# Andrew Goldsborough

Supervisor: Rudolf A. Roemer

Group: Disordered Quantum Systems (DisQS)

Department: Physics

### Research

My long term research aim is to use tensor networks to study disordered quantum systems in two dimensions. In order to do this we propose the use of tree tensor networks (TTN) [1] or the multi-scale entanglement renormalization ansatz (MERA) [2] on a two dimensional lattice system.

We are currently studying [3] the holographic properties of a self-assembling inhomogeneous tree tensor network based on the SDRG algorithm of Hikihara et. al. [4]. This numerical RG can be viewed as a means of constructing a tensor network based on the local disorder of the couplings, building a non-homogeneous tree tensor network (TTN).

We can then use algorithms like those described in [1] to efficiently calculate properties such as correlation functions and entanglement entropy. We show that after disorder averaging, correlation functions scale with the path length through the tensor network and entanglement entropy is related to the number of bonds that have to be cut to separate the two regions as suggested by Evenbly and Vidal [5]. This suggests that the TTN describes an effective conformal field theory (CFT) on the boundary of a holographic bulk.

Published in Physical Review B (open access)

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

Entanglement is one of the properties that differentiates classical many-body systems from quantum ones and it has recently become a valuable tool in the interpretation of quantum phases [6]. We show that using ITensor DMRG [7] it it possible to construct an accurate full phase diagram for the disordered Bose-Hubbard model by concentrating on the error on the entanglement [8]. We show the phase diagrams for densities 1, 0.5 and 2.

Published in EPL (open access)

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

Our other work includes the analysis of leaf-to-leaf path lengths in tree graphs. These are analogous to correlation functions in certain tensor network wavefunctions. We have found an analytic result for the average leaf-to-leaf path length in m-ary tree graphs and generalised it to all statistical moments [9].

Published in Physical Review E (open access)

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

We have also analysed the set of Catalan tree graphs [10] and found that the average leaf-to-leaf path length for a separation *r* is equal to the average depth of a leaf at lattice position *m*.

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

To view posters and lecture slides click here.

#### Indexes

h-index: 2

Erdös number: 5

#### References

[1] L. Tagliacozzo, G. Evenbly, and G. Vidal, Phys. Rev. B 80, 235127 (2009).

[2] G. Vidal, Phys. Rev. Lett. 99, 220405 (2007).

[3] A. M. Goldsborough, and R. A. Roemer, Phys. Rev. B 89, 214203 (2014).

[4] T. Hikihara, A. Furusaki, and M. Sigrist, Phys. Rev. B 60, 12116 (1999).

[5] G. Evenbly and G. Vidal, J. Stat. Phys. 145, 891 (2011).

[6] H. Li, and F. D. M Haldane, Phys. Rev. Lett., 101 010504 (2008).

[7] ITensor library version: 0.2.3. http://itensor.org/.

[8] A. M. Goldsborough, and R. A. Roemer, arXiv:1503.02973 [cond-mat.dis-nn].

[9] A. M. Goldsborough, S. A. Rautu, and R. A. Roemer, arXiv:1406.4079.

## Andrew Goldsborough

**e-mail:** a.goldsborough@warwick.ac.uk

**Office:** PS001

**Write to:** Department of Physics, University of Warwick, Coventry, CV4 7AL