The Bethe Ansatz


Let's consider the Lieb-Liniger model,

\begin{equation*} H_{LL} = \int_0^L dx \left\{ \Psi^\dagger (x) (-\partial_x^2) \Psi(x) + c \Psi^\dagger(x) \Psi^\dagger(x) \Psi(x) \Psi(x) - \mu \Psi^\dagger(x) \Psi(x) \right\} \end{equation*}

with canonical equal-time commutation relations \(\left[ \Psi(x), \Psi^\dagger (x') \right] = \delta(x-x')\).

Notations used here: system length is \(L\), number of particles is \(N = \int_0^L dx \Psi^\dagger (x) \Psi(x)\) (previously, the integer \(N\) was the number of sites, which now gets replaced by the system length given by the continuous parameter \(L\); don't confuse these for each other).

We define the Fourier transforms 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) \end{equation*}

so \(\left[ \Psi_k, \Psi^\dagger_{k'} \right] = L \delta_{k k'}\) and the Hamiltonian is

\begin{equation*} H_{LL} = \frac{1}{L} \sum_k (k^2 - \mu) \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}. \end{equation*}

Density-density correlations (dynamical structure factor)

The dynamical structure factor in our notations is then

\begin{equation*} {\boldsymbol S}^{\rho \rho} (k, \omega) = \frac{1}{L} \int_0^L dx dx' e^{-ik (x - x')} \int_{-\infty}^\infty dt e^{i\omega t} \langle \frac{1}{2}\left[ \rho(x,t), \rho(x',0) \right] \rangle \end{equation*}

For the density operator, we write

\begin{equation*} \rho_k \equiv \int_0^L dx e^{-ikx} \rho(x) = \int_0^L dx e^{-ikx} \Psi^\dagger (x) \Psi(x) = \frac{1}{L} \sum_{k_1} \Psi^\dagger_{k_1} \Psi_{k_1 + k}. \end{equation*}

For the f-sumrule applied to the dynamical structure factor, we get

\begin{equation*} \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{\rho \rho} (k, \omega) = \frac{-1}{2L} \langle \left[ \left[ H, \rho_k \right], \rho_{-k} \right] \rangle. \end{equation*}

Let's calculate the right-hand side of the f-sumrule. Start with

\begin{equation*} \left[ H, \rho_k \right] = \frac{1}{L} \sum_{k_1} \left\{ \frac{1}{L} \sum_{k_2} (k_2^2 -\mu) \left[ \Psi^\dagger_{k_2} \Psi_{k_2}, \Psi^\dagger_{k_1} \Psi_{k_1 + k} \right] + \frac{c}{L^3} \sum_{k_3 k_4 q} \left[ \Psi^\dagger_{k_3 + q} \Psi^\dagger_{k_4 - q} \Psi_{k_4} \Psi_{k_3}, \Psi^\dagger_{k_1} \Psi_{k_1 + k} \right] \right\} \end{equation*}

The commutator in the first term is

\begin{equation*} \Psi^\dagger_{k_2} \Psi_{k_1 + k} L \delta_{k_1 k_2} - \Psi^\dagger_{k_1} \Psi_{k_2} L \delta_{k_2, k_1 + k} \end{equation*}

so the first term becomes

\begin{align*} \frac{1}{L} &\sum_{k_2} (k_2^2 - \mu) \left( \Psi^\dagger_{k_2} \Psi_{k_2 + k} - \Psi^\dagger_{k_2 - k} \Psi_{k_2} \right) \nonumber \\ &= \frac{1}{L} \sum_{k_2} (k_2^2 - (k_2 + k)^2) \Psi^\dagger_{k_2} \Psi_{k_2 + k} \nonumber \\ &= \frac{1}{L} \sum_{k_1} (-k)(k+2k_1) \Psi^\dagger_{k_1} \Psi_{k_1 + k}. \end{align*}

The commutator in the second term is

\begin{equation*} \Psi^\dagger_{k_1} \left[ \Psi^\dagger_{k_3 + q} \Psi^\dagger_{k_4 - q}, \Psi_{k_1 + k} \right] \Psi_{k_4} \Psi_{k_3} + \Psi^\dagger_{k_3 + q} \Psi^\dagger_{k_4 - q} \left[ \Psi_{k_4} \Psi_{k_3}, \Psi^\dagger_{k_1} \right] \Psi_{k_1 + k} \end{equation*}

Calculating this and performing the sums over \(k_1\), \(k_3\) and \(k_4\) shows that this whole second term vanishes. We thus get

\begin{equation*} \left[ H, \rho_k \right] = \frac{1}{L} \sum_{k_1} (-k)(k+2k_1) \Psi^\dagger_{k_1} \Psi_{k_1 + k}. \end{equation*}

Going further (using \((\rho_k)^\dagger = \rho_{-k}\)),

\begin{align*} \left[ \left[ H, \rho_k \right], \rho_{-k} \right] &= \frac{1}{L^2} \sum_{k_1} (-k)(k+2k_1) \sum_{k_2} \left[ \Psi^\dagger_{k_1} \Psi_{k_1 + k}, \Psi^\dagger_{k_2} \Psi_{k_2 - k} \right] \nonumber \\ &= \frac{1}{L} \sum_{k_1} (-k) (k + 2k_1) \left( \Psi^\dagger_{k_1} \Psi_{k_1} - \Psi^\dagger_{k_1 + k} \Psi_{k_1 + k} \right) \nonumber \\ &= \frac{1}{L} \sum_{k_1} (-k)[k + 2k_1 - (-k + 2k_1)] \Psi^\dagger_{k_1} \Psi_{k_1} \nonumber \\ &= -2k^2 \frac{1}{L} \sum_{k_1} \Psi^\dagger_{k_1} \Psi_{k_1} = -2N k^2 \end{align*}

We thus obtain the explicit f-sumrule for the dynamical structure factor,

\begin{equation} \int_{-\infty}^\infty \frac{d\omega}{2\pi} \omega ~{\boldsymbol S}^{\rho \rho} (k, \omega) = \frac{N}{L} k^2. \tag{l.dsf.fsr}\label{l.dsf.fsr} \end{equation}

This allows to check the overall intensity for fixed values of \(k\).

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24