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Critical properties of the XXZ model with long-range interactions on the double chain

The $XXZ$ model $(s = 1/2)$ in a transverse field on a double chain with a uniform long-range interaction among the $z$ components of the spins is considered. The nearest-neighbour interactions are restricted to the components in the $xy$ plane and to the spins within the same chain leg, such that the Hamiltonian is given by $H = -\sum_{m=1}^{2} J_{m} \sum_{j=1}^{N}(S_{m,j}^{x}S_{m, j+1}^{x} + S_{m,j}^{y}S_{m,j+1}^{y}) - \frac{I}{N}\sum_{m,n=1}^{2} \sum_{j,k=1}^{N}S_{m,j}^{z}S_{n,k}^{z}-h\sum_{m=1}^{2} \sum_{j=1}^{N}S_{m,j}^{z}$, where $N$ is the number of sites of the lattice and $m,n$ $(m,n = 1, 2)$ label the chain legs. The model is solved exactly by introducing the Jordan-Wigner and integral Gaussian transformations, which map the Hamiltonian in a non-interacting fermion system and corresponds to an extension of the model recently studied by the authors for a single chain. The equation of state is obtained in closed form, and the critical classical (at $T > 0$) and quantum (at $T = 0$) behaviours can be determined exactly. The quantum critical surface is determined in the space generated by the transverse field and interaction parameters, and the crossover lines separating the different critical regimes are also obtained. It is also shown that, differently from the results obtained for the single chain, the system can present multiple quantum transitions.