The Bethe Ansatz

Specializing the Lieb-Liniger Hamiltonian l.h1 to two particles gives the simple Hamiltonian

\begin{equation} H_{\small LL}^{\small (2)} = -\partial_{x_1}^2 - \partial_{x_2}^2 + 2c \delta (x_1 - x_2). \tag{l.h1n2}\label{l.h1n2} \end{equation}

For the moment, we consider that the particles live on the real line, and thus \((x_1, x_2) \in \mathbb{R}^2\). Our interaction parameter \(c\) can for now also take any real value, \(c>0\) representing repulsive interactions, and \(c<0\) attractive ones.

As is familiar from basic quantum mechanics, the contact interaction between the two particles brings about a cusp (discontinuity in the first derivative) in the wavefunction at the point where coordinates coincide. This is most easily expressed by moving to relative coordinates \(x_+ = \frac{x_1 + x_2}{2}, x_- = x_1 - x_2\), integrating l.se1 over a vanishingly small inverval \(x_- = [-\epsilon, \epsilon]\), and taking the limit \(\epsilon \rightarrow 0^+\), yielding the condition

\begin{equation} -\left.\frac{1}{2} (\partial_{x_1} - \partial_{x_2}) \Psi_2 \right|_{x_1 -x_2 = -0^+}^{x_1 - x_2 = 0^+} + c \Psi_2 \bigr |_{x_1 = x_2} = 0. \tag{l.bc}\label{l.bc} \end{equation}

Since we are dealing with bosons, the physical Hilbert space is restricted to completely symmetric functions, meaning that we require \(\Psi_2 (x_1, x_2) = \Psi_2 (x_2, x_1)\). In that case, condition l.bc can be simplified to

\begin{equation} \left(\partial_{x_2} - \partial_{x_1} - c \right) \Psi_2 (x_1, x_2) \bigr|_{x_2 - x_1 = 0^+} = 0. \tag{l.d}\label{l.d} \end{equation}

The wavefunctions thus indeed acquire a jump in their first derivative when relative particle positions change sign, and the size of this jump is the value of the interaction strength \(c\).

In order to display explicit solutions of the Schrödinger equation, two routes can be followed. Namely, one can simply solve l.h1n2 in the whole domain \((x_1, x_2) \in \mathbb{R}^2\) starting from a general enough initial form (Ansatz) for the form of the wavefunctions. Alternately, one can restrict to the fundamental domain

\begin{equation*} D_2: x_1 < x_2, \hspace{10mm} \mbox{with boundary} \hspace{10mm} \partial D_2: x_2 = x_1 + 0^+ \end{equation*}

and solve the multivariable boundary value problem

\begin{equation*} \left( H_{\small LL}^{\small (2)} - E_2 \right) \Psi_2 (x_1, x_2) \bigr|_{(x_1, x_2) \in D_2} = 0, \hspace{10mm} \left(\partial_{x_2} - \partial_{x_1} - c \right) \Psi_2 (x_1, x_2) \bigr|_{(x_1, x_2) \in \partial D_2} = 0, \end{equation*}

the extension from \(D_2\) to \(\mathbb{R}^2\) being effectuated by invoking the symmetry properties of the wavefunctions under particle exchange.

Let us follow this second route. The projection of the Hamiltonian l.h1n2 to \(D_2\) is then simply the two-particle Laplacian \(-\partial_{x_1}^2 - \partial_{x_2}^2\), which is simply solved in terms of free waves (note: one might also consider a zero mode solution of the form \(\alpha + \beta x\), with constant \(\alpha\) and \(\beta\). The first is excluded by the interaction term; the second, by the boundary conditions (if in finite size) or the normalizability condition (if on the real line)).

Given two generic complex numbers \(\lambda_1, \lambda_2\) representing quasimomenta (postponing until later the discussion of whether these numbers are restricted to the real line or not), one can construct two plane wave solutions

\begin{equation*} e^{i \lambda_1 x_1 + i \lambda_2 x_2} \hspace{5mm} \mbox{and}\hspace{5mm} e^{i \lambda_2 x_1 + i \lambda_1 x_2} \end{equation*}

having the same energy and total momentum, namely

\begin{equation*} E_2 = \lambda_1^2 + \lambda_2^2, \hspace{5mm} P_2 = \lambda_1 + \lambda_2. \end{equation*}

