The Bethe Ansatz

We now consider the general \(M\) subspace \({\cal H}^N_M\). This subspace is spanned by the \({\scriptstyle \left(\begin{array}{c} N \\ M \end{array} \right)}\) states

\[ |j_1, j_2, \ldots, j_M \rangle \equiv \prod_{a=1}^M S^-_{j_a} |0\rangle, \hspace{5mm} {\scriptstyle 1 \leq j_1 < j_2 \ldots < j_M\leq N}. \]

We adopt the Ansatz

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

parametrizing the wavefunction in terms of \({\scriptstyle \left(\begin{array}{c} N \\ M \end{array} \right)}\) amplitudes. As in the \(M=2\) case c.h.s.2, we will define the amplitudes for indices outside this set to facilitate bookkeeping.

In view of our detailed work for \(M=2\) case in c.h.s.2, we shall be quite succinct here.

In the bulk

As a generalization of xxz.s2bulk, all projections of the Schrödinger equation in which only xxz.hbulk acts nontrivially are written as

\begin{align*} &{\scriptstyle 1 < j_1 < j_2 \ldots < j_M < N ~(\text{in bulk;}~{\scriptstyle \left(\substack{N-2\\M}\right)}~\text{equations})}: \nonumber\\ &~\frac{J}{2} \sum_{a=1}^M \sum_{\sigma = \pm1} \Psi_{j_1, \ldots, j_a +\sigma, \ldots, j_M} = (E_M + MJ\Delta) \Psi_{j_1, \ldots, j_M} \tag{xxz.sbulk}\label{xxz.sbulk} \end{align*}

in which we allow multiple neighbouring \(j_a, j_{a+1} = j_a + 1\) on the right-hand side (and thus single coordinate coincidences on the left-hand side) by following Bethe's idea of defining amplitudes for null states with pairwise coinciding indices (occurring on the left-hand side), generalizing xxz.s2x to

\[ \Psi_{j_1, \ldots, j, j, \ldots, j_M} + \Psi_{j_1, \ldots, j+1, j+1, \ldots, j_M} = 2\Delta \Psi_{j_1, \ldots, j, j+1, \ldots, j_M}. \tag{xxz.sx}\label{xxz.sx} \]

Multiple neighbouring coordinates are covered by this construction. For example, let us look at a trio of neighbouring spins. Let \(b\) be an index such that \(1 < b < M\), and let \(j_{b-1} = j-1, j_b = j, j_{b+1} = j+1\) with \(j_{b-2} < j-2, j_{b+2} > j+2\). We then have

\begin{align*} & \underbrace{\sum_{\substack{a=1\\a\neq b-1, b, b+1}}^M \sum_{\sigma = \pm1} \Psi_{j_1, \ldots, j_a +\sigma, \ldots, j_M}}_{\equiv S_b} + \nonumber\\ &~~+ \Psi_{j_1, \ldots, j_{b-2}, j-2, j, j+1, j_{b+2},\ldots, j_M} + \Psi_{j_1, \ldots, j_{b-2}, j-1, j, j+2, j_{b+2},\ldots, j_M} \nonumber\\ &\hspace{10mm}= \frac{2}{J}(E_M + (M-2) J\Delta) \Psi_{j_1, \ldots, j_M} \\[0.5em] &\hspace{20mm}\downarrow\\[0.5em] &S_b + \Psi_{j_1, \ldots, j_{b-2}, j-2, j, j+1, j_{b+2},\ldots, j_M} + \nonumber\\[0.5em] &~~{\color{NavyBlue} \Psi_{j_1, \ldots, j_{b-2}, j, j, j+1, j_{b+2},\ldots, j_M} + \Psi_{j_1, \ldots, j_{b-2}, j-1, j-1, j+1, j_{b+2},\ldots, j_M}} \nonumber \\[0.5em] &~~{\color{NavyBlue} \Psi_{j_1, \ldots, j_{b-2}, j-1, j+1, j+1, j_{b+2},\ldots, j_M} + \Psi_{j_1, \ldots, j_{b-2}, j-1, j, j, j_{b+2},\ldots, j_M}} \nonumber \\[0.5em] &~~+ \Psi_{j_1, \ldots, j_{b-2}, j-1, j, j+2, j_{b+2},\ldots, j_M} \nonumber \\[0.5em] &\hspace{10mm}= \frac{2}{J}(E_M + (M-2) J\Delta {\color{NavyBlue} + 2J\Delta}) \Psi_{j_1, \ldots, j_M}. \end{align*}

Amplitudes for higher numbers of neighbouring spins are easily seen to follow a similar pattern, so xxz.sx covers all bulk cases.

