The Bethe Ansatz

The logic we have followed while first was to make the simplest nontrivial choice at each step. This leaves us with many pathways to follow when seeking other instances of Bethe Ansatz-solvable models.

We can in fact go back to the initial general considerations we started from at the beginning of the Algebraic Bethe Ansatz section. There, we introduced a complex variable \(\lambda\) which we called spectral parameter. When looking for our first explicit \(R\)-matrix of the simplest form in a.R.se, we assumed functions \(b, c\) to be analytic and bounded as the spectral parameter went to infinity. These functions had no periodicity in \(\lambda\). Complex functions can however also be (singly) periodic, and doubly periodic (triply periodic functions cannot exist (Jacobi 1835)). Let us thus consider singly-periodic functions for \(b, c\) and see where a logic as in a.R.se leads us.

Let us again make \(b, c\) single-variable functions \(b(\lambda), c(\lambda)\). We now require these functions to be periodic (say in the imaginary direction), and to remain bounded in the other (real) direction. Inspired by Rr, we can try to generalize the "function" \(\lambda\) to one which displays imaginary-direction periodicity, for example \(\sinh \lambda\) (the linear function then being reobtained in the limit of small spectral parameters). We thus attempt the form

\begin{equation} R (\lambda) = \left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & b (\lambda) & c (\lambda) & 0 \\ 0 & c (\lambda) & b (\lambda) & 0 \\ 0 & 0 & 0 & 1 \end{array} \right), \hspace{1cm} b (\lambda) = \frac{\sinh \lambda}{\sinh (\lambda + \eta)}, \hspace{1cm} c (\lambda) = \frac{\sinh \eta}{\sinh (\lambda + \eta)}. \tag{a.r}\label{a.r} \end{equation}

One can verify that this choice still satisfies constraints bc12 (from the inversion relation RRe1) and bc345 (from the Yang-Baxter relation YB). We thus have a different, bona-fide \(R\) matrix to play with, which (following tradition) we call the trigonometric \(R\)-matrix.

The logic of the Algebraic Bethe Ansatz can again be applied, following the previous example of the rational (\(XXX\)) case. First, the entirety of the operator algebra a.R.s.o remains unchanged, since this is only dependent on the morphology of the \(R\) matrix, which is unchanged as compared to the rational case. Adopting the same convention for the reference state (pseudovacuum) pv, all the considerations about eigenstates of the transfer matrix in a.R.s.t and about dual states in a.R.s.d also remain unchanged.

Changes however occur while following the finding an explicit model logic. Choosing again the homogeneous limit with \(\xi = \eta/2\), but leaving \(\eta\) here as a free parameter, adR1 yields

\begin{equation} a (\lambda) = 1, \hspace{1cm} d(\lambda) = b(\lambda - \eta/2)^N = \left[ \frac{\sinh (\lambda - \eta/2)}{ \sinh (\lambda + \eta/2)} \right]^N. \tag{a.ad}\label{a.ad} \end{equation}

The Bethe equations BER1 take the form

\begin{equation*} \left[\frac{\sinh(\lambda_j + \eta/2)}{\sinh (\lambda_j - \eta/2)} \right]^N = \prod_{k (\neq j) = 1}^M \frac{\sinh (\lambda_j - \lambda_k + \eta)}{\sinh(\lambda_j - \lambda_k - \eta)}. \end{equation*}

The eigenvalue of the transfer matrix tauev becomes

\begin{equation} \tau (\lambda | \{ \lambda_j \}) = \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda + \eta)}{\sinh(\lambda_j - \lambda)} + \left[\frac{\sinh(\lambda - \eta/2)}{\sinh(\lambda + \eta/2)}\right]^N \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda - \eta)}{\sinh(\lambda_j - \lambda)}. \tag{a.te}\label{a.te} \end{equation}

The first few conserved charges can be computed as

\begin{equation} P = i \ln \tau (\lambda)|_{\lambda = \eta/2} \tag{a.q1}\label{a.q1} \end{equation}

giving eigenvalue

\begin{equation} P (\{ \lambda_j \}_M) = \sum_{j=1}^M k(\lambda_j), \hspace{5mm} k(\lambda) = i\ln \frac{\sinh(\lambda + \eta/2)}{\sinh(\lambda - \eta/2)}. \tag{a.q1e}\label{a.q1e} \end{equation}

The Hamiltonian, under an appropriate choice of the constant prefactor, becomes that of the periodic \(S=1/2\) Heisenberg chain with anisotropic coupling,

\begin{equation} H_{XXZ} = \frac{\sinh \eta}{2} \frac{d}{d\lambda} \ln \tau (\lambda) |_{\lambda = \eta/2} = \sum_{j=1}^N \left[S^x_j S^x_{j+1} + S^y_j S^y_{j+1} + \Delta (S^z_j S^z_{j+1} - \frac{1}{4}) \right] \tag{a.q2}\label{a.q2} \end{equation}

in which \(\Delta = \cosh \eta\). This is precisely the model which we treated using coordinate Bethe Ansatz in section c.h.

