We present an algorithm to simulate two-dimensional quantum lattice systems in the thermodynamic limit. Our approach builds on the projected entangled-pair state algorithm for finite lattice systems [F. Verstraete and J.I. Cirac, cond-mat/0407066] and the infinite time-evolving block decimation algorithm for infinite onedimensional lattice systems [G. Vidal, Phys. Rev. Lett. 98, 070201 (2007)]. The present algorithm allows for the computation of the ground state and the simulation of time evolution in infinite two-dimensional systems that are invariant under translations. We demonstrate its performance by obtaining the ground state of the quantum Ising model and analysing its second order quantum phase transition. PACS numbers:Strongly interacting quantum many-body systems are of central importance in several areas of science and technology, including condensed matter and high-energy physics, quantum chemistry, quantum computation and nanotechnology. From a theoretical perspective, the study of such systems often poses a great computational challenge. Despite the existence of well-stablished numerical techniques, such as exact diagonalization, quantum monte carlo [1], the density matrix renormalization group [2] or series expansion [3] to mention some, a large class of two-dimensional lattice models involving frustrated spins or fermions remain unsolved.Fresh ideas from quantum information have recently led to a series of new simulation algorithms based on an efficient representation of the lattice many-body wave-function through a tensor network. This is a network made of small tensors interconnected according to a pattern that reproduces the structure of entanglement in the system. Thus, a matrix product state (MPS) [4], a tensor network already implicit in the density matrix renormalization group, is used in the time-evolving block decimation (TEBD) algorithm to simulate time evolution in one-dimensional lattice systems [5], whereas a tensor product state [6] or projected entangled-pair state (PEPS) [7] is the basis to simulate two-dimensional lattice systems. In turn, the multi-scale entanglement renormalization ansatz accuratedly describes critical and topologically ordered systems [8].In this work we explain how to modify the PEPS algorithm of Ref.[7] to simulate two-dimensional lattice systems in the thermodynamic limit. By addressing an infinite system directly, the infinite PEPS (iPEPS) algorithm can analyse bulk properties without dealing with boundary effects or finite-size corrections. This is achieved by generalizing, to two dimensions, the basic ideas underlying the infinite TEBD (iTEBD) [9]. Namely, we exploit translational invariance (i) to obtain a very compact PEPS description with only two independent tensors and (ii) to simulate time evolution by just updating these two tensors. We describe the essential new ingredients of the iPEPS algorithm, which is based on numerically solving a transfer matrix problem with an MPS. We then use it to compute the ground state of the quantum Ising model with...

In a recent contribution ͓P. Corboz, R. Orús, B. Bauer, and G. Vidal, Phys. Rev. B 81, 165104 ͑2010͔͒ fermionic projected entangled-pair states ͑PEPSs͒ were used to approximate the ground state of free and interacting spinless fermion models, as well as the t-J model. This paper revisits these three models in the presence of an additional next-nearest hopping amplitude in the Hamiltonian. First we explain how to account for next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS algorithms based on simulating time evolution. Then we present benchmark calculations for the three models of fermions and compare our results against analytical, mean-field, and variational Monte Carlo results, respectively. Consistent with previous computations restricted to nearest neighbor Hamiltonians, we systematically obtain more accurate ͑or better converged͒ results for gapped phases than for gapless ones.

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We present a study of the hard-core Bose-Hubbard model at zero temperature on an infinite square lattice using the infinite projected entangled pair state algorithm ͓J. Jordan, R. Orús, G. Vidal, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 101, 250602 ͑2008͔͒. Throughout the whole phase diagram our values for the groundstate energy, particle density, and condensate fraction accurately reproduce those previously obtained by other methods. We also explore ground-state entanglement, compute two-point correlators, and conduct a fidelitybased analysis of the phase diagram. Furthermore, for illustrative purposes we simulate the response of the system when a perturbation is suddenly added to the Hamiltonian.

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