Thema der Dissertation:
Topological Phases, Fixed-Point Models, Extended TQFT, and Fault-Tolerant Quantum Computation - Tensors in Spacetime
Topological Phases, Fixed-Point Models, Extended TQFT, and Fault-Tolerant Quantum Computation - Tensors in Spacetime
Abstract: This thesis studies the classification of topological phases of matter in terms of tensors associated to discrete spacetimes. We show that all algebraic descriptions of topological order can be reduced to a single property, namely combinatorial topological invariance. This invariance yields a set of equations for the tensors forming a description of a phase, which we call a topological tensor scheme (tTS). tTS formalize the whole spectrum of topological phase classifications: They can describe intrinsic bosonic topological, SPT and SET, symmetry-breaking, or fermionic topological orders, as well as superselection sectors of topological boundaries, anyons, twist defects, corners, and all other kinds of defects. Further, the framework can describe both microscopic fixed-point models, where the invariance is implemented as recellulation, as well as extended TQFT, where the invariance is implemented by gluing.
In contrast to much of the established literature, we do not merely postulate the higher-categorical structures classifying topological phases, but derive them from a coarse and simple ansatz for topological invariance. We further investigate to what extent these ansatzes are universal in the sense that they can emulate any other arbitrarily complicated ansatz for topological invariance. We find that established ansatzes are universal only under the condition that there exists a topological boundary, and construct a new ansatz that is universal independent of the existence of a topological boundary. This provides a promising route to reconcile microscopic fixed-point models with chiral phases of matter, which is one of the major unsettled questions of the field.
We illustrate many of the above ansatzes by concrete examples. Most notably, we look at the family of models arising from cohomology theory, including twisted gauge theories. In particular, we give an efficient systematic procedure to calculate arbitrary defects of spacetime dimensions 0, 1, and 2 in such models.
As an application, we show how to construct dynamic error-correcting codes from fixed-point path integrals, by measuring defects or 1-form symmetries of these path integrals. As an example, we demonstrate that the toric code, the subsystem toric code, as well as the recently developed CSS and honeycomb Floquet codes are secretly the same code up to microscopic equivalence. We also showcase this application by constructing two new codes, namely a Floquet version of the 3+1-dimensional toric code, as well as a dynamic code for the double-semion phase.
In contrast to much of the established literature, we do not merely postulate the higher-categorical structures classifying topological phases, but derive them from a coarse and simple ansatz for topological invariance. We further investigate to what extent these ansatzes are universal in the sense that they can emulate any other arbitrarily complicated ansatz for topological invariance. We find that established ansatzes are universal only under the condition that there exists a topological boundary, and construct a new ansatz that is universal independent of the existence of a topological boundary. This provides a promising route to reconcile microscopic fixed-point models with chiral phases of matter, which is one of the major unsettled questions of the field.
We illustrate many of the above ansatzes by concrete examples. Most notably, we look at the family of models arising from cohomology theory, including twisted gauge theories. In particular, we give an efficient systematic procedure to calculate arbitrary defects of spacetime dimensions 0, 1, and 2 in such models.
As an application, we show how to construct dynamic error-correcting codes from fixed-point path integrals, by measuring defects or 1-form symmetries of these path integrals. As an example, we demonstrate that the toric code, the subsystem toric code, as well as the recently developed CSS and honeycomb Floquet codes are secretly the same code up to microscopic equivalence. We also showcase this application by constructing two new codes, namely a Floquet version of the 3+1-dimensional toric code, as well as a dynamic code for the double-semion phase.
Time & Location
Dec 04, 2024 | 02:00 PM