Final Report Summary - CONDMATH (Mathematical Problems in Superconductivity and Bose-Einstein Condensation)
The project CondMath has been concerned with the mathematical understanding of certain phenomena from modern quantum physics. An important focus area has been the mathematical theory of superconductivity. Superconductivity is well described by a non-linear model proposed by Ginzburg and Landau. The question that we study is the behavior of superconducting materials when submitted to strong external magnetic fields. It is known from experiments that the following scenario takes place (for superconductors of Type II).
For weak magnetic fields, the material will remain in the globally superconducting state. At a certain critical strength of the external magnetic field, HC1, superconductivity will be broken at pointlike singularities called vortices. When increasing the strength of the magnetic field further to a second critical field, HC2, superconductivity will be completely destroyed in the interior and only remain in a narrow boundary region. The third and final critical field, HC3, is where superconductivity is completely destroyed, also at the boundary.
The results of the project give a precise understanding of this scenario for field strengths starting below HC2 and up to HC3. This gives a full mathematical understanding of physical phenomena described already by deGennes and coworkers in the 1950s. The analysis is interesting and challenging both in 2 and 3 dimensions. For simplicity, let us describe the situation for a ball-shaped sample in 3D subjected to a constant magnetic field pointing from ‘south’ to ‘north’ on the ball. Starting from very strong magnetic fields and decreasing the intensity, there will be a well-defined value of HC3 (calculated by members of CondMath) where superconductivity will appear. It will first appear in a narrow region near the ‘equator’ of the ball. When decreasing the intensity of the magnetic field, superconductivity will remain localized to the boundary of the sample but a progressively larger ‘tropical’ region, i.e. a symmetric region around the ‘equator’, will carry superconductivity. For a given magnetic field strength, we can also prescribe the size of this ‘tropical’ region. The second critical field, HC2, is both where the ‘tropical’ region reaches the ‘poles’ of the ball and where superconductivity starts to build up in the interior of the ball. For magnetic field strengths slightly below HC2, superconductivity will be uniformly weak in the interior, but will slowly increase, still in a uniform way, i.e. every small volume of the ball (away from the boundary) will carry the same amount of superconductivity (to leading order).
When the magnetic field is decreased far below HC2 we reach another interesting regime, the vortex regime, where other mathematical techniques are necessary, but this is outside the scope of CondMath.
For weak magnetic fields, the material will remain in the globally superconducting state. At a certain critical strength of the external magnetic field, HC1, superconductivity will be broken at pointlike singularities called vortices. When increasing the strength of the magnetic field further to a second critical field, HC2, superconductivity will be completely destroyed in the interior and only remain in a narrow boundary region. The third and final critical field, HC3, is where superconductivity is completely destroyed, also at the boundary.
The results of the project give a precise understanding of this scenario for field strengths starting below HC2 and up to HC3. This gives a full mathematical understanding of physical phenomena described already by deGennes and coworkers in the 1950s. The analysis is interesting and challenging both in 2 and 3 dimensions. For simplicity, let us describe the situation for a ball-shaped sample in 3D subjected to a constant magnetic field pointing from ‘south’ to ‘north’ on the ball. Starting from very strong magnetic fields and decreasing the intensity, there will be a well-defined value of HC3 (calculated by members of CondMath) where superconductivity will appear. It will first appear in a narrow region near the ‘equator’ of the ball. When decreasing the intensity of the magnetic field, superconductivity will remain localized to the boundary of the sample but a progressively larger ‘tropical’ region, i.e. a symmetric region around the ‘equator’, will carry superconductivity. For a given magnetic field strength, we can also prescribe the size of this ‘tropical’ region. The second critical field, HC2, is both where the ‘tropical’ region reaches the ‘poles’ of the ball and where superconductivity starts to build up in the interior of the ball. For magnetic field strengths slightly below HC2, superconductivity will be uniformly weak in the interior, but will slowly increase, still in a uniform way, i.e. every small volume of the ball (away from the boundary) will carry the same amount of superconductivity (to leading order).
When the magnetic field is decreased far below HC2 we reach another interesting regime, the vortex regime, where other mathematical techniques are necessary, but this is outside the scope of CondMath.