Studying topological order and Ising criticality with tensor network algorithms

Tensor network algorithms have emerged as a new approach in simulating strongly correlated quantum many-body systems. In this thesis, we focus on the infinite projected entangled state (iPEPS), a tensor network ansatz for two-dimensional quantum models in the thermodynamic limit. This thesis contains the results out of two main research lines. In the first line we develop a new approach where we use iPEPS to detect topologically ordered phases of matter. We test this method by simulating the topological phase transition known to occur in the Kitaev toric code model. The second research line uses iPEPS to simulate the Shastry-Sutherland model in the vicinity of the plaquette phase. The most striking result is that the known dimer-plaquette transition extends into a finite temperature Ising critical point. This transition is found to be similar to the familiar liquid-gas transition observed in water. Remarkably, a similar critical point is observed in the specific heat measurements of SrCu2(BO3)2, a compound very well described by the Shastry-Sutherland model. The iPEPS calculations are found to be in quantitative agreement with the experimental data.