We thus begin by setting the form

\begin{equation} \Psi_2^{\small (a)} (x_1, x_2 | \lambda_1, \lambda_2) \bigr|_{(x_1, x_2) \in D_2} = S_1 e^{i \lambda_1 x_1 + i \lambda_2 x_2} + S_2 e^{i \lambda_2 x_1 + i \lambda_1 x_2} \tag{l.psi2a}\label{l.psi2a} \end{equation}

for some as yet undetermined complex amplitudes \(S_1, S_2\). Substituting this form in l.d immediately leads to the restriction

\begin{equation} \frac{S_2}{S_1} = - \frac{c + i (\lambda_1 - \lambda_2)}{c - i (\lambda_1 - \lambda_2)} \tag{l.s}\label{l.s} \end{equation}

which must be fulfilled in order to solve the Schrödinger equation on the boundary \(\partial D_2\).

Let us now reconsider the question of whether our quasimomenta can indeed take values in the unrestricted field of complex numbers. From the form of the wavefunction l.psi2a, we can see that we must impose \(\mbox{Im}~ \lambda_i \leq 0\) if we require that \(\Psi_2\) remains finite as we take \(x_1 \rightarrow -\infty\). Similarly, we must impose \(\mbox{Im}~ \lambda_i \geq 0\) if the same is to hold true when we take \(x_2 \rightarrow \infty\). This argumentation of course does not apply to a finite system. We will discuss this `reality' issue in further details later on, assuming for the moment that the quasimomenta are indeed real numbers.

Since the ratio l.s is of the form \(a/a^*\), we can write conveniently it as a phase,

\begin{equation*} \frac{S_2}{S_1} = -e^{i\phi(\lambda_1- \lambda_2)}, \end{equation*}

where

\begin{equation} \phi(\lambda) = \frac{1}{i} \ln \frac{c + i\lambda}{c - i\lambda} = 2 ~\mbox{atan}~ \frac{\lambda}{c}. \tag{l.phi}\label{l.phi} \end{equation}

This allows us to write our two-particle wavefunction (up to an arbitrary phase, and postponing the question of normalization until later) as

\begin{equation*} \Psi_2^{\small (a)} (x_1, x_2 | \lambda_1, \lambda_2) \bigr|_{(x_1, x_2) \in D_2} = e^{i \lambda_1 x_1 + i \lambda_2 x_2 - \frac{i}{2} \phi(\lambda_1 - \lambda_2)} - e^{i \lambda_2 x_1 + i \lambda_1 x_2 + \frac{i}{2} \phi(\lambda_1 - \lambda_2)}. \end{equation*}

The domain of applicability of this form can then simply be extended to the whole domain \((x_1, x_2) \in \mathbb{R}^2\) by invoking complete symmetry under coordinate exchange, yielding the wavefunction

\begin{equation*} \Psi_2^{\small (a)} (x_1, x_2 | \lambda_1, \lambda_2) = sgn(x_2 - x_1)~\left\{e^{i \lambda_1 x_1 + i \lambda_2 x_2 - \frac{i}{2}sgn(x_2 - x_1) \phi(\lambda_1 - \lambda_2)} - e^{i \lambda_2 x_1 + i \lambda_1 x_2 + \frac{i}{2} sgn(x_2 - x_1) \phi(\lambda_1 - \lambda_2)} \right\}, \end{equation*}

or more aesthetically

\begin{equation*} \Psi_2^{\small (a)} (x_1, x_2 | \lambda_1, \lambda_2) = sgn(x_2 - x_1) \sum_{P \in \pi_2} (-1)^{[P]} e^{i \lambda_{P_1} x_1 + i \lambda_{P_2} x_2 - i sgn(x_2 - x_1) \phi (\lambda_{P_1} - \lambda_{P_2})/2}. \end{equation*}

Here, the summation is over the set \(\pi_2\) of permutations of the set of integers \((1,2)\). This form will be readily generalizable to arbitrary particle numbers, in which case we will call it the Bethe Ansatz.

Note that this wavefunction, manifestly symmetric in coordinates, is manifestly anti-symmetric in quasimomenta:

\begin{equation*} \Psi_2^{\small (a)} (x_1, x_2 | \lambda_1, \lambda_2) = - \Psi_2^{\small (a)} (x_1, x_2 | \lambda_2, \lambda_1). \end{equation*}

