The Bethe Ansatz

The Lieb-Liniger model c.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{}\label{} \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{}\label{} \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{}\label{} \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.

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24