The energy eigenvalue of a state is

\begin{equation} E (\{ \lambda_j \}_M) = \sum_{j=1}^M e_1 (\lambda_j), \hspace{5mm} e_1(\lambda) = \frac{\sinh^2 \eta}{\cosh 2\lambda - \cosh \eta}. \tag{a.q2e}\label{a.q2e} \end{equation}

Regimes

The trigonometric \(R\)-matrix construction above allows us in fact to cover a variety of different regimes of physical interest depending on the value of the anisotropy \(\Delta\). Throughout these notes, we shall make use of the following terminology and notational conventions:

Regime	\(\hat{z}\) order	Region	Parametrization	Representation	Domain	Rapidities
Axial antiferro	\(\uparrow\downarrow\)	\(\Delta > 1\)	\(\eta \rightarrow \eta\)	\(\Delta = \cosh \eta\)	\(\eta > 0\)	\(\lambda \rightarrow i\lambda\)
Planar antiferro	\(\uparrow\downarrow\) -like	\(0 < \Delta < 1\)	\(\eta \rightarrow -i\zeta\)	\(\Delta = \cos \zeta\)	\(0 < \zeta < \pi/2\)	\(\lambda \rightarrow \lambda\)
Planar ferro	\(\uparrow\uparrow\) -like	\(-1 < \Delta < 0\)	\(\eta \rightarrow -i(\pi - \check{\zeta})\)	\(\Delta = -\cos \check{\zeta}\)	\(0 < \check{\zeta} < \pi/2\)	\(\lambda \rightarrow i\pi/2 + \lambda\)
Axial ferro	\(\uparrow\uparrow\)	\(\Delta < -1\)	\(\eta \rightarrow -i\pi - \check{\eta}\)	\(\Delta = -\cosh \check{\eta}\)	\(\check{\eta} > 0\)	\(\lambda \rightarrow i (\pi/2 + \lambda)\)

Note that the ferro/antiferro qualifier indicates how the \(\hat{z}\) components of spins prefer to align with each other locally. The entire \( -1 \leq \Delta \leq 1\) regime is gapless and has no true long-range order. Correlations along \(\hat{z}\) are nonetheless locally antiferromagnetic for \(0 < \Delta \leq 1\) and ferromagnetic for \(-1 \leq \Delta < 0\). Correlations in the \(\hat{x}-\hat{y}\) plane are locally antiferromagnetic throughout the gapless regime. Despite the risk of confusion to the uninitiated, we take the liberty of talking of planar (anti)ferro regimes without implying that there is any order present.

We can thus compile the following set of basic equations for the various regimes.

Axial antiferromagnet \(\Delta > 1\)

\begin{equation*} a(\lambda) = 1, ~~b(\lambda) = \frac{\sin\lambda}{\sin(\lambda - i\eta)}, ~~c(\lambda) = \frac{\sinh \eta}{\sin(\lambda -i\eta)}, ~~d(\lambda) = \left[\frac{\sin(\lambda + i\frac{\eta}{2})}{\sin(\lambda -i\frac{\eta}{2})}\right]^N \end{equation*} \begin{equation*} \left[ \frac{\sin (\lambda_j + i\eta/2)}{\sin(\lambda_j - i\eta/2)} \right]^N = \prod_{k \neq j}^M \frac{\sin(\lambda_j - \lambda_k + i\eta)}{\sin(\lambda_j - \lambda_k - i\eta)} \end{equation*} \begin{equation} \tau (\lambda | \{ \lambda_j \}) = \prod_{j=1}^M \frac{\sin(\lambda_j - \lambda - i\eta)}{\sin(\lambda_j - \lambda)} + \left[\frac{\sin(\lambda + i\eta/2)}{\sin(\lambda -i \eta/2)}\right]^N \prod_{j=1}^M \frac{\sin(\lambda_j - \lambda +i\eta)}{\sin(\lambda_j - \lambda)}. \end{equation} \begin{equation*} k (\lambda) = \frac{1}{i} \ln \frac{\sin(\lambda + i\eta/2)}{\sin(\lambda - i\eta/2)} \end{equation*} \begin{equation*} e (\lambda) = \frac{-\sinh^2 \eta}{\cosh \eta - \cos 2\lambda_j} \end{equation*}

