The Bethe Ansatz

Let us now go further towards defining actual models using this logic. Now that the \(R\)-matrix is fully specified, all commutation relations ABCDcr are fixed. We have however not yet defined the transfer matrix tau or rather the monodromy matrix TeABCD besides these conditions on the \(A,B,C,D\) operators. Neither have we defined the Hilbert space itself.

Let us start by looking at constructing a monodromy matrix explicitly. We are thus looking for a representation of \(T\) in RTTeTTR. One obvious possible route comes from looking at YB: if we simply take the monodromy matrix to be constructed directly from the \(R\)-matrix, we will be able to satisfy RTTeTTR. The \(R\)-matrix we have defined is however a matrix in \({\cal A}\otimes{\cal A}\), where we have chosen the auxiliary space to be two-dimensional. This suggests to start from basic two-dimensional Hilbert spaces isomorphic to auxiliary space, \({\cal H}_j \sim {\cal A}\), for a set of labels \(j\).

Let us define a matrix \(L_j\) living in \({\cal A} \otimes {\cal H}_j\) as

\begin{equation} L_j (\lambda, \xi_j) = R_{aj} (\lambda - \xi_j) \tag{LeR}\label{LeR} \end{equation}

where the subscript \(aj\) on the \(R\)-matrix means that the matrix acts in \({\cal A} \otimes {\cal H}_j\), and in which we have introduced parameters \(\xi_j\) called inhomogeneity parameters. We can then construct a nontrivial Hilbert space by tensoring these basic Hilbert spaces, \({\cal H} = \otimes_{j=1}^N {\cal H}_j\). We can then construct a monodromy matrix as

\begin{equation} T(\lambda) = L_N (\lambda, \xi_N) L_{N-1} (\lambda, \xi_{N-1}) ... L_2 (\lambda, \xi_2) L_1 (\lambda, \xi_1). \tag{TprodL}\label{eq:TisprodL} \end{equation}

This matrix obeys RTTeTTR by construction, in view of its construction as a product of \(R\)-matrices LeR and the Yang-Baxter relation YB.

We then have explicitly (using Rr and LeR in TprodL)

\begin{align*} T(\lambda) &= \left[b(\lambda - \xi_N) {\bf 1} + c(\lambda - \xi_N) \mathbb{P} \right]_{aN} \left[ b(\lambda - \xi_{N-1}) {\bf 1} + c(\lambda - \xi_{N-1}) \mathbb{P} \right]_{aN-1} ... \left[ ... \right]_{a1} \nonumber \\ &= (\lambda - \xi_N + \eta)^{-N} \left[ (\lambda - \xi_N) {\bf 1} + \eta \mathbb{P} \right]_{aN} \left[ ... \right]_{aN-1} ... \left[ ... \right]_{a1}. \end{align*}

If we define without loss of generality the pseudovacuum \(| 0 \rangle\) as the vector \((1, 0, 0, ...)\), direct calculations show that this choice of monodromy matrix then imposes

\begin{equation} a(\lambda) = 1, \hspace{1cm} d(\lambda) = \prod_{j=1}^N b(\lambda - \xi_j). \tag{adR1}\label{adR1} \end{equation}

Let us from now on put \(\xi_j = \xi ~\forall j\) (homogeneous limit) for simplicity, so \(d(\lambda) = b(\lambda - \xi)^N\). Evaluating the monodromy matrix at \(\lambda \rightarrow \xi\) gives

\begin{equation*} T (\xi) = \mathbb{P}_{aN} \mathbb{P}_{aN-1} ... \mathbb{P}_{a1} = \mathbb{P}_{a1} \mathbb{P}_{1N} \mathbb{P}_{1N-1} ... \mathbb{P}_{12} \end{equation*}

where we have used the property \(\mathbb{P}_{ab} \mathbb{P}_{bc} = \mathbb{P}_{ac} \mathbb{P}_{ab}\) of the permutation operator to shift \(\mathbb{P}_{a1}\) from the right to the left. Taking the trace over auxiliary space, we get the transfer matrix evaluated at \(\lambda \rightarrow \xi\):

\begin{equation*} \tau (\xi) = \left(\mbox{Tr}_{\cal A} \mathbb{P}_{a1} \right) \mathbb{P}_{1N} \mathbb{P}_{1N-1} ... \mathbb{P}_{12} = \mathbb{P}_{1N} \mathbb{P}_{1N-1} ... \mathbb{P}_{12} \equiv U \end{equation*}

