Impurity effect and vortex cluster phase in mesoscopic type-1.5 superconductors

1- clearly written
2-easily reproducible
3-focused on a simple model

1-mathematically naive
2- not very comprehensive

This paper presents a numerical study of vortex solutions of a simple two-component Ginzburg Landau theory in a rectangular domain with prescribed impurities. Energy minimizers are found by solving the gradient flow equation for the energy functional for large times. An advantage of this method is that the PDEs can be formulated in a standard computational package (COMSOL) implementing a finite element method. This is not the first paper to take this approach, but still, going through the formulation in detail is potentially useful for the condensed matter theory community. The main advantage of finite elements (ability to accommodate irregular domains) is not relevant here since the work considers only rectangles. A disadvantage of the numerical approach is that gradient flow is extremely slow to converge - conjugate gradient methods or arrested Newton flow are many times faster.

The focus of the paper is unclear: are we studying impurities, or boundary effects? It would be informative to compare the results on mesoscopic domains with no impurities and varying sizes and shapes, and on large domains with impurities. Restricting the study to the simplest TCGL model is justified I think: the parameter space of the full model is too large to be surveyed.

Since boundary effects are important for the results here, I'm a bit troubled by the authors' choice of boundary condition. It's true that the conditions assumed ensure that no supercurrent passes through the boundary, but they are much stronger than is required by that condition. They are not gauge invariant, and they impose that each "component" of the supercurrent (associated with each condensate) is confined separately. For the simple model studied here, which has U(1) x U(1) symmetry, and hence separately conserved supercurrents, this may well be justified, but for two component GL models in general this strikes me as a very artificial assumption. The authors justify their choice by citation to the literature, but having followed the thread back 3 links I still haven't found a derivation of them. Given the importance (presumably) of boundary effects, I think a derivation of the boundary conditions from physical/mathematical principles is needed.

The figures refer to "evolution" of physical quantities. This is misleading as it is unrelated to the "time evolution" used to generate the solutions.

1- derive the boundary conditions from first principles
2- extend the numerical investigation to separte out boundary effects and impurity effects.
3- clarify the meaning of "evolution"

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The paper studies two-band superconductors in the presence of impurities in two dimensions. The authors numerically solve time-dependent Ginzburg-Landau equations for two-band superconductors to obtain stationary solutions. They analyze the magnetic flux and superconducting order parameter distributions for one isotropic, two isotropic, and one non-isotropic impurity with $C_4$ symmetry in the presence of the external magnetic field. For each case, the authors demonstrate solutions corresponding to type-1, type-1.5, and type-2 superconductors. In the second regime, the authors observe vortex clustering around the impurities, whose pattern is different for each of the three impurity types.

The paper contains a detailed description of the numerical algorithm and provides enough information to reproduce the demonstrated results. However, the authors do not fully utilize their opportunity to investigate the properties of their system.
First of all, the role of the impurities was not thoroughly analyzed. To improve this, the authors may, for example, consider how impurities and their strength affect the transitions between the Meissner, vortex cluster, and vortex lattice phases.
Secondly, the authors do not discuss how they can distinguish effects caused by impurities and boundaries. I suggest the authors compare the results for various system sizes and aspect ratios to separate the boundary effects.

In addition to this, there are several minor points that need to be clarified.
The choice of the value for the parameter $t$ (lines 153, 167, 175) requires some motivation.
The authors call the phase, demonstrated in Figs. 1a, 2a, 3a, a "Meissner phase". However, there is a nonzero magnetic field density on the impurities in this phase. This contradiction should be clarified.

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This paper addresses a topic of significant interest: the interplay between \ defects and vortices, specifically in the context of two-component superconductors. While thousands of studies explore this relationship in single-component systems, very few have extended this analysis to two-component models. This is an important gap in the literature, especially given the growing number of materials experimentally identified as type-1.5 superconductors. These discoveries underscore the need for theoretical investigations into vortex structures in two-component systems under the influence of pinning. The strength of the paper is that it focuses on this underexplored yet promising research area.

For a paper to be published in a good journal, it is required, in my opinion, to present a more systematic study.

While the paper addresses an important and underexplored topic, it lacks the systematic analysis required for publication in a top-tier journal. A more comprehensive study could include investigations of various defect configurations and models of disorder. For instance, vortex clustering can arise not only from attractive intervortex interactions but also as a consequence of pinning or disorder. The authors could explore whether there exists a critical disorder strength, or density of pinning centers, where the role of attractive interactions becomes particularly significant.

Additionally, a comparative analysis of type-II and type-1.5 vortex systems under different disorder models would add valuable context and broader relevance to the study. The inclusion of both correlated and uncorrelated disorder is also interesting.

With the current material, the paper is suitable for publication in some journal, but not yet at the level of a leading one. A more systematic approach, as outlined above, would strengthen its impact and contribution to the field.

The authors state: "As we know, each condensate in two-band superconductors is predicted to support vortex excitation with fractional quantum flux [4, 5]." However, this statement is problematic. Reference [4] does not actually consider fractional vortices; instead, it deals with an infinitely thin loop. For such a configuration, the enclosed flux is effectively zero. The misinterpretation in [4] arises from the assumption that the phase winding can differ from an integer multiple of 2𝜋, which is incorrect.

Additionally, the authors should update their discussion to acknowledge recent experimental advances. A Science paper published in 2023 and two arXiv preprints from 2024 report experimental observations of fractional vortices.

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