Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices

We develop a new methodology to contract tensor networks within the corner transfer matrix renormalization group approach for a wide range of two-dimensional lattice geometries. We discuss contraction algorithms on the example of triangular, kagome, honeycomb, square-octagon, star, ruby, square-hexagon-dodecahedron, and dice lattices. As benchmark tests, we apply the developed method to the classical Ising model on different lattices and observe a remarkable agreement of the results with the available from the literature. The approach also shows the necessary potential to be applied to various quantum lattice models in a combination with the wave-function variational optimization schemes.

Figure 1

Overview of different lattices for which we construct (generalize), discuss, and apply the CTMRG algorithms: (a) triangular, (b) kagome, (c) honeycomb, (d) square-octagon, (e) star, (f) square-hexagon-dodecahedron, (g) ruby, and (h) dice.

Figure 2

Triangular lattice: (a) Construction of the transfer-matrix network and notation of the bulk rank-6 tensor $A$ (we omit its indices for simplicity); (b) definition of the boundary MPS (bMPS); (c) iterative update of the individual tensors holding within bMPS.

Figure 3

Triangular lattice: (a) Definition of the corner matrix ${\tilde{C}}_3$ on the intersection of two boundary MPS and its update; (b) update procedure of the corner matrix ${\tilde{C}}_3$, which results in the new corner matrix $C_3$; (c) the second update step, which transforms $C_3$ back into ${\tilde{C}}_3$; (d) the projectors $P$ can be chosen from the truncated eigendecomposition of the matrix $C_3$. The matrix $C_3$ can be assumed to be always diagonal.

Figure 4

Triangular lattice: (a) Definition of the tensor $R$, which is factorized into two tensors $O$, and the possible ambiguity (gauge choice) in the factorization due to the arbitrary orthogonal matrix $W$, which can be inserted in the factorized index. (b) Ambiguity in the factorization leads to the transformation of the corner matrix ${\tilde{C}}_3$. We can fix the gauge by the condition, that the corner matrix ${\tilde{C}}_3$ is also diagonal. (c) The definition of the new tensor $U$, which can be shown (from the point (b)) to be isometric. (d) We can define the new tensor $N$ as the tensor $R$ weighted by the corner matrices. The eigendecomposition of $N$ results in both isometry $U$ and the diagonal corner matrix ${\tilde{C}}_3$. (e) The original bMPS tensor $O$ can be found from the isometry $U$ by the inverse transformation. This ends the update step for the tensor $O$ and simultaneously finds the updated corner matrix ${\tilde{C}}_3$.

Figure 5

Triangular lattice: (a) Definition of the corner tensor $T$, as the intersection of two bMPS with $2\pi /3$ angle; (b) update rule for the tensor $T$; (c) environment of the bulk tensor $A$ in terms of the corner tensor $T$.

Figure 6

Kagome lattice: (a) Definition of the boundary MPS on the kagome lattice and the structure of the transfer matrix. (b) Local conditions on the tensors of the bMPS, which guarantee approximate holding of the above eigenvector condition. To fulfill this, we introduce the (isometrical) projectors $P_3$ and $P_6$ and determine them later from the corner matrices update rules. (c) The bulk tensor $A_{ij}^{kl}$ with the dashed lines indicating to the reflection symmetry axes.

Figure 7

Kagome lattice: (a) Definition of the corner matrix $C_6$ as an intersection of two bMPS with the angle $2\pi /3$ between them. (b) The update rule for the matrix $C_6$; the isometrical projector $P_6$ can be chosen in the way that the updated matrix $C_6$ will be diagonal. (c) The definition of the corner matrix $C_3$ as an intersection point between two bMPS with the angle $\pi /3$; (d) the matrix $C_3$ can be expressed as the square of the matrix $C_6$; (e) still, the matrix $C_3$ has a different update rule: here, we define the new corner matrix $C_3^{\prime }$ (in green) from the matrix $C_3$; (f) yet another corner matrix $C_3^{″}$ (gray) can be obtained from $C_3^{\prime }$. The diagonalization of this matrix leads to the projector $P_3$. (g) One can arrive back to the $C_3$ matrix from $C_3^{″}$. (h) The illustration of the equality in the point (d) between the matrices $C_3$ and $C_6^2$. It is always possible to change the crossing of two bMPS with the angle $\pi /3$ into two crossings with an additional bMPS, where the new crossings are at the angle $2\pi /3$.

