My research centres around using tensor networks to model quantum many-body systems. It is closely related to the fields of condenced matter physics and quantum information theory. A comprehensive overview of my PhD work can be found in my thesis (link below), which provides an overview of the field as well as the technical details of my work.

Tensor networks and geometry for the modelling of disordered quantum many-body systems

A more concise and approachable summary article appeared in the Autumn 2016 IoP Computational Physics Group Newsletter as a result of being runner up in the IoP CPG thesis prize.

Below is a list of my publications.

Andrew M. Goldsborough and Glen Evenbly

Published in Physical Review B - Selected for *Editors' Suggestion*

preprint: https://arxiv.org/abs/1708.07652

Here, we combine the concepts of strong disorder renormalisation (SDRG) and the multi-scale entanglement renormalisation ansatz (MERA) to create a self-assembling, *disordered
MERA*.

Previous SDRG based methods have only been *one-hit,* in that the renormalisation is performed once. On the other hand most tensor network methods perform a variational update but do
not take into consideration the inhomogeneity of the system. Furthermore, variational energy minimisation on a tensor network has a tendency to preferentially include short range information at the
expense of long range correlation. Here we create a tensor network algorithm that combines the structural advantages of SDRG as well as performing a variational update that retains the long range
entanglement that is key to understanding the physics of the disordered system.

All code used is available on my GitHub.

Andrew M. Goldsborough and Rudolf A. Römer

Published in EPL (open access)

Preprint: http://arxiv.org/abs/1503.02973

Appeared in the EPL *Highlights of 2015 *collection

Entanglement is one of the key properties that separates *quantum* from *classical* systems* *and has recently become a powerful tool in the analysis of quantum phases.
Unlike traditional observables, entanglement entropy and the *entanglement spectrum *is a non-local property, so can contains information about the wavefunction as a whole. We use an
MPS based density matrix renormalisation group (DMRG) algorithm from the ITensor library to construct the phase diagram of the disordered Bose-Hubbard model in 1D by looking only at entanglement. The
calculation of which is essentially free as it is found as part of the DMRG update scheme.

The source code used to generate the data is on my GitHub.

Andrew M. Goldsborough, Jonathan M. Fellows, Matthew Bates, S. Alex Rautu, George Rowlands, Rudolf A. Römer

Preprint: http://arxiv.org/abs/1502.07893

This paper applies a graph theory approach to find the leaf-to-leaf distances in inhomogeneous tree graphs as a means of exploring the correlation functions of the disordered tensor network SDRG.
We work with the Catalan trees, which is the set of all possible unique binary trees. Using generating functions we show that the average distance of leaves of separation *r* is
equal to the average *depth *of a leaf r sites in from the left of tree. Furthermore we produce an analytic formula for the average leaf-to-leaf distance as a function of leaf
separation and study its asymptotic properties.

Andrew M. Goldsborough, S. Alex Rautu, and Rudolf A. Römer

Published in Physical Review E (open access)

Preprint: http://arxiv.org/abs/1406.4079

Two-point correlation functions in 1D tensor networks have been seen to decay exponentially in the number of tensors that connect the two operators. This results in the exponential decay inherent with matrix product states (MPS) and power law decay of the multi-scale entanglement renormalisation ansatz (MERA). As we were studying tree tensor networks (TTNs), we asked; what should we expect for regular TTNs? This paper is the explicit calculation of average distances within tree graphs and the generalisation to any splitting and any moment of the distribution. We show that, as expected, TTNs also omit a power law correlation function on average, but there are deviations due to the self-similar nature of the tree graph.

Andrew M. Goldsborough and Rudolf A. Römer

Published in Physical Review B (open access)

preprint: http://arxiv.org/abs/1401.4874

The aim of this paper is twofold: firstly to develop a tensor network variant of the Strong Disorder Renormalisation Group (SDRG) of Ma, Dasgupta and Hu, and secondly to explore the relationship between tensor networks and geometry for network with a non-trivial structure.

The paper describes how the numerical SDRG algorithm of Hikihara at al. can be seen to *self-construct* an inhomogeneous tree tensor network where the structure is set by the
disorder in the system. We then show that properties such as the entanglement entropy and correlation functions are related to the geometry of the tensor network. This highlights an important fact
about tensor networks; that the physical properties that the network can capture are highly dependent on its structure.

The code used in this paper is available on my GitHub.