At the boundary

Direct application of xxz.h with xxz.htbc and projecting onto all cases where at least one site at the boundary is touched yields the following equations:

\begin{align*} &{\scriptstyle 1 = j_1 < j_2 < j_3 \ldots < j_M < N ~(j_1~\text{at boundary},~\text{rest in bulk;}~{\scriptscriptstyle \left( \substack{N-2\\M-1} \right)}~\text{equations})}: \nonumber\\ &~\frac{J}{2} \left[ e^{i\alpha} \Psi_{j_2,\ldots,j_M, N} + \Psi_{2, j_2,\ldots,j_M} + \sum_{a=2}^M \sum_{\sigma = \pm1} \Psi_{1, j_2, \ldots, j_a +\sigma, \ldots, j_M} \right] \\[0.5em] &\hspace{5mm}= (E_M + MJ\Delta) \Psi_{1, j_2, \ldots, j_M}, \\[1em] &{\scriptstyle 1 < j_1 < j_2 \ldots < j_{M-1} < j_M = N ~(j_M~\text{at boundary},~\text{rest in bulk};~{\scriptscriptstyle \left( \substack{N-2\\M-1} \right)}~\text{equations})}: \nonumber \\ &~\frac{J}{2} \left[ \sum_{a=1}^{M-1} \sum_{\sigma = \pm1} \Psi_{j_1, \ldots, j_a +\sigma, \ldots, N} + \Psi_{j_1,\ldots,j_{M-1}, N-1} + e^{-i\alpha} \Psi_{1, j_1,\ldots,j_{M-1}} \right] \\[0.5em] &\hspace{5mm} = (E_M + MJ\Delta) \Psi_{j_1, \ldots, j_{M-1}, N},\\[1em] &{\scriptstyle 1 = j_1 < j_2 \ldots < j_{M-1} < j_M = N ~(j_1 \& j_M~\text{at boundary},~\text{rest in bulk};~{\scriptscriptstyle \left( \substack{N-2\\M-2} \right)}~\text{equations})}: \nonumber \\ &~\frac{J}{2} \left[ \Psi_{2, j_2,\ldots, j_{M-1}, N} + \Psi_{1,j_2,\ldots,j_{M-1}, N-1} + \sum_{a=2}^{M-1} \sum_{\sigma = \pm1} \Psi_{1, j_2, \ldots, j_a +\sigma, \ldots, j_{M-1}, N} \right] \\[0.5em] &\hspace{5mm} = (E_M + (M-1)J\Delta) \Psi_{1, j_2, \ldots, j_{M-1}, N}. \tag{xxz.sbdry}\label{xxz.sbdry} \end{align*}

Together with xxz.sbulk this constitutes a set of \(\left( \substack{N\\M} \right)\) equations corresponding to the dimensionality of this subspace.

