Geometric Criterion for Solvability of Lattice Spin Systems

Exactly solvable models have been played important roles in the understanding of physics in strongly correlated systems. In particular, exactly solvable lattice spin models have revealed many important phenomena. For instance, solving the 2d Ising model exactly, Onsager Onsager (1944) showed the presence of ferromagnetic phase transition in spin systems for the first time, which is one of milestones in statistical physics. Since Onsager’s work, other lattice spin models were solved exactly, such as the Potts model, the hard-hexagon model, and so on Baxter (2016); Wu (1971); Kadanoff and Wegner (1971). More recently, exactly solvable models also have disclosed exotic quantum phases in strongly correlated systems, such as spin liquid phases with non-abelian anyon excitations Kitaev (2006).

Quantum solvable lattice spin models are classified into three types. The first one has a Hamiltonian of which terms commute with each other, which includes the 2d Kitaev’s toric code Kitaev (2006); Kitaev and Laumann (2009), the X-cube model Castelnovo et al. (2010); Nandkishore and Hermele (2019), and so on. The second one has special symmetries such as Lie groups or quantum groups. This type includes the 1d Heisenberg model and the XXZ model Pasquier and Saleur (1990). Then, the last one can be transformed into free-fermion systems Jordan and Wigner (1928); Nambu (1995); Lieb et al. (1961); Niemeijer (1967); Katsura (1962); Perk et al. (1975); Minami (2016, 2017); Imamura and Katsura ; Prosko et al. (2017); Kaufman and Onsager (1949); Perk et al. (1975, 1984); Perk (2017); Kitaev (2006); Feng et al. (2007); Chen and Hu (2007); Chen and Nussinov (2008); Kitaev and Laumann (2009); Minami (2019). For instance, both the 1d XY model and the 1d transverse field Ising model can be converted into free-fermion systems by using the Jordan-Wigner transformation. Another example is the Kitaev’s honeycomb lattice model, which is transformed into a free fermion system by adapting a redundant representation of spins with Majorana operators.

In this paper, we present a simple criterion for the third type of solvability of lattice spin systems. Our criterion is based on the graph theory and the simplicial homology. For a lattice spin system with an algebra with a graphical representation, we show that the null space of the adjacency matrix of the graph provides conserved quantities of the system. Furthermore, when the graph belongs to a class of simplicial complexes, we reveal that the Hamiltonian is mapped to a bilinear form of Majorana fermions, from which the full spectrum of the system is obtained. We also find a relation between the conserved quantities and the first homology group of the graph. Based on the relation, we interpret the conserved quantities as flux excitations. We apply our criterion for several known solvable spin systems including the 1d transverse-field Ising chain, the 1d XY model, the 2d Kitaev honeycomb model, and the 3d diamond lattice model. We also present new solvable models on a 1d tri-junction, 2d and 3d fractal lattices, and the 3d cubic lattice.

The rest of this paper is organized as follows. In Sec. II, we present the main results. We introduce lattice models which satisfy a class of algebras. Representing the algebra in the form of a graph, we present Theorems that give the criterion of solvability in terms of the graph theory and the simplicial homology. In Sec. III, we illustrate our criterion by applying it to the 1d transverse-field Ising model, the XY model, the Kitaev honeycomb model and so on. We also provide new solvable models in Sec.IV In Sec. V, we present proofs of Theorems in Sec.II. We finally give discussions in Sec.VI.

First, we present our main results in this paper, which are summarized in three Theorems. The proofs of these Theorems will be given in Sec.V.

In this paper, we consider a class of Hamiltonians $H$ that satisfy the following properties.

• •

  $H$ has the form of $H=\sum_{j=1}^{n}\lambda_{j}h_{j}$ with coefficients $\lambda_{j}\in\mathbb{R}$ and operators $h_{j}$ $(j=1,\dots,n)$.

• •

  The operators $h_{j}$ obey $h_{j}^{2}=1$, $h_{j}^{\dagger}=h_{j}$, and $h_{j}h_{k}=\epsilon_{jk}h_{k}h_{j}$ with $\epsilon_{ij}=\pm 1$.

The second property requires that $h_{j}$s commute or anti-commute with each other. The operators $h_{j}$ generate an algebra ${\cal A}$ on $\mathbb{C}$, which we call the bond algebra (BA) Nussinov and Ortiz (2009); Cobanera et al. (2011). To represent the BA ${\cal A}$ visually, we introduce a graph ${\cal G(A)}$ as follows.

• •

  Put $n$ vertices in general position and place $h_{i}$ on the $i$-th vertex.

• •

  When $h_{i}$ and $h_{j}$ anti-commute (commute) with each other, we draw (do not draw) a line between the vertices with $h_{j}$ and $h_{k}$.

The resulting graph compactly encodes the information of the commutativity among $h_{j}$s. We call the graph ${\cal G(A)}$ as commutativity graph (CG) Wang and Hazzard (2019) of $\cal{A}$. The CG ${\cal G(A)}$ has an algebraic representation with an adjacency matrix $M({\cal A})$. The adjacency matrix $M({\cal A})$ is a real symmetric $n\times n$ matrix of which elements indicate whether pairs of vertices are adjacent or not in ${\cal G(A)}$: The diagonal elements of $M({\cal A})$ are zero and the $(i,j)$-component is chosen to be $1$ ($0$) if $i$- and $j$-th vertices in ${\cal G(A)}$ are connected (not connected) by a line. The multiplication and the addition for $M({\cal A})$ are defined as a matrix on the binary field $\mathbb{F}_{2}$, i.e. a matrix with entries $0$ or $1$, which satisfy $0+0=0$, $0+1=1$, $1+0=1$, and $1+1=0$.

