The Bethe Ansatz

Solving the Schrödinger equation c.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{}\label{} \end{align}

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

\begin{equation} \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} \end{equation}

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

\begin{equation} \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} \end{equation}

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, 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

\begin{equation} \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{}\label{} \end{equation}

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

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

or in logarithmic form,

\begin{equation} 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} \end{equation}

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

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

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 \(\Delta = 1\).

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24