Similarly to what we did for \(M=1,2\), we introduce "ghost" sites, here for convenience at \(j=-1, 0, N+1, N_2\), and extend our set of amplitudes accordingly. Let us proceed as we did for xxz.s2bdry2, 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 1 = j_1 < j_2 < j_3 \ldots < j_M < N ~(j_1~\text{at boundary},~\text{rest in bulk;}~{\scriptscriptstyle \left( \substack{N-2\\M-1} \right)}~\text{equations})}: \nonumber\\ &~\underbrace{\color{ForestGreen} \Psi_{0, j_2, \dots, j_M}}_{\color{BrickRed} e^{i\alpha} \Psi_{j_2,\ldots,j_M, N}} + \Psi_{2, j_2,\ldots,j_M} + \sum_{a=2}^M \sum_{\sigma = \pm1} \Psi_{1, j_2, \ldots, j_a +\sigma, \ldots, j_M} \\ &\hspace{10mm}= \frac{2}{J}(E_M + MJ\Delta) \Psi_{1, j_2, \ldots, j_M}, \\[1em] &{\scriptstyle 1 < j_1 < j_2 \ldots < j_{M-1} < j_M = N ~(j_M~\text{at boundary},~\text{rest in bulk};~{\scriptscriptstyle \left( \substack{N-2\\M-1} \right)}~\text{equations})}: \nonumber \\ &~\sum_{a=1}^{M-1} \sum_{\sigma = \pm1} \Psi_{j_1, \ldots, j_a +\sigma, \ldots, N} + \Psi_{j_1,\ldots,j_{M-1}, N-1} + \underbrace{\color{ForestGreen} \Psi_{j_1, \ldots, j_{M-1}, N+1}}_{\color{BrickRed} e^{-i\alpha} \Psi_{1, j_1,\ldots,j_{M-1}}} \\ &\hspace{10mm}= \frac{2}{J}(E_M + MJ\Delta) \Psi_{j_1, \ldots, j_{M-1}, N},\\[1em] &{\scriptstyle 1 = j_1 < j_2 \ldots < j_{M-1} < j_M = N ~(j_1 \& j_M~\text{at boundary},~\text{rest in bulk};~{\scriptscriptstyle \left( \substack{N-2\\M-2} \right)}~\text{equations})}: \nonumber \\ &~{\color{NavyBlue} \Psi_{0, j_2,\ldots, j_{M-1}, N} +} \Psi_{2, j_2,\ldots, j_{M-1}, N} + \Psi_{1,j_2,\ldots,j_{M-1}, N-1} + {\color{NavyBlue} \Psi_{1, j_2,\ldots,j_{M-1}, N+1} +} \\ &\hspace{5mm}+ \sum_{a=2}^{M-1} \sum_{\sigma = \pm1} \Psi_{1, j_2, \ldots, j_a +\sigma, \ldots, j_{M-1}, N} = \frac{2}{J}(E_M + (M-1)J\Delta {\color{NavyBlue} +J\Delta}) \Psi_{1, j_2, \ldots, j_{M-1}, N} \\[1em] &\hspace{20mm}\text{either}~\downarrow \times e^{i\alpha}\\ &~\underbrace{\color{ForestGreen} \Psi_{0, 0, j_2,\ldots, j_{M-1}}}_{\color{BrickRed} e^{i\alpha} \Psi_{0, j_2,\ldots, j_{M-1}, N}} + \underbrace{\color{ForestGreen} \Psi_{0, 2, j_2,\ldots, j_{M-1}}}_{\color{BrickRed} e^{i\alpha} \Psi_{2, j_2,\ldots, j_{M-1}, N}} + \underbrace{\color{ForestGreen} \Psi_{-1, 1,j_2,\ldots,j_{M-1}}}_{\color{BrickRed} e^{i\alpha} \Psi_{1,j_2,\ldots,j_{M-1}, N-1}} + \underbrace{\color{ForestGreen} \Psi_{1, 1, j_2,\ldots,j_{M-1}}}_{\color{BrickRed} e^{i\alpha} \Psi_{1, j_2,\ldots,j_{M-1}, N+1}} + \\ &\hspace{5mm}+ \sum_{a=2}^{M-1} \sum_{\sigma = \pm1} \underbrace{\color{ForestGreen} \Psi_{0, 1, j_2, \ldots, j_a +\sigma, \ldots, j_{M-1}}}_{\color{BrickRed} e^{i\alpha} \Psi_{1, j_2, \ldots, j_a +\sigma, \ldots, j_{M-1}, N}} = \frac{2}{J}(E_M + MJ\Delta) \underbrace{\color{ForestGreen} \Psi_{0,1, j_2, \ldots, j_{M-1}}}_{\color{BrickRed} e^{i\alpha} \Psi_{1, j_2, \ldots, j_{M-1}, N}}\\[1em] &\hspace{20mm}\text{or}~\downarrow \times e^{-i\alpha}\\ &~\underbrace{\color{ForestGreen} \Psi_{j_2,\ldots, j_{M-1}, N, N}}_{\color{BrickRed} e^{-i\alpha} \Psi_{0, j_2,\ldots, j_{M-1}, N}} + \underbrace{\color{ForestGreen} \Psi_{j_2,\ldots, j_{M-1}, N, N+2}}_{\color{BrickRed} e^{-i\alpha} \Psi_{2, j_2,\ldots, j_{M-1}, N}} + \underbrace{\color{ForestGreen} \Psi_{j_2,\ldots,j_{M-1}, N-1, N}}_{\color{BrickRed} e^{-i\alpha} \Psi_{1,j_2,\ldots,j_{M-1}, N-1}} + \underbrace{\color{ForestGreen} \Psi_{j_2,\ldots,j_{M-1}, N+1, N+1}}_{\color{BrickRed} e^{-i\alpha} \Psi_{1, j_2,\ldots,j_{M-1}, N+1}} + \\ &\hspace{5mm}+ \sum_{a=2}^{M-1} \sum_{\sigma = \pm1} \underbrace{\color{ForestGreen} \Psi_{j_2, \ldots, j_a +\sigma, \ldots, j_{M-1}, N, N+1}}_{\color{BrickRed} e^{-i\alpha} \Psi_{1, j_2, \ldots, j_a +\sigma, \ldots, j_{M-1}, N}} = \frac{2}{J}(E_M + MJ\Delta) \underbrace{\color{ForestGreen} \Psi_{j_2, \ldots, j_{M-1}, N, N+1}}_{\color{BrickRed} e^{-i\alpha} \Psi_{1, j_2, \ldots, j_{M-1}, N}}\\ \tag{xxz.sbdry2}\label{xxz.sbdry2} \end{align*}

