A more precise analogy is this: The degrees of freedom of the critical long-wavelength theory annihilate a nontrivial quantum vacuum in our case, while the critical theory of absorbing-state transitions in the directed percolation universality class act on a trivial vacuum.

This follows from straightforward manipulations of $\gamma$ matrices: $$\begin{gathered} (V_l^{†}{V}_r+V_r^{†}{V}_l)={\alpha }_{\mu }^l{\alpha }_{\nu }^r{\gamma }_{\mu }{\gamma }_{\nu }+{\alpha }_{\nu }^r{\alpha }_{\mu }^l{\gamma }_{\nu }{\gamma }_{\mu } \\ =2{\alpha }_{\mu }^l{\delta }_{\mu \nu }{\alpha }_{\nu }^r+{\alpha }_{\mu }^l{\alpha }_{\nu }^r{\sigma }_{\mu \nu }+{\alpha }_{\nu }^r{\alpha }_{\mu }^l{\sigma }_{\nu \mu } \\ =2{\alpha }_{\mu }^l{\delta }_{\mu \nu }{\alpha }_{\nu }^r+({\alpha }_{\mu }^l{\alpha }_{\nu }^r-{\alpha }_{\nu }^r{\alpha }_{\mu }^l){\sigma }_{\mu \nu } \\ =2{\alpha }_{\mu }^l{\delta }_{\mu \nu }{\alpha }_{\nu }^r, \end{gathered}$$where ${\sigma }_{\mu \nu }=\frac{1}{2}[{\gamma }_{\mu },{\gamma }_{\nu }]$.

Moving from the lattice to the continuum changes the topology of the base manifold. However, all symmetry classes can be captured in continuum Dirac models [63]; see Ref. [50] for a discussion of Dirac stationary states. In particular, the microscopic model considered here has a continuum limit preserving the topological properties, as the winding density is concentrated around zero momentum.

To do this explicitly, consider the basis where ${\gamma }_{d+1}={\sigma }_z\otimes \mathbb{1}$, and then the chiral representation can be obtained by considering a unitary transformation $\mathrm{\Lambda }=\exp (\mathrm{i}[\pi /4]{\sigma }_y\otimes \mathbb{1})$ acting on the $\gamma$ matrices: $$\begin{gathered} {\gamma }_{d+1}\to {\mathrm{\Lambda }}^{-1}{\gamma }_{d+1}\mathrm{\Lambda }={\sigma }_x\otimes \mathbb{1}, \\ {\gamma }_i\to {\mathrm{\Lambda }}^{-1}{\gamma }_i\mathrm{\Lambda }={\gamma }_{\alpha }\forall i=1,\dots ,d, \\ {\gamma }_{d+2}\to {\mathrm{\Lambda }}^{-1}{\gamma }_{d+2}\mathrm{\Lambda }=-{\sigma }_z\otimes \mathbb{1}. \end{gathered}$$

In the absence of dissipation and/or presence of strong translation invariance, the collective excitations of the bosons are instead described by propagating sound modes.

Since the product of the generators ${\gamma }_{d+2}\sim \mathbb{1}$ is trivial in odd dimensions, the quadratic action in odd dimensions can also be interpreted as a representative of near-critical topological insulators in class A.

Note that the presence of the $d_{d+1}$ coefficient, while compatible with the symmetries of the microscopic model, renders $D$ nonpositive semidefinite over a certain momentum range $q$ and can be set to zero by hand to ensure stability of the dark state.

There are two equivalent ways of formulating the action of thermal symmetry on the fields; cf. Ref. [37]. Here, we chose the one that emphasizes the similarity to the action of time reversal on field operators.