An extension of the projected entangled-pair states (PEPS) algorithm to infinite systems, known as the iPEPS algorithm, was recently proposed to compute the ground state of quantum systems on an infinite two-dimensional lattice. Here we investigate a modification of the iPEPS algorithm, where the environment is computed using the corner transfer matrix renormalization group (CTMRG) method, instead of using one-dimensional transfer matrix methods as in the original proposal. We describe a variant of the CTMRG that addresses different directions of the lattice independently, and use it combined with imaginary time evolution to compute the ground state of the two-dimensional quantum Ising model. Near criticality, the modified iPEPS algorithm is seen to provide a better estimation of the order parameter and correlators.
An extension of the projected entangled-pair states (PEPS) algorithm to infinite systems, known as the iPEPS algorithm, was recently proposed to compute the ground state of quantum systems on an infinite two-dimensional lattice. Here we investigate a modification of the iPEPS algorithm, where the environment is computed using the corner transfer matrix renormalization group (CTMRG) method, instead of using one-dimensional transfer matrix methods as in the original proposal. We describe a variant of the CTMRG that addresses different directions of the lattice independently, and use it combined with imaginary time evolution to compute the ground state of the two-dimensional quantum Ising model. Near criticality, the modified iPEPS algorithm is seen to provide a better estimation of the order parameter and correlators.
An extension of the projected entangled-pair states (PEPS) algorithm to infinite systems, known as the iPEPS algorithm, was recently proposed to compute the ground state of quantum systems on an infinite two dimensional lattice. Here we investigate a modification of the iPEPS algorithm, where the environment is computed using the corner transfer matrix renormalization group (CTMRG) method, instead of using one-dimensional transfer matrix methods as in the original proposal. We describe a variant of the CTMRG that addresses different directions of the lattice independently, and use it combined with imaginary time evolution to compute the ground state of the two dimensional quantum Ising model. Near criticality, the modified iPEPS algorithm is seen to provide a better estimation of the order parameter and correlators.
The concept of the corner transfer matrix (CTM) was first discovered by Baxter in 1968, when he derived a set of variational matrix equations, which for finite matrix sizes, could be numerically solved to obtain a sequence of approximations for the statistical mechanical properties of a system of monomers and dimers on a rectangular lattice [1]. It was not until 1978, however, in a seminal paper entitled "Variational Approximations for Square Lattice Models in Statistical Mechanics" [4], that Baxter outlined his CTM variational method, which brought to light the potential power of the former objects to obtain numerics and series expansions for unsolved models and to calculate the order parameter of solved ones. Subsequent numerical work led to the realisation that the method, though general, was not very efficient; and increasing efficiency required making model specific modifications, which restricted the transferability of the resulting algorithm to other models. The CTM variational method was thus not widely adopted, despite holding much promise. More recently, however, Nishino and Okunishi discovered that White's density matrix renormalisation group (DMRG) algorithm [55, 56] could be efficiently extended to study two-dimensional classical lattice models, if the density matrix was approximated by Baxters CTMs. Numerical tests of their algorithm, the corner transfer matrix renormalisation group (CTMRG) method, were met with much success. Notable among these is its implementation within the infinite projected entangled-pair states (iPEPS) algorithm of Orus and Vidal to simulate the ground state of infinite two-dimensional quantum systems. In this thesis we review CTM derived variational methods: Baxter's original CTM iterative method, and developments of the CTM concept within the DMRG algorithm by Nishino and Okunishi, which led to the CTMRG method to calculate critical phenomena of classical and quantum systems. This will begin with elucidating the theoretical formalisms of both methods in two dimensions and their generalisations to three dimensions; followed by review of important application, namely, within the iPEPS algorithm of Orus and Vidal to numerically study infinite planar quantum systems.
Projected entangled-pair states (PEPS) have become a powerful tool for studying quantum many-body systems in the condensed matter and quantum materials context, particularly with advances in variational energy optimization methods. A key challenge within this framework is the computational cost associated with the contraction of the two-dimensional lattice, crucial for calculating state vector norms and expectation values. The conventional approach, using the corner transfer matrix renormalization group (CTMRG), involves combining two tensor network layers, resulting in significant time and memory demands. In this work, we introduce an alternative split-CTMRG algorithm, which maintains separate PEPS layers and leverages modified environment tensors, reducing computational complexity while preserving accuracy. Benchmarks on quantum lattice models demonstrate substantial speedups for variational energy optimization, rendering this method valuable for large-scale PEPS simulations.
We develop a QR-based corner transfer matrix renormalization group (CTMRG) framework for contracting infinite projected entangled-pair states (iPEPS) on honeycomb lattices. Our method explicitly uses the lattice's native C3v symmetry at each site, generalizing QR-based acceleration (previously limited to square lattices) to enable efficient and stable contractions. This approach achieves order-of-magnitude speedups over conventional singular value decomposition (SVD)-based CTMRG while maintaining high numerical precision. Comprehensive benchmark calculations for the spin-1/2 Heisenberg and Kitaev models demonstrate higher computational efficiency without sacrificing accuracy. We further employ our method to study the Kitaev-Heisenberg model, where we provide numerical evidence for the universal 1/r^4 decay of the dimer-dimer correlation function within the quantum spin liquid (QSL) phase. Our work establishes a framework for extending QR-based CTMRG to other lattice geometries, opening new avenues for studying exotic quantum phases with tensor networks.
A typical quantum state obeying the area law for entanglement on an infinite two-dimensional (2D) lattice can be represented by a tensor network Ansatz, known as an infinite projected entangled pair state (iPEPS), with a finite bond dimension D. Its real-imaginary time evolution can be split into small time steps. An application of a time step generates a new iPEPS with a bond dimension k times the original one. The new iPEPS does not make optimal use of its enlarged bond dimension kD; hence, in principle, it can be represented accurately by a more compact Ansatz, preferably with the original D. In this work we show how the more compact iPEPS can be optimized variationally to maximize its overlap with the new iPEPS. To compute the overlap we use the corner-transfer-matrix renormalization group. By simulating sudden quench of the transverse field in the 2D quantum Ising model with the proposed algorithm, we provide a proof of principle that real-time evolution can be simulated with iPEPS. A similar proof is provided with the same model for imaginary-time evolution of purification of its thermal states.
In this paper we explore the practical use of the corner transfer matrix and its higher-dimensional generalization, the corner tensor, to develop tensor network algorithms for the classical simulation of quantum lattice systems of infinite size. This exploration is done mainly in one and two spatial dimensions (1D and 2D). We describe a number of numerical algorithms based on corner matrices and tensors to approximate different ground-state properties of these systems. The proposed methods also make use of matrix product operators and projected entangled pair operators and naturally preserve spatial symmetries of the system such as translation invariance. In order to assess the validity of our algorithms, we provide preliminary benchmarking calculations for the spin-1/2 quantum Ising model in a transverse field in both 1D and 2D. Our methods are a plausible alternative to other well-established tensor network approaches such as iDMRG and iTEBD in 1D, and iPEPS and TERG in 2D. The computational complexity of the proposed algorithms is also considered and, in 2D, important differences are found depending on the chosen simulation scheme. We also discuss further possibilities, such as 3D quantum lattice systems, periodic boundary conditions, and real-time evolution. This discussion leads us to reinterpret the standard iTEBD and iPEPS algorithms in terms of corner transfer matrices and corner tensors. Our paper also offers a perspective on many properties of the corner transfer matrix and its higher-dimensional generalizations in the light of novel tensor network methods.