The Bethe Ansatz

Let us proceed further by considering the \(M = 2\) subspace \({\cal H}^N_2\). As we shall see, this leads to a rich structure with substantial (though overcomable) complications, and provides some necessary building blocks for generalization to higher values of \(M\).

This subspace is spanned by the \({\scriptstyle \left(\begin{array}{c} N \\ 2 \end{array} \right) = N(N-1)/2}\) states

\[ |j_1, j_2\rangle \equiv S^-_{j_1} S^-_{j_2} |0\rangle, \hspace{10mm} {\scriptstyle 1 \leq j_1 < j_2 \leq N}. \]

We adopt the Ansatz

\[ ~| \Psi_2 \rangle = \sum_{1 \leq j_1 < j_2 \leq N} \Psi_{j_1, j_2} \,| j_1, j_2 \rangle \]

parametrizing the wavefunction in terms of \(N(N-1)/2\) amplitudes \(\Psi_{j_1, j_2}\) for \(1 \leq j_1 < j_2 \leq N\). Amplitudes \(\Psi_{j_1, j_2}\) for \(j_1 > j_2\) are not required; that said, in view of \([S^-_{j_1}, S^-_{j_2}] = 0\), it would be natural to define them as symmetrical with respect to exchange of coordinates, since any antisymmetrical part would be associated to a null state. Separately, amplitudes \(\Psi_{j,j}\) are also associated to null states \((S^-_j)^2 | 0\rangle\) so their value is physically inconsequential; we will (as Bethe did) exploit this fact below to simplify bookkeeping.

We will here consider the Schrödinger equation for xxz.h but with a slightly generalized boundary term, namely one with a twist, implemented by the unitary rotation \(R^z_N (\alpha) \equiv e^{-i\alpha S^z_N}\):

\begin{align*} H_{\scriptscriptstyle bdry} &= h_{N,1} (\alpha) \equiv R^z_N (\alpha) \,h_{N,1} \,(R^z_N (\alpha))^{-1} \\ &= J \left[\frac{1}{2} \left(e^{i\alpha} S^+_1 S^-_N + e^{-i\alpha} S^-_1 S^+_N \right) + \Delta S^z_1 S^z_N\right]. \tag{xxz.htbc}\label{xxz.htbc} \end{align*}

The periodic case is recovered by setting \(\alpha = 0\). The twist parameter will prove useful for future considerations.

Projecting the Schrödinger equation for xxz.h onto bras \(\langle j_1, j_2|\) yields a number of different cases, depending on where our "particles" (downturned spins) are located.

In the bulk

For \(j_1, j_2\) in the bulk such that only xxz.hbulk acts nontrivially, two cases arise. The first is when the two down spins are separated,

\begin{align*} &{\scriptstyle (i)~1 < j_1; ~j_1+1 < j_2; ~j_2 < N ~(\text{separated, in bulk;}~(N-3)(N-4)/2~\text{equations})}: \nonumber\\ &~\frac{J}{2} \left[ \Psi_{j_1 - 1, j_2} + \Psi_{j_1 + 1, j_2} + \Psi_{j_1, j_2 - 1} + \Psi_{j_1, j_2 + 1} \right] = (E_2 + 2J\Delta) \Psi_{j_1, j_2}. \tag{xxz.s2i}\label{xxz.s2i} \end{align*}

The second case covers neighbouring down spins:

\begin{align*} &{\scriptstyle (ii)~1 < j_1; ~j_1 + 1 (= j_2) < N ~(\text{neighbouring, in bulk;}~(N-3)~\text{equations})}: \nonumber\\ &~\frac{J}{2} \left[ \Psi_{j_1 - 1, j_1+1} + \Psi_{j_1, j_1+2} \right] = (E_2 + J\Delta) \Psi_{j_1, j_1+1}. \tag{xxz.s2ii}\label{xxz.s2ii} \end{align*}

