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The secret of short-range entangled states: How does the SPT order protect symmetry in quantum physics?
As scientists delve deeper into quantum physics, their classification and understanding of quantum states become increasingly complex. Recently, symmetry-protected topological order (SPT order) has become a research hotspot and has aroused widespread discussion in the physics community. This order represents a special state of a quantum system at low temperatures, with obvious characteristics and quantum properties, which are of great significance for physics research and practical applications.
Symmetry-protected topological order (SPT order) is a quantum state based on symmetry, in which even small changes cannot smoothly deform into other states without undergoing phase changes.
The definition of SPT order is based on two key features: first, for a given symmetry, different SPT states cannot be smoothly transformed while maintaining the symmetry. Second, these states can be transformed into trivial product states in a phase-change-free manner when symmetry is broken. In short, SPT order exhibits the characteristics of short-range entanglement in physical systems, which makes these states unable to form long-range entanglement, thus showing a clear distinction from other topological states.
The short-range entangled state has the property of being just a trivial topological order, which can also be called a symmetry-protected "trivial" order.
Further exploring the characteristic properties of SPT order, we find that the boundary effective theory of these states must have scalar anomalies or mixed gravitational potential anomalies. This means that no matter how one cuts the sample to form the boundaries, the boundaries of SPT states are either gapless or multiply degenerate, and that pure gapped boundaries are impossible for nontrivial SPT states. In addition, if the boundary exhibits a gap-degenerate state, this degeneracy may be caused by spontaneous symmetry breaking or intrinsic topological order.
After introducing the concept of quantum entanglement, we understand the relationship between the SPT state and the intrinsic topological order. The intrinsic topological order represents a state of long-range entanglement, while the SPT state maintains short-range entanglement. Although both may have the ability to protect gapless boundary excitations in some cases, the toughness properties of these two states are different. The boundary excitations of the intrinsic topological order are more resistant to local perturbations due to their topological protection characteristics; while the boundary excitations of the SPT order are only stable to local perturbations that do not destroy the symmetry.
For example, in a 2+1D spin-orbit coupling system, both the spin Hall derivative and the quantum Hall derivative exhibit different quantization characteristics, which are closely related to the existence of SPT order.
The applications and examples of SPT order are also very rich. The earliest example can be traced back to the Haldane phase, which corresponds to an odd-spin chain. The Haldane phase is protected by the SO(3) spin rotational symmetry. In contrast, the Haldane phase of an even-spin chain does not possess this symmetry-protected topological order. In addition, topological insulators of non-interacting fermions are also a well-known SPT phase, which is protected by U(1) and time-reversal symmetries. In contrast, fractional quantum Hall states do not belong to SPT states. They are states with intrinsic topological order and have the characteristics of long-range entanglement.
In the process of systematically studying SPT phases, scientists used the cohomology theory of groups to classify them. All gapped zero-temperature phases can be divided into two categories: long-range entangled phases and short-range entangled phases. The short-range entangled phase can be further divided into the symmetry breaking phase, the SPT phase and their mixed phases. This series of studies not only expanded our understanding of quantum phases, but also predicted many new states of quantum matter, including interacting topological insulators and superconductors.
With further exploration of the SPT order, scientists have gained new insights into the complete classification of one-dimensional quantum phases. The study shows that all one-dimensional gapped quantum states are short-range entangled, which means that in the absence of symmetry violation, all these states belong to the same phase - the trivial product state. Through this system, we can better understand the relationship between various quantum phases and demonstrate the richness of quantum systems under interaction.
The research on SPT order is still continuing in depth around the world, which will not only enrich our understanding of quantum physics, but also may lead to new technologies and applications in the future. Do you also feel a desire to explore and understand such a complex and wonderful quantum world?
