Strongly-entangled fermionic, two-dimensional systems pose an extremely hard problem for numerical methods. The extension of the Density Matrix Renormalisation Group (DMRG) to two dimensions is relatively straightforward, but amplifies all previously-known difficulties, such as rate of convergence, number of basis states kept and parallelisation.
The project started with firstly improving the rate of convergence of DMRG in large-scale calculations, secondly implementing efficient methods to minimise the number of states kept and thirdly parallelising the DMRG method and implementation to a large number of cores. We have presented a method to improve the rate of convergence of DMRG and its scaling behaviour and implemented the exploitation of non-Abelian symmetries (such as SU(2)-Spin) as well as large-scale parallelisation.
The main focus of the project is now the application of the improved DMRG to numerically difficult problems, such as long-range time evolutions or two- dimensional lattices (e.g. the Kagomé spin lattice). With other members of the group, DMRG is applied as the impurity solver for DMFT calculations using imaginary and real time evolution as well as a verification method for results obtained from CORE.
During the first part of the PhD project, the highly versatile SyTen tensor network toolkit has been implemented in C++ which is currently also employed in other contexts. This toolkit could potentially be extended to implement truly two- dimensional tensor network approaches such as (i)PEPS or MERA. Work on the numerical treatment of Matrix Product Operators, also in the infinite limit of iDMRG, is currently ongoing and expected to further simplify the application of DMRG and the tensor network toolkit in particular to arbitrary problems.
 Phys. Rev. B, 95 035129: Claudius Hubig, Ian P. McCulloch and Ulrich Schollwöck: Generic construction of efficient matrix product operators
 arXiv:1609.08518 (submitted to PRX): Zi Cai, Claudius Hubig and Ulrich Schollwöck: Universal long-time behavior of stochastically driven interacting quantum systems
 Phys. Rev. B, 91 155115: Claudius Hubig, Ian P. McCulloch, Ulrich Schollwöck and F. Alexander Wolf: Strictly single-site DMRG algorithm with subspace expansion
2016: Munich Quantum Symposium: A Generic Algorithm for the Construction of Efficient Matrix Product Operators
2016: International Summer School on Computational Quantum Materials, Sherbrooke: Generic Algorithm for the Construction of Efficient Matrix Product Operators
2016: DPG Frühjahrstagung Regensburg: DMRG on Binary Tree Tensor Networks
2016: Entanglement in Strongly Correlated Systems, Benasque: DMRG on Binary Tree Tensor Networks
2016: Fritz Haber Institute, Berlin: Symmetries in Tensor Networks and Subspace Expansion with DMRG
2015: WEH-Workshop Isolated Quantum Many-Body Systems out of Equilibrium, Bad Honnef: Spin Excitations in a Staggered Magnetic Field
2015: Advanced Computational Methods for Strongly Correlated Quantum Systems, Würzburg: A Strictly Single-Site DMRG Algorithm with Subspace Expansion
2014: Munich Quantum Day: Algorithmic Advancements in DMRG
Ian P. McCulloch, University of Queensland
Fabian Essler, University of Oxford
Teresa Reinhard, MPSD Hamburg