Figure 8

Kagome lattice: (a) Definition of the new rank-4 tensor $R$ to be factorized and the update rule of the matrix $C_3^{\prime }$. (b) The tensor $N$ as a weighted tensor $R$ and its eigendecomposition (eig.) from which we define the new matrix $C_3^{\prime }$ and the isometry $U$. (c) The reverse transformation between the bMPS local tensor and the isometry $U$.

Figure 9

Honeycomb lattice: (a) Definition of the transverse matrix and boundary MPS. (b) The update procedure for the bMPS tensors. (c) The definition of the tensor $C_6$. (d) The update procedure for the tensor $C_6$. (e) Definition and update for the corner matrix $C_3$. (g) Consistency condition for the corner matrices $C_3$ and $C_6$.

Figure 10

Honeycomb lattice: (a) Definition of the two-site bMPS and transfer matrix for the case of two-site unit cell. (b) The update procedure for the bMPS tensors, where we insert the projectors $P_L$ and $P_R$ ($P_L{P}_R=1$). (c) The updates of the corner matrices $C_{6,A}$ and $C_{6,B}$. (d) Biorthogonalization procedure (involving QR decomposition and SVD), which allows to define the projectors $P_L$ and $P_R$ from the corner matrices.

Figure 11

Square-octagon lattice: (a) Definition of the first transfer matrix and the respective bMPS. (b) The local updates of bMPS tensors for the first type of transfer matrix. (c) The second type of transfer matrix on the square-octagon lattice and the corresponding bMPS. (d) The local update rules for the bMPS local tensors of the second type.

Figure 12

Square-octagon lattice: (a) Intersection point of two bMPS of the first kind defines the corner tensor $T$ (symmetric upon reflections). (b) The update rule for the corner tensor $T$ and the corner matrix $C_4$. The isometry $W$ can be chosen to diagonalize the corner matrix $C_4$. (c) The corner matrix $C_8$ must be inserted at the intersection point of two different types of bMPS. (d) The update step of the corner matrix $C_8$ enlarges its dimensions, which can be then truncated back with the SVD decomposition, which also defines the projectors $U$ and $V$. (e) The factorization step of the update for the second bMPS. The factorization is performed in the same way as for the kagome lattice.

Figure 13

Star lattice: (a) Definition of the first transfer matrix on the star lattice and the respective bMPS. (b) The local updates of bMPS tensors of the first type of the transfer matrix.

Figure 14

Star lattice: (a) Definition of the corner matrix $C_6$ as an intersection point between two bMPS with the angle $\pi /3$. The density matrix on the dodecahedrons can be cast in the form ${\rho }_d=C_6^6$. (b) The update step of the corner matrix $C_6$. (c) The definition of the corner matrix $C_3$ as an intersection point between two bMPS with the angle $2\pi /3$. (d) The consistency condition between the corner matrices $C_6$ and $C_3$, similar to the kagome lattice. (e) The factorization step in the bMPS tensors update determining the matrix $C_3^{\prime }$, which describes the internal correlations inside the triangles. (f) The update step for the matrix $C_3^{\prime }$. (g) The final step of the update of $C_3^{\prime }$, which results in the corner matrix $C_3^{″}$. The isometrical projector $W$ is chosen to diagonalize the new corner matrix.

Figure 15

Star lattice: (a) Definition of the second bMPS. (b) Update rule for the boundary tensor, which requires the new isometric projector $K$. (c) The projector can be determined from the new corner matrix $C_{12}$, which is a square root of the previously introduced corner matrix $C_6$. Its update rule can be viewed as a modification of the rule in Fig. 13, but results here in two different isometries by using SVD of the enlarged matrix $C_{12}$.

Figure 16

SHD lattice: (a) Definition of the first transfer matrix on the 4-6-12 lattice and the respective bMPS. (b) The local updates of bMPS tensors for the first type of the transfer matrix.