(using \(\mbox{Tr}_{\cal A} \mathbb{P}_{aj} = {\bf 1}_j\)), where \(U\) is the cyclic shift operator. Since \(U^N = 1\), this operator has eigenvalues \(e^{ik}\) with \(k = \frac{2\pi}{N} n\), \(n \in \mathbb{Z}\). We thus construct our first integral of motion using Qn,

\begin{equation} Q_0 = \ln \tau (\xi) = i P \tag{RrQ0}\label{RrQ0} \end{equation}

where \(P\) is the generator of translations, and thus is our momentum operator.

For the next integral of motion, we need

\begin{equation*} \frac{d}{d\lambda} \ln \tau (\lambda) |_{\lambda = \xi}. \end{equation*}

We have the building block

\begin{equation*} \frac{d}{d\lambda} L_j (\lambda, \xi)|_{\lambda = \xi} = \frac{1}{\eta} \left[ {\bf 1} - \mathbb{P} \right]_{aj} \end{equation*}

so

\begin{align*} \eta \frac{d}{d\lambda} \tau (\lambda) |_{\lambda = \xi} &= \mbox{Tr}_{\cal A} \left( \left[ {\bf 1} - \mathbb{P}\right]_{aN} \mathbb{P}_{aN-1} ... \mathbb{P}_{a1} + \mathbb{P}_{aN} \left[ {\bf 1} - \mathbb{P}\right]_{aN-1}\mathbb{P}_{aN-2} ... \mathbb{P}_{a1} + ... \right. \nonumber \\ &\hspace{5mm}\left. + \mathbb{P}_{aN} ... \mathbb{P}_{a3} \left[ {\bf 1} - \mathbb{P}\right]_{a2} \mathbb{P}_{a1} + \mathbb{P}_{aN} ... \mathbb{P}_{a2} \left[ {\bf 1} - \mathbb{P}\right]_{a1} \right) \nonumber \\ &= (\mbox{Tr}_{\cal A} \mathbb{P}_{a1}) \left( \left[ {\bf 1} - \mathbb{P}\right]_{1N} \mathbb{P}_{1N-1}...\mathbb{P}_{12} + \mathbb{P}_{1N} \left[ {\bf 1} - \mathbb{P}\right]_{1N-1} \mathbb{P}_{1N-2}...\mathbb{P}_{12} + ... \right. \nonumber \\ &\hspace{5mm}\left. + \mathbb{P}_{1N} ... \mathbb{P}_{13} \left[ {\bf 1} - \mathbb{P}\right]_{12} \right) + \mathbb{P}_{NN-1} \mathbb{P}_{NN-2} ... \mathbb{P}_{N2} \left[ {\bf 1} - \mathbb{P}\right]_{N1} (\mbox{Tr}_{\cal A} \mathbb{P}_{aN}) \end{align*}

(in the second line, we have treated the last term slightly differently). Now because of tauc, we have \(\left[ \frac{d}{d\lambda} \tau(\lambda), \tau(\lambda') \right] = 0\), and therefore can write

\begin{equation*} \eta \frac{d}{d\lambda} \ln \tau (\lambda) |_{\lambda = \xi} = \eta ~\tau^{-1} (\lambda) \frac{d}{d\lambda} \tau(\lambda) |_{\lambda = \xi} = \sum_{j=1}^N \mathbb{P}_{j j+1} - {\bf 1}_{j j+1}. \end{equation*}

