 studying states in quantum spin systems with boundary conditions were made using Poisson point processes; see \cite{Aiz1, Aiz2}. In their papers, M. Aizenman and B. Nachtergaele proposed the notion of quasi-states, a linear function that has a positive restriction to an abelian subalgebra. Although promising, no relation with KMS states has been further investigated since then. In \cite{Wer}, M. Fannes and R. F. Werner raised concerns about the validity of what they called a DLR inclusion, even suggesting some counterexamples to constructing such a theory for quantum systems. But in recent years, M. Cha, P. Naaijkens, and B. Nachtergaele \cite{Pieter} have characterized the ground states of Kitaev models using a suitable notion of boundary operators, reigniting our interest and showing that maybe a DLR theory could be developed. By analyzing the monograph by Gruber, Hintermann, and Merlini \cite{Gruber}, we noticed that the transformation group studied there had the usual spin algebra as its $\text{C}^*$-algebra, giving us a possible "quantum space" to act as a configuration space for a DLR theory. Thus by combining random representations and the groupoid model for the $\text{C}^*$-algebra of the quantum spin system, we could find a suitable generalization of the DLR equations for the quantum setting. In this paper, we show some basic properties that a quantum specification should have and show that the states compatible with it are KMS states. In fact, DLR and conformal measures already were considered in the setting of groupoid $\text{C}^*$-algebra algebras exploring the connections with the results in thermodynamic formalism on countable Markov shifts, see \cite{BEFR, BEFR2}. 
 
Our paper is divided as follows. In Section 2, we give some basic definitions, state some known-results on groupoid and their C*-algebras, and about Poisson point processes representations applied to short-range models in quantum spin systems. 
In section 3, we introduce the notions finite volume quantum Gibbs states depending on paths. We also introduce a definition of quantum specification for groupoids and show that the finite volume quantum Gibbs states satisfy them. In Section 4, we discuss how the sates introduced in previous sections relate to the known KMS states for short-range interactions. 	
	\section{Preliminary Results}
	Following \cite{Gruber}, we introduce the state space as a group. Let $G_q \subset \mathbb{S}^1$ be the subgroup of all $q$ roots of unity, i.e.,
	\[
	G_q = \{e^{2\pi i\frac{k}{q}}: k=0,1, \dots, q-1\}.
	\]
	For simplicity, we choose $q=2$. With this assumption, we recover the usual state space $\{-1,+1\}$ for Ising spin systems. The Pontryagin dual of $G_q$ can be readily shown to be isomorphic to itself. Indeed, just notice that a character $\chi$ on $G_q$ must also be a root of unit since $1=\chi(1) = \chi(e^{\frac{2\pi i}{q}})^q $. Thus there must exist $k$ such that $\chi(e^{\frac{2\pi i}{q}}) = e^{2\pi i \frac{k}{q}}$. Moreover, the group $G_q$ is isomorphic to $\mathbb{Z}_q$, the group of integers modulo $q$. Define the action $\alpha: \mathbb{Z}_q \times G_q \rightarrow G_q$ given by
	\[
	\alpha(m, e^{2\pi i \frac{k}{q}}) = e^{2\pi i \frac{k+m}{q}}.
	\]
	With this action, we can define the transformation groupoid $G_q \rtimes_\alpha \mathbb{Z}_q$. This is a compact Hausdorff \'{e}tale groupoid with respect to the product topology. The following proposition follows by standard methods:
	\begin{proposition}
		$C^*(G_q \rtimes_\alpha \mathbb{Z}_q) \simeq M_q(\mathbb{C})$.