We now proceed as Bethe originally did (see equation 6 of 1931.Bethe.ZP.71), by noticing that if we were to add

\[ \Psi_{j_1+1,j_1+1} + \Psi_{j_1, j_1} \]

to the left-hand side of xxz.s2ii, then it would become like the left-hand side of xxz.s2i for \(j_2=j_1+1\). Since these are precisely the free-to-set null state amplitudes mentioned above, we are thus at liberty to choose

\begin{align*} &{\scriptstyle 2 \leq j \leq N-2 ~(\text{null, in bulk;}~(N-3)~\text{equations})}: \nonumber\\ &~\Psi_{j + 1, j + 1} + \Psi_{j, j} = 2\Delta \Psi_{j, j +1} \tag{xxz.s2x}\label{xxz.s2x} \end{align*}

by which all instances of xxz.s2ii become of the same form as xxz.s2i, namely (highlighting our \( {\color{NavyBlue} \text{neutral additions}} \))

\begin{align*} &{\scriptstyle (ii)~1 < j_1; ~j_1 + 1 (= j_2) < N ~(\text{neighbouring, in bulk;}~(N-3)~\text{equations})}: \nonumber\\ &~\frac{J}{2} \left[ \Psi_{j_1 - 1, j_1+1} + {\color{NavyBlue} \Psi_{j_1+1,j_1+1} + \Psi_{j_1,j_1} +} \Psi_{j_1, j_1+2} \right] = (E_2 + J\Delta + {\color{NavyBlue} J\Delta}) \Psi_{j_1, j_1+1}. \tag{xxz.s2ii2}\label{xxz.s2ii2} \end{align*}

Our two bulk cases thus merge into the extended bulk system

\begin{align*} &{\scriptstyle 1 < j_1 < j_2 < N ~(\text{in bulk;}~(N-2)(N-3)/2~\text{equations})}: \nonumber\\ &~\frac{J}{2} \left[ \Psi_{j_1 - 1, j_2} + \Psi_{j_1 + 1, j_2} + \Psi_{j_1, j_2 - 1} + \Psi_{j_1, j_2 + 1}\right] = (E_2 + 2J\Delta) \Psi_{j_1, j_2}. \tag{xxz.s2bulk}\label{xxz.s2bulk} \end{align*}

At the boundary

Let us now deal with the Schrödinger equation projections involving xxz.hpbc. These yield the following cases:

\begin{align*} &{\scriptstyle (iii)~1 = j_1, 2 < j_2 < N ~(j_1~\text{at boundary},~j_2~\text{separated in bulk;}~(N-3)~\text{equations})}: \nonumber\\ &~\frac{J}{2} \left[ e^{i\alpha} \Psi_{j_2, N} + \Psi_{2, j_2} + \Psi_{1, j_2 - 1} + \Psi_{1, j_2 + 1}\right] = (E_2 + 2J\Delta) \Psi_{1, j_2}, \nonumber \\[1em] &{\scriptstyle (iv)~1 < j_1 < N-1, j_2 = N ~(j_1~\text{separated in bulk},~j_2~\text{at boundary};~(N-3)~\text{equations})}: \nonumber \\ &~\frac{J}{2} \left[ \Psi_{j_1 - 1, N} + \Psi_{j_1 + 1, N} + \Psi_{j_1, N - 1} + e^{-i\alpha} \Psi_{1, j_1}\right] = (E_2 + 2J\Delta) \Psi_{j_1, N}, \nonumber \\[1em] &{\scriptstyle (v)~j_1 = 1, j_2 = 2 ~(\text{stacked at left boundary})}: \nonumber \\ &~\frac{J}{2} \left[ e^{i\alpha} \Psi_{2, N} + \Psi_{1, 3} \right] = (E_2 + J\Delta) \Psi_{1, 2}, \nonumber \\[1em] &{\scriptstyle (vi)~j_1 = N-1, j_2 = N ~(\text{stacked at right boundary})}: \nonumber \\ &~\frac{J}{2} \left[ \Psi_{N-2, N} + e^{-i\alpha} \Psi_{1, N-1} \right] = (E_2 + J\Delta) \Psi_{N-1, N}, \nonumber \\[1em] &{\scriptstyle (vii)~j_1 = 1, j_2 = N ~(\text{neighbours across boundary})}: \nonumber \\ &~\frac{J}{2} \left[ \Psi_{2, N} + \Psi_{1, N-1} \right] = (E_2 + J\Delta) \Psi_{1, N}. \tag{xxz.s2bdry}\label{xxz.s2bdry} \end{align*}