Planar antiferromagnet \(0 < \Delta < 1\)

\begin{equation*} a(\lambda) = 1, ~~b(\lambda) = \frac{\sinh\lambda}{\sinh(\lambda - i\zeta)}, ~~c(\lambda) = \frac{i\sin \zeta}{\sinh(\lambda -i\zeta)}, ~~d(\lambda) = \left[\frac{\sinh(\lambda + i\frac{\zeta}{2})}{\sinh(\lambda -i\frac{\zeta}{2})}\right]^N \end{equation*} \begin{equation*} \left[ \frac{\sinh (\lambda_j + i\zeta/2)}{\sinh(\lambda_j - i\zeta/2)} \right]^N = \prod_{k \neq j}^M \frac{\sinh(\lambda_j - \lambda_k + i\zeta)}{\sinh(\lambda_j - \lambda_k - i\zeta)} \end{equation*} \begin{equation} \tau (\lambda | \{ \lambda_j \}) = \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda -i \zeta)}{\sinh(\lambda_j - \lambda)} + \left[\frac{\sinh(\lambda +i \zeta/2)}{\sinh(\lambda -i \zeta/2)}\right]^N \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda +i \zeta)}{\sinh(\lambda_j - \lambda)}. \end{equation} \begin{equation*} k (\lambda) = \frac{1}{i} \ln \frac{\sinh(\lambda + i\zeta/2)}{\sinh(\lambda - i\zeta/2)} \end{equation*} \begin{equation*} e (\lambda) = \frac{-\sin^2 \zeta}{\cosh 2\lambda_a - \cos \zeta} \end{equation*}

Planar ferromagnet \(-1 < \Delta < 0\)

\begin{equation*} a(\lambda) = 1, ~~b(\lambda) = \frac{-\cosh\lambda}{\cosh(\lambda + i\check{\zeta})}, ~~c(\lambda) = \frac{\sin\check{\zeta}}{\cosh(\lambda +i\check{\zeta})}, ~~d(\lambda) = \left[-\frac{\sinh(\lambda - i\frac{\check{\zeta}}{2})}{\sinh(\lambda +i\frac{\check{\zeta}}{2})}\right]^N \end{equation*} \begin{equation*} (-1)^N \left[ \frac{\sinh (\lambda_j + i\check{\zeta}/2)}{\sinh(\lambda_j - i\check{\zeta}/2)} \right]^N = \prod_{k \neq j}^M \frac{\sinh(\lambda_j - \lambda_k + i\check{\zeta})}{\sinh(\lambda_j - \lambda_k - i\check{\zeta})} \end{equation*} \begin{equation} \tau (\lambda | \{ \lambda_j \}) = (-1)^M \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda +i \check{\zeta})}{\sinh(\lambda_j - \lambda)} + (-1)^M\left[-\frac{\sinh(\lambda -i \check{\zeta}/2)}{\sinh(\lambda +i \check{\zeta}/2)}\right]^N \prod_{j=1}^M \frac{\sinh(\lambda_j - \lambda -i \check{\zeta})}{\sinh(\lambda_j - \lambda)}. \end{equation} \begin{equation*} k (\lambda) = \pi + \frac{1}{i} \ln \frac{\sinh(\lambda + i\check{\zeta}/2)}{\sinh(\lambda - i\check{\zeta}/2)} \end{equation*} \begin{equation*} e (\lambda) = \frac{\sin^2 \check{\zeta}}{\cosh 2\lambda_a - \cos \check{\zeta}} \end{equation*}