Here, as per the \(M=2\) case, we did 3 things: first, we defined any amplitude outside the fundamental domain \(1 \leq j_1 < j_2 \ldots < j_M \leq N\) from amplitudes inside it, by (multiply, if needed) applying the periodicity rule

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

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

\begin{align*} &\Psi_{0, j_2, \ldots, j_{M-1}, N} + \Psi_{1, j_2, \ldots, j_{M-1}, N+1} = \Delta \Psi_{1, j_2, \ldots, j_{M-1}, N} \\ &~\rightarrow~ \Psi_{0,0, j_2, \ldots, j_{M-1}} + \Psi_{1,1, j_2, \ldots, j_{M-1}} = \Delta \Psi_{0,1, j_2, \ldots, j_{M-1}}. \end{align*}

The result is that all our \(\left(\substack{N\\M}\right)\) equations xxz.sbulk & xxz.sbdry now take the same form, 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 \(M\)-downturned spins sector is thus expressed as the following system for the \(\left(\substack{N\\M}\right) + N\left(\substack{N-1\\M-2}\right) = \left(\substack{N+1\\M}\right) \frac{N+(M-1)^2}{N+1}\) unknowns \(\Psi_{j_1,\ldots, j_M}\) in the fundamental domain \(1\leq j_1 \leq j_2 \ldots \leq j_M \leq N\) (with periodicity constraints used to shift any over/underflowing coordinates back into the fundamental domain:

\begin{align*} &{\scriptstyle \circledS:~~1 \leq j_1 < j_2 < \ldots < j_M \leq N ~(\text{Schrödinger})}: \nonumber\\ &~\frac{J}{2} \sum_{a=1}^M \sum_{\sigma = \pm1} \Psi_{j_1, \ldots, j_a +\sigma, \ldots, j_M} = (E_M + 2J\Delta) \Psi_{j_1, \ldots, j_M}, \nonumber \\[1em] &{\scriptstyle \varnothing: ~~0 \leq j < N, ~j_a < j_{a+1}, j_a \neq j,j+1 ~(\text{null state constraints})}: \nonumber\\ &\Psi_{j_1, \ldots, j, j, \ldots, j_M} + \Psi_{j_1, \ldots, j+1, j+1, \ldots, j_M} = 2\Delta \Psi_{j_1, \ldots, j, j+1, \ldots, j_M} \nonumber \\[1em] &{\scriptstyle \circlearrowleft: ~~1 \leq j_1 \leq j_2 < \ldots \leq j_M \leq N ~(\text{periodicity constraints})}: \nonumber\\ &\Psi_{j_2, \ldots, j_{M}, j_1+N} = e^{-i\alpha} \Psi_{j_1, \ldots, j_{M}}. \tag{xxz.s}\label{xxz.s} \end{align*}

