The topological order of a $(2+1)\mathrm{D}$ topological phase of matter is characterized by its chiral central charge and a unitary modular tensor category that describes the universal fusion and braiding properties of its anyonic quasiparticles. I discuss the topologically invariant quantities associated with these and identify ones that are useful for determining the topological order. I propose a variety of physical experiments that probe these quantities and detail the relation of the measured data to the topological invariants.
The topological order of a (2+1)D topological phase of matter is characterized by its chiral central charge and a unitary modular tensor category that describes the universal fusion and braiding properties of its anyonic quasiparticles. I discuss the topologically invariant quantities associated with these and identify ones that are useful for determining the topological order. I propose a variety of physical experiments that probe these quantities and detail the relation of the measured data to the topological invariants.
This thesis aims at concluding the classification results for topological phases with symmetry in 2+1 dimensions. The main result is that topological phases are classified by a triple of unitary braided fusion categories E⊂C⊂M plus the chiral central charge c. Here E is a symmetric fusion category, E=Rep(G) for boson systems or E=sRep(Gf) for fermion systems, consisting of the representations of the symmetry group and describing the local excitations with symmetry; C is the category of all the quasiparticle excitations in the bulk, containing E as its Müger center; M is a minimal modular extension of C, that also includes the gauged symmetry defects. We also study the stacking of topological phases with symmetry and two types of anyon condensations based on such classification.
The modern conception of phases of matter has undergone tremendous developments since the first observation of topologically ordered state in fractional quantum Hall systems in the 1980s. Topological orders are exotic states of matter characterized by patterns of long-range entanglement that lie beyond Landau's symmetry breaking paradigm. In 2+1 dimensions, these phases support signature anyonic excitations with fractional statistics or even non-Abelian braiding effects. One important theoretical question is how we should characterize these phases. In 2+1 dimensions, plenty of evidence suggests that the \emph{modular matrices}, S and T, suffice to characterize the order completely. It is thus of equal importance to find out if these quantities are experimental observables. Here we demonstrate how such measurements can indeed be achieved using only the given Hamiltonian, allowing one to identify the corresponding topological order using a nuclear magnetic resonance simulator. This is a first experimental realization of these fundamental signatures of a topological order; it opens up new future avenues toward identifying more generic topological orders based purely on experimental measurements.
Topological orders are new phases of matter beyond Landau symmetry breaking. They correspond to patterns of long-range entanglement. In recent years, it was shown that in 1+1D bosonic systems there is no nontrivial topological order, while in 2+1D bosonic systems the topological orders are classified by a pair: a modular tensor category and a chiral central charge. In this paper, we propose a partial classification of topological orders for 3+1D bosonic systems: If all the point-like excitations are bosons, then such topological orders are classified by unitary pointed fusion 2-categories, which are one-to-one labeled by a finite group G and its group 4-cocycle ω4∈H4[G;U(1)] up to group automorphisms. Furthermore, all such 3+1D topological orders can be realized by Dijkgraaf-Witten gauge theories.
We examine the interplay of symmetry and topological order in 2+1 dimensional fermionic topological phases of matter. We define fermionic topological symmetries acting on the emergent topological effective theory described using braided tensor category theory. Connecting this to the Gf fermionic symmetry of the microscopic physical system, we characterize and classify symmetry fractionalization in fermionic topological phases. We find that the physical fermion provides constraints that result in a tiered structure of obstructions and classification of fractionalization with respect to the physical fermions, the quasiparticles, and the vortices. The fractionalization of the (bosonic) symmetry G=Gf/Z2f on the physical fermions is essentially the central extension of G by the Z2f fermion parity conservation that yields the fermionic symmetry Gf. We develop an algebraic theory of Gf symmetry defects for fermionic topological phases using G-crossed braided tensor category theory. This formalism allows us to fully characterize and classify 2+1 dimensional fermionic symmetry enriched topological phases with on-site unitary fermionic symmetry group Gf. We first apply this formalism to extract the minimal data specifying a general fermionic symmetry protected topological phase, and demonstrate that such phases with fixed Gf form a group under fermionic stacking. Then we analyze general fermionic symmetry enriched topological phases and find their classification is given torsorially by the classification of the symmetry fractionalization of quasiparticles combined with the classification of fermionic symmetry protected topological phases. We illustrate our results by detailing a number of examples, including all the invertible fermionic topological phases.
Topological phases are gapped quantum phases of matter classified beyond the paradigm of Landau's symmetry breaking theory.In particular, topologically ordered phases are topological phases with topological order.Unlike the conventional symmetry-breaking order described by local order parameter, topological order is characterized by the excitations.For example, in (2+1)D topologically ordered phases, there are particle-like excitations called anyons.In contrast, in (3+1)D topologically ordered phases, there are both particle-like and loop-like excitations.These excitations exhibit braiding statistics, meaning that carrying one excitation adiabatically around another transform the ground state by a phase (or generally a unitary transformation).The braiding statistics of these excitations serves as a crucial ingredient in characterizing the topological order in these phases.One important aspect is to understand the braiding statistics in topologically ordered phases.While the anyonic braiding statistics in (2+1)D topologically ordered phases is well established, the braiding statistics in (3+1)D topologically ordered phases is far from clear.In (3+1)D, the most well-understood braiding statistics is the particle-loop braiding statistics, where a particle winds around a loop excitation acquires an Aharonov Bohm phase.In the thesis, we introduce a new braiding statistics, which we called the Borromean-Rings braiding statistics.It is a particleloop-loop braiding statistics where a particle travels around two unlinked loops acquires a braiding phase.Unlike the particle-loop braiding, the particle trajectory in the Borromean-Rings braiding is not linked with any of the two loops.Another important aspect is to probe the topological order in topologically ordered phases.One usual way to probe the topological order is to examine the properties of the phases on various orientable surfaces.For example, for (2+1)D topologically ordered phases, the number of degenerate ground states on Torus counts the number of anyons.Also, ground state transformation under twisting the torus encodes the braiding data.In the thesis, we attempt to probe the topological order by putting the phases on non-orientable surfaces.We find that the ground state degeneracy and the ground state transformation reveals a special kind of anyons, called the parity symmetric anyons, which show a lost of orientation in their braiding statistics.They in turn distinguish different topological order.Even without topological order, there can be non-trivial topological phases in the presence of global symmetry.These topological phases are called the symmetry protected topological phases.For example, (3+1)D fermionic matter with time reversal symmetry and charge conservation symmetry has a non-trivial phase, namely the topological I would like to give my sincere thanks to my advisor Shinsei Ryu.Throughout my PhD study, he taught me a lot about condensed matter physics, particularly on the forefront subjects in topological phases.Under his supervision, I worked on different abstract yet inspiring topics on topological phases.His guide is always clear with mathematical interpretation and physical insight.His advice not only guided me though the research, but also shaped my views and thinking.From my advisor, I learnt the way to convey difficult ideas
We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry G. The key is to use the so-called symmetric fusion category E to describe the symmetry. Here, E=sRep(Z2f) describing particles in a fermionic product state without symmetry, or E=sRep(Gf)[E=Rep(G)] describing particles in a fermionic (bosonic) product state with symmetry G. Then, topological orders with symmetry E are classified by nondegenerate unitary braided fusion categories over E, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to p+ip fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill \ensuremath{-}1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge c=41, whose only topological excitation has non-Abelian statistics with spin s=41 and quantum dimension d=1+2.