and see that $$H_F=-\frac{i}2\sum_{x}\psi^{\dag a\alpha}_x  \, U_{x,ab} \, \psi^{b\alpha}_{x+1}+h.c. + m\sum_x \, (-1)^x\, \psi^{\dag a\alpha}_x \, \left(\sigma_3\right)_{\alpha\beta} \, \psi^{a\beta}_x.  \label{eq:H_F_stag}$$ Here, while the first term is invariant under a $U(2)$ group, the mass term is invariant only under a $U(1)\times U(1)$. In our actual calculations we will drop the second component of this spin-diagonalized Hamiltonian (the $\alpha=2$), and thus work with staggered fermions [@Susskind]. To go back to Dirac fermions is easy and for our purpose it is useful to note that the continuum chiral condensate and baryon number local densities, at position $X$, $\left(\bar \psi\psi\right)(X)$ and $\left(\psi^\dag\psi\right)(X)$, are given, up to overall renormalization factors, by [@Susskind] $$\begin{aligned}
\left(\psi^\dag \psi\right)(X)&=&\frac1{\sqrt{2}}\left(\psi^\dag_x \psi_x + \psi^\dag_{x-1} \psi_{x-1} \right),\label{psibarpsi}\\
\left(\bar \psi \psi\right)(X)&=&\frac1{\sqrt{2}}\left(\psi^\dag_x \psi_x - \psi^\dag_{x-1} \psi_{x-1} \right),\label{psidagpsi}\end{aligned}$$ where we take $X/a=x$, and $x$ denotes an even site.[^5]

Axial gauge fixing {#Haxial}
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In this section we show how to fix the axial gauge. Since axial gauge fixing in the Hamiltonian approach is less familiar than it is in the Euclidean approach we begin with an explanation of the general strategy.

Temporal gauge fixing left us with a Hamiltonian that is invariant under time-independent gauge transformations discussed in the previous section. With no external charges 