Periodic Reporting for period 2 - SEQUAM (Symmetries and Entanglement in Quantum Matter)
Reporting period: 2022-04-01 to 2023-09-30
A central topic in physics is the study of different phases of matter, that is, the different ways in which complex systems can organize, and the resulting behavior they exhibit. Familiar examples are, e.g. water vs. ice, or magnetic vs. non-magnetic behavior of metals. The key to understanding these different behaviors are symmetries: By studying the symmetries of the physical laws which describe a given system and the way in which the system orders relative to those symmetries, we can classify the different phases of matter and characterize the way in which transitions between them happen. E.g. ice - unlike water - breaks the translational symmetry of the underlying physical laws, and magnetic materials break rotational symmetry.
This understanding has been challenged by the discovery of novel exotic phases - "quantum matter" - which organize in ways which cannot be understood in terms of symmetry breaking. Rather, the system displays non-trivial ordering in its complex global quantum correlations - quantum entanglement. These systems hold great potential for novel applications, such as measurement devices with unprecented precision, or as building blocks for powerful quantum computers. Given their technological importance, a full understanding of the different quantum materials and the physics they can exhibit, in analogy to the theory of symmetry breaking, is all the more pressing, as it would allow us to explore the entire range of future quantum materials, and use their full potential.
The goal of the project SEQUAM - "Symmetries and entanglement in quantum matter" - is to establish a systematic and comprehensive framework which allows to reconcile the notion of local symmetries and ordering relative to them with the global entanglement ordering displayed by quantum materials. A central tool is the formalism of tensor networks, which form a highly versatile language for the description of complex quantum systems, highlighting the role played by entanglement, but are also very successfully applied in other fields such as data science or artificial intelligence. Tensor networks provide us with direct access to the entanglement degrees of freedom in a local fashion, and are thus ideally suited to reconcile local symmetries and global entanglement. The resulting framework will provide us with the necessary tools to understand the different types of exotic order which quantum materials can exhibit, as well as with tools to probe this behavior in simulations and experiments, and enable us to identify novel useful applications of such exotic quantum materials. We will apply this framework to a wide range of systems which appear in condensed matter and high energy physics, or are realizable in quantum simulators, e.g. with cold gases.
The results of the project SEQUAM will give a unified understanding of unconventional phases, based on physical symmetries and the resulting entanglement order. It will yield their physical manifestations, numerical probes for their detection, and simple ways to realize and probe these models in experimental scenarios, and thus significantly advance our ability to understand, study, and realize complex quantum phases.
This understanding has been challenged by the discovery of novel exotic phases - "quantum matter" - which organize in ways which cannot be understood in terms of symmetry breaking. Rather, the system displays non-trivial ordering in its complex global quantum correlations - quantum entanglement. These systems hold great potential for novel applications, such as measurement devices with unprecented precision, or as building blocks for powerful quantum computers. Given their technological importance, a full understanding of the different quantum materials and the physics they can exhibit, in analogy to the theory of symmetry breaking, is all the more pressing, as it would allow us to explore the entire range of future quantum materials, and use their full potential.
The goal of the project SEQUAM - "Symmetries and entanglement in quantum matter" - is to establish a systematic and comprehensive framework which allows to reconcile the notion of local symmetries and ordering relative to them with the global entanglement ordering displayed by quantum materials. A central tool is the formalism of tensor networks, which form a highly versatile language for the description of complex quantum systems, highlighting the role played by entanglement, but are also very successfully applied in other fields such as data science or artificial intelligence. Tensor networks provide us with direct access to the entanglement degrees of freedom in a local fashion, and are thus ideally suited to reconcile local symmetries and global entanglement. The resulting framework will provide us with the necessary tools to understand the different types of exotic order which quantum materials can exhibit, as well as with tools to probe this behavior in simulations and experiments, and enable us to identify novel useful applications of such exotic quantum materials. We will apply this framework to a wide range of systems which appear in condensed matter and high energy physics, or are realizable in quantum simulators, e.g. with cold gases.
The results of the project SEQUAM will give a unified understanding of unconventional phases, based on physical symmetries and the resulting entanglement order. It will yield their physical manifestations, numerical probes for their detection, and simple ways to realize and probe these models in experimental scenarios, and thus significantly advance our ability to understand, study, and realize complex quantum phases.
The key results of the work carried out in the project SEQUAM up to now have been as follows:
1. We have established a powerful framework which allows us to characterize the most general way in which symmetries in quantum materials can act on the entanglement, linking it to intricate mathematical structures, and broadly explored the ramifications of these different symmetry actions in the types of exotic quantum order which these systems can display. This forms a key step to a full classification of exotic quantum phases.
2. We have demonstrated that it is not always necessary to enforce exact symmetries, but that these symmetries can emerge in complex quantum materials; and we have established a set of tools to study and characterize such emergent symmetries. This significantly advances our understanding of symmetries in quantum materials, and our ability to characterize them.
3. We have established a range of tools to identify and characterize exotic entangled quantum phases and transitions between them in numerical simulations and beyond. These tools make use of the direct access to the entanglement degrees of freedom provided by tensor networks, and thus allow to go significantly beyond other probes.
4. We have developed a set of approaches to prepare exotic quantum states in realistic setups, and to investigate the exotic quantum order which they exhibit through practical probes, further advancing our ability to experimentally realize novel quantum materials.
1. We have established a powerful framework which allows us to characterize the most general way in which symmetries in quantum materials can act on the entanglement, linking it to intricate mathematical structures, and broadly explored the ramifications of these different symmetry actions in the types of exotic quantum order which these systems can display. This forms a key step to a full classification of exotic quantum phases.
2. We have demonstrated that it is not always necessary to enforce exact symmetries, but that these symmetries can emerge in complex quantum materials; and we have established a set of tools to study and characterize such emergent symmetries. This significantly advances our understanding of symmetries in quantum materials, and our ability to characterize them.
3. We have established a range of tools to identify and characterize exotic entangled quantum phases and transitions between them in numerical simulations and beyond. These tools make use of the direct access to the entanglement degrees of freedom provided by tensor networks, and thus allow to go significantly beyond other probes.
4. We have developed a set of approaches to prepare exotic quantum states in realistic setups, and to investigate the exotic quantum order which they exhibit through practical probes, further advancing our ability to experimentally realize novel quantum materials.
Each of the results discussed above presents a significant advancement beyond state-of-the-art methods. The comprehensive framework for describing symmetries on the level of the entanglement opens up the route for a full characterization of all unconventional quantum phases subject to symmetries; this classification will be extensively explored in the second part of the project. The tools developed to characterize emergent symmetries present an entirely novel tool to characterize enigmatic systems such as deconfined quantum critical points with unprecedented accuracy. The novel order parameters developed to fingerprint unconventional quantum phases provide access to a wealth of information about quantum systems, far beyond what existing approaches can achieve, by probing systems right where the entanglement sits. And finally, the approaches developed to probe unconventional quantum phases through simple measurements provide a hitherto non-existing tool to unequivocally identify exotic quantum order through simple and realizable probes. Together with the steps taken in the second part of the project, these results will give a unified understanding of novel quantum materials, allow to explore the full range of unconventional behavior they can exhibit, and provide us with powerful numerical probes for their simulation and characterization, simple ways to realize and probe these models in experimental scenarios, and thus overall significantly advance our ability to understand, study, and realize complex quantum phases.