Note that (iii)-(vii) represent \({\scriptstyle 2N-3}\) equations. Together with the \({\scriptstyle (N-2)(N-3)/2}\) equations xxz.s2bulk, we thus have \({\scriptstyle N(N-1)/2}\) equations for the same number of unknowns, namely the physical state amplitudes \(\Psi_{j_1,j_2}, ~{\scriptstyle 1 \leq j_1 < j_2 \leq N}\).

Our boundary equations look like a mess. To simplify bookkeeping, as per the \(M=1\) case, we introduce "ghost" sites at \(j=0, N+1\) and rewrite our system in a better form, highlighting the \({\color{BrickRed} \text{replaced}}, ~\text{their}~{\color{ForestGreen} \text{replacements}}~\text{and the}~{\color{NavyBlue} \text{neutral additions}}\) we operate:

\begin{align*} &{\scriptstyle (iii)~1 = j_1, 2 < j_2 < N ~(j_1~\text{at boundary},~j_2~\text{separated in bulk;}~(N-3)~\text{equations})}: \nonumber\\ &~\frac{J}{2} \left[ \underbrace{\color{ForestGreen}\Psi_{0,j_2}}_{\color{BrickRed} e^{i\alpha} \Psi_{j_2, N}} + \Psi_{2, j_2} + \Psi_{1, j_2 - 1} + \Psi_{1, j_2 + 1}\right] = (E_2 + 2J\Delta) \Psi_{1, j_2}, \nonumber \\[1em] &{\scriptstyle (iv)~1 < j_1 < N-1, j_2 = N ~(j_1~\text{separated in bulk},~j_2~\text{at boundary};~(N-3)~\text{equations})}: \nonumber \\ &~\frac{J}{2} \left[ \Psi_{j_1 - 1, N} + \Psi_{j_1 + 1, N} + \Psi_{j_1, N - 1} + \underbrace{\color{ForestGreen} \Psi_{j_1, N+1}}_{\color{BrickRed} e^{-i\alpha} \Psi_{1, j_1}}\right] = (E_2 + 2J\Delta) \Psi_{j_1, N}, \nonumber \\[1em] &{\scriptstyle (v)~j_1 = 1, j_2 = 2 ~(\text{stacked at left boundary})}: \nonumber \\ &~\frac{J}{2} \left[ \underbrace{\color{ForestGreen} \Psi_{0,2}}_{\color{BrickRed} e^{i\alpha} \Psi_{2, N}} + {\color{NavyBlue} \Psi_{2, 2} + \Psi_{1,1}} + \Psi_{1, 3} \right] = (E_2 + J\Delta {\color{NavyBlue} +J\Delta}) \Psi_{1, 2}, \nonumber \\[1em] &{\scriptstyle (vi)~j_1 = N-1, j_2 = N ~(\text{stacked at right boundary})}: \nonumber \\ &~\frac{J}{2} \left[ \Psi_{N-2, N} + {\color{NavyBlue} \Psi_{N,N} + \Psi_{N-1,N-1}} + \underbrace{\color{ForestGreen} \Psi_{N-1,N+1}}_{\color{BrickRed} e^{-i\alpha} \Psi_{1, N-1}} \right] = (E_2 + J\Delta {\color{NavyBlue} +J\Delta}) \Psi_{N-1, N}, \nonumber \\[1em] &{\scriptstyle (vii)~j_1 = 1, j_2 = N ~(\text{neighbours across boundary})}: \nonumber \\ &~\frac{J}{2} \left[ {\color{NavyBlue} \Psi_{0,N} +} \Psi_{2, N} + \Psi_{1, N-1} {\color{NavyBlue} + \Psi_{1,N+1}} \right] = (E_2 + J\Delta {\color{NavyBlue} +J\Delta}) \Psi_{1, N},\\ &\hspace{20mm}\text{either}~\downarrow \times e^{i\alpha}\\ &~\frac{J}{2} \left[ \overbrace{\color{ForestGreen} \Psi_{0,0}}^{\color{BrickRed} e^{i\alpha} \Psi_{0,N}} + \overbrace{\color{ForestGreen} \Psi_{0,2}}^{\color{BrickRed} e^{i\alpha} \Psi_{2,N}} + \overbrace{\color{ForestGreen} \Psi_{-1,1}}^{\color{BrickRed} e^{i\alpha} \Psi_{1, N-1}} + \overbrace{\color{ForestGreen} \Psi_{1,1}}^{\color{BrickRed} e^{i\alpha} \Psi_{1,N+1}} \right] = (E_2 + 2J\Delta) \overbrace{\color{ForestGreen} \Psi_{0,1}}^{\color{BrickRed} e^{i\alpha} \Psi_{1, N}}\\ &\hspace{24mm}\text{or}~\downarrow \times e^{-i\alpha}\\ &~\frac{J}{2} \left[ \overbrace{\color{ForestGreen} \Psi_{N,N}}^{\color{BrickRed} e^{-i\alpha} \Psi_{0,N}} + \overbrace{\color{ForestGreen} \Psi_{N,N+2}}^{\color{BrickRed} e^{-i\alpha} \Psi_{2,N}} + \overbrace{\color{ForestGreen} \Psi_{N-1,N+1}}^{\color{BrickRed} e^{-i\alpha} \Psi_{1, N-1}} + \overbrace{\color{ForestGreen} \Psi_{N+1,N+1}}^{\color{BrickRed} e^{-i\alpha} \Psi_{1,N+1}} \right] = (E_2 + 2J\Delta) \overbrace{\color{ForestGreen} \Psi_{N,N+1}}^{\color{BrickRed} e^{-i\alpha} \Psi_{1, N}}. \tag{xxz.s2bdry2}\label{xxz.s2bdry2} \end{align*}

