Higher-order Tensor Renormalization Group Research Articles

Spectroscopy with the tensor renormalization group method

We present a spectroscopy scheme for the lattice field theory by using the tensor renormalization group method combining with the transfer matrix formalism. By using the scheme, we cannot only compute the energy spectrum for the lattice theory but also determine quantum numbers of the energy eigenstates. Furthermore, the wave function of the corresponding eigenstate can also be computed. The first step of the scheme is to coarse grain the tensor network of a given lattice model by using the higher order tensor renormalization group, and then after making a matrix corresponding to a transfer matrix from the coarse-grained tensors, its eigenvalues are evaluated to extract the energy spectrum. Second, the quantum number of the eigenstates can be identified by a selection rule that requires to compute matrix elements of an associated insertion operator. The matrix elements can be represented by an impurity tensor network and computed by the coarse-graining scheme. Moreover, we can compute the wave function of the energy eigenstate by putting the impurity tensor at each point in space direction of the network. Additionally, the momentum of the eigenstate can also be identified by computing appropriate matrix elements represented by the tensor network. As a demonstration of the new scheme, we show the spectroscopy of the (1+1)d Ising model and compare it with exact results. We also present a scattering phase shift obtained from two-particle state energy using Lüscher’s formula. Published by the American Physical Society 2024

Phase diagram of generalized XY model using the tensor renormalization group

We use the higher-order tensor renormalization group method to study the two-dimensional generalized XY model that admits integer and half-integer vortices. This model is the deformation of the classical XY model and has a rich phase structure consisting of nematic, ferromagnetic, and disordered phases and three transition lines belonging to the Berezinskii-Kosterlitz-Thouless and Ising class. We explore the model for a wide range of temperatures, T, and the deformation parameter, Δ, and compute specific heat along with integer and half-integer magnetic susceptibility, finding both Berezinskii-Kosterlitz-Thouless-like and Ising-like transitions and the region where they meet. Published by the American Physical Society 2024

Tensor network methods for extracting conformal field theory data from fixed-point tensors and defect coarse graining.

We present a comprehensive study on extracting conformal field theory data using tensor network methods, especially, from the fixed-point tensor of the linearized tensor renormalization group (lTRG) for the classical two-dimensional Ising model near the critical temperature. Utilizing two different methods, we extract operator scaling dimensions and operator product expansion coefficients by introducing defects on the lattice and by employing the fixed-point tensor. We also explore the effects of pointlike defects in the lattice on the coarse-graining process. We find that there is a correspondence between coarse-grained defect tensors and conformal states obtained from the lTRG fixed-point equation. We also analyze the capabilities and limitations of our proposed coarse-graining scheme for tensor networks with pointlike defects, including graph-independent local truncation (GILT) and higher-order tensor renormalization group (HOTRG). Our results provide a better understanding of the capacity and limitations of the tensor renormalization group scheme in coarse-graining defect tensors, and we show that GILT+HOTRG can be used to give accurate two- and four-point functions under specific conditions. We also find that employing the minimal canonical form further improves the stability of the RG flow.

Higher-order tensor renormalization group study of the J_{1}-J_{2} Ising model on a square lattice.

Phase transitions of the J_{1}-J_{2} Ising model on a square lattice are studied using the higher-order tensor renormalization group (HOTRG) method. This system involves a competition between the ferromagnetic interaction J_{1} and antiferromagnetic interaction J_{2}, and in previous studies, weak first-order and second-order transitions were observed near the ratio g=J_{2}/|J_{1}|=1/2. It has also been suggested that the universality class of the second-order phase transition connected to the first-order transition line for g>1/2 belongs to the Ashkin-Teller class, which is characterized by a continuously varying critical exponent with g, as predicted by field-theoretical and other studies. Our results, based on the HOTRG calculations for significantly larger sizes, indicate that the region of the first-order transition is marginally narrower than that in previous studies. Furthermore, it is suggested that the region where the critical exponent changes does not necessarily coincide with the Ashkin-Teller region.

Tensor network based finite-size scaling for two-dimensional Ising model

We propose a scheme to perform tensor network based finite-size scaling analysis for two-dimensional classical models. In the tensor network representation of the partition function, we use higher-order tensor renormalization group (HOTRG) method to coarse grain the weight tensor. The renormalized tensor is then used to construct the approximated transfer matrix of an infinite strip of finite width. By diagonalizing the transfer matrix we obtain the correlation length, the magnetization, and the energy density which are used in finite-size scaling analysis to determine the critical temperature and the critical exponents. As a benchmark we study the two-dimensional classical Ising model. We show that the critical temperature and the critical exponents can be accurately determined. With HOTRG bond dimension $D=70$, the absolute errors of the critical temperature $T_c$ and the critical exponent $\nu$, $\beta$ are at the order of $10^{-7}, 10^{-5}$, $10^{-4}$ respectively. Furthermore, the results can be systematically improved by increasing the bond dimension of the HOTRG method. Finally, we study the length scale induced by the finite cut-off in bond dimension and elucidate its physical meaning in this context.

