# The Bethe Ansatz

#### Solving the Schrödinger equationc.h.s

Let us now construct the eigenstates of the $$XXZ$$ Hamiltonian xxz.h (or the equivalent xxz.hp). We proceed sector by sector in magnetization.

0 down spins

The fully (up) polarized sector is trivial: $$S^z_{\rm tot} = \frac{N}{2}$$, and this sub-Hilbert space is spanned by the single state xxz.r of zero energy, $$H |0 \rangle = 0$$.

1 down spin

Only slightly less trivial is the $$M = 1$$ subsector. Let us use the Ansatz $$|\Psi_1 \rangle = \sum_{j=1}^N \Psi_1 (j) | j \rangle$$ in which $$| j \rangle = S^-_j | 0 \rangle$$ and $$\Psi_1 (j + N) = \Psi_1 (j)$$ to ensure periodicity. Projecting the Schrödinger equation $$H |\Psi_1 \rangle = E_1 |\Psi_1 \rangle$$ onto the bra $$\langle j |$$ yields the conditions $\frac{J}{2} \left(\Psi_1 (j - 1) + \Psi_1 (j + 1)\right) = (E_1 + J\Delta) \Psi_1 (j),$ which are solved by the free wave Ansatz $$\Psi_1(j) = e^{i k j}$$ provided the energy is

\begin{equation*} E_1 = J (\cos k -\Delta). \end{equation*}

Periodicity quantizes the momentum according to $$e^{ikN} = 1$$. The allowed values of momentum are therefore given by

\begin{equation*} k = 2\pi \tilde{I}/N, \hspace{1cm} \tilde{I} = 0, 1, ..., N - 1 \end{equation*}

(or any equivalent coverage of the Brillouin zone). For lattice size $$N$$, there are thus $$N$$ linearly independent solutions, a number corresponding to the Hilbert space dimensionality $$\left( \begin{array}{c} N \\ 1 \end{array} \right) = N$$. This simple Ansatz therefore generates all the wavefunctions in this subspace.

2 down spins

The $$M = 2$$ subsector is less trivial, but provides the necessary building blocks for generalization to arbitrary $$M \leq \frac{N}{2}$$. We look for an eigenstate of the form $$| \Psi_2 \rangle = \sum_{j_1 < j_2} \Psi_2 (j_1, j_2) | j_1, j_2 \rangle$$ where $$|j_1, j_2\rangle = S^-_{j_1} S^-_{j_2} |0\rangle$$. Projecting the Schrödinger equation onto the bra $$\langle j_1, j_2|$$ yields the bulk conditions (note that $$| j_1, j_2 \rangle$$ is identically zero for $$j_1 = j_2$$ mod $$N$$, so $$\Psi(j, j)$$ can actually be chosen arbitrarily)

\begin{align} \frac{J}{2} \left[ \Psi_2 (j_1 - 1, j_2) + \Psi_2 (j_1, j_2 - 1) + \Psi_2 (j_1 + 1, j_2) + \Psi_2 (j_1, j_2 + 1)\right] \nonumber \\ = (E_2 + 2J\Delta) \Psi_2 (j_1, j_2), \hspace{0.5cm} 2 < j_1 + 1 < j_2 < N, \nonumber \\ \frac{J}{2} \left[ \Psi_2 (j_1, j_2 + 1) + \Psi_2 (j_1 - 1, j_2) \right] = (E_2 + J\Delta) \Psi_2 (j_1, j_2), \nonumber \\ 2 < j_1 + 1 = j_2 < N. \tag{xxz.se}\label{xxz.se} \end{align}

To solve these, we generalize the $$M = 1$$ case and look for a wavefunction in the form of a free wave Ansatz,

$$\Psi_2 (j_1, j_2) = A_{12} e^{i k_1 j_1 + i k_2 j_2} + A_{21} e^{i k_2 j_1 + i k_1 j_2}, \hspace{5mm} 1 \leq j_1 < j_2 \leq N. \tag{xxz.psi2}\label{xxz.psi2}$$

