# The Bethe Ansatz

#### The Lieb-Liniger modelc.l.l

Let us now specialize to the case of a single species of bosonic particles. We will thus use field operators $$\psi^\dagger (x)$$, $$\psi(x)$$ corresponding respectively to the creation/annihilation of a single particle at position $$x$$, and obeying the canonical equal-time commutation relations

\begin{equation} \left[ \psi(x_1), \psi^\dagger (x_2) \right] = \delta (x_1 - x_2), \hspace{10mm} \left[ \psi(x_1), \psi (x_2) \right] = 0 = \left[ \psi^\dagger(x_1), \psi^\dagger (x_2) \right]. \tag{l.cr}\label{l.cr} \end{equation}

The Hamiltonian we will consider takes the interaction potential $$V(x)$$ to be zero-range (ultralocal), $$V(x) \equiv 2c \delta(x)$$. For convenience, units are chosen such that $$\hbar = 1$$ and $$2m = 1$$. Taking the particle number to be fixed and equal to $$N$$, this yields the (first-quantized version of the) Lieb-Liniger Hamiltonian

\begin{equation} H_{\small LL}^{\small (N)} = \sum_{j=1}^N -\frac{\partial^2}{\partial_{x_j}^2} + 2c \sum_{j_1 < j_2} \delta (x_{j_1} - x_{j_2}) \tag{l.h1}\label{l.h1} \end{equation}

whose second-quantized version is (written as the integral of a local, Hermitian Hamiltonian density)

\begin{equation} \hat{H}_{\small LL} = \int dx ~{\cal H}_{\small LL} (x), \hspace{10mm} {\cal H}_{\small LL} (x) = \partial_x \psi^{\dagger}(x) \partial_x \psi(x) + c ~\psi^{\dagger}(x) \psi^{\dagger}(x) \psi(x) \psi(x). \tag{l.h2}\label{l.h2} \end{equation}

The equation of motion for the field operators is

\begin{equation} i\partial_t \psi = \left[ \hat{H}_{\small LL}, \psi \right] = -\partial_x^2 \psi + 2c \psi^{\dagger} \psi \psi \tag{nls}\label{nls} \end{equation}

and is known as the nonlinear Schrödinger equation. The particle number and total momentum operators

\begin{equation} \hat{N} = \int dx~ \psi^{\dagger}(x) \psi(x), \hspace{1cm} \hat{P} = -\frac{i}{2} \int dx \left\{ \psi^{\dagger}(x) \partial_x \psi(x) - \left[\partial_x \psi^{\dagger}(x) \right] \psi(x) \right\} \tag{l.np}\label{l.np} \end{equation}

give us the two simplest conservation laws \todo{Write these laws as integrals of local densities; show they are locally conserved?}

\begin{equation*} \left[ \hat{H}_{\small LL}, \hat{N} \right] = 0, \hspace{10mm} \left[ \hat{H}_{\small LL}, \hat{P} \right] = 0. \end{equation*}

Defining the Fock vacuum $$|0\rangle$$ and its dual $$\langle 0 |$$ as

\begin{equation} \psi (x) |0\rangle = 0, \hspace{1cm} \langle 0 | \psi^{\dagger} (x) = 0, \hspace{1cm} \langle 0 | 0 \rangle = 1, \tag{l.fv}\label{l.fv} \end{equation}

we can parametrize states in the Fock space having a fixed particle number $$N$$ in terms of a complex-valued, space-dependent amplitude $$\Psi_N (x_1, ..., x_N)$$,

\begin{equation} | \Psi_N \rangle = \int dx_1 ... dx_N \Psi_N (x_1, ..., x_N) \psi^{\dagger} (x_1) ... \psi^{\dagger} (x_N) | 0 \rangle. \tag{l.psi}\label{l.psi} \end{equation}

The time-independent Schrödinger equation for this quantum field theory,

\begin{equation} \hat{H}_{\small LL} |\psi_N \rangle = E_N | \psi_N \rangle \tag{l.se}\label{l.se} \end{equation}

is then equivalent to the quantum mechanical problem

\begin{equation} H_{\small LL}^{\small (N)} \Psi_N ({\bf x}) = E_N \Psi_N ({\bf x}). \tag{l.se1}\label{l.se1} \end{equation}
Derivation \begin{align*} H_{LL} | \Psi_N \rangle = \int_0^L dx \int_0^L dx_1 ... dx_N \Psi_N (x_1, ..., x_N) \left\{ \partial_x \Psi^{\dagger}(x) \partial_x \Psi(x) + \right.\nonumber \\ \left. + c \Psi^{\dagger}(x) \Psi^{\dagger}(x) \Psi(x) \Psi(x) \right\} \Psi^{\dagger} (x_1) ... \Psi^{\dagger} (x_N) | 0 \rangle \end{align*}

