From bc7a39cd7a158510c05dbc3e03550e86c3041fba Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Attila=20Szab=C3=B3?= <attila.szabo@physics.ox.ac.uk>
Date: Sat, 8 May 2021 21:26:13 +0100
Subject: [PATCH] Parity symmetry

---
 docs/docs/representation_theory.md | 21 +++++++++++++++++++--
 1 file changed, 19 insertions(+), 2 deletions(-)

diff --git a/docs/docs/representation_theory.md b/docs/docs/representation_theory.md
index 68c3e0b5c..023955cbb 100644
--- a/docs/docs/representation_theory.md
+++ b/docs/docs/representation_theory.md
@@ -144,10 +144,27 @@ $$ \begin{aligned}
 \chi(s) &= 0\ \textrm{otherwise}.
 \end{aligned} $$
 
+## Other symmetries
+
+In addition to space-group symmetries, many systems have discrete on-site symmetries one would like to impose to improve the quality of wave functions.
+
+A case in point is the $\mathbb{Z}_2$ parity symmetry $\hat \pi = \prod_i \hat\sigma_i^x$ of spin-1/2 systems.
+Usually, one does not try to enforce the full SU(2) spin rotation symmetry of a Heisenberg model [^vieijra]. Instead, we focus its U(1) subgroup generated by $S^z$: constraining the total $S^z$ of the system is straightforward in the usual computational basis.
+However, one can show that wave functions with SU(2) spin quantum number $s$ and $S^z = 0$ are also eigenstates of parity with eigenvalue $(-1)^s$. Imposing this eigenvalue allows one to distinguish the singlet and triplet sectors of antiferromagnets, and hence extract the spin gap.
+
+[^vieijra]: However, neural quantum states with full SU(2) symmetry have recently been [introduced](https://arxiv.org/abs/2104.14869).
+
+Parity symmetry fits neatly in the framework we use for the space group, for it too maps each $\sigma^z$ basis state $|\boldsymbol\sigma\rangle$ to precisely one other, $|-\boldsymbol\sigma\rangle$. This commutes with every space-group symmetry, hence the total symmetry group is a direct product of the space group and the $\mathbb{Z}_2$ group generated by $\hat\pi$. NetKet already offers functionality for generating the direct product of two groups, which could be used to construct the product table used by `DenseEquivalent`; expanding `DenseSymm` should be straightforward, too.
+
+The character table of direct products of too groups is a "direct product" of those of the factors. That is, irreps of the space-parity group are described by an irrep of the space group (characters: $\chi_0(s)$) and a parity eigenvalue $p=\pm1$, from which
+$$ \begin{aligned}
+\chi(s) &= \chi_0(s); & \chi(\hat\pi s) &= p\chi_0(s).
+\end{aligned} $$
+
 ## From character tables to symmetric wave functions
 
 Characters of symmetry groups can be used directly to construct wave functions that transform according to the corresponding irrep. Namely, it follows from the orthogonality properties of irreps that
-$$ \mathcal{P}_\alpha |\psi\rangle = \frac{d_\alpha}{|S|} \sum_{g\in S} \chi_\alpha(g)^* g |\psi\rangle $$
-is the projection of the wave function $|\psi\rangle$ onto the part of the Hilbert space that transforms according to the irrep $\alpha$ of dimension $d_\alpha$.
+$$ \mathcal{P}_\alpha |\psi\rangle = \frac{d_\alpha}{|G|} \sum_{g\in G} \chi_\alpha(g)^* g |\psi\rangle $$
+is the projection of the wave function $|\psi\rangle$ onto the part of the Hilbert space that transforms according to the irrep $\alpha$ of the symmetry group $G$, which has dimension $d_\alpha$.
 
 This motivates creating a network architecture that is able to generate the components of the wave functions $g|\psi\rangle$ for all group elements at once. NetKet offers this functionality in all its glory through *group convolutional neural networks,* which can be constructed with arbitrary width, depth, and activation functions. All that is needed is a final layer that combines them with the prefactors $\chi(g)^*$.
\ No newline at end of file
-- 
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