GREEN1	Loops and Gauge Freedom in Two Dimensional Tensor Networks
Type	Theoretical
#students	1
Orientation	Why is the scientific problem of interest at all? Hilbert space is big! For a system consisting of $N$ spin halves, there are $2^N$ basis states. This exponential scaling with system size underpins many of the most interesting and counterintuitive aspects of quantum mechanics. However, it turns out that in many cases, only a vanishingly small fraction of Hilbert space is visited by most systems. In recent years, a lot of progress has been made towards understanding what part of Hilbert space one should concentrate upon and how to parametrise it[1].
How	How is the research going to shed light on the given problem?. Tensor networks in particular provide a set of variational states that parametrise a sub-region of Hilbert space with a certain amount of quantum mechanical entanglement. In one dimension, where the relevant tensor networks are called matrix product states, there exists an efficient way to approximate the ground state of a given Hamiltonian[2]. There are also numerically efficient  algorithms to simulate the time-dependence of such states[3,4].  In higher dimensions, such efficient algorithms are not known at present. One of the reasons for this is the rather non-local way in which information is contained in the tensor network state. The problem is already found in going from a one-dimensional system with open boundary conditions to one on a ring. The number of computational steps required to simulate the time evolution of the open boundary condition case scales as $N^3$, whereas naively simulating the evolution of the N-site ring scales as $N^6$[5]. The problems of simulating on higher dimensional lattices stems from the proliferation of loops.
What	What is the specific thing that the student will do, and how does it fit inside the overall project? In this project we will focus on trying to understand the time evolution of states on the ring. One way round the inefficient scaling is to squash the ring to a double line and to write a matrix product state for this doubled line. The Hamiltonian at the ends of this line will be modified in order to accommodate the fact that the double line actually forms a loop. There will also be a new gauge symmetry associated with the $N/2$ ways in which squash the loop to a line. By studying this representation of states on a loop and the gauge symmetries that arise, we will attempt to gain useful insights into the symmetries present in higher dimensions.
Special Knowledge
Supervisor	Prof Andrew G Green Andrew.green@ucl.ac.uk
References (optional)	[1] D. Perez-Garcia, F. Verstraete, M. M. Wolf and J. I. Cirac arXiv:quant-ph/0608197 [2] S. R. White. Phys. Rev. Lett 69, 2863 (1992) [3] G. Vidal, Phys.Rev. Lett 98, 070201 (2007) [4]J utho Haegeman et al PHys. Rev. Lett 107, 070601 (2011) [5]  P. Pippan, S. R. White and Hans Gerd Evertz Phys Rev B 081103(R ) (2010)