Tensor Network Simulation Methods for Open Quantum Lattice Models

A complex quantum system cannot be perfectly isolated from its surroundings and is typically subject to incoherent processes. Dissipation and/or an external drive can move the system away from thermal equilibrium to a non-equilibrium regime. Often, dissipation is an unwanted feature which is minimised as much as possible, while in others cases, it can be harnessed to stabilise interesting phases of matter. The subject of this thesis is the development of tensor network techniques to probe the dynamics and steady state properties of many-body open quantum systems. Our theoretical understanding of many-body open quantum systems is greatly aided by numerical techniques. However, numerical methods are remarkably limited by the exponential growth of many-body Hilbert spaces. Tensor network methods are a class of numerical techniques which aim to circumvent the exponential growth of Hilbert space by representing the quantum state as a network of tensors. Doing so allows for an efficient representation and manipulation of the quantum state. In the first part of this thesis, a tensor network method is presented in a Cluster Mean Field framework. This method integrates a one-dimensional Lindblad master equation by dividing the system into finite sized clusters, each represented by a tensor network. The effective master equation is integrated in real time using a sweeping Time Evolving Block Decimation algorithm and the method is used to investigate the steady properties of a dissipative Jaynes-Cummings-Hubbard model with a two-photon drive where a finite size scaling of the cluster sizes allows for comparison with equilibrium models. The simulation of two-dimensional open quantum lattice models are the subject of the second part of the thesis. The Infinite Projected Entangled Pair Operator is used as an ansatz for the density matrix of a system on an infinite square lattice. The key development is a method to optimise the truncation of enlarged tensor bonds in a way which is appropriate for mixed states. The method is tested against exactly solvable cases and literature results. In the final chapter, the algorithm is applied to a dissipative anisotropic XY-model and revealing the nature of a transition parameterised by the strength of dissipation