Wavefunctions for which \(\lambda_1 = \lambda_2\) thus identically vanish and do not represent bona fide eigenstates. This is a manifestation of an underlying Pauli-like principle, which will be determinantal in the classification of eigenstates to be performed later. That said, this antisymmetry under permutation of quasimomenta is artificial, and carries no direct physical meaning. The only real constraint is that the wavefunction forms an irreducible representation of the permutation group of quasimomenta. We can thus just as well choose our wavefunction to be symmetric under quasimomenta exchanges,

\begin{equation*} \Psi_2^{\small (s)} (x_1, x_2 | \lambda_1, \lambda_2) = sgn (\lambda_2 - \lambda_1) \Psi_2^{\small (a)} (x_1, x_2 | \lambda_1, \lambda_2). \end{equation*}

This choice will in fact turn out to be more natural in view of our later considerations of the Algebraic Bethe Ansatz. We will thus take our final form for the \(N=2\) wavefunction to be

\begin{equation} \Psi_2 (x_1, x_2 | \lambda_1, \lambda_2) = sgn(x_2 - x_1) sgn (\lambda_2 - \lambda_1) \sum_{P \in \pi_2} (-1)^{[P]} e^{i \lambda_{P_1} x_1 + i \lambda_{P_2} x_2 - i sgn(x_2 - x_1) \phi (\lambda_{P_1} - \lambda_{P_2})/2}. \tag{l.psi2}\label{l.psi2} \end{equation}

As a final comment, one could have found the same wavefunction using the first route mentioned above, starting from the symmetry-inspired Ansatz

\begin{eqnarray} \Psi_2^{\small (a)} (x_1, x_2 | \lambda_1, \lambda_2) = \theta (x_2 - x_1) \left\{ S_1 e^{i \lambda_1 x_1 + i \lambda_2 x_2} + S_2 e^{i \lambda_2 x_1 + i \lambda_1 x_2} \right\} + \nonumber \\ \theta (x_1 - x_2) \left\{ S_2 e^{i \lambda_1 x_1 + i \lambda_2 x_2} + S_1 e^{i \lambda_2 x_1 + i \lambda_1 x_2} \right\}. \tag{l.psi2t}\label{l.psi2t} \end{eqnarray}

By applying the differential operator form l.h1, using the properties of the \(\delta\) function, one can directly verify that this is indeed a solution of the time-independent Schrödinger equation l.se1 provided the coefficients \(S_i\) satisfy the constraint l.s.

Derivation

Verification that l.psi2t solves the Schrödinger equation:

\begin{align*} \partial_{x_1} \Psi_2 (x_1, x_2) = -\delta (x_2 - x_1) (S_1 + S_2) e^{i (\lambda_1 + \lambda_2) x_1} + \delta (x_1 - x_2) (S_1 + S_2) e^{i(\lambda_1 + \lambda_2) x_1} + \nonumber \\ + \theta (x_2 - x_1) \left\{ i\lambda_1 S_1 e^{i \lambda_1 x_1 + i \lambda_2 x_2} + i\lambda_2 S_2 e^{i \lambda_2 x_1 + i \lambda_1 x_2} \right\} + \theta (x_1 - x_2) \left\{ i\lambda_1 S_2 e^{i \lambda_1 x_1 + i \lambda_2 x_2} + i\lambda_2 S_1 e^{i \lambda_2 x_1 + i \lambda_1 x_2} \right\} \nonumber \\ = \theta (x_2 - x_1) \left\{ i\lambda_1 S_1 e^{i \lambda_1 x_1 + i \lambda_2 x_2} + i\lambda_2 S_2 e^{i \lambda_2 x_1 + i \lambda_1 x_2} \right\} + \theta (x_1 - x_2) \left\{ i\lambda_1 S_2 e^{i \lambda_1 x_1 + i \lambda_2 x_2} + i\lambda_2 S_1 e^{i \lambda_2 x_1 + i \lambda_1 x_2} \right\} \end{align*} \begin{align*} -\partial_{x_1}^2 \Psi_2 (x_1, x_2) = \delta (x_1 - x_2) \left\{ i (\lambda_1 - \lambda_2) (S_1 - S_2) e^{i (\lambda_1 + \lambda_2) x_1} \right\} + \nonumber \\ + \theta (x_2 - x_1) \left\{ \lambda_1^2 S_1 e^{i \lambda_1 x_1 + i \lambda_2 x_2} + \lambda_2^2 S_2 e^{i \lambda_2 x_1 + i \lambda_1 x_2} \right\} + \nonumber \\ + \theta (x_1 - x_2) \left\{ \lambda_1^2 S_2 e^{i \lambda_1 x_1 + i \lambda_2 x_2} + \lambda_2^2 S_1 e^{i \lambda_2 x_1 + i \lambda_1 x_2} \right\} \end{align*}