ular in iPESS. Previous results have shown that using energy [7]. Let us mention that for all the iPESS simula- three-site iPESS without symmetries produces reason- tions that we performed, the extrapolation in the dis- ably good numbers for the ground state energy [5]. For carded weight was not possible because the discarded the SU (2)-symmetric simulations, however, we need to weight was always too small. The spin-spin correlators resort to the six-site unit cell in order to accommodate for each link of the unit cell are shown in Fig. 7, for consistently the SU (2) quantum numbers on all the in- the non-symmetric 3-PESS, the non-symmetric 6-PESS dices of the symmetric TN ansatz. Since the physical and the SU (2)-invariant 6-PESS respectively. While the sites carry spin-1=2, the geometry and the unit cell force 3-site unit cell produces a state that seems compatible us to use mixed spins (integer and half-integer) on the with a quantum spin liquid, the 6-site unit cells seem bond indices of the iPESS. to produce valence-bond crystal structures with strong We computed the ground state energy of the model and weak links, thus breaking invariance under trans- for a 3-site and a 6-site unit cell without symmetry, and lations and lattice rotations. We observe, in any case, for a 6-site unit cell with SU (2) symmetry. The results that the valence bond crystal tends to melt when we in- are shown in Fig. 6. The symmetric results are com- crease the bond dimension of the iPESS ansatz, both patible with those obtained without symmetries, with for the non-symmetric and the SU (2)-invariant simula- an algebraic convergence of the ground state energy as tions, thus slowly recovering translation invariance (see a function of 1=D (see Fig. 6(a)), in turn reinforc- Fig. 8 for plots of the correlator skewness as D in- e e ing the observation that the ground state is a quan- creases). Interestingly, we also observe in the gure that tum spin liquid and therefore SU (2)-invariant. How- the non-symmetric simulations tend to melt faster than -0.2169 -0.2169 -0.2207 -0.2207 -0.2205 -0.2208 -0.5577 -0.0907 -0.2430 +0.0124 5 lower variational energy than the non-symmetric simula- tions. Moreover, both in the non-symmetric and symmet- ric cases we observe in Figs. 9(c,d) a structure of spin- spin correlators in the unit cell that seems compatible with that of a quantum spin liquid, which is also com- patible with the algebraic behaviour of the ground state energy with D in Fig. 9(a). The energies are, however, dicult to extrapolate to in nite bond dimension, and hence we cannot be sure whether this is the true nature of the ground state. But we can claim that, to the best of our calculations, here the SU (2)-invariant iPESS with a 3-site unit cell produces the best variational energy for the ground state, which seems compatible with a quan- tum spin liquid. Moreover, we computed the expectation value of the chiral correlators S  (S  S ) on all tri- i j k angles, and obtained exactly 0 everywhere, in turn also compatible with the structure of a non-chiral quantum spin liquid. FIG. 9: (Color online) Ground state energy of the spin-2 KHAF, as a function of (a) 1=D and (b) 1=N , with the Model No symmetry SU (2) red line denoting the extrapolation to in nite bond dimen- sion. Notice that in this case, this extrapolation is just for s = 1 BLBQ (7; 0:3188) (6; 19:5; 0:3108) completeness since the numbers still do not show convergence s = 1=2 KHAF (10;0:4348) (7; 17:75;0:4349) for the achievable bond dimensions due to the large local spin s = 2 KHAF (10;4:7975) (5; 19;4:8227) at every site. (c,d) Spin-spin correlation hS S i on each link i j of the unit cell for the non-symmetric and the SU (2)-invariant TABLE I: Ground state energies obtained for the maxi- 3-PESS respectively. mum achievable bond dimension for the bilinear-biquadratic (BLBQ) and KHAF models that we considered. We show (D; e ) for non-symmetric simulations and (D ; D ; e ) for 0 sym e 0 the SU (2)-symmetric ones. The observed pattern of local SU (2)-invariant ones, with D the non-symmetric bond di- correlations for the non-symmetric 6-site iPESS in Fig. 7 mension, D the symmetric bond dimension and D the sym e is expected, since TN simulations tend to trade symme- e ective bond dimension when using SU (2) averaged for all try for injectivity of the target state. For the symmetric bonds, which can be integer or fractional. 6-site iPESS the correlators are even more skewed pre- sumably due to non-uniform e ective bond dimensions Conclusions.