Exactly solvable one-dimensional quantum models with gamma matrices

In this paper we write exactly solvable generalizations of one-dimensional quantum $XY$ and Ising-like models by using $2^d$-dimensional gamma matrices as the degrees of freedom on each site. We show that these models result in quadratic Fermionic Hamiltonians with Jordan-Wigner-like transformations. We illustrate the techniques using a specific case of four-dimensional gamma matrices and explore the quantum phase transitions present in the model.

Figure 1

Sublattice structure for generalized $XY\mathrm{\Gamma }$ models. Each lattice site (gray block), denoted by the index $a$, consists of $d$ sublattice points (white circles), marked by the index $i$.

Figure 2

Jordan string of ${\mathrm{\Gamma }}^{2d+1}$ operators corresponding to the fermionic operator ${\chi }_a^{\mu }$, spanning the lattice sites $1,2,...,a-1$.

Figure 3

Interactions in the Hamiltonian for $d=2$. Each lattice site $a$ (gray block) can house four $\mathrm{\Gamma }$ matrices $\{{\mathrm{\Gamma }}_a^1,{\mathrm{\Gamma }}_a^2,{\mathrm{\Gamma }}_a^3,{\mathrm{\Gamma }}_a^4\}, a=1,2,...,N$. The lines in the diagram represent the couplings $J_{\mu \nu }$ involving ${\mathrm{\Gamma }}_a^{\mu }$ and ${\mathrm{\Gamma }}_{a+1}^{\nu }, \mu ,\nu \in \{1,2,3,4\}$ [see (17)].

Figure 4

Variation of (a) $\partial E_g/\partial h$ and (b) ${\partial }^2{E}_g/\partial h^2$ as functions of $h$, setting $\mu =2, {\mathcal{A}}_{12}=1, {\mathcal{A}}_{13}=1, {\mathcal{A}}_{14}=1, {\mathcal{A}}_{34}=1, {\mathcal{S}}_{11}=1, {\mathcal{S}}_{33}=1, {\mathcal{S}}_{13}=1$, and ${\mathcal{S}}_{14}=1$. The quantity ${\partial }^2{E}_g/\partial h^2$ exhibits a divergence at $h_c=0.350781$, indicating a quantum phase transition.