The equation for $$j_1 + 1 < j_2$$ gives

\begin{equation*} E_2 = J(\cos k_1 + \cos k_2 - 2\Delta). \end{equation*}

The second equation then yields $\frac{A_{21}}{A_{12}} = - \frac{1 + e^{i (k_1 + k_2) } - 2\Delta e^{i k_2}}{1 + e^{i (k_1 + k_2)} - 2\Delta e^{i k_1}} \equiv -e^{i\phi(k_1, k_2)}$ where we have defined the scattering phase shift function

$$\phi (k_1, k_2) =2 \mbox{atan} \frac{\Delta \sin \frac{k_1 - k_2}{2}}{\cos \frac{k_1 + k_2}{2} - \Delta \cos \frac{k_1 - k_2}{2}} \tag{xxz.phi}\label{xxz.phi}$$

which will play an immensely important role in the following. Using the phase shift function, the (unnormalized) trial wavefunction can now be written

\begin{equation*} \Psi_2 (j_1, j_2) = e^{i k_1 j_1 + i k_2 j_2 - \frac{i}{2} \phi(k_1, k_2)} - e^{i k_2 j_1 + i k_1 j_2 + \frac{i}{2} \phi(k_1, k_2)}. \end{equation*}

We do not have our eigenstates yet: in addition to the bulk conditions xxz.se, there are also four equations of boundary type which are equivalent to the periodicity conditions

\begin{equation*} \Psi_2(j_2, j_1 + N) = \Psi_2 (j_1, j_2), \hspace{1cm} \Psi_2(j_2 - N, j_1) = \Psi_2 (j_1, j_2). \end{equation*}

These conditions yield the so-called Bethe equations, constraining the allowed values of the bare momenta $$k_i$$:

\begin{equation*} e^{i k_1 N} = - e^{-i \phi(k_1, k_2)}, \hspace{1cm} e^{i k_2 N} = -e^{+i \phi(k_1, k_2)}. \end{equation*}

For classifying the solutions, it is more conveninent (as was done for $$M = 1$$) to take the logarithm:

\begin{equation*} N k_1 + \phi (k_1, k_2) = 2\pi \tilde{I}_1, \hspace{1cm} N k_2 - \phi(k_1, k_2) = 2\pi \tilde{I}_2 \end{equation*}

where $$\tilde{I}$$ are half-odd integers.

General $$M$$

For $$XXZ$$ with $$M \leq N/2$$ down spins, the Bethe Ansatz reads

$$\Psi_M(j_1, ..., j_M) = \prod_{M \geq a > b \geq 1} sgn(j_a - j_b) \times \sum_{P_M} (-1)^{[P]} e^{i \sum_{a=1}^M k_{P_a} j_a + \frac{i}{2} \sum_{M \geq a > b \geq 1} sgn(j_a - j_b) \phi (k_{P_a}, k_{P_b})} \tag{xxz.ba}\label{xxz.ba}$$

where the two-particle scattering phase shift is given by xxz.phi. This is an eigenstate provided the bare momenta $$k_a$$ fulfill the set of $$M$$ coupled Bethe equations

$$e^{ik_a N} = (-1)^{M-1} e^{-i \sum_b \phi (k_a, k_b)}, \hspace{0.3cm} a = 1, ..., M, \tag{xxz.be}\label{xxz.be}$$

or in logarithmic form,

$$N k_a + \sum_b \phi (k_a, k_b) = 2\pi \tilde{I}_a, \hspace{0.5cm} a = 1, ..., M, \tag{xxz.bel}\label{xxz.bel}$$

where $$\tilde{I}_a$$ are half-odd integers if $$M$$ is even, and integers if $$M$$ is odd.

The energy and momentum of such an eigenstate are

$$E_M = J \sum_{a=1}^M (\cos k_a - \Delta), \hspace{10mm} P = \sum_a k_a. \tag{xxz.ep}\label{xxz.ep}$$

The physics of Bethe Ansatz-solvable spin-$$1/2$$ chains depends crucially on the value of the anisotropy parameter $$\Delta$$. We begin by a thorough examination of the most important case, namely the isotropic antiferromagnet case with $$\Delta = 1$$.