Figure 17

SHD lattice: (a) Definition of the corner matrix $C_6$ as an intersection point between two bMPS with the angle $\pi /3$. (b) We use a representation of the corner matrix $C_6$ as a square of the corner matrix $C_{12}$, which can be defined as an intersection of two different types of bMPS. Here, we use $C_{12}$ as a conventional tool in the calculations. We also define the corner matrix $C_3$ as a square of the corner matrix $C_6$. (c) The illustration of the role of the corner matrix $C_3$ as an intersection of two bMPS. (d) We do not work with the matrix $C_3$ explicitly, but use its implicit representation in terms of the matrices $C_6$. This allows us to find the first isometric projector $U$ and the updated corner matrix $C_6^{\prime }$ through SVD of the enlarged matrix $C_6$. Note that the isometry $K$ in SVD is auxiliary in this version of the algorithm; we do not employ it hereafter, as all other isometries $K$. (e) The illustration of the nonsymmetric factorization step. First, we can express full lattice in terms of corner matrices and bMPS tensors. Then, we can use exact factorization of the rank-4 tensor, where we do not perform any truncation. The enlarged index is shown with two lines. We express this density matrix, which represents the full lattice as a product of two matrices $C_L$ and $C_R$. To obtain the projectors, we apply the biorthogonalization procedure to the tensors $C_L$ and $C_R$. The obtained projectors are then applied to truncate the enlarged factorized indices. We can also update the matrices $C_{12}$ and $C_6^{\prime }$. The update is carried out with QR decompositions. Note that the new matrices $C_6^{″}$ and $C_{12}^{\prime }$ are not symmetric or diagonal. (f) Another step of the matrix $C_6^{″}$ update, which results in the second isometry $V$ and another corner matrix $C_3^{″\prime }$. (g) Analogous update step for the matrix $C_{12}^{\prime }$ gives back the matrix $C_{12}$ and also the isometrical projector $M$. (h) To obtain the final projector $W$, we construct the corner matrix $C_4$ of the squares in the 4-6-12 lattice from the bMPS tensors and the third power of the matrix $C_{12}$. The SVD of this enlarged corner allows us to obtain the last projector $W$ and also the matrix $C_4$, which corresponds to correlations internal to the squares.

Figure 18

Ruby lattice: (a) Definition of the first transfer matrix and the respective bMPS. (b) The local updates of bMPS tensors for the first type of transfer matrix. Note the similarity of the updates and conventions with Fig. 16.

Figure 19

Ruby lattice: (a) Definition of the corner matrix $C_6$ inside the hexagon as an intersection point between two bMPS with the angle $\pi /3$. (b) The definition of the corner matrix $C_3$ inside the hexagon as an intersection point between two bMPS with the angle $2\pi /3$. (c) The consistency condition between the corner matrices $C_3$ and $C_6$. (d) The first update step of the corner matrix $C_3$ allows us to find the first isometric projector $U$ and the updated corner matrix $C_3^{\prime }$. (e) The illustration of the nonsymmetric factorization step. First, we can express the lattice in terms of the corner matrices and bMPS tensors. Then, we employ the exact factorization of the rank-4 tensor, where we do not perform any truncation. The enlarged index is shown with two lines. Next, we write this density matrix, which represents the lattice as a product of two matrices $C_L$ and $C_R$. To obtain the projectors we apply the biorthogonalization procedure to the tensors $C_L$ and $C_R$. The obtained projectors are then applied to truncate the enlarged factorized indices. We can also update the matrices $C_6$ and $C_3^{\prime }$. Note that the new matrices $C_3^{″}$ and $C_6^{\prime }$ are not diagonal. (f) The first factorization step is performed in a similar way to factorizations on different lattices. The main difference is that the matrix $C_3^{″}$ is no longer diagonal. To proceed further, we factorize this matrix with the Cholesky decomposition. This decomposition works only for the positive matrices, which is a major limitation for this type of algorithm. After the decomposition, we can perform a factorization step. (g) Another factorization step, which is performed analogously to the previous point. The only difference is that we obtain a new $C_6$ matrix as a byproduct of factorization. (h) The final projector can be found from the SVD of the effective corner matrix $C_4$. This decomposition requires $C_6^{3/2}$, thus sets an additional positivity condition.

Figure 20

Dice lattice: (a) Definition of the transfer matrix on the dice lattice and the respective bMPS. (b) The local updates of bMPS tensors.

Figure 21

Dice lattice: (a) Definition of the corner matrix $C_3$. (b) Definition of the corner matrix $C_6$. (c) The growth step for the corner matrix $C_3$. (d) The update steps for the corner matrix $C_6$, which result in the new matrix $C_6$ with a larger dimension. (e) The contracted tensor network in terms of the larger matrices $C_3$ and $C_6$, which can be written as a product of the matrices $C_L$ and $C_R$. We apply the biorthogonalization to $C_L$ and $C_R$ to obtain the projectors $P_L$ and $P_R$. (f) The symmetric factorization step. Initially, we perform a factorization without truncating the factorized index (indicated with two lines). We can write a contracted tensor network in terms of factorized tensors and corner matrices. The index can be truncated using the isometry obtained from the eigendecomposition of the density matrix (or its half denoted as $K$).