Here, we did 3 things: first, we defined any amplitude outside the fundamental domain \(1 \leq j_1 < j_2 \leq N\) from amplitudes inside it, by (multiply, if needed) applying the periodicity rule

\[ \Psi_{j_2, j_1+N} = e^{-i\alpha} \Psi_{j_1, j_2}, \hspace{10mm} j_1 \leq j_2 \leq j_1 + N. \]

Second, we extended the null state constraints xxz.s2x to also cover the cases \(j=1, N-1\). Third, we imposed (in \((vii)\)) the boundary version of the null state constraint, and rewrote it using the periodicity conditions:

\begin{equation*} \Psi_{0, N} + \Psi_{1, N+1} = \Delta \Psi_{1, N} ~\rightarrow~ \Psi_{0,0} + \Psi_{1,1} = \Delta \Psi_{0,1}. \end{equation*}

The result is that all our \(\left(\substack{N\\2}\right)\) equations xxz.s2bulk & xxz.s2bdry now take the same form as xxz.s2i for extended coverage of the (still ordered) coordinates.

The Schrödinger problem

The time-independent Schrödinger problem of Hamiltonian xxz.h with xxz.htbc in the two-downturned spins sector is thus expressed as the following system for the \(\left(\substack{N+1\\2}\right)\) unknowns \(\Psi_{j_1,j_2}, 1\leq j_1 \leq j_2 \leq N\) (with periodicity constraints used to shift any over/underflowing coordinates back into the fundamental domain \(1 \leq j \leq N\)):

\begin{align*} &{\scriptstyle \circledS:~~1 \leq j_1 < j_2 \leq N ~(\text{Schrödinger})}: \nonumber\\ &~\frac{J}{2} \sum_{\sigma=\pm} \left[ \Psi_{j_1 +\sigma, j_2} + \Psi_{j_1, j_2 + \sigma} \right] = (E_2 + 2J\Delta) \Psi_{j_1, j_2}, \nonumber \\[1em] &{\scriptstyle \varnothing:~~ 0 \leq j < N ~(\text{null state constraints})}: \nonumber\\ &~\Psi_{j, j} + \Psi_{j+1, j+1} = \Delta \Psi_{j, j+1}, \nonumber \\[1em] &{\scriptstyle \circlearrowleft:~~ j_1 \leq j_2 \leq j_1 + N ~(\text{periodicity constraints})}: \nonumber\\ &\Psi_{j_2, j_1+N} = e^{-i\alpha} \Psi_{j_1, j_2}. \tag{xxz.s2}\label{xxz.s2} \end{align*}

The Schrödinger part of the problem \(\circledS\) could not look friendlier: a trivial-looking Hamiltonian equation for free dynamics on the lattice. The periodicity conditions \(\circlearrowleft\) are similarly just like those of free models. Amazingly, all the complexities of this integrable model are hidden in constraints \(\varnothing\) imposed by null states.

The Bethe Ansatz

Now that our problem is cleanly formulated, let us seek its solution. To find amplitudes satisfying xxz.s2 \(\circledS\), which is just like a set of equations for free particles, we generalize the \(M = 1\) case to a two-particle free wave Ansatz involving two as yet unspecified quasimomenta \(k_1, k_2\):

\begin{equation} \Psi_{j_1, j_2} \rightarrow \Psi_{j_1, j_2}(k_1, k_2) \equiv A_{\scriptscriptstyle 12} \,e^{i k_1 j_1 + i k_2 j_2} + A_{\scriptscriptstyle 21} \,e^{i k_2 j_1 + i k_1 j_2}, \hspace{10mm} {\scriptstyle j_1 \leq j_2} \tag{xxz.a2}\label{xxz.a2} \end{equation}

(note that we explicitly allow evaluating this on the ghost sites, and beyond, provided the coordinates remain ordered (with equality allowed)). All instances of xxz.s2 \(\circledS\) are then seen to be satisfied provided the energy is the sum of xxz.e1 for both involved quasimomenta,

\begin{equation*} E_2 = e_1(k_1) + e_1(k_2) = J(\cos k_1 + \cos k_2 - 2\Delta). \tag{xxz.e2}\label{xxz.e2} \end{equation*}

Let us now consider the null state constraints xxz.s2 \(\varnothing\). Substituting xxz.a2 in, one can see that all instances are satistied provided the amplitudes \(A_{\scriptscriptstyle 12}, A_{\scriptscriptstyle 21}\) obey the single constraint

\[ A_{\scriptscriptstyle 12} (1 + e^{i (k_1 + k_2) } - 2\Delta e^{i k_2}) + A_{\scriptscriptstyle 21} (1 + e^{i (k_1 + k_2)} - 2\Delta e^{i k_1}) = 0. \tag{xxz.A2c}\label{xxz.A2c} \]

Postponing questions about normalization to a later stage, we can thus always choose

\begin{align*} A_{\scriptscriptstyle 12} &= +(1 + e^{i (k_1 + k_2)} - 2\Delta e^{i k_1}) \\ A_{\scriptscriptstyle 21} &= -(1 + e^{i (k_1 + k_2)} - 2\Delta e^{i k_2}) \end{align*}

and write our (unnormalized) prototype (not yet fully proven) wavefunction amplitudes as a sum over permutations,

\begin{align*} &{\scriptstyle j_1 \leq j_2}: \nonumber\\ &~\Psi_{j_1,j_2} (k_1, k_2) = \sum_{P \in S_2} A_{\scriptscriptstyle P} (k_1, k_2) \,e^{i \sum_{i=1}^2 j_i k_{P_i}} \tag{xxz.ba2}\label{xxz.ba2} \end{align*}

