Infinite matrix product states, conformal field theory, and the Haldane-Shastry model

We generalize the matrix product states method using the chiral vertex operators of conformal field theory and apply it to study the ground states of the XXZ spin chain, the $J_1\text{−}{J}_2$ model and random Heisenberg models. We compute the overlap with the exact wave functions, spin-spin correlators, and the Renyi entropy, showing that critical systems can be described by this method. For rotational invariant ansatzs we construct an inhomogenous extension of the Haldane-Shastry model with long-range exchange interactions.

Figure 1

(a) Overlap of the variational ansatz, Eq. (5), with the exact GS wave function of Hamiltonian (7) in the sector with $S^z=0$ for $N=20$ spins; (b) optimal value of $\alpha$. The dashed line shows the curve $\Delta =-\text{cos}\left(2\pi \alpha \right)$.

Figure 2

(Color online) [(a)–(c)] Renyi entropy $S_L^{\left(2\right)}$ for a chain with $N=200$ sites and several values of $\alpha ∊\left(0,1/2\right]$. The dashed line shows the CFT prediction for $c=1$. [(d) and (e)] Correlator $C_n^{zz}$ for several values of $\alpha$. In (d) the stars indicate the exact result.

Figure 3

(Color) $J_1\text{−}{J}_2$ model (a) values of ${\varphi }_0$ and $\delta$ for which the energy is minimal; (b) overlap of the iMPS state with the exact GS. The values of the $z_n$ in the complex plane are qualitatively depicted for each region.

Figure 4

(Color) Renyi entropy $S_L^{\left(2\right)}$ as a function of $x_L=\text{log}\left[\text{sin}\left(\pi L/N\right)\right]$. We have subtracted from the entropy $\text{log}\left(2\right)x_L/3$ in order to compare with the prediction (Refs. 20, 21). Upper (red), medium (green), and lower (blue) curves correspond to $\delta =0.75,0.5,0.1$. The dashed line corresponds to $x_L/4$. In (b) we have averaged over consecutive points and plotted for low values of $x_L$.