Entanglement Order Parameters

One of the key achievements of modern physics is to give us a unified understanding of phases of matter -- for instance, water can appear as solid ice, liquid water, or gaseous vapor. The key insight, pioneered by Lev Landau in the early 20th century, has been that phases differ by the way in which they behave relative to the symmetries of the underlying physical laws: For instance, while water respects the translational symmetry of the physical laws, ice breaks translation in space by forming a regular crystalline lattice. This ordering, which breaks a given symmetry, can be detected by socalled order parameters, which have turned out to form extremely powerful tools not only in distinguishing phases of matter, but also in understanding their relation and transitions between them. Modern quantum materials have challenged this understanding: These systems termed topologically ordered can organize in ways which cannot be detected through order parameters, but are rather characterized by global orderings in their quantum correlations entanglement. At the same time, these exotic phases hold big promises for applications such as high-precision measurement devices, or as a way to store and process information in quantum computers. In the light of these promises, a comprehensive understanding of these phases, connecting them to the powerful framework of order parameters, is highly desirable. The goal of this project is to construct a systematic framework to design and subsequently determine order parameters which are capable of detecting both conventional ordering and exotic ordering in the entanglement. By design, the framework will treat these seemingly different phenomena on an equal footing, and thus give a unified way to address conventional order, topological order, as well as exotic systems where those two types of order interplay. This will provide us with a powerful framework to analyze exotic topologically ordered phases, both theoretically and numerically, far beyond what has been possible with existing methods. It will give us access to a wealth of additional information to analyze the behavior of unconventional quantum materials, and thus lead to novel insights into the use of these systems in fields such as quantum computing, or quantum metrology and sensing.

Our understanding of the different phases of matter - such as ice, water, and steam - goes back to the pioneering work of Lev Landau in the 1930s. Landau realized that phases and transitions between them are governed by symmetry: the underlying physical laws may possess certain symmetries, but the state of the system can break them. This symmetry breaking can be detected through order parameters, simple quantities that are non-zero in one phase and vanish in another. This idea is remarkably powerful, providing not only a way to distinguish phases, but also a framework to quantitatively characterize transitions, for instance by extracting universal fingerprints known as critical exponents. This picture was fundamentally challenged by the discovery of unconventional, or "topological", phases of matter, recognized by the 2016 Nobel Prize to Haldane, Kosterlitz, and Thouless. These phases cannot be detected by any local order parameter. Instead, they are characterized by global patterns in the quantum correlations - the entanglement - of the many-body system. This is what makes them so interesting: topologically ordered systems can serve as quantum memories and as platforms for quantum computation, protected against noise by the very absence of local distinguishability. However, the same property makes them difficult to study - without a suitable order parameter, there is no quantitative probe to characterize these phases and transitions between them. In the FWF Quantum Austria project "Entanglement Order Parameters", we developed a comprehensive framework for order parameters that operate directly at the level of entanglement, which we termed entanglement order parameters. The central tool is the language of tensor networks, which describe quantum many-body states by decomposing them into networks of small building blocks - tensors - encoding both the physical and the entanglement degrees of freedom. Crucially, the symmetries of the physical system as well as the global entanglement structure are reflected in local symmetry properties of these tensors. Our entanglement order parameters detect ordering with respect to these symmetries, thereby probing the entanglement structure of the system directly. The framework treats conventional symmetry-breaking order and exotic topological order on a unified footing, capturing the rich interplay between the two. We classified the possible behaviors of entanglement order parameters and related them to physical properties of the underlying quantum phases. Through duality mappings to statistical mechanics and gauge theory, we extracted universal signatures at topological phase transitions, including information not accessible through conventional methods. We also developed practical tools for determining entanglement order parameters in numerical simulations and experiments with quantum simulators. Overall, the results of this project significantly advance our ability to identify, characterize, and understand the rich variety of quantum phases displayed by strongly correlated systems.