A topological order is a new quantum phase that is beyond Landau's symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that $R$- and $F$-matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the $R$- and $F$-matrices be experimentally measured? By using quantum simulations based on a toric code model with boundaries and state-of-the-art technology, we show that the braidings, i.e. the $R$-matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The $F$-matrices can also be measured in a scattering quantum circuit involving the fusion of three anyons in two different orders. Thus we provide an experimental protocol for measuring the unique identifiers of topological order.
A topological order is a new quantum phase that is beyond Landau's symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that R- and F-matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the R- and F-matrices be experimentally measured? By using quantum simulations based on a toric code model with boundaries and state-of-the-art technology, we show that the braidings, i.e. the R-matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The F-matrices can also be measured in a scattering quantum circuit involving the fusion of three anyons in two different orders. Thus we provide an experimental protocol for measuring the unique identifiers of topological order.
ABSTRACT Topological order is a new quantum phase that is beyond Landau’s symmetry-breaking paradigm. Its defining features include robust degenerate ground states, long-range entanglement and anyons. It was known that R and F matrices, which characterize the fusion-braiding properties of anyons, can be used to uniquely identify topological order. In this article, we explore an essential question: how can the R and F matrices be experimentally measured? We show that the braidings, i.e. the R matrices, can be completely determined by the half braidings of boundary excitations due to the boundary-bulk duality and the anyon condensation. The F matrices can also be measured by comparing the quantum states involving the fusion of three anyons in two different orders. Thus we provide a model-independent experimental protocol to uniquely identify topological order. By using quantum simulations based on a toric code model with boundaries encoded in three- and four-qubit systems and state-of-the-art technology, we obtain the first experimental measurement of R and F matrices by means of an NMR quantum computer at room temperature.
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 phases are exotic states of matters which do not order locally yet exhibit a global ordering in their entanglement, making their characterization a challenging task. In this work, the authors construct an explicit holographic duality between topological phases in the bulk and the structure of the entanglement spectrum at its boundary, thereby establishing a one-to-one correspondence between different bulk topological orders and symmetry-breaking and symmetry-protected orderings of the entanglement. This allows classification and detection of topological phases holographically through ``entanglement spectroscopy'' at the boundary, and construction of order parameters that measure the condensation and confinement of anyons, and thus permits characterization of the nature of topological phase transitions.
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 particle-loop-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 insulators. While the bulk of the topological insulators behaves like trivial insulators, the surface is anomalous with odd number of massless dirac fermions. While most of the understanding of topological insulators is developed upon the free fermion picture, such phase survives under interaction. In the thesis, we derive hydrodynamic field theory for the topological insulators by functional bosonization. Such formulation is robust against interaction. In addition, by suitably incorporating electron interactions, we predict new topological phases, called the fractional topological insulators. Besides, the surface theory of topological insulators also justify the famous duality between free massless dirac fermion and QED_3.
Topological orders can be used as media for topological quantum computing-a promising quantum computation model due to its invulnerability against local errors. Conversely, a quantum simulator, often regarded as a quantum computing device for special purposes, also offers a way of characterizing topological orders. Here, we show how to identify distinct topological orders via measuring their modular S and T matrices. In particular, we employ a nuclear magnetic resonance quantum simulator to study the properties of three topologically ordered matter phases described by the string-net model with two string types, including the Z_{2} toric code, doubled semion, and doubled Fibonacci. The third one, non-Abelian Fibonacci order is notably expected to be the simplest candidate for universal topological quantum computing. Our experiment serves as the basic module, built on which one can simulate braiding of non-Abelian anyons and ultimately, topological quantum computation via the braiding, and thus provides a new approach of investigating topological orders using quantum computers.
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