Using the identity

\begin{equation} \sum_a \sigma^a_j \otimes \sigma^a_l = 2 \mathbb{P}_{j l} - {\bf 1}_{j l}, \tag{sigsig2Pm1}\label{sigsig2Pm1} \end{equation}

we thus get

\begin{equation*} \eta \frac{d}{d\lambda} \ln \tau (\lambda) |_{\lambda = \xi} = \sum_{j=1}^N \frac{1}{2} \left[ \sigma_j \cdot \sigma_{j+1} - 1 \right] \end{equation*}

so our second conserved charge is

\begin{equation} Q_1 = \frac{\eta}{2} \frac{d}{d\lambda} \ln \tau (\lambda) |_{\lambda = \xi} = \sum_{j=1}^N {\bf S}_j \cdot {\bf S}_{j+1} - \frac{1}{4} = H_{XXX}. \tag{RrQ1}\label{RrQ1} \end{equation}

We have thus derived the Heisenberg model from basic algebraic principles. Such an equation is called a trace identity: a conserved charge is obtained from taking derivatives of the generating function given, the latter being the log of the transfer matrix.

States of the form prodBpv diagonalize the transfer matrix (and thus all conserved charges simultaneously) provided the Bethe equations BER1 are satisfied for the functions Rr and adR1, i.e. if

\begin{equation*} \left[\frac{\lambda_j - \xi + \eta}{\lambda_j - \xi} \right]^N = \prod_{l (\neq j) = 1}^M \frac{\lambda_j - \lambda_k + \eta}{\lambda_j - \lambda_k - \eta} \end{equation*}

(note however that we have not yet dealt with the important question of completeness). The eigenvalue of the transfer matrix on these states is

\begin{equation*} \tau (\lambda | \{ \lambda_j \}) = \prod_{j=1}^M \frac{\lambda_j - \lambda + \eta}{\lambda_j - \lambda} + \left[\frac{\lambda - \xi}{\lambda - \xi + \eta}\right]^N \prod_{j=1}^M \frac{\lambda_j - \lambda - \eta}{\lambda_j - \lambda}. \end{equation*}

The eigenvalue equation for the momentum operator is thus

\begin{equation*} P | \{ \lambda_j \}_M \rangle = P (\{ \lambda_j \}) | \{ \lambda_j \}_M \rangle, \hspace{1cm} P (\{ \lambda_j \}) = -i \ln \tau(\xi | \{ \lambda_j \}) = -i \sum_{j=1}^M \ln \frac{\lambda_j - \xi + \eta}{\lambda_j - \xi}. \end{equation*}

For the energy, we can similarly use the trace identity RrQ1 to get

\begin{equation*} E (\{ \lambda_j \}) = \frac{\eta}{2} \frac{d}{d\lambda} \ln \tau(\lambda | \{ \lambda_j \} )|_{\xi} = \frac{\eta^2}{2} \sum_{j=1}^M \frac{1}{(\lambda_j - \xi) (\lambda_j - \xi + \eta)}. \end{equation*}

We have not yet specified the \(R\)-matrix parameter \(\eta\) or the inhomogeneity parameter \(\xi\). Our rapidity is until now an arbitrary parametrization whose scale and origin we can choose at will, so the value of \(\eta\) is immaterial (as long as it doesn't vanish, in which case our \(R\)-matrix would become trivial). On the other hand, we want our momentum and energy to be manifestly real. A convenient choice is thus to take

\begin{equation} \eta = i, \hspace{2cm} \xi = \eta/2 = i/2. \label{eq:etaxi} \end{equation}

In the next section, we collect for convenience our final form for these equations.

General remarks

As we went along, we made a number of choices which simplified the construction. We can of course make these choices otherwise. One particularly important choice was to select the \(L\) operators such that they became the permutation matrix when evaluated at a specific value of the spectral parameter. This allowed us to extract the translation operator; the trace identities we used, involving the logarithm of a product of \(L\) operators, then ultimately led to conserved charges (e.g. the Hamiltonian) expressed in terms of a sum of (quasi-)local terms. This is physically reasonable, but in no way necessary to construct a Bethe Ansatz solvable model.