First term:

\begin{align*} \int_0^L dx \int_0^L dx_1 ... dx_N \Psi_N (x_1, ..., x_N) \partial_x \Psi^{\dagger}(x) \partial_x \left\{ \sum_{j=1}^N \delta(x - x_j) \prod_{l \neq j} \Psi^{\dagger}(x_l) \right\}|0\rangle \nonumber \\ = \int_0^L dx \int_0^L dx_1 ... dx_N \Psi_N (x_1, ..., x_N) \left\{ -\partial_x^2 \Psi^{\dagger}(x) \sum_{j=1}^N \delta(x - x_j) \prod_{l \neq j} \Psi^{\dagger}(x_l) \right\}|0\rangle \nonumber \\ = \int_0^L dx \int_0^L dx_1 ... dx_N \Psi_N (x_1, ..., x_N) \left\{ -\sum_{j=1}^N \delta(x - x_j) \partial_{x_j}^2 \right\} \prod_{l=1}^N \Psi^{\dagger}(x_l) |0\rangle \nonumber \\ = \int_0^L dx_1 ... dx_N \left\{ -\sum_{j=1}^N \partial_{x_j}^2 \Psi_N (x_1, ..., x_N) \right\} \prod_{l=1}^N \Psi^{\dagger}(x_l) |0\rangle \nonumber \\ \end{align*}

Second term:

\begin{align*} \int_0^L dx \int_0^L dx_1 ... dx_N \Psi_N (x_1, ..., x_N) c \Psi^{\dagger}(x) \Psi^{\dagger}(x) \Psi(x) \Psi(x) \prod_{l=1}^N \Psi^{\dagger}(x_l) |0\rangle \nonumber \\ = \int_0^L dx \int_0^L dx_1 ... dx_N \Psi_N (x_1, ..., x_N) ~c \Psi^{\dagger}(x) \Psi^{\dagger}(x) \sum_{j,k; j\neq k} \delta (x - x_j) \delta (x - x_k) \prod_{l \neq j,k} \Psi^{\dagger} (x_l) |0\rangle \nonumber \\ = \int_0^L dx \int_0^L dx_1 ... dx_N \Psi_N (x_1, ..., x_N) ~c \times \sum_{j,k; j\neq k} \Psi^{\dagger}(x_j) \Psi^{\dagger}(x_k) \delta (x - x_j) \delta (x_j - x_k) \prod_{l \neq j,k} \Psi^{\dagger} (x_l) |0\rangle \nonumber \\ = \int_0^L dx_1 ... dx_N \left\{c \sum_{j,k; j\neq k} \delta (x_j - x_k) \Psi_N (x_1, ..., x_N) \right\} \prod_{l=1}^N \Psi^{\dagger}(x_l) |0\rangle \nonumber \\ = \int_0^L dx_1 ... dx_N \left\{2c \sum_{j < k} \delta (x_j - x_k) \Psi_N (x_1, ..., x_N) \right\} \prod_{l=1}^N \Psi^{\dagger}(x_l) |0\rangle \end{align*}

The interaction term prevents a straightforward solution of the Schrödinger equation for the Lieb-Liniger model using the standard tools of many-body theory. To gain some insights into the difficulties, let us proceed as follows. Taking the particles to be confined to an interval of length $$L$$, Fourier transforms of the fields are defined as

\begin{equation} \psi (x) = \frac{1}{L} \sum_k e^{ikx} \psi_k, \hspace{10mm} \psi_k = \int_0^L dx e^{-ikx} \psi (x) \tag{l.f}\label{l.f} \end{equation}

giving canonical equal-time commutators

\begin{equation} \left[ \psi_k, \psi^\dagger_{k'} \right] = L \delta_{k k'}. \tag{l.crf}\label{l.crf} \end{equation}

For definiteness, we here impose periodic boundary conditions on the wavefunctions, meaning that the momenta are given by

\begin{equation*} k \in \{ k_n \}, \hspace{10mm} k_n = \frac{2\pi}{L} n, \hspace{10mm} n \in \mathbb{Z}. \end{equation*}

The Lieb-Liniger Hamiltonian is then

\begin{equation} \hat{H}_{\small LL} = \frac{1}{L} \sum_k k^2 \psi^\dagger_k \psi_k + \frac{c}{L^3} \sum_{k_1 k_2 q} \psi^\dagger_{k_1 + q} \psi^{\dagger}_{k_2 - q} \psi_{k_2} \psi_{k_1}. \tag{l.hf}\label{l.hf} \end{equation}

The interaction term thus induces transitions between momentum occupation modes which are of equal amplitude irrespective of the momentum transfer. Trying to apply perturbation theory in the interaction around $$c=0$$, we face many difficulties associated to the fact that we are perturbing around a macroscopically degenerate state (all bosons in the $$k=0$$ mode). The next subsection provides an improved (though still inexact) line of attack. Except where otherwise noted, all content is licensed under a Creative Commons Attribution 4.0 International License.

Created: 2023-06-07 Wed 16:02