Final Report Summary - PEPS-PROPOSAL (New techniques in the simulation of quantum many-body systems in two dimensions: methods and applications)
Overview of work and results:
(1) Establishment of new numerical methods to simulate the physics of quantum many-body systems in two dimensions:
Throughout this project, Dr Orus has produced a number of papers in which he provided important improvements in the development of the so-called iPEPS algorithm to study two-dimensional (2D) quantum lattice systems. In particular, he has extended the method to deal with plaquette interactions in order to study models exhibiting the so-called topological order. This has been done in Physical Review Letters 106, 107203 (2011) and New Journal of Physics 14 025005 (2012), in collaboration with the groups of Prof. J. Vidal at CNRS (Paris) and Dr Kai P. Schmidt at TU Dortmund. Furthermore, Dr Orus has also extended the methods to simulate infinite 2D systems using projected entangled pair operators to deal with five-body spin interactions. This has been done in the context of the investigation of the robustness under perturbations of the so-called cluster Hamiltonian in Physical Review A 8, 022317 (2012) and arXiv:1211.4054 (submitted to Physical Review Letters), in collaboration with the group of Dr Kai P. Schmith at TU Dortmund. Dr Orus has also proposed some new methods to simulate 1D and 2D quantum lattice systems using corner transfer matrices and generalisations thereof. This work, which proposes a number of new algorithms, has been published in Physical Review B 85, 205117 (2012). Finally, Dr Orus has also further contributed in the use of iPEPS algorithms to simulate infnite 2D fermionic lattice systems. Regarding this, a study of interacting spinless fermions on the honeycomb lattice has been carried out together with Dr P. Corboz, Prof. S. Capponi, Prof. A. Lauchli and Dr B. Bauer, and published in Europhysics Letters 98, 27005 (2012).
(2) Use these methods to investigate problems of key importance in physics:
Dr Orus has employed the above methods to study problems such as the robustness of topoligical order, as well as the phase diagram of the cluster Hamiltonian and its usefulness for universal quantum computation. Regarding the robustness of topological order, the phase diagrams of Kitaev's toric code model in arbitrary magnetic field was elucidated for the first time in Physical Review Letters 106, 107203 (2011), and for the case of a generalisation of the toric code to Z3 symmetry the phase diagram was again numerically determined in New Journal of Physics 14 025005 (2012). These studies were important, since they proved that topologically-ordered phases of matter are robust under external perturbations and can therefore be found in real-life scenarios. Regarding the cluster Hamiltonian, Dr Orus determined the phase diagram of the model in arbitrary magnetic field and in the presence of Ising-like perturbations, identifying the regions that are useful for the so-called 'measurement based quantum computation'. Thanks to this study, it has been understood that states useful for quantum computation correspond mostly to a very characteristic phase of matter, the so-called 'cluster-phase'. It has also been found that this phase is in fact quite large, which brings positive news for future experimental implementations of a quantum computer. Finally, Dr Orus has contributed to further understand the physics of interacting spinless fermions on a Honeycomb lattice. This study showed that some previous works proposing some topological ground states for the model were actually wrong, and that the true ground state undergoes phase separation (recently published as Europhysics Letters 98, 27005 (2012)).
The above results are important also in the context of frustrated antiferromagnetism as well as for bosonic 2D models. Specifically, some frustrated antiferromagnets are believed to have the same type of topological order as the toric code (the so-called gapped Z2 spin liquids), and some bosonic models are also equivalent to the studied spin models.
Moreover, Dr Orus has also produced important insights in the study of the entanglement properties of 2D systems. In particular, together with Prof. T. Chieh Wei, he has proposed the so-called 'topological geometric entanglement' as an indicator of the existence of topological order in 2D quantum lattice systems. This contribution has appeared in aXiv:1108.1537 as a preprint, and is currently under consideration for publication.