Local and global magnetization on the Sierpiński carpet.

The phase transition of the classical Ising model on the Sierpiński carpet, which has the fractal dimension log_{3}^{}8≈1.8927, is studied by an adapted variant of the higher-order tensor renormalization group method. The second-order phase transition is observed at the critical temperature T_{c}^{}≈1.478. Position dependence of local functions is studied through impurity tensors inserted at different locations on the fractal lattice. The critical exponent β associated with the local magnetization varies by two orders of magnitude, depending on lattice locations, whereas T_{c}^{} is not affected. Furthermore, we employ automatic differentiation to accurately and efficiently compute the average spontaneous magnetization per site as a first derivative of free energy with respect to the external field, yielding the global critical exponent of β≈0.135.

Calculation of critical exponents on fractal lattice Ising model by higher-order tensor renormalization group method.

The critical behavior of the Ising model on a fractal lattice, which has the Hausdorff dimension log_{4}12≈1.792, is investigated using a modified higher-order tensor renormalization group algorithm supplemented with automatic differentiation to compute relevant derivatives efficiently and accurately. The complete set of critical exponents characteristic of a second-order phase transition was obtained. Correlations near the critical temperature were analyzed through two impurity tensors inserted into the system, which allowed us to obtain the correlation lengths and calculate the critical exponent ν. The critical exponent α was found to be negative, consistent with the observation that the specific heat does not diverge at the critical temperature. The extracted exponents satisfy the known relations given by various scaling assumptions within reasonable accuracy. Perhaps most interestingly, the hyperscaling relation, which contains the spatial dimension, is satisfied very well, assuming the Hausdorff dimension takes the place of the spatial dimension. Moreover, using automatic differentiation, we have extracted four critical exponents (α, β, γ,andδ) globally by differentiating the free energy. Surprisingly, the global exponents differ from those obtained locally by the technique of the impurity tensors; however, the scaling relations remain satisfied even in the case of the global exponents.

Improved local truncation schemes for the higher-order tensor renormalization group method

The higher-order tensor renormalization group is a tensor-network method providing estimates for the partition function and thermodynamical observables of classical and quantum systems in thermal equilibrium. At every step of the iterative blocking procedure, the coarse-grid tensor is truncated to keep the tensor dimension under control. For a consistent tensor blocking procedure, it is crucial that the forward and backward tensor modes are projected on the same lower-dimensional subspaces. In this paper we present two methods, the SuperQ and the iterative SuperQ method, to construct tensor truncations that reduce or even minimize the local approximation errors, while satisfying this constraint.

Grassmann higher-order tensor renormalization group approach for two-dimensional strong-coupling QCD

We present a tensor-network approach for two-dimensional strong-coupling QCD with staggered quarks at nonzero chemical potential. After integrating out the gauge fields at infinite coupling, the partition function can be written as a full contraction of a tensor network consisting of coupled local numeric and Grassmann tensors. To evaluate the partition function and to compute observables, we develop a Grassmann higher-order tensor renormalization group method, specifically tailored for this model. During the coarsening procedure, the blocking of adjacent Grassmann tensors is performed analytically, and the total number of Grassmann variables in the tensor network is reduced by a factor of two at each coarsening step. The coarse-site numeric tensors are truncated using higher-order singular value decompositions. The method is validated by comparing the partition function, the chiral condensate and the baryon density computed with the tensor method with exact analytical results on small lattices up to volumes of 4×4. For larger volumes, we present first tensor results for the chiral condensate as a function of the mass and volume, and observe that the chiral symmetry is not broken dynamically in two dimensions. We also present tensor results for the number density as a function of the chemical potential, which hint at a first-order phase transition.

Tensor Network Calculation of the Logarithmic Correction Exponent in the XY Model

We study the logarithmic correction to the scaling of the first Lee-Yang (LY) zero in the classical $XY$ model on square lattices by using tensor renormalization group methods. In comparing the higher-order tensor renormalization group (HOTRG) and the loop-optimized tensor network renormalization (LoopTNR), we find that the entanglement filtering in LoopTNR is crucial to gaining high accuracy for the characterization of the logarithmic correction, while HOTRG still proposes approximate bounds for the zero location associated with two different bond-merging algorithms of the higher-order singular value decomposition and the oblique projectors. Using the LoopTNR data computed up to the system size of $L=1024$ in the $L \times L$ lattices, we estimate the logarithmic correction exponent $r = -0.0643(9)$ from the extrapolation of the finite-size effective exponent, which is comparable to the renormalization group prediction of $r = -1/16$.