Using $M({\cal A})$, we present our first main result. A product $h_{j_{1}}h_{j_{2}}\cdots h_{j_{k}}$ conserves if it commutes with any $h_{j}$ in $H$. We find that such conserved quantities in ${\cal A}$ can be counted by using the adjacency matrix $M({\cal A})$. More precisely, we have Theorem 1: {itembox}[l]Theorem 1 Let $\cal{A}$ be the BA of a Hamiltonian $H=\sum_{j=1}^{n}\lambda_{j}h_{j}$, $\cal{G(A)}$ be the corresponding CG of $\cal{A}$, and $M({\cal A})$ be the adjacency matrix of ${\cal G(A)}$. Then, the dimension of the kernel space of $M({\cal A})$ coincides with the total number of conserved quantities in the form of $h_{j_{1}}\cdots h_{j_{k}}$.

The CG also enables us to characterize the BA geometrically. For this purpose, we adapt the notion of simplex: A $d$-simplex is a $d$-dimensional polyhedron having the minimal number of vertices, namely $d+1$ vertices. For instance, a 0-simplex is a vertex, a 1-simplex is a line, a 2-simplex is a triangle, a 3-simplex is a tetrahedron, and so on. In particular, we consider a special set of simplices, which we call point-connected simplices: Let us consider a set of simplices $S=\{s_{1},\dots,s_{m}\}$ and let $V$ be a set consisting of all vertices of $s_{\alpha}\in S$ ($\alpha=1,\dots,m$). Then, we call $S$ as point-connected if $S$ is connected and any pair of $s_{\alpha},s_{\beta}\in S$ ($\alpha\neq\beta$) having a non-empty intersection shares only a single vertex $v\in V$ (Namely $s_{\alpha}\cap s_{\beta}=\{v\}$). Furthermore, we call $S$ as single-point-connected if any vertex $v\in V$ is shared by at most two different $s_{\alpha}$s. Adding all faces of $s_{\alpha}\in S$ $(\alpha=1,\dots,m)$ into $S$, we obtain a simplicial complex $K(S)$, which we dub single-point-connected simplicial complex (SPSC). See Fig. 1.

Now we describe Theorem 2.

Remarks are in order. (i) Without loss of generality, we can assume that any vertex $v$ of $s_{\alpha}\in S$ is shared by another $s_{\beta}\in S$ ($\beta\neq\alpha$): If not, we can add $v$ itself into $S$ as a 0-simplex to meet the assumption. (ii) Under this assumption, the Majorana operator $\varphi_{\alpha}$ in Theorem 2 can be assigned to the simplex $s_{\alpha}\in S$. Then, $\varphi_{\alpha}$ and $\varphi_{\beta}$ in Eq.(4) are given by those on the simplices that share the vertex with $h_{j}$. (iii) The sign factors $\epsilon_{\alpha\beta}$ in Eq.(4) are determined as follows. First, we use a sign ambiguity in Majorana operators: We can multiply $\varphi_{\alpha}$ by $-1$ without changing the (anti-)commutation relations between them. Using this gauge transformation, we can change the $m-1$ relative signs between $\varphi_{\alpha}$, which enables us to erase $m-1$ $\epsilon_{\alpha\beta}$s. There still, however, remain $n-m+1$ $\epsilon_{\alpha\beta}$s. The following Theorem 3 tells us that these remaining sign factors are determined by conserved quantities.

[l]Theorem 3 Let ${\cal A}$ be the BA obeying the same assumption of Theorem 2. Then, $K(S)$ has independent $n-m+1$ non-contractible loops as a simplicial complex on $\mathbb{F}_{2}$. Correspondingly, there exist $n-m+1$ conserved quantities that determine the remaining $n-m+1$ sign factors.

It should be noted here that for each non-contractible loop, there remains a sign factor that cannot be removed by the gauge transformation. To count the number of independent non-contractible loops in $K(S)$, we calculate the homology group $H_{q}(K(S))$ of $K(S)$. As we shall show in Sec.V, a straightforward calculation shows that $H_{q\geq 2}(K(S))=0$ and ${\rm dim}H_{1}(K(S))=n-m+1$ when $K(S)$ is a SPSC. The latter result implies that $K(S)$ has $n-m+1$ independent non-contractible loops. We also find that each loop gives a conserved quantity: Take non-contractible loops as small as possible, then the product of all $h_{j}$s on each loop gives a conserved quantity. Furthermore, we find that the conserved quantity reduces to the sign factor on the loop by rewriting it in terms of Majorana fermions in Eq.(4).

Theorems 2 and 3 imply that $H$ is solvable as a free Majorana system: We can obtain the full spectrum of $H$ just by diagonalizing the free Majorana Hamiltonian.

We summarize the relation between the original spin model, the CG, the SPSC, and the free-fermion representation in Table 1.

In this section, we apply our theory to known solvable models, which confirms the validity of our criterion. There are also a lot of solvable lattice models by our method. For example, we have checked our method in models in Refs. Minami (2019, 2017); Nussinov and Ortiz (2009); Shi et al. (2009); Imamura and Katsura ; Prosko et al. (2017); Lee et al. (2007); Yu and Wang (2008); Chen and Kapustin (2019).