Spin Topological Field Theory and Fermionic Matrix Product States

TL;DR

The paper develops a spin-TQFT-based framework that connects 2d spin topological orders to fermionic MPS, enabling a concrete classification of 1d fermionic SRE phases with symmetry. It shows that annulus evaluations yield generalized MPS in NS and Ramond sectors, and that stacking corresponds to the supertensor product, with the nontrivial Majorana chain arising from ${ m C ell}(1)$. The fermionic SRE phases with symmetry $({rak G},p)$ are classified by pairs $(oldsymbol{eta},oldsymbol{eta})$ and, in the split case, a $oldsymbol{ u}$ parameter, with a group law that matches spin cobordism data $ ext{Ω}^2_{ ext{Spin}}(BG_b)$; in the non-split case the extra ${rak G}$-Spin structure enters via a modified cocycle condition. The results unify fermionic MPS, spin-TQFT, and cobordism-based classifications, providing explicit state-sum constructions, Hamiltonians for trivial and nontrivial SRE phases, and an explicit framework for equivariant fermionic MPS and their stackings.

Abstract

We study state-sum constructions of G-equivariant spin-TQFTs and their relationship to Matrix Product States. We show that in the Neveu-Schwarz, Ramond, and twisted sectors, the states of the theory are generalized Matrix Product States. We apply our results to revisit the classification of fermionic Short-Range-Entangled phases with a unitary symmetry G and determine the group law on the set of such phases. Interesting subtleties appear when the total symmetry group is a nontrivial extension of G by fermion parity.

Figures (13)

• Figure 1: Black arrows are edge orientations, and red arrows are special edges. All of the spin signs are $-1$ except possibly the one on the $N$-to-$1$ edge, which is $+1$ in the NS sector and $-1$ in the R sector.
• Figure 2:
• Figure 3: The cylinder partition sum $Z(C)$ factors as a signed sum of four colored diagrams: $\sigma(\beta_1)C_1+\sigma(\beta_2)C_2+\sigma(\beta_3)C_3+\sigma(\beta_4)C_4=C_1+\eta C_2+C_3-\eta C_4$. Magenta lines indicate odd edges.
• Figure 4: $\left\langle{\psi_\text{even}}\right\rvert \IfNoValueTF{}{}{} =\sigma(\beta_1) \left\langle{\psi_1}\right\rvert \IfNoValueTF{}{}{} +\sigma(\beta_2) \left\langle{\psi_2}\right\rvert \IfNoValueTF{}{}{} = \left\langle{\psi_1}\right\rvert \IfNoValueTF{}{}{} +\eta \left\langle{\psi_2}\right\rvert \IfNoValueTF{}{}{}$
• Figure 5: $\left\langle{\psi_\text{odd}}\right\rvert \IfNoValueTF{}{}{} =\sigma(\beta_3) \left\langle{\psi_3}\right\rvert \IfNoValueTF{}{}{} +\sigma(\beta_4) \left\langle{\psi_4}\right\rvert \IfNoValueTF{}{}{} = \left\langle{\psi_3}\right\rvert \IfNoValueTF{}{}{} +\eta \left\langle{\psi_4}\right\rvert \IfNoValueTF{}{}{}$