Figure 22

(a) The general construction of the tensor network on the lattice (here, triangular) by placing tensors $\delta$ on the sites and matrices $W$ on the bonds. (b) For the ferromagnetic models we can find the unique positive symmetric square root of the bond matrix $W$. These square roots can be absorbed into the tensors $\delta$, forming the tensor network of identical rotationally and reflection-symmetric tensors $A$. (c) For the bipartite antiferromagnetic model in the external field, the partition function can be determined by placing identical tensors $x$ on the lattice sites and bond matrices $W$ on the links. (d) For the antiferromagnetic model we do not have a unique symmetric decomposition of the matrix $W$, but we can choose an arbitrary one (e.g., based on SVD) and then absorb the resulting matrices $L$ and $R$ into different site tensors, forming the tensor network with two-site unit cell. (e) Example of the transformation mapping the ruby lattice into the SHD one. Note that here we assume the symmetric factorization of the tensor $A$ on the ruby lattice. If this factorization is not available, then one can employ a nonsymmetric factorization, but the mapping results in the two-site unit cell on the SHD lattice. (g) The mapping from the SHD to kagome lattice.

Figure 23

Triangular lattice: The corner matrix spectrum in the ordered phase of the Ising model, normalized by the gap $\mathrm{\Delta }$ between the first and second eigenvalues. The bond dimension of the corner matrix $\chi =120$ and the inverse temperature $\beta =0.28$.

Figure 24

Triangular lattice: Dependence of the magnetization $\sigma$ and entanglement gap $\mathrm{\Delta }$ on the inverse temperature $\beta$ at $\chi =200$.

Figure 25

Kagome lattice: The corner matrix spectrum in the disordered phase of the Ising model, normalized by the gap $\mathrm{\Delta }$ between the first and second eigenvalues. The bond dimension $\chi =90$ and the inverse temperature $\beta =0.45$, which is close to the transition point.

Figure 26

Kagome lattice: Dependence of the magnetization $\sigma$ and entanglement gap $\mathrm{\Delta }$ on the inverse temperature $\beta$ at $\chi =200$. The exact critical value ${\beta }_c=0.46657$, while our estimate is ${\beta }_c=0.4666\left(1\right)$.

Figure 27

Kagome lattice: The logarithm of the Frobenious norm of the difference between two matrices $C_3$ and $C_6^2$ (preliminary normalized). The parameters are $\chi =90$ and $\beta =0.45$.

Figure 28

Square-octagon lattice: Convergence of the norm of the difference between the tensors $T$ and $C_{8,i}^2{O}_{ijk}$ (preliminarily normalized). The parameters are $\chi =100$ and $\beta =0.69$. The norm of the difference converges to the value around ${10}^{-8}$.

Figure 29

Square-octagon lattice: The corner matrix spectrum in the disordered phase of the Ising model, normalized by the gap $\mathrm{\Delta }$ between the first and second eigenvalues. The parameters are $\chi =100$ and $\beta =0.69$.

Figure 30

Square-octagon lattice: Dependence of the magnetization $\sigma$ and entanglement gap $\mathrm{\Delta }$ on the inverse temperature $\beta$ at $\chi =200$. The estimated critical value ${\beta }_c=0.6950\left(1\right)$.

Figure 31

Star lattice: Convergence of the norm of the difference between two definitions of the matrix $C_3$ (preliminarily normalized). The parameters are $\chi =120$ and $\beta =0.78$.

Figure 32

Star lattice: The corner matrix spectrum in the disordered phase of the Ising model, normalized by the gap $\mathrm{\Delta }$ between the first and second eigenvalues. The parameters are $\chi =120$ and $\beta =0.78$.

Figure 33

Star lattice: Dependence of the magnetization $\sigma$ and entanglement gap $\mathrm{\Delta }$ on the inverse temperature $\beta$ at $\chi =200$. The estimated critical value ${\beta }_c=0.8120\left(1\right)$.

Figure 34

Ruby lattice: Convergence of the maximal difference between the ten largest eigenvalues of the corner matrices $C_6^{\prime }$ and $C_6^{″\prime }$ (preliminary normalized). The parameters are $\chi =40$ and $\beta =0.46$. The fluctuations on the plot correspond to the consecutive increases of $\chi$.

Figure 35

Ruby lattice: The corner matrix spectrum in the disordered phase of the Ising model, normalized by the gap $\mathrm{\Delta }$ between the first and second eigenvalues. The parameters are $\chi =90$ and $\beta =0.466$, which is close to the critical point.

Figure 36

Ruby lattice: Dependence of the magnetization $\sigma$ and entanglement gap $\mathrm{\Delta }$ on the inverse temperature $\beta$ at ${\chi }_{\text{max}}=90$. The estimated critical value is ${\beta }_c=0.4666\left(1\right)$.