Let us now discuss the solution to this problem.

The Bethe Ansatz

Following Bethe, we posit

\begin{align*} &{\scriptstyle j_1 \leq j_2 \leq \ldots \leq j_M}: \\ &~\Psi_{j_1, \ldots, j_M} \rightarrow \Psi_{j_1, \ldots, j_M} (k_1, \ldots, k_M) = \sum_{P \in S_M} A_{\scriptscriptstyle P} (k_1,\ldots, k_M) \,e^{i \sum_{i=1}^M k_{P_i} j_i}, \tag{xxz.ba}\label{xxz.ba} \end{align*}

in which \(S_M\) is the symmetric group of \(M\) elements. Note that we allow evaluation of these amplitudes for any ordered (up to equality) set of coordinates.

All instances of xxz.sbulk \(\circledS\) are solved by setting the energy to the sum of individual contributions xxz.e1 (as we did in xxz.e2 ),

\begin{equation*} E_M = \sum_{a=1}^M e_1(k_a) = J \sum_{a=1}^M (\cos k_a - \Delta). \tag{xxz.e}\label{xxz.e} \end{equation*}

To solve xxz.sx \(\varnothing\), we start by rewriting it as

\[ \sum_P A_P \,e^{i(k_{P_a} + k_{P_{a+1}})j} \left( 1 + e^{ik_{P_a} + ik_{P_{a+1}}} - 2\Delta e^{ik_{P_{a+1}}} \right) e^{i \sum_{b\neq a,a+1} k_{P_b} j_b} = 0. \]

Letting \(P_{a,b}\) be the pairwise permutation operation for indices \(a,b\), a sufficient condition to solve the above constraint is a simple generalization of xxz.A2c,

\begin{equation*} A_P \left( 1 + e^{ik_{P_a} + ik_{P_{a+1}}} - 2\Delta e^{ik_{P_{a+1}}} \right) + A_{P P_{a,a+1}} \left( 1 + e^{ik_{P_a} + ik_{P_{a+1}}} - 2\Delta e^{ik_{P_a}} \right) = 0. \tag{xxz.Ac}\label{xxz.Ac} \end{equation*}

Modulo a constant factor, the solution is

\begin{equation*} A_P (\{ k_1, \ldots, k_M\}) \equiv (-1)^{[P]} \prod_{1 \leq a < b \leq M} \left(1 + e^{ik_{P_a} + i k_{P_b}} - 2\Delta e^{ik_{P_a}}\right). \tag{xxz.A}\label{xxz.A} \end{equation*}

Finally, let us deal with the periodicity constraints xxz.s \(\circlearrowleft\). Substituting xxz.ba in yields the Bethe equations

\begin{align*} e^{ik_a N + i\alpha} &= \prod_{b\neq a} -\frac{1 + e^{i (k_a + k_b)} - 2\Delta e^{i k_a}}{1 + e^{i (k_a + k_b)} - 2\Delta e^{i k_b}} = (-1)^{M-1} e^{-i\sum_{b=1}^M \phi(k_a, k_b)},\\ & a = 1, \ldots, M \tag{xxz.be}\label{xxz.be} \end{align*}

where the two-particle scattering phase shift \(\phi\) is given by xxz.phi (we have used \(\phi(k, k) = 0\) to have a full sum in the exponential on the right-hand side).

The logarithmic form of the Bethe equations is

\begin{equation} N k_a + \sum_b \phi (k_a, k_b) + \alpha = 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 \(N-M\) is even, and integers if \(N-M\) is odd.