(3) Contribute towards the development of a software package for the simulation of strongly correlated 2D quantum many-body systems:
In the course of developing the above numerical methods, Dr Orus has ellaborated a number of new functions and subroutines that expand the range of utility of the algorithms based on PEPS and tensor networks to simulate 2D quantum lattice systems. The goal is that, in the long term, these functions will be rewritten in a user-friendly and optimised way in order to make them available to the general public. In fact, researchers at the National Taiwan University in Taipei have already expressed a very strong interest in implementing open-source versions of these functions for running simulations with graphical processing units (GPUs), in turn enhancing the efficiency of the calculations by a very large factor (around 30 times faster). Regarding this, Dr Orus is currently in contact with the researchers at this University, in order to provide them with the proper advice and insight.
(4) Extra contributions: independently of the above achievements, let us mention that Dr Orus has also produced a number of results in the context of the simulation and entanglement properties of 1D quantum lattice systems. These results are a basis for further developments towards the study of 2D systems, since they also make use of the tensor network language employed in the 2D algorithms, and allow to understand some of the boundary problems that one may encounter when simulating higher-dimensional systems. These works have been published in Physical Review B 83, 201101(R) (2011), Physical Review Letters 107, 077204 (2011), Physical Review B 84, 064409 (2011) and Physical Review B 84, 140407(R) (2011). Moreover, Dr Orus is currently working on some introductory notes to the methods developed throughout this project to simulate 2D systems, and which will be available to the community in the near future.
Overview of socio-economic impact of the project:
Thanks to this project, it has been possible to:
(i) Develop new algorithmic techniques for the simulation of quantum many-body systems. This, in turn, will find many future applications in e.g. material science. In fact, researchers from the graduate school of excellence in material science MAINZ, at the University of Mainz, have already expressed their interest in learning about these new methods to simulate materials.
(ii) Understand better 2D topological order and cluster states. This will help in the construction of a future quantum computer (both topologically based and measurement-based), with its corresponding socio-economic benefits.
Let us stress that the socio-economic importance and impact of the results of this project have also been recognised by other institutions. Above we mentioned the examples of the graduate school of excellence MAINZ and the University of Taiwan. But this research has also been recognised by the University of Mainz, the Ecole Polytechnique Federale Lausanne, and Royal Holloway University of London, where Dr Orus has attended interviews for a number of academic positions. In the end, the position offered to Dr Orus at the level of Junior Professor by the University of Mainz reflects the long-term impact of the results in this project and willing commitment of other institutions.
(1) Establishment of new numerical methods to simulate the physics of quantum many-body systems in two dimensions:
Throughout this project, Dr Orus has produced a number of papers in which he provided important improvements in the development of the so-called iPEPS algorithm to study two-dimensional (2D) quantum lattice systems. In particular, he has extended the method to deal with plaquette interactions in order to study models exhibiting the so-called topological order. This has been done in Physical Review Letters 106, 107203 (2011) and New Journal of Physics 14 025005 (2012), in collaboration with the groups of Prof. J. Vidal at CNRS (Paris) and Dr Kai P. Schmidt at TU Dortmund. Furthermore, Dr Orus has also extended the methods to simulate infinite 2D systems using projected entangled pair operators to deal with five-body spin interactions. This has been done in the context of the investigation of the robustness under perturbations of the so-called cluster Hamiltonian in Physical Review A 8, 022317 (2012) and arXiv:1211.4054 (submitted to Physical Review Letters), in collaboration with the group of Dr Kai P. Schmith at TU Dortmund. Dr Orus has also proposed some new methods to simulate 1D and 2D quantum lattice systems using corner transfer matrices and generalisations thereof. This work, which proposes a number of new algorithms, has been published in Physical Review B 85, 205117 (2012). Finally, Dr Orus has also further contributed in the use of iPEPS algorithms to simulate infnite 2D fermionic lattice systems. Regarding this, a study of interacting spinless fermions on the honeycomb lattice has been carried out together with Dr P. Corboz, Prof. S. Capponi, Prof. A. Lauchli and Dr B. Bauer, and published in Europhysics Letters 98, 27005 (2012).