Axial ferromagnet \(\Delta < -1\)

\begin{equation*} a(\lambda) = 1, ~~b(\lambda) = \frac{\cos\lambda}{\cos(\lambda + i\check{\eta})}, ~~c(\lambda) = \frac{i\sinh \check{\eta}}{\cos(\lambda +i\check{\eta})}, ~~d(\lambda) = \left[-\frac{\sin(\lambda - i\frac{\check{\eta}}{2})}{\sin(\lambda +i\frac{\check{\eta}}{2})}\right]^N \end{equation*} \begin{equation*} (-1)^N \left[ \frac{\sin (\lambda_j + i\check{\eta}/2)}{\sin(\lambda_j - i\check{\eta}/2)} \right]^N = \prod_{k \neq j}^M \frac{\sin(\lambda_j - \lambda_k + i\check{\eta})}{\sin(\lambda_j - \lambda_k - i\check{\eta})} \end{equation*} \begin{equation} \tau (\lambda | \{ \lambda_j \}) = (-1)^M\prod_{j=1}^M \frac{\sin(\lambda_j - \lambda + i\check{\eta})}{\sin(\lambda_j - \lambda)} + (-1)^M\left[-\frac{\sin(\lambda - i\check{\eta}/2)}{\sin(\lambda +i \check{\eta}/2)}\right]^N \prod_{j=1}^M \frac{\sin(\lambda_j - \lambda -i\check{\eta})}{\sin(\lambda_j - \lambda)}. \end{equation} \begin{equation*} k (\lambda) = \pi + \frac{1}{i} \ln \frac{\sin(\lambda + i\check{\eta}/2)}{\sin(\lambda - i\check{\eta}/2)} \end{equation*} \begin{equation*} e (\lambda) = \frac{\sinh^2 \check{\eta}}{\cosh \check{\eta} - \cos 2\lambda_j} \end{equation*}

Duality relation between \(\Delta\) and \(-\Delta\)

The above equations display the relation between antiferromagnetic and ferromagnetic cases, which is related to the following unitary transformation rotating each other spin by \(\pi\) along \(\hat{z}\):

\[ D \equiv \prod_{a=1}^{\lfloor N/2 \rfloor} e^{-i\pi S^z_{2a-1}} \]

This transformation acts on the Hamiltonian as

\begin{equation*} D H (\Delta; \kappa) D^{-1} = -H(-\Delta; (-1)^N\kappa) \end{equation*}

in which \(\kappa = 1\) (resp. \(-1\)) represent periodic (resp. anti-periodic) boundary conditions. For an even-length chain, the spectra of antiferromagnetic and ferromagnetic versions are thus precisely the flipped versions of each other, and in particular the highest-energy state of the antiferromagnetic chain is then the ground state of the ferromagnetic version (and conversely).

Note that under the same transformation, each quasiparticle picks up another shift of \(\pi\) in momentum.

Limiting cases

Isotropic antiferromagnet \(\Delta = 1\)

The antiferromagnetic \(XXX\) model can be recovered by taking \(\lambda \rightarrow \eta \lambda\) and \(\eta \rightarrow 0^+\) starting from the axial antiferromagnet, or alternately \(\lambda \rightarrow \zeta \lambda\) and \(\zeta \rightarrow 0^+\) in the planar antiferromagnet. This yields

\begin{equation*} \left[ \frac{\lambda_j + i/2}{\lambda_j - i/2} \right]^N = \prod_{k \neq j}^M \frac{\lambda_j - \lambda_k + i}{\lambda_j - \lambda_k - i} \end{equation*} \begin{equation*} k (\lambda) = \frac{1}{i} \ln \frac{\lambda + i/2}{\lambda - i/2} \end{equation*} \begin{equation*} e (\lambda) = \frac{-2}{4 \lambda_j^2 + 1} \end{equation*}

Isotropic ferromagnet \(\Delta = -1\)

The ferromagnetic \(XXX\) model can be recovered by taking \(\lambda \rightarrow \eta \lambda\) and \(\eta \rightarrow 0^+\) starting from the axial ferromagnet, or alternately \(\lambda \rightarrow \check{\zeta} \lambda\) and \(\check{\zeta} \rightarrow 0^+\) in the planar ferromagnet. This yields

\begin{equation*} (-1)^N \left[ \frac{\lambda_j + i/2}{\lambda_j - i/2} \right]^N = \prod_{k \neq j}^M \frac{\lambda_j - \lambda_k + i}{\lambda_j - \lambda_k - i} \end{equation*} \begin{equation*} k (\lambda) = \pi + \frac{1}{i} \ln \frac{\lambda + i/2}{\lambda - i/2} \end{equation*} \begin{equation*} e (\lambda) = \frac{2}{4 \lambda_j^2 + 1} \end{equation*}

Ising antiferromagnet \(\Delta \rightarrow \infty\)

Ising ferromagnet \(\Delta \rightarrow -\infty\)