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The periodic table of topological invariants is an application of topology to physics. It indicates the group of topological invariant for topological insulators and superconductors in each dimension and in each discrete symmetry class.[1]
There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices. They are defined by three symmetries of the Hamiltonian , (where , and , are the annihilation and creation operators of mode , in some arbitrary spatial basis) : time reversal symmetry, particle hole (or charge conjugation) symmetry, and chiral (or sublattice) symmetry.
Chiral symmetry is a unitary operator , that acts on , as a unitary rotation (,) and satisfies ,. A Hamiltonian possesses chiral symmetry when , for some choice of (on the level of first-quantised Hamiltonians, this means and are anticommuting matrices).
Time reversal is an antiunitary operator , that acts on , (where , is an arbitrary complex coefficient, and , denotes complex conjugation) as ,. It can be written as where is the complex conjugation operator and is a unitary matrix. Either or . A Hamiltonian with time reversal symmetry satisfies , or on the level of first-quantised matrices, , for some choice of .
Charge conjugation is also an antiunitary operator which acts on as , and can be written as where is unitary. Again either or depending on what is. A Hamiltonian with particle hole symmetry satisfies , or on the level of first-quantised Hamiltonian matrices, , for some choice of .
I believe that charge conjugation and chiral symmetries map between creation and annihilation operators. Otherwise there is no difference between the two anti-unitary classes. Not confident enough to make these changes myself, the reference I'm looking at, also contains inconsistencies. Link below:
Ludwig, Andreas W W (2016-12-01). "Topological phases: classification of topological insulators and superconductors of non-interacting fermions, and beyond". Physica Scripta. T168: 014001. doi: 10.1088/0031-8949/2015/T168/014001. Retrieved 2023-11-28. 2001:1BA8:401:7E:CD5E:CD38:DD29:AFEB ( talk) 10:42, 6 December 2023 (UTC)
![]() | This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
|
The periodic table of topological invariants is an application of topology to physics. It indicates the group of topological invariant for topological insulators and superconductors in each dimension and in each discrete symmetry class.[1]
There are ten discrete symmetry classes of topological insulators and superconductors, corresponding to the ten Altland–Zirnbauer classes of random matrices. They are defined by three symmetries of the Hamiltonian , (where , and , are the annihilation and creation operators of mode , in some arbitrary spatial basis) : time reversal symmetry, particle hole (or charge conjugation) symmetry, and chiral (or sublattice) symmetry.
Chiral symmetry is a unitary operator , that acts on , as a unitary rotation (,) and satisfies ,. A Hamiltonian possesses chiral symmetry when , for some choice of (on the level of first-quantised Hamiltonians, this means and are anticommuting matrices).
Time reversal is an antiunitary operator , that acts on , (where , is an arbitrary complex coefficient, and , denotes complex conjugation) as ,. It can be written as where is the complex conjugation operator and is a unitary matrix. Either or . A Hamiltonian with time reversal symmetry satisfies , or on the level of first-quantised matrices, , for some choice of .
Charge conjugation is also an antiunitary operator which acts on as , and can be written as where is unitary. Again either or depending on what is. A Hamiltonian with particle hole symmetry satisfies , or on the level of first-quantised Hamiltonian matrices, , for some choice of .
I believe that charge conjugation and chiral symmetries map between creation and annihilation operators. Otherwise there is no difference between the two anti-unitary classes. Not confident enough to make these changes myself, the reference I'm looking at, also contains inconsistencies. Link below:
Ludwig, Andreas W W (2016-12-01). "Topological phases: classification of topological insulators and superconductors of non-interacting fermions, and beyond". Physica Scripta. T168: 014001. doi: 10.1088/0031-8949/2015/T168/014001. Retrieved 2023-11-28. 2001:1BA8:401:7E:CD5E:CD38:DD29:AFEB ( talk) 10:42, 6 December 2023 (UTC)