Language
Country/Area
No result found
From Haldan phase to topological insulator: How did the mysterious world of SPT order arise?
With the deepening of quantum physics research, scientists' understanding of matter has become more refined. Especially for the properties of zero-temperature quantum states, one of the emerging concepts is the Symmetry-Protected Topological (SPT) order. The rise of this concept has opened up a new horizon for the classification of matter in the world of quantum physics.
SPT order is an order in a quantum state with symmetry and limited energy gap, and has unique physical properties.
The definition of SPT sequence contains two main characteristics. On the one hand, different SPT states with the same symmetry cannot deform smoothly without phase change; on the other hand, if the symmetry is broken during the deformation process, then these states can deform into the same state without phase change. Any product status. This allows the SPT order not only to exist in bosonic systems, but also to be found in fermion systems, forming the concepts of bosonic SPT order and fermion SPT order.
In this context, some scholars have introduced the concept of quantum entanglement into their explanation, referring to the SPT state as a short-range entangled state with symmetry. This is in contrast to the topological order of long-range entanglement, which is not related to the famous EPR paradox.
Characteristic attributes of SPT sequence
The boundary effective theory of non-trivial SPT states will always have pure quantum anomalies or mixed gravity anomalies, which also gives them the property of being gapless or degenerate under any form of sample boundary. In particular, for non-trivial SPT states, a gapless non-degenerate boundary cannot be formed.
If the boundary is a gapless degenerate state, then this degeneration may be caused by spontaneous symmetry breaking and/or intrinsic topological order.
For example, in the non-trivial 2+1-dimensional SPT state, monotonic defects carry non-trivial statistics and fractional quantum numbers of the symmetry group. This shows the profound connection between the boundaries of the SPT order and the internal topological properties.
The relationship between SPT order and intrinsic topological order
SPT states are short-range entangled, while intrinsic topological order is long-range entangled. Although both can sometimes protect gapless boundary excited states, their sources of stability differ. The gapless boundary excited states in the intrinsic topological order are stable to any local perturbation, while the gapless boundary excited states in the SPT order are stable only to those local perturbations that do not break the symmetry.
The gapless boundary excited states in the SPT order are protected by symmetry, while the intrinsic topological order is topologically protected.
The rise of SPT order is not only a theoretical breakthrough, but also inspires the prediction of many new quantum states. In particular, the research on bosonic topological insulators and topological superconductors has made SPT order an active field in modern condensed matter physics.
Group cohomology theory of SPT phase
When quantum states are partitioned at zero temperature, the dynamics of the SPT phase lose spontaneous symmetry, leading to profound connections with group cohomology theory. Researchers found that these (d + 1)-dimensional SPT states can be classified by group cohomology.
For bosonic SPT phases with pure quantum anomaly boundaries, these phases can be calibrated by the following group homology categories:
H^{d+1}[G,U(1)]
This enables the scientific community to gain in-depth understanding of the characteristics of various SPT phases through mathematical tools, thereby accurately classifying 1D, 2D and higher-dimensional quantum states.
Complete classification of one-dimensional interacting quantum phases
In the process of exploring SPT order, researchers found that there is no intrinsic topological order in 1D systems, and all 1D compact quantum states are short-range entangled. According to this discovery, when the Hamiltonian value has no symmetry, these quantum states are classified as arbitrary product states.
If Hamiltonian has symmetry, the quantum phase of the 1D condensed matter can be the symmetry breaking phase, the SPT phase or their mixed state. This new understanding allows us to more systematically classify all one-dimensional compact quantum phases.
Faced with the expansion of various characteristics of SPT sequences and related knowledge, future research in this field will continue. So, will the SPT sequence become the key to uncovering more unknown quantum worlds?
