Vortex stability in pseudo-Hermitian theories

Abstract

Pseudo-Hermitian (including $\mathcal{PT}$-symmetric) field theories support phenomenology that cannot be replicated in standard Hermitian theories. We describe a concrete example in which the vortex solutions that are realised in a prototypical pseudo-Hermitian field theory exhibit a novel metastability, despite the model parameters residing within the naively stable regime of exact antilinear symmetry of the vacuum theory. This instability is identified analytically and confirmed through numerical simulations, and it arises from the small breaking of the underlying antilinear symmetry of the pseudo-Hermitian theory due to the presence of the topological defect. This leads to spacetime-dependent parameters in the equations of motion governing fluctuations around the vortex, inducing non-trivial exceptional-point structures and complex frequencies within their spectrum. Aside from offering intriguing possibilities for cosmology, this result serves to illustrate the ability to produce long-lived metastable configurations in pseudo-Hermitian field theories of relevance beyond cosmology and high energy physics.

Figures (6)

• Figure 1: The profiles of string solutions for two different parameter sets. Both cases have $m_1^2=-1$, $\lambda_1=\lambda_2=1$ but parameter set P1 has $m_2^2=1$ and $m_3^2=0.1$, while parameter set P2 has $m_2^2 = 4$ and $m_3^2 = 1.5$. The vortex solutions for parameter set P1 have been presented previously in ref. Begun:2022ufc.
• Figure 2: Two of the eigenfunctions for parameter set P1 with $m=0$, normalised so that $\int r{\rm d}r \mathbf{v}^\dagger\mathbf{v} = 1$. One (a) corresponds to an instability with eigenvalue $\Lambda = 1.592-0.022i$ and the other (b) has the nearby eigenvalue of $\Lambda = 1.601$. As $m=0$, the $\tilde{\delta f_a}^*$ functions are identical to the ones presented here, so we have omitted them from the plots. The shaded band corresponds to the narrow region within which the linearised system can be described as locally non-Hermitian and it is clear that the unstable mode is concentrated around this zone. In contrast, the stable mode is more spread out despite having a similar typical oscillation length scale.
• Figure 3: Evolution of Re$(\phi_1)$ at $x = y = \frac{1}{2}\Delta x$ relative to the core of the string in both parameter sets. In the lower panel of (a), we also show the fractional difference between our prediction from the stability analysis and the results shown in the top panel. The P1 string is clearly unstable and matches the expectation from our stability analysis very well, albeit with a growing difference developing between the two. This difference only reaches a maximum of $0.01$ by $t=500$, which is about $8$ light-crossing times. In contrast, the P2 string appears stable, even when tested over very long timescales as presented here. Note that the other components of the fields have very similar behaviour.
• Figure 4: We show how the position of the P2 string along the $x=z=0$ line changes throughout a 3D simulation where we have set up the initial conditions to represent the string solution but sinusoidally displaced as a function of $z$. Even in this scenario with a manually applied perturbation, the string shows no signs of instability.
• Figure 5: Convergence testing of the complex eigenvalue, $\Lambda = 1.592 - 0.022i$, in parameter set P1, clearly showing that the value has converged to a good degree. For all other eigenvalues of our analysis for this vortex solution, the imaginary component converged to a value that could not confidently be distinguished from zero. We tested changing both the physical size of the grid, while leaving $\Delta r = 0.02$ (variable size), and changing the lattice spacing while keeping $N_r\Delta r = 160$ (variable spacing). We define $A = N_r/4000$ in the former case and $A = 0.04/\Delta r$ in the latter so that the two tests can be displayed on the same plot.