Interpretation of Θ

Although it happens to be true that the kagome half-integer spin example has no tunable Θ𝑅 indices, Θ𝑅 indices do appear in general quantum systems.

In fact, the Θ𝑅indices and the 𝜒𝑅 indices generally appear even when the 𝐼𝐺𝐺 is trivial. For instance, we could consider a system on the kagome lattice with no on-site symmetry (i.e., remove the spin 𝑆𝑈 (2) rotation and the time-reversal symmetry in our main example), and consequently the minimal required 𝐼𝐺𝐺 is trivial. Assuming

𝐼𝐺𝐺 being trivial in this system, we will not have the 𝜂 indices but still have the 𝜒 indices. The calculation procedure of transformation rules almost remains the same as before if we simply limit all the 𝜂’s to be identity. Eventually we will arrive at Eq.(3.55) replacing all the 𝜇𝑅 by +1. Note that there is no 𝜂-ambiguities to tune away the signs for the square roots as in the half-integer spin case. In this system, apart from the 𝜒 indices, we do have two tunable Θ indices in the PEPS classification: Θ𝐶6(𝑢) = ±1 and Θ𝜎(𝑢) = ±1.

Different Θ𝑅indices can be viewed as different symmetry quantum numbers (for ei- ther on-site symmetries or space group symmetries) carried by each site tensor. These quantum numbers of the site tensors, generally speaking, directly contribute to the quantum numbers of a finite size sample. The physics of Θ𝑅 indices is similar to the physics of the so-called “fragile Mott insulator” discussed by Yao and Kivelson[165]. And similar indices in one-dimensional matrix product states have been investigated recently[52]. For instance, in the fragile Mott insulator example[165], a Mott insulator wavefunction is constructed on the checkerboard lattice which carries nontrivial point group quantum numbers on the odd-by-odd unit cell lattices. This distinguishes the fragile Mott insulator from trivial insulators which carries trivial quantum numbers on the same lattices. And such nontrivial quantum numbers can be traced back to the quantum numbers carried by the wavefunction on every square cluster on the checkerboard lattice. If one tries to use a site tensor in PEPS to represent the square cluster wavefunction, it is clear that this site tensor forms a nontrivial representation of the point group symmetry.

The physical meaning of 𝜒𝑅 may be more well-known. These are generalizations of the symmetry fractionalizations in the 2d AKLT model[1]. Let’s firstly briefly describe the PEPS construction of the 𝑆𝑂(3) symmetric spin-2 AKLT state on the square lattice. In this construction, each virtual leg forms a spin-1/2 projective rep- resentation of the 𝑆𝑂(3) symmetry group of the spin-2 system. Each site tensor is given by the only singlet state formed by the physical spin-2 and the four virtual spin-1/2’s, and each bond tensor is formed by the only spin singlet formed by the two spin-1/2’s on the two ends of the bond. Such an AKLT wavefunction can be shown

to be the unique gapped ground state of the AKLT Hamiltonian on the square lattice with periodic boundary conditions[53].

However, when the system has an open boundary, one needs to specify a symmetric boundary condition. But one encounters the following problem: each site tensor on the boundary has only three virtual spin-1/2’s and it is impossible for form a spin- singlet with the physical spin-2. Basically each site on the boundary can be viewed as a half-integer spin — which is a projective representation of the original 𝑆𝑂(3) group. One sometimes calls this phenomena as the symmetry fractionalization in 2d in the absence of topological orders. When coupled together along a translational symmetric edge, the low energy dynamics of the edge states can be effectively described by a translational symmetric half-integer spin chain, which would give a gapless excitation spectrum assuming no spontaneous translational symmetry breaking. Clearly, in the PEPS construction, the origin of such symmetry fractionalization behavior is due to the fact that projective representations appear in the virtual legs.

For an on-site symmetry 𝑅, this is exactly the physics that 𝜒𝑅 captures. For in- stance, the 𝜒𝒯 index appearing in the kagome example is really about the projective representations of the symmetry group 𝑆𝑈 (2) × 𝒯 on the virtual legs. As mentioned before, when 𝜒𝒯 = 1, the half-integer (integer) spins on the virtual legs form Kramer doublet (singlet) under the time-reversal transformation. This is the usual represen- tation of 𝑆𝑈 (2) × 𝒯 . However when 𝜒𝒯 = −1, the half-integer (integer) spins on the virtual legs form Kramer singlet (doublet) under the time-reversal transformation. This is a nontrivial projective representation of 𝑆𝑈 (2) × 𝒯 . We expect that 𝜒𝒯 = −1 would give rise to nontrivial signatures in entanglement spectra and physical edge states.

For a spatial symmetry 𝑅, the physical meaning of 𝜒𝑅is less obvious. But it’s one- dimensional analog has been investigated in the context of matrix product states[23, 118, 109, 108]. In our example, the 𝜒𝜎 is capturing similar physics in 2d kagome lattice, which basically describes how the tensor network forms possible projective representations of the spatial reflection. We speculate that nontrivial 𝜒𝜎 would give rise to signatures in entanglement spectra when the partition of the system respects

the 𝜎 reflection.

In summary, Θ𝑅 is capturing local contributions to symmetry group quantum numbers, and 𝜒𝑅 is capturing the symmetry fractionalizations not due to topological orders.