Symmetries in physics (specifically condensed matter physics)

Symmetries play a big role in physics. Some symmetries are translation symmetry, rotation symmetry, time translation symmetry, timereversal symmetry etc.

It seems that in condensed matter physics the role of timereversal symmetry, particle hole symmetry and chiral symmetry is different from the role of the other symmetries. Is this true? What makes them so special?

• For example I read: when you have for example rotation symmetry then there an unitary operator for which: U^(dagger) H U = H and the Hamiltonian can be written in a block diagonal form. It is claimed this can be done for any symmetry. But then another example is given where you have spinful timereversal symmetry and then the Hamiltonian cannot be written in block diagonal form. So it seems that time reversal symmetry is special. – Marnix Apr 21 '16 at 9:18

The time reversal and chiral symmetry are special because they are antiunitary symmetries, in contrast to the other unitary symmetries like translation and rotation symmetries. Antiunitary symmetry operation involves complex conjugation of the wave function of the system, which is a non-trivial operation beyond unitary transforms.

Unitary symmetries are simpler to deal with because the symmetry operation $U$ commutes with the Hamiltonian $H$ (as $UHU^\dagger=H$), which means the Hamiltonian can be block-diagonalized in the irreducible representations of the symmetry group. Within each representation, one can just study the small block-Hamiltonian labeled by the symmetry quantum number. For example, if the system has translation symmetry, the Hamiltonian can be block-diagonalized by Fourier transform to the momentum space. Each block-Hamiltonian is labeled by the momentum quantum number. Then one can further study the effect of other symmetry operations in each momentum block.

However with antiunitary symmetries, such block diagonalization is no longer possible. Different states connected by time-reversal symmetry should be put together and analyzed as a whole. There no separation of representations by symmetry quantum numbers for antiunitary symmetries, which make the antiunitary symmetries non-trivial and harder to handle.

When it comes to the classification of symmetry protected topological states in condensed matter physics, it is usually assumed that the unitary symmetries have been treated by transforming the Hamiltonian to the block-diagonal basis of their irreducible representations. Then the antiunitary symmetries are the only symmetries that are left for more detailed case-by-case analysis, which eventually leads to the famous ten-fold-way classification (http://arxiv.org/abs/0912.2157).

• Thank you. I found out that particle hole symmetry also corresponds with an anti-unitary operator. – Marnix Apr 22 '16 at 9:41
• @Marnix The issue is subtle for particle-hole. In the many-body basis, particle-hole symmetry is a unitary symmetry. But for single-particle free fermion systems, particle-hole symmetry "looks like" an antiunitary symmetry, because complex conjugation of the single-particle wave function is involved. That is why particle-hole symmetry is also "special" for free fermion systems, even though it is a unitary symmetry in the many-body sense. – Everett You Apr 22 '16 at 17:27
• @EverettYou Chiral symmetry is unitary not anti-unitary(if I am not mistaken). I think you mean to say time reversal and particle -hole symmetry. By the way good answer. – L.K. Apr 3 '17 at 13:16
• @L.K. Chiral symmetry is unitary in the single-particle Hilbert space and is anti-unitary in the many-body Hilbert space. Particle-hole symmetry is anti-unitary in the single-particle Hilbert space and is unitary in the many-body Hilbert space. The fact that the notions of unitary/anti-unitary switch for these two symmetries has been discussed in details in section III.A.2 of a nice review paper by A. Ludwig (arxiv.org/pdf/1512.08882.pdf) – Everett You Apr 9 '17 at 17:59
• @L.K. By the way, when we say that the symmetry is unitary/anti-unitary, by default, we refer to the many-body Hilbert space. That is why we typically say that the chiral symmetry is anti-unitary. – Everett You Apr 9 '17 at 18:02