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Symmetry-protected topological order: Why does it change our understanding of matter?
In phase research in physics, symmetry protected topological order (SPT) is a new state of matter. This state not only has the particularity of quantum mechanics, but also has a profound impact on our understanding of the nature of matter. . This state involves some symmetries and a finite energy difference in zero-temperature quantum mechanical systems. From a macroscopic perspective, the SPT sequence shows how to distinguish different states of matter while maintaining symmetry, but these states cannot be continuously deformed without phase change.
The SPT state is a short-range entangled state with symmetry, which is in sharp contrast to the topological order of long-range entanglement.
The sequences protected by these symmetries are relatively stable even in different substances. This protection promotes the dialectical relationship between SPT states and other types of topological order, both in bosonic and fermion systems. For many physicists, the discovery of the SPT sequence is not only a challenge to existing theories, but also a window to a new state of matter.
For a certain non-trivial SPT state, the effective theory at the boundary always has a pure gauge anomaly or a mixed gauge-gravity anomaly. This makes the boundaries of the SPT state either gapless or degenerate, and is therefore thought-provoking in terms of the structure of matter.
A non-trivial SPT state cannot have a gapped non-degenerate boundary, which gives us a new understanding of the boundary behavior of matter.
SPT order appears in many well-known physical phenomena, such as the Haldane phase, and non-interacting fermion topological insulators. These examples not only demonstrate the diversity of these states, but also prompt researchers to seek new physical implementations. These findings mean we have more concrete ways to understand and exploit quantum states.
Further, the difference between SPT states and (intrinsic) topological order is that the latter involves long-range entanglement and is generally able to maintain the stability of its boundaries independent of local perturbations. For example, in intrinsic topological order, boundary excitations are topologically protected, whereas in SPT order symmetry needs to be maintained to maintain this stability.
In the SPT order, we see that the excitation of the boundary is only a product under the protection of symmetry, rather than an expensive structure brought about by the real topology.
Furthermore, molecular defects carry non-trivial statistical properties and fractional quantum numbers in the non-trivial two-dimensional SPT state, which particularly contributes to the emergence of new quantum material concepts. These quantum defects not only exhibit complex behaviors related to symmetry, but are also indispensable elements for studying SPT order.
Theoretically, the understanding of symmetry-protected topological order can also be inspired by group conformal theory. Using the concept of quantum entanglement, we can divide all gapped zero-temperature phases into two categories: long-range entangled phases (with intrinsic topological order) and short-range entangled phases (without intrinsic topological order). This classification helps physicists more accurately locate the properties and application potential of various quantum materials.
Group homology theory provides a powerful tool for the classification of SPT phases, helping us to systematically understand the different states of quantum matter.
In the future, through in-depth understanding of SPT states, scientists will be able to predict new quantum matter states, such as bosonic topological insulators and topological superconductors, which may change all our understanding of matter, especially Applications in fields such as materials science and quantum computing. Of course, this also means that we need to explore more possibilities to completely change our view of matter.
The study of SPT sequences is not only an important cutting-edge topic in physics, but also the cornerstone for us to understand and shape future technologies, so as to better understand the nature and laws of the universe. So, how many unknown physical phenomena do we still have to wait for? Shall we reveal it?