(2) Use these methods to investigate problems of key importance in physics:
Dr Orus has employed the above methods to study problems such as the robustness of topoligical order, as well as the phase diagram of the cluster Hamiltonian and its usefulness for universal quantum computation. Regarding the robustness of topological order, the phase diagrams of Kitaev's toric code model in arbitrary magnetic field was elucidated for the first time in Physical Review Letters 106, 107203 (2011), and for the case of a generalisation of the toric code to Z3 symmetry the phase diagram was again numerically determined in New Journal of Physics 14 025005 (2012). These studies were important, since they proved that topologically-ordered phases of matter are robust under external perturbations and can therefore be found in real-life scenarios. Regarding the cluster Hamiltonian, Dr Orus determined the phase diagram of the model in arbitrary magnetic field and in the presence of Ising-like perturbations, identifying the regions that are useful for the so-called 'measurement based quantum computation'. Thanks to this study, it has been understood that states useful for quantum computation correspond mostly to a very characteristic phase of matter, the so-called 'cluster-phase'. It has also been found that this phase is in fact quite large, which brings positive news for future experimental implementations of a quantum computer. Finally, Dr Orus has contributed to further understand the physics of interacting spinless fermions on a Honeycomb lattice. This study showed that some previous works proposing some topological ground states for the model were actually wrong, and that the true ground state undergoes phase separation (recently published as Europhysics Letters 98, 27005 (2012)).
The above results are important also in the context of frustrated antiferromagnetism as well as for bosonic 2D models. Specifically, some frustrated antiferromagnets are believed to have the same type of topological order as the toric code (the so-called gapped Z2 spin liquids), and some bosonic models are also equivalent to the studied spin models.
Moreover, Dr Orus has also produced important insights in the study of the entanglement properties of 2D systems. In particular, together with Prof. T. Chieh Wei, he has proposed the so-called 'topological geometric entanglement' as an indicator of the existence of topological order in 2D quantum lattice systems. This contribution has appeared in aXiv:1108.1537 as a preprint, and is currently under consideration for publication.
(3) Contribute towards the development of a software package for the simulation of strongly correlated 2D quantum many-body systems:
In the course of developing the above numerical methods, Dr Orus has ellaborated a number of new functions and subroutines that expand the range of utility of the algorithms based on PEPS and tensor networks to simulate 2D quantum lattice systems. The goal is that, in the long term, these functions will be rewritten in a user-friendly and optimised way in order to make them available to the general public. In fact, researchers at the National Taiwan University in Taipei have already expressed a very strong interest in implementing open-source versions of these functions for running simulations with graphical processing units (GPUs), in turn enhancing the efficiency of the calculations by a very large factor (around 30 times faster). Regarding this, Dr Orus is currently in contact with the researchers at this University, in order to provide them with the proper advice and insight.
(4) Extra contributions: independently of the above achievements, let us mention that Dr Orus has also produced a number of results in the context of the simulation and entanglement properties of 1D quantum lattice systems. These results are a basis for further developments towards the study of 2D systems, since they also make use of the tensor network language employed in the 2D algorithms, and allow to understand some of the boundary problems that one may encounter when simulating higher-dimensional systems. These works have been published in Physical Review B 83, 201101(R) (2011), Physical Review Letters 107, 077204 (2011), Physical Review B 84, 064409 (2011) and Physical Review B 84, 140407(R) (2011). Moreover, Dr Orus is currently working on some introductory notes to the methods developed throughout this project to simulate 2D systems, and which will be available to the community in the near future.
Overview of socio-economic impact of the project:
Thanks to this project, it has been possible to:
(i) Develop new algorithmic techniques for the simulation of quantum many-body systems. This, in turn, will find many future applications in e.g. material science. In fact, researchers from the graduate school of excellence in material science MAINZ, at the University of Mainz, have already expressed their interest in learning about these new methods to simulate materials.
(ii) Understand better 2D topological order and cluster states. This will help in the construction of a future quantum computer (both topologically based and measurement-based), with its corresponding socio-economic benefits.
Let us stress that the socio-economic importance and impact of the results of this project have also been recognised by other institutions. Above we mentioned the examples of the graduate school of excellence MAINZ and the University of Taiwan. But this research has also been recognised by the University of Mainz, the Ecole Polytechnique Federale Lausanne, and Royal Holloway University of London, where Dr Orus has attended interviews for a number of academic positions. In the end, the position offered to Dr Orus at the level of Junior Professor by the University of Mainz reflects the long-term impact of the results in this project and willing commitment of other institutions.