Studies of Infinite Two-Dimensional Quantum Lattice Systems with Projected Entangled Pair States

Determining the properties of quantum many-body systems is a central challenge in modern physics. Being able to determine the macroscopic properties of a system from its microscopic description would hasten progress in many fields of science and technology. However, we currently lack the tools to solve such problems generally, or even to develop a theoretical intuition about how many systems might behave. From a simple Hamiltonian description of the system, one may obtain complex, highly correlated collective behaviour. Computational techniques have played a major part in the effort to determine properties of quantum many-body systems. However, as the total degrees of freedom in the system scales exponentially in the system size, numerical diagonalization of the Hamiltonian quickly becomes computationally intractable and one must develop more efficient approximate techniques to explore the system. Present numerical methods such as quantum Monte Carlo and series expansion have provided insight into many systems of interest, but are also held back by fundamental difficulties. In this thesis, we focus on tensor networks, a relatively new ansatz for representing quantum many-body states. Tensor networks are motivated by two ideas from quantum information: firstly, that quantum entanglement is the source of the immense difficulty of simulating quantum systems classically, and secondly that the ground states of certain Hamiltonians exist in a low-entanglement region of the entire Hilbert space. The strength of tensor networks is that they provide a systematic way of representing this class of low-entanglement quantum states. In particular, this thesis describes the iPEPS algorithm for computing the ground states of infinite, two-dimensional quantum lattice systems based on the Projected Entangled Pair States (PEPS) ansatz. We then benchmark the algorithm by computing the phase diagrams of several systems that have been studied with other techniques. Lastly, we apply our algorithm to problems that are not well solved by current approaches, such as frustrated spin systems