Hydrodynamics of a quantum vortex in the presence of twist – CORRIGENDUM

In Foresti & Ricca (2020) (hereafter referred to as FR20) we derived a modified form of the Gross–Pitaevskii equation for a defect subject to twist. A mistake was introduced by the wrong use of the operator $\widetilde {\boldsymbol {\nabla }}=\boldsymbol {\nabla }-\mathrm {i}\boldsymbol {\nabla }\theta _{tw}$. By repeating the same calculations we can see that the mGPE (2.6) must be replaced by the following equation:

(0.1)\begin{align} \partial_t\psi_1 &= \frac{\mathrm{i}}{2} \nabla^{2}\psi_1 + \frac{\mathrm{i}}{2}\left(1-|\psi_{tw}|^{2}-|\boldsymbol{\nabla} \theta_{tw}|^{2}\right)\psi_1+\mathrm{i}(\partial_t\theta_{tw})\psi_1 \nonumber\\ &\quad + \frac{1}{2}\nabla^{2}\theta_{tw}\psi_1 + \boldsymbol{\nabla}\theta_{tw}\boldsymbol{\cdot}\boldsymbol{\nabla}\psi_1. \end{align}

Note the extra terms that come from the broken symmetry of the theory under superposition of a local phase.

The Hamiltonian (3.1) then becomes

(0.2)\begin{equation} H_{tw} = \tfrac{1}{2} {{\boldsymbol{p}}}^{2} - \tfrac{1}{2}(1 - |\psi_{tw}|^{2}) + V_{tw}, \end{equation}

where ${{\boldsymbol {p}}} = -\mathrm {i}\boldsymbol{\nabla}$ is the momentum operator, and

(0.3)\begin{equation} V_{tw}=\frac{\mathrm{i}}{2}\nabla^{2}\theta_{tw} + \frac{1}{2}\left|\boldsymbol{\nabla}\theta_{1}\right|^{2} -\partial_t\theta_{tw} - \boldsymbol{\nabla}\theta_{tw}\boldsymbol{\cdot}{{\boldsymbol{p}}} \end{equation}

is the twist potential. It can be directly verified that the above Hamiltonian is also non-Hermitian.

The energy expectation value $E_{tw}$ is given by the contribution of the unperturbed state $\psi _0$ and twist. Since the twist contribution is linear in $\psi _1$, it can be obtained from the expectation value of $V_{tw}$ and the kinetic part that depends on $\theta _{tw}$; thus, (3.5) must be replaced by

(0.4)\begin{align} E_{tw} &= \int \left[\left(\frac{1}{2} |\boldsymbol{\nabla}\theta_{tw}|^{2} -\partial_t\theta_{tw} + \frac{\mathrm{i}}{2} \nabla^{2}\theta_{tw}\right)|\psi_{1}|^{2}+ \mathrm{i}\boldsymbol{\nabla}\theta_{tw}\boldsymbol{\cdot}\boldsymbol{\nabla}\psi_{1} \right. \nonumber\\ &\quad +\frac{1}{2}\left. |\boldsymbol{\nabla}\psi_{1}|^{2} -\frac{1}{2} |\psi_{1}|^{2} + \frac{1}{4}|\psi_{1}|^{4}\right] {{\rm d}}V. \end{align}

Upon application of the Madelung transform $\psi _{1} = \sqrt \rho \exp (\mathrm {i} \chi _{1})$, taking $\boldsymbol {\nabla }\theta _{tw}\boldsymbol {\cdot }\boldsymbol {\nabla }\rho = 0$ in the neighborhood of the defect, we have

(0.5)\begin{align} E_{tw} &= \int \left[\left(\frac{1}{2} |\boldsymbol{\nabla}\theta_{tw}|^{2} -\partial_t\theta_{tw}- \boldsymbol{\nabla}\theta_{tw}\boldsymbol{\cdot}\boldsymbol{\nabla}\chi_{1} + \frac{1}{2} |\boldsymbol{\nabla}\psi_{1}|^{2} -\frac{1}{2}+\frac{1}{4}|\psi_{1}|^{2}\right) \right. \nonumber\\ &\quad + \frac{\mathrm{i}}{2} \left. \nabla^{2}\theta_{tw}\right]|\psi_{1}|^{2} \,{{\rm d}}V. \end{align}

As in FR20, the imaginary term above makes the Hamiltonian non-Hermitian, and the twisted state remains unstable. Following what is done in FR20 (§ 3), by the same procedure we obtain the correct dispersion relation

(0.6)\begin{equation} \nu = \frac{1}{2}\left[\left(|\boldsymbol{k}|^{2} - 2\boldsymbol{\nabla}\theta_{tw}\boldsymbol{\cdot}\boldsymbol{k} + |\boldsymbol{\nabla}\theta_{tw}|^{2}-1-2\partial_t\theta_{tw}\right)+\frac{\mathrm{i}}{2}\nabla^{2}\theta_{tw}\right]. \end{equation}

The instability criterion of § 3 remains unaltered.

Since injection of negative twist is given by a rotation of the twist phase opposite to the vortex orientation, if we replace $\theta _{tw}\rightarrow -\theta _{tw}$ we evidently have instability when $\nabla ^{2}\theta _{tw} < 0$ as $t \rightarrow \infty$.