Figure 37

SHD lattice: Convergence of the maximal difference between the ten largest eigenvalues of the corner matrices $C_6^{\prime }$ and $C_6^{″\prime }$ (preliminary normalized). The parameters are $\chi =40$ and $\beta =0.71$. The fluctuations on the plot correspond to the consecutive increases of $\chi$.

Figure 38

SHD lattice: The corner matrix spectrum in the disordered phase of the Ising model, normalized by the gap $\mathrm{\Delta }$ between the first and second eigenvalues. The parameters are $\chi =105$ and $\beta =0.719$.

Figure 39

SHD lattice: Dependence of the magnetization $\sigma$ and entanglement gap $\mathrm{\Delta }$ on the inverse temperature $\beta$ at ${\chi }_{\text{max}}=90$. The exact value of the inverse critical temperature ${\beta }_c^{\left(\mathrm{ex}\right)}\approx 0.71951019$, while according to our estimates ${\beta }_c=0.7195\left(1\right)$.

Figure 40

Dice lattice: Dependence of the local magnetizations ${\sigma }_{A,B}$ on two types of lattice sites $A,B$ and entanglement gap $\mathrm{\Delta }$ on the inverse temperature $\beta$ at ${\chi }_{\text{max}}=200$. The estimated critical value of the inverse temperature is ${\beta }_c=0.4157\left(1\right)$.

Figure 41

Dice lattice: The $T^{\prime }$-tensor spectrum in the ordered phase of the Ising model, normalized by the gap $\mathrm{\Delta }$ between the first and second eigenvalues. The parameters are $\chi =45$ and $\beta =0.417$.

Figure 42

Dice lattice: The $T^{\prime }$-tensor spectrum in the disordered phase of the Ising model, normalized by the gap $\mathrm{\Delta }$ between the first and second eigenvalues. The parameters are $\chi =75$ and $\beta =0.415$.

Figure 43

Honeycomb lattice: Dependence of the local magnetizations ${\sigma }_A$ and ${\sigma }_B$ on the inverse temperature $\beta$ at $x=0.5$ and ${\chi }_{\text{max}}=120$.

Figure 44

Honeycomb lattice: Convergence of the maximal difference between $C_3=\left(R_A{R}_B\right)\left(C_{6,A}{C}_{6,B}\right)$ and $C_{6,A}{C}_{6,B}$. Both products are preliminary normalized by their respective maximal elements. The parameters are $\chi =56$ and $a=0.255$.

Figure 45

Honeycomb lattice: Convergence of the maximal absolute value of the antisymmetrization of the tensor $R_B{C}_{6,B}$ (both $R_B$ and $C_{6,B}$ are normalized by their respective norms). The parameters are $\chi =56$ and $a=0.255$.

Figure 46

Triangular lattice: (a) Transfer matrix for the nonpositive bulk tensor and the corresponding bMPS, which consists of two different boundary tensors. (b) The update rule for the boundary tensors, which mimics the steps shown in Fig. 2. The difference is that the factorization is nonsymmetric and projectors are not isometric. (c) The modified update step for the corner matrix $C_3$ (which is also no longer symmetric). The projectors $P_L$ and $P_R$ can be obtained, e.g., from the biorthogonalization of the environments with enlarged $C_3$. (d) The nonsymmetric factorization step. First, as in the positive case, we define the tensor $R$. Then, we perform an arbitrary factorization of this tensor into two rank-3 tensors $L$ and $R$. We do not perform truncation in this step, which is illustrated with the double-line links for the enlarged not-truncated factorization index. The tensors $L$ and $R$ are then absorbed into the corner matrix. The new corner matrix has an enlarged bond dimension, which is truncated back with the projectors $P_L^{\prime }$ and $P_R^{\prime }$, which we also obtain from biorthogonalization. The same projectors can be also used to truncate the factorization index in $L$ and $R$ back to the bMPS bond dimension.

Figure 47

Dependence of the magnetization $\sigma$ and entanglement gap $\mathrm{\Delta }$ on the inverse temperature $\beta$ for the $q$-Potts model on triangular lattice. The parameters and exact values of the inverse critical temperature are: (a) isotropic case with $J_1=J_2=J_3=1,q=4, \chi =100$, and ${\beta }_c\approx 0.6931$; (b) isotropic case with $J_1=J_2=J_3=1,q=6, \chi =100$, and ${\beta }_c\approx 0.7866$ (note that the transition is of the first order for $q>4$); (c) anisotropic case with $q=4, \chi =50, J_1=1, J_2=2, J_3=3$, and ${\beta }_c\approx 0.3535$.