	\end{proposition}

	\begin{comment}
	\begin{proof}
		Since the groupoid is finite, every function $f:G_q \rtimes_\alpha \mathbb{Z}_q \rightarrow \mathbb{C}$ is continuous. More than that, each function can be written as
		\[
		f = \sum_{(m,e^{2\pi i \frac{k}{q}}) \in G_q \rtimes_\alpha \mathbb{Z}_q} c_{m,k} \delta_{(m,e^{2\pi i \frac{k}{q}})},
		\]
		where $c_{m,k}\in\mathbb{C}$ and $\delta_{(m,e^{2\pi i \frac{k}{q}})}$ is the delta function. Then, to define the isomorphism we just need to define it on the deltas. Let $\pi:C(G_q \rtimes_\alpha \mathbb{Z}_q \rightarrow \mathbb{C})\rightarrow M_q(\mathbb{C})$ be defined by
		\[
		\pi(\delta_{(m,e^{2\pi i \frac{k}{q}})}) = e_{m,k}.
		\]
		where $e_{m,k}$ is the matrix with entry $1$ only on the $m$-th row and $k$-th column and extend it linearly. Notice that
		\[
		\pi(\delta_{(m,e^{2\pi i \frac{k}{q}})}^*) = \pi(\delta_{(q-m,e^{2\pi i \frac{k'}{q}})}) = e_{q-m, k'}
		\]
		where $k'= k+m \mod q$. But this is exactly $e_{m,k}*$, thus $\pi(f^*)= \pi(f)^*$. Moreover, since we have
		\begin{align*}
			\delta_{(m',e^{2\pi i \frac{k'}{q}})}*\delta_{(m,e^{2\pi i \frac{k}{q}})}(n, e^{2\pi i \frac{j}{q}}) &= \sum_{\substack{n' \in \mathbb{Z}_q \\ n'+j' = n + j \mod q}} \delta_{(m',e^{2\pi i \frac{k'}{q}})}(n',e^{2\pi i \frac{j'}{q}})\delta_{(m,e^{2\pi i \frac{k}{q}})}(n'-n,e^{2\pi i \frac{j}{q}}) \\
			&=\delta_{m'+k'= n + j \mod q}\delta_{(m,e^{2\pi i \frac{k}{q}})}(m'-n,e^{2\pi i \frac{j}{q}}) \\
			&=\delta_{m'+k'= m'-m + k \mod q},
		\end{align*}
		giving us $e_{m',k'}e_{m,k}= \delta_{k',m}e_{m',k}$.
	\end{proof}
	\end{comment}
	For each subset $\Lambda \subset \mathbb{Z}^d$, consider
	\[
	\Omega_\Lambda = \prod_{x\in \Lambda}\mathbb{Z}_n \quad \text{and} \quad \widehat{G_\Lambda} = \bigoplus_{x \in \Lambda} \mathbb{Z}_n.
	\]
	When $n=2$ we will consider $\Omega_{\Lambda} = \{-1,+1\}^\Lambda$ and use the multiplicative notation for $\mathbb{Z}_2$ since this is related to the more familiar case of two-valued spins in classical statistical mechanics. When $\Lambda = \mathbb{Z}^d$, we will write $\Omega_{\mathbb{Z}^d} \coloneqq \Omega$ and $G_{\mathbb{Z}^d}\coloneqq G$. Let the action $\alpha_\Lambda : \Omega_\Lambda \times G_\Lambda \rightarrow G_\Lambda$ be given by
	\[
	\alpha(\sigma,g) = (g_x\sigma_x)_{x\in \Lambda}, 
	\] 
	where we used implicitly that $g = \oplus_{x\in \Lambda}g_x$ and that $g_x\sigma_x$ is the usual action of $\mathbb{Z}_n$ in $\{0,1,\dots,n\}$. For the case $n=2$, this action is known as spin-flip. Our groupoid then will be the transformation groupoid 
	\[
	\mathcal{G}_\Lambda = \Omega_\Lambda \rtimes_{\alpha_\Lambda} G_\Lambda.
	\]
	Again, we will make the identification $\mathcal{G}\coloneqq \mathcal{G}_{\mathbb{Z}^d}$. We will denote the elements of the groupoid by different greek letters, to distinguish from the usual notation for configurations in classical statistical mechanics, e.g., $\sigma_\Lambda