The Altland-Zirnbauer classification of random matrices is based on three symmetries: time-reversal, charge conjugation, and a third which is sometimes referred to as "chiral" or "sublattice" symmetry, which is satisfied when a unitary matrix $\Omega$ exists that anticommutes with the Hamiltonian. This usage seems to have very little to do with the concept of chirality in relativistic quantum field theories (the chirality operator $\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3$ commutes with the Hamiltonian of a massless fermion and anticommutes with a mass term $m\gamma^0$). Where did this terminology originate?

EDIT: I am not asking why chiral symmetry is a symmetry or what it means (according to wikipedia it comes from the greek word for 'hand' which is appropriate because it refers to left or right handedness in most contexts). The question is what does this have to do with the definition of chiral symmetry used for the Altland-Zirnbauer classification of random matrices, i.e. a unitary operator which anticommutes with the single particle Hamiltonian. The most basic model this shows up in is the SSH model, where it has nothing to do with left or right handedness. I want to know where this language comes from. This is not a question about physics, it's a question about the origin of terminology used in a specific area of physics. The meaning in field theory contexts is clear to me, so I would appreciate answers which refer to random matrices or the periodic table of topological invariants. I am unable to find any papers or textbooks which explain this terminology and all the experts in the field I have spoken to say they do not know where it came from and heavily prefer the term "sublattice symmetry" specifically in order to avoid confusion with the meaning of chiral symmetry in other contexts. If the use of terminology seems to be so unrelated to its obvious meaning to the extent that people avoid such usage due to possible confusion, why did people use it in the first place?

  • $\begingroup$ I don't know the answer, but in general chirality is the characteristic of a thing whose a mirror image is not reproducible with a combination of rotation and translation, and more specifically, which side of the mirror image it is (with respect to an arbitrary convention). If you hold your left hand to a mirror, the reflection is a right hand; the left hand and the mirror image of the right hand have "left" character, while the right hand and the mirror image of the left hand have "right" character. $\endgroup$
    – g s
    Mar 31, 2022 at 6:24
  • $\begingroup$ That "leftness" or "rightness" is what I would expect a chirality operator to return, whereas I would expect a test for chiral symmetry to be a test for mirror symmetry, and return "true" or "false", not "left" or "right". $\endgroup$
    – g s
    Mar 31, 2022 at 6:26
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    $\begingroup$ Does this answer your question? Why particle hole symmetry and chiral symmetry are called symmetries? $\endgroup$
    – Fabian
    Mar 31, 2022 at 7:32
  • $\begingroup$ No it does not. In many contexts, including relativistic QFT, molecular physics, etc. chirality refers to the distinction between left-handedness and right-handedness, e.g. for elementary particles the operator $\gamma^5$ commutes with the Hamiltonian for left-handed or right-handed fermions. However, for the Altland-Zirnbauer classification, chiral symmetry is used to refer to any unitary operator which anticommutes with the single-particle Hamiltonian. (Half the time this is just called sublattice symmetry because that's what it physically means, eg in the SSH model.) $\endgroup$
    – tommyli
    Apr 1, 2022 at 12:32
  • $\begingroup$ and in a material such as hexagonal boron nitride chiral symmetry is broken because the sublattices are inequivalent. What does that have to do with left- or right-handedness? $\endgroup$
    – tommyli
    Apr 1, 2022 at 12:34

1 Answer 1


First we should ask what happens when we consider symmetries that are not conserved quantities. Let us take your example from particle physics and work out how one would use non-Abelian chiral symmetry transformations. Let me introduce you to two simple examples first: A massive spinor coupled to an abelian gauge potential $A_\mu$, where it acts on fields like $\psi \to U_{\rm AB} \psi$, or more interestingly another simple case of a complex scalar field interacting with an abelian gauge potential, where it acts like $\phi_i \to U_{ij} \phi_j$. Both these cases share the same symmetries: Charge conservation (if you treat the spinor as real) and global phase rotations for all components (if you treat the complex scalar as real). The reason these examples are so trivial compared to your question about Dirac fermions in condensed matter is that I have neglected Lorentz invariance. Now let's add some Lorentz indices $a$, and include them into our transformation rule such that it still conserves charge $$\psi_i(\mathbf x)\stackrel{U_{ab}}{\longrightarrow}\sum_j[U^\dagger(\mathbf x)]_{ij}\psi_j(\mathbf y)\,.$$ We then find that this transformation leaves invariant $$S=\int d^{\,3}x\,\frac{1}{2}\,F_{\mu\,ab}^2+\frac{\epsilon_{abc}}{4}\,\nabla_\mu A^{a\,b}_\nu\,\nabla_\mu A^{c\,\nu}-m\,\bar{\psi}_i (\sigma_a)_{ij}\,\partial_\mu \psi^{aj}(\sigma_b)_{jk}(C^{-1})_{kl}^b \,\partial^\mu\bar{\psi}_{li}\,,$$ which makes perfect sense for Abelian gauge theories because now under local translations/rotations/Lorentz transformations we can distinguish between electric/magnetic charges ($Q=T=0$) from other charges ($Q=T=\pm 1$). For Dirac fermions at low energy this distinction becomes unimportant though since all three charges become degenerate (up to higher order corrections), so their electromagnetic interactions decouple just like they did classically. To make sure there are no confusions about how exactly locality works here I added an explicit index contraction which cannot be omitted otherwise. For instance if we had euclideanised space then nothing would prevent us from using local phase rotations such as $$e^{i \, 2\, n \,\pi}\,(n=1,2,...,N)$$ while they don't affect anything if one includes all other quantum numbers such as color etc., but here things get really confusing since rotations won't preserve electric charge anymore. But nonetheless these kind of symmetries would be perfectly fine for particles whose electric charge was always zero. So what happened? We just found out something called parity breaking due to spontaneous symmetry breaking caused by radiative corrections. In any case our conclusion must be that local chiral symmetries don't play any role at low energies if they don't conserve electrical charge or parity.


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