Excited states in spin chains from conformal blocks

We develop a method of constructing excited states in one-dimensional spin chains which are derived from the $\text{SU}{\left(2\right)}_1$ Wess-Zumino-Witten conformal field theory (CFT) using a parent Hamiltonian approach. The resulting systems are equivalent to the Haldane-Shastry model. In our ansatz, correlation functions between primary fields correspond to the ground state of the spin system, whereas excited states are obtained by insertion of descendant fields. Our construction is based on the current algebra of the CFT and emphasizes the close relation between the spectrum of the spin system and the underlying CFT. This general structure might imply that the method could be applied to a wider range of model systems.

Figure 1

The Riemann sphere with $N$ spins at the equator (unit circle) and two additional spins, one at the north pole ($z=0$) and one at the south pole ($z=\infty$).

Figure 2

Analytically calculated energy levels and their spin content as a function of the number of current operators $k$. $\tilde{E}$ is defined as $\tilde{E}=(E-(N-1)k)/k$ for $k>0$ and $\tilde{E}=E=0$ for $k=0$. The three rows in the boxes next to the energy level correspond to the calculated values of (1) the energy $\tilde{E}$ (2) the spin content (3) a value for the number of spins $N^{\prime }$ so that the corresponding state is null for all systems with $N\le N^{\prime }$ spins. The inequalities indicate for which $N$ the condition $E\ge 0$ is satisfied. For a given number of spins $N$, all eigenvalues that lie in bands where the inequality is not satisfied correspond to null states. By applying additional current operators to these null states, we could identify further null states that do not violate the energy condition.

Figure 3

Numerically calculated spectra for the odd-$N$ spin chain ($N=7$, left panels) and the even-$N$ spin chain ($N=8$, right panels). The upper panels show data obtained by applying current operators to the states ${\psi }_0^{s_{\infty }}$ and ${\psi }_0$, respectively. In the lower panels, the current operators were applied to the states ${\psi }_0^{s_0}$ and ${\psi }_0^{s_0{s}_{\infty }}$, respectively. The horizontal axis shows the number of current operators $k$ that are present in the corresponding states. The numbers above the levels indicate their degeneracy. Notice that the spectrum is the same in the upper and lower plots, but a given state does not necessarily appear at the same number of current operators.

Figure 4

Maximal number of current operator insertions $k_{\max }$ of order $-1$ needed to obtain the spectrum of the Hamiltonian dependent on the number of spins $N$. The data are consistent with $k_{\max }={(N/2)}^2$.