- In Table I we make a comparison of the caused by the mixture of integer and half-integer repre- computed ground state energies for the maximum achiev- sentations. This e ect is expected to vanish in the limit able bond dimensions, for the three models considered of large bond dimensions, and also explains the slower here, and for non-symmetric and SU (2)-symmetric sim- melting in Fig. 8. ulations. We conclude that implementing SU (2) symme- Finally, we computed the ground state energy of the try in 2d TN algorithms usually produces better energies spin-2 KHAF for a 3-site unit cell without and with than non-symmetric simulations, but the performance SU (2) symmetry. Unlike in the spin-1=2 case, the fact depends on the speci cs of the model and in particular that we have spin-2 in the physical indices allows us to use on the gap of the phase being targeted. For the spin-2 the 3-site unit cell (this, in fact, is true for all integer-spin model, the SU (2) simulations point towards a quantum Heisenberg models on the Kagome lattice). The results spin liquid as a plausible ground state. are shown in Fig. 9. This time, due to the large dimension of the physical spin at every site, we cannot reach values Acknowledgments.- We acknowledge Andreas of D as large as for the spin-1=2 case. However, the ef- Haller, Saeed Jahromi, Matteo Rizzi, and Sukhbinder sym fective bond dimension is larger for the spin-2 KHAF, as Singh for insightful discussions. We also acknowledge shown in Table I. We see in Figs. 9(a,b) that the SU (2)- DFG funding through project GZ OR 381/3-1, as well invariant simulations can handle larger D and produce as the MAINZ Graduate School of Excellence. [1] R. Orus,  Ann. Phys. 349, 117 (2014); R. Orus,  Nat. Rev. [4] J. Jordan, R. Orus,  G. Vidal, F. Verstraete, J. I. Cirac, Phys. 1, 538 (2019). Phys. Rev. Lett. 101, 250602 (2008); J. Jordan, R. Orus, [2] S. R. White, Phys. Rev. Lett.69, 28632866 (1992); S. R. G. Vidal, Phys. Rev. B 79, 174515 (2009). White, Phys. Rev. B 48, 10345 (1992). [5] Z. Y. Xie, J. Chen, J. F. Yu, X. Kong, B. Normand, T. [3] F. Verstraete, J. I. Cirac, cond-mat/0407066. Xiang, Phys. Rev. X 4, 011025 (2014). 6 [6] A. Kshetrimayum, T. Picot, R. Orus,  D. Poilblanc, Phys. Capponi, D. Poilblanc, Phys. Rev. B 98, 184409 (2018). Rev. B 94, 235146 (2016); A. Kshetrimayum, C. Balz, B. [11] P. Schmoll, S. Singh, M. Rizzi, R. Orus,  Annals of Physics Lake, J. Eisert, arXiv:1904.00028; S. S. Jahromi, R. Orus,  419, 168232 (2020) D. Poilblanc, F. Mila, arXiv:1912.10756. [12] P. Schmoll, A. Haller, M. Rizzi, R. Orus,  Phys. Rev. B [7] S. Depenbrock, I. P. McCulloch, U. Schollw ock, Phys. 99, 205121 (2019). Rev. Lett. 109, 067201 (2012). [13] H. C. Jiang, Z. Y. Weng, T. Xiang, Phys. Rev. Lett. 101, [8] H. C. Jiang, Z. Y. Weng, D. N. Sheng, Phys. Rev. Lett. 090603 (2008). 101, 117203 (2008); S. Yan, D. A. Huse, S. White, Sci- [14] M. Lubasch, I. Cirac, M.C. Banuls, ~ Phys. Rev. B 90, ence 332, 1173 (2011); S. Depenbrock, I. P. McCulloch, 064425 (2014); H. N. Phien, J. A. Bengua, H. D. Tuan, U. Schollw ock, Phys. Rev. Lett. 109, 067201 (2012); G. P. Corboz, R. Orus,  Phys. Rev. B 92, 035142 (2015). Evenbly, G. Vidal, Phys. Rev. Lett. 104, 187203 (2010); [15] L. Vanderstraeten, J. Haegeman, P. Corboz, F. Ver- T. Picot, M. Ziegler, R. Orus,  D. Poilblanc, Phys. Rev. straete, Phys. Rev. B 94, 155123 (2016); P. Corboz, B 93, 060407 (2016). Phys. Rev. B 94, 035133 (2016). [9] I. McCulloch, M. Gulacsi, Eur. Phys. Lett. 57, 852 [16] R. J. Baxter, Physica A 106, pp18-27 (1981); R. J. Bax- (2002); S. Singh, H.-Q. Zhou, G. Vidal, New J. Phys. ter, Exactly Solved Models in Statistical Mechanics, Aca- 033029 (2010). demic Press, London, (1982); R. J. Baxter, J. Math. [10] T. Liu, W. Li, A. Weichselbaum, J. von Delft, G. Su, Phys. 9, 650 (1968); R. J. Baxter, J. Stat. Phys. 19, 461 Phys. Rev. B 91, 060403 (2015); W. Li, A. Weichsel- (1978); T. Nishino, K. Okunishi, J. Phys. Soc. Jpn. 65, baum, J. von Delft, H.-H. Tu, Phys. Rev. B 91, 224414 pp. 891-894 (1996); T. Nishino, K. Okunishi, J. Phys. (2015); J.-W. Mei, J.-Y. Chen, H. He, X.-G. Wen, Phys. Soc. Jp. 66, 3040 (1997); R. Orus,  G. Vidal, Phys. Rev. Rev. B 95, 235107 (2017); S.-J. Ran, W. Li, S.-S. Gong, B 80, 094403 (2009); R. Orus,  Phys. Rev. B 85, 205117 A. Weichselbaum, J. von Delft, G. Su, Phys. Rev. B (2012); L. Vanderstraeten, M. Mari'en, F. Verstraete, J. 97, 075146 (2018); M. Mambrini, R. Orus,  D. Poilblanc, Haegeman, Phys. Rev. B 92, 201111 (2015). Phys. Rev. B 94, 205124 (2016); D. Poilblanc, M. Mam- [17] I. Niesen, P. Corboz, SciPost Phys. 3, 030 (2017). brini, Phys. Rev. B 96, 014414 (2017); C. Hubig, SciPost [18] P. Corboz, Phys. Rev. B 93, 045116 (2016). Phys. 5, 047 (2018) ; J.-Y. Chen, L. Vanderstraeten, S.

Journal

Condensed Matter – arXiv (Cornell University)

Published: May 6, 2020

There are no references for this article.