(in which \(S_2\) is the symmetric group of 2 elements) with amplitudes

\begin{equation*} A_{\scriptscriptstyle P} = \sum_{P \in S_2} (-1)^P (1 + e^{i (k_{P_1} + k_{P_2})} - 2\Delta e^{i k_{P_1}}). \tag{xxz.A2}\label{xxz.A2} \end{equation*}

Finally, let us deal with the periodicity conditions xxz.s2 \(\circlearrowleft\). Substituting xxz.a2 in gives the requirements

\[ A_{\scriptscriptstyle 12} = A_{21} e^{ik_1 N + i\alpha} = A_{21} e^{-ik_2 N -i\alpha} \]

representing nontrivial quantization conditions (called Bethe equations) for \(k_1, k_2\) since the \(A\) coefficients are themselves dependent on these parameters. Explicitly,

\begin{align*} e^{i(k_1 + k_2) N} &= e^{-2i\alpha}, \\ \left(1 + e^{ik_1 + ik_2} - 2\Delta e^{ik_1} \right) &= -\left(1 + e^{ik_1 + ik_2} - 2\Delta e^{ik_2} \right) e^{ik_1 N + i\alpha}. \end{align*}

The first equation is expressing the fact that total momentum is quantized (modulo a shift from \(\alpha\)) in units of \(\frac{2\pi}{N}\).

Remark: since the scattering amplitudes are not a function of the difference in quasimomenta (this being a consequence of the lack of Galilean invariance of our theory), the twist \(\alpha\) cannot simply be reabsorbed by a shift of the momenta.

We can also equivalently write the Bethe equations (assuming the \(A_P\) to be nonzero) in the more traditional form

\begin{align*} e^{ik_1 N + i\alpha} &= \frac{A_{12}}{A_{21}} = -\frac{1 + e^{i (k_1 + k_2)} - 2\Delta e^{i k_1}}{1 + e^{i (k_1 + k_2)} - 2\Delta e^{i k_2}}, \\ e^{ik_2 N + i\alpha} &= \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}}. \tag{xxz.be2}\label{xxz.be2} \end{align*}

We will refer to this as the exponential form of the Bethe equations. Noticing that for real \(k\) the \(A\) ratio is unimodular invites us to define the scattering phase shift function

\begin{equation} \phi (k_1, k_2) =\frac{1}{i} \ln \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}} =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}

in terms of which the Bethe equations become

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

For many purposes including classifying and labelling the solutions, it is more convenient (as was done for \(M = 1\)) to take the logarithmic form of the Bethe equations:

\begin{equation*} N k_1 + \phi (k_1, k_2) +\alpha = 2\pi \tilde{I}_1, \hspace{1cm} N k_2 - \phi(k_1, k_2) +\alpha = 2\pi \tilde{I}_2 \tag{xxz.bel2}\label{xxz.bel2} \end{equation*}

where \(\tilde{I}\) are half-odd integers if \(N\) is even, and integers if \(N\) is odd.

So, we have an eigenstate basis?

This is all very well, but up to now, we have left an important question aside:

• Are all wavefunctions of the form xxz.ba2?

To answer this, we need to carefully count how many distinct wavefunctions the Bethe Ansatz produces, namely to determine the set of pairs \((k_1, k_2)\) solving the Bethe equations and leading to admissible wavefunctions. Only then can we assert whether we have a basis for \({\cal H}^N_2\) or not.

Classifying Bethe eigenstates is an extremely rewarding hobby. Bethe himself (temporarily putting away his stamp collection) took immense care (in his original article 1931.Bethe.ZP.71) to construct and count the eigenstates for the isotropic \(XXX\) case with periodic boundary conditions. In fact, carelessness ended up costing Bloch (in his earlier paper 1930.Bloch.ZP.61, after equation (8b)) the chance to give his name to the Ansatz.