I am currently reading the following review paper:
(1) Two Dimensional Model as a Soluble Problem for Many Fermions by Schultz et. al.
Equation (3.2), which is reproduced below, introduces the Jordan-Wigner transformations. $n$ corresponds to the site number. Suppose there are $N$ sites.
$$
c_n = \prod_{j = 1}^{n – 1} \exp(i\pi \sigma^+_j \sigma^-_j) \sigma^-_n \hspace{0.95in} c^\dagger_n = \sigma^+_n \prod_{j = 1}^{n – 1}\exp(-i\pi \sigma^+_j \sigma^-_j)
$$
Equations (3.12a) and (3.12b) provide some boundary conditions for these new operators $c$ and $c^\dagger$. These are produced below.
$$
c_{N+1} = c_1 (\text{periodic}) \hspace{0.95in} c_{N+1} = -c_1 (\text{anti-periodic})
$$
I am trying to explore what these boundary conditions on $c$ imply about the boundary conditions for spins. To explore this, I used the following
\begin{align}
\sigma^+_n \sigma^-_n &= \frac{1}{4}\big[ \sigma_n^x + i \sigma_n^y \big]\big[ \sigma_n^x – i \sigma_n^y \big] = \frac{1}{2}[1 + \sigma^z_n] \\
\exp(i\pi\sigma^z_n/2) &= i\sigma^z_n
\end{align}
to conclude
\begin{align}
c_n &= \prod_{j = 1}^{n – 1} \big[- \sigma_j^z\big]\sigma^-_n\\
c^\dagger_n &= \sigma^+_n \prod_{j = 1}^{n – 1}\big[ – \sigma^z_j\big]
\end{align}
Thus we see that $c_{N+1} = (-1)^N\prod_{j = 1}^{N} \big[\sigma_j^z\big]\sigma^-_{N+1}$. Now it seems that $c_1 = \sigma_1^-$ First Question: Is my expression for $c_1$ correct?
Now suppose $N$ is even. Hence $c_{N+1} = \prod_{j = 1}^{N} \big[\sigma_j^z\big]\sigma^-_{N+1}$ Second Question: Can I say that $\prod_{j = 1}^{N} \big[\sigma_j^z\big] = 1$? If so, why? If not, why not?
In general: What Does Periodic Boundary Conditions for Fermions Say about Boundary Conditions for Spins?
Best Answer
First, indeed $c_1=\sigma_1^-$ is correct.
Now to the general question. Note that $\sigma_n^z=(-1)^{c_n^\dagger c_n}=1-2c_n^\dagger c_n$. By inverting the transformation, we can write
$$ \sigma_n^-=\prod_{j=1}^{n-1}(-1)^{c_n^\dagger c_n} c_n. $$
In particular, take $n=N+1$, it becomes $\sigma_{N+1}^-=\prod_{j=1}^N (-1)^{c_j^\dagger c_j} c_{N+1}$.
Now observe that $(-1)^{\sum_{j=1}^N c_j^\dagger c_j}=(-1)^{N_f}$, where $N_f=\sum_{j=1}^N c_j^\dagger c_j$ is a conserved quantity, called the fermion parity, for any fermionic Hamiltonian with local interactions, so we can treat it as a c-number. In the spin representation, we have $$ (-1)^{N_f}=\prod_{j=1}^N \sigma_j^z $$ Again this must be a global symmetry of the spin model, otherwise the Jordan-Wigner transformation does not work.
Suppose we impose the following boundary condition on the fermion: $c_{N+1}=s c_1$ where $s=\pm 1$ for periodic/anti-periodic BCs. Then the boundary condition on the $\sigma_n^\pm$ operator becomes
$$ \sigma_{N+1}^\pm = (-1)^{N_f} s \sigma_1^\pm $$
This relation defines the map between boundary conditions of the fermion and spin operators. If you impose PBC on $c$'s, the spins can still have P/AP BC depending on the total fermion parity/$Z_2$ charge.