A parallel computing method for the higher order tensor renormalization group

In this paper, we propose a parallel computing method for the Higher Order Tensor Renormalization Group (HOTRG) applied to a d-dimensional (d≥2) simple lattice model. Sequential computation of the HOTRG requires O(χ4d−1) computational cost, where χ is bond dimension, in a step to contract indices of tensors. When we simply distribute elements of a local tensor to each process in parallel computing of the HOTRG, frequent communication between processes occurs. The simplest way to avoid such communication is to hold all the tensor elements in each process, however, it requires O(χ2d) memory space. In the presented method, placement of a local tensor element to more than one process is accepted and sufficient local tensor elements are distributed to each process to avoid communication between processes during considering computation step. For the bottleneck part of computational cost, such distribution is achieved by distributing elements of two local tensors to χ2 processes according to one of the indices of each local tensor which are not contracted during considering computation. In the case of d≥3, computational cost in each process is reduced to O(χ4d−3) and memory space requirement in each process is kept to be O(χ2d−1).

Residual Entropy of the Cubic Dimer Model: The Higher Order Tensor Renormalization Group Approach

Residual Entropy of the Cubic Dimer Model: The Higher Order Tensor Renormalization Group Approach

Ising ferromagnets and antiferromagnets in an imaginary magnetic field.

We study classical Ising spin-1/2 models on a two-dimensional (2D) square lattice with ferromagnetic or antiferromagnetic nearest-neighbor interactions, under the effect of a pure imaginary magnetic field. The complex Boltzmann weights of spin configurations cannot be interpreted as a probability distribution, which prevents application of standard statistical algorithms. In this work, the mapping of the Ising spin models under consideration onto symmetric vertex models leads to real (positive or negative) Boltzmann weights. This enables us to apply accurate numerical methods based on the renormalization of the density matrix, namely, the corner transfer matrix renormalization group and the higher-order tensor renormalization group. For the 2D antiferromagnet, varying the imaginary magnetic field, we calculate with high accuracy the curve of critical points related to the symmetry breaking of magnetizations on the interwoven sublattices. The critical exponent β and the anomaly number c are shown to be constant along the critical line, equal to their values β=1/8 and c=1/2 for the 2D Ising model in a zero magnetic field. The 2D ferromagnets behave in analogy with their 1D counterparts defined on a chain of sites, namely, there exists a transient temperature which splits the temperature range into its high-temperature and low-temperature parts. The free energy and the magnetization are well defined in the high-temperature region. In the low-temperature region, the free energy exhibits singularities at the Yang-Lee zeros of the partition function and the magnetization is also ill-defined: It varies chaotically with the size of the system. The transient temperature is determined as a function of the imaginary magnetic field by using the fact that from the high-temperature side both the first derivative of the free energy with respect to the temperature and the magnetization diverge at this temperature.

J_{1}-J_{2} fractal studied by multirecursion tensor-network method.

We generalize a tensor-network algorithm to study the thermodynamic properties of self-similar spin lattices constructed on a square-lattice frame with two types of couplings, J_{1}^{} and J_{2}^{}, chosen to transform a regular square lattice (J_{1}^{}=J_{2}^{}) onto a fractal lattice if decreasing J_{2}^{} to zero (the fractal fully reconstructs when J_{2}^{}=0). We modified the higher-order tensor renormalization group (HOTRG) algorithm for this purpose. Single-site measurements are performed by means of so-called impurity tensors. So far, only a single local tensor and uniform extension-contraction relations have been considered in HOTRG. We introduce 10 independent local tensors, each being extended and contracted by 15 different recursion relations. We applied the Ising model to the J_{1}^{}-J_{2}^{} planar fractal whose Hausdorff dimension at J_{2}^{}=0 is d^{(H)}=ln12/ln4≈1.792. The generalized tensor-network algorithm is applicable to a wide range of fractal patterns and is suitable for models without translational invariance.

Bond-weighted tensor renormalization group

We propose an improved tensor renormalization-group (TRG) algorithm, the bond-weighted TRG (BTRG). In BTRG, we generalize the conventional TRG by introducing bond weights on the edges of the tensor network. We show that BTRG outperforms the conventional TRG and the higher-order tensor renormalization group with the same bond dimension, whereas its computation time is almost the same as that of TRG. Furthermore, BTRG can have nontrivial fixed-point tensors at an optimal hyperparameter. We demonstrate that the singular value spectrum obtained by BTRG is invariant under the renormalization procedure in the case of the two-dimensional Ising model at the critical point. This property indicates that BTRG performs the tensor contraction with high accuracy whereas keeping the scale-invariant structure of tensors.

Global optimization of tensor renormalization group using the corner transfer matrix

A tensor network renormalization algorithm with global optimization based on the corner transfer matrix is proposed. Since the environment is updated by the corner transfer matrix renormalization group method, the forward-backward iteration is unnecessary, which is a time-consuming part of other methods with global optimization. In addition, a further approximation reducing the order of the computational cost of contraction for the calculation of the coarse-grained tensor is proposed. The computational time of our algorithm in two dimensions scales as the sixth power of the bond dimension while the higher-order tensor renormalization group and the higher-order second renormalization group methods have the seventh power. We perform benchmark calculations in the Ising model on the square lattice and show that the time-to-solution of the proposed algorithm is faster than that of other methods.

Higher-Order Tensor Renormalization Group Approach to Lattice Glass Model

In this study, the higher-order tensor renormalization group (HOTRG) method is applied to a lattice glass model that has local constraints on the occupation number of neighboring particles represented by many-body interactions. This model exhibits first- and second-order transitions depending on a certain model parameter. The results obtained by using the HOTRG method for the model were confirmed to be consistent with those obtained by a Markov-chain Monte Carlo (MCMC) method for systems of relatively small sizes. The transition points are accurately estimated by the HOTRG calculation for the systems of large sizes, which is challenging to perform using the MCMC method. These results demonstrate that the HOTRG method can be an efficient method for studying systems with many-body interactions.

Finite-size scaling analysis of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki states

Using tensor network methods, we perform finite-size scaling analysis to study the parameter-induced phase transitions of two-dimensional deformed Affleck-Kennedy-Lieb-Tasaki states. We use higher-order tensor renormalization group method to evaluate the moments and the correlations. Then, the critical point and critical exponents are determined simultaneously by collapsing the data. Alternatively, the crossing points of the dimensionless ratios are used to determine the critical point, and the scaling at the critical point is used to determine the critical exponents. For the transition between the disordered AKLT phase and the ferromagnetic ordered phase, we demonstrate that both the critical point and the exponents can be determined accurately. Furthermore, the values of the exponents confirm that the AKLT-FM transition belongs to the 2D Ising universality class. We also investigate the Berezinskii-Kosterlitz-Thouless transition from the AKLT phase to the critical XY phase. In this case we show that the critical point can be located by the crossing point of the correlation ratio.

Anisotropic tensor renormalization group

We propose a new tensor renormalization group algorithm, Anisotropic Tensor Renormalization Group (ATRG), for lattice models in arbitrary dimensions. The proposed method shares the same versatility with the Higher-Order Tensor Renormalization Group (HOTRG) algorithm, i.e., it preserves the lattice topology after the renormalization. In comparison with HOTRG, both of the computation cost and the memory footprint of our method are drastically reduced, especially in higher dimensions, by renormalizing tensors in an anisotropic way after the singular value decomposition. We demonstrate the ability of ATRG for the square lattice and the simple cubic lattice Ising models. Although the accuracy of the present method degrades when compared with HOTRG of the same bond dimension, the accuracy with fixed computation time is improved greatly due to the drastic reduction of the computation cost.

Logarithmic finite-size scaling correction to the leading Fisher zeros in the p-state clock model: A higher-order tensor renormalization group study.

We investigate the finite-size-scaling (FSS) behavior of the leading Fisher zero of the partition function in the complex temperature plane in the p-state clock models of p=5 and 6. We derive the logarithmic finite-size corrections to the scaling of the leading zeros which we numerically verify by performing the higher-order tensor renormalization group (HOTRG) calculations in the square lattices of a size up to 128×128 sites. The necessity of the deterministic HOTRG method in the clock models is noted by the extreme vulnerability of the numerical leading zero identification against stochastic noises that are hard to be avoided in the Monte Carlo approaches. We characterize the system-size dependence of the numerical vulnerability of the zero identification by the type of phase transition, suggesting that the two transitions in the clock models are not of an ordinary first- or second-order type. In the direct FSS analysis of the leading zeros in the clock models, we find that their FSS behaviors show excellent agreement with our predictions of the logarithmic corrections to the Berezinskii-Kosterlitz-Thouless ansatz at both of the high- and low-temperature transitions.