arXiv subjects

M. Correggi

Publications and source records attributed to M. Correggi.

At least 19 recordsLinked to original sources

arXiv2026

Existence of Bulk Vortices in Superconductors with Strong Magnetic Fields

We study the vortex formation in extreme type-II superconductors immersed in strong magnetic fields in the framework of the the Ginzburg-Landau theory. We focus on the regime where superconductivity survives in the bulk of the material but the magnetic field penetrates the sample, i.e., for applied field much larger than the first critical one, but below the transition to surface superconductivity. Through a two-scale vortex construction, we obtain precise estimates for the vortex distribution and prove the existence of isolated defects with non-trivial winding numbers. In this respect, our work provides the first rigorous mathematical proof of the existence of isolated vortices for fields comparable to the second critical one.

math-ph↗

arXiv2024

Quantum Point Charges Interacting with Quasi-classical Electromagnetic Fields

We study effective models describing systems of quantum particles interacting with quantized (electromagnetic) fields in the quasi-classical regime, i.e., when the field's state shows a large average number of excitations. Once the field's degrees of freedom are traced out on factorized states, the reduced dynamics of the particles' system is described by an effective Schrödinger operator keeping track of the field's state. We prove that, under suitable assumptions on the latter, such effective models are well-posed even if the particles are point-like, that is no ultraviolet cut-off is imposed on the interaction with quantum fields.

math-ph↗

arXiv2021

Microscopic Derivation of Time-dependent Point Interactions

We study the dynamics of the three-dimensional polaron - a quantum particle coupled to bosonic fields - in the quasi-classical regime. In this case the fields are very intense and the corresponding degrees of freedom can be treated semiclassically. We prove that in such a regime the effective dynamics for the quantum particles is approximated by the one generated by a time-dependent point interaction, i.e., a singular time-dependent perturbation of the Laplacian supported in a point. As a by-product, we also show that the unitary dynamics of a time-dependent point interaction can be approximated in strong operator topology by the one generated by time-dependent Schrödinger operators with suitably rescaled regular potentials.

math-ph↗

arXiv2021

Quasi-Classical Dynamics

We study quantum particles in interaction with a force-carrying field, in the quasi-classical limit. This limit is characterized by the field having a very large number of excitations (it is therefore macroscopic), while the particles retain their quantum nature. We prove that the interacting microscopic dynamics converges, in the quasi-classical limit, to an effective dynamics where the field acts as a classical environment that drives the quantum particles.

math-ph↗

arXiv2019

A Quantum Model of Feshbach Resonances

We consider a quantum model of two-channel scattering to describe the mechanism of a Feshbach resonance. We perform a rigorous analysis in order to count and localize the energy resonances in the perturbative regime, i.e., for small inter-channel coupling, and in the non-perturbative one. We provide an expansion of the effective scattering length near the resonances, via a detailed study of an effective Lippmann-Schwinger equation with energy-dependent potential.

math-ph↗

arXiv2018

Hamiltonians for Two-Anyon Systems

We study the well-posedness of the Hamiltonian of a system of two anyons in the magnetic gauge. We identify all the possible quadratic forms realizing such an operator for non-interacting anyons and prove their closedness and boundedness from below. We then show that the corresponding self-adjoint operators give rise to a one-parameter family of extensions of the naive two-anyon Schrödinger operator. We finally extend the results in presence of a two-body radial interaction.

math-ph↗

arXiv2018

Magnetic Schrödinger Operators as the Quasi-Classical Limit of Pauli-Fierz-type Models

We study the quasi-classical limit of the Pauli-Fierz model: the system is composed of finitely many non-relativistic charged particles interacting with a bosonic radiation field. We trace out the degrees of freedom of the field, and consider the classical limit of the latter. We prove that the partial trace of the full Hamiltonian converges, in resolvent sense, to an effective Schrödinger operator with magnetic field and a corrective electric potential that depends on the field configuration. Furthermore, we prove the convergence of the ground state energy of the microscopic system to the infimum over all possible classical field configurations of the ground state energy of the effective Schrödinger operator.

math-ph↗

arXiv2018

Local Density Approximation for Almost-Bosonic Anyons

We discuss the average-field approximation for a trapped gas of non-interacting anyons in the quasi-bosonic regime. In the homogeneous case, i.e., for a confinement to a bounded region, we prove that the energy in the regime of large statistics parameter, i.e., for "less-bosonic" anyons, is independent of boundary conditions and of the shape of the domain. When a non-trivial trapping potential is present, we derive a local density approximation in terms of a Thomas-Fermi-like model.

math-ph↗

arXiv2016

On the Third Critical Speed for Rotating Bose-Einstein Condensates

We study a two-dimensional rotating Bose-Einstein condensate confined by an anharmonic trap in the framework of the Gross-Pitaevksii theory. We consider a rapid rotation regime close to the transition to a giant vortex state. It was proven in [M. Correggi {\it et al}, {\it J. Math. Phys. \textbf{53}(2012)] that such a transition occurs when the angular velocity is of order $ \varepsilon ^{-4}$, with $ \varepsilon ^{-2} $ denoting the coefficient of the nonlinear term in the Gross-Pitaevskii functional and $ \varepsilon \ll 1 $ (Thomas-Fermi regime). In this paper we identify a finite value $ Ω_{\mathrm{c}} $ such that, if $ Ω= Ω_0/\varepsilon ^4 $ with $ Ω_0 > Ω_{\mathrm{c}} $, the condensate is in the giant vortex phase. Under the same condition we prove a refined energy asymptotics and an estimate of the winding number of any Gross-Pitaevskii minimizer.

math-ph↗

arXiv2016

Surface Superconductivity in Presence of Corners

We consider an extreme type-II superconducting wire with non-smooth cross section, i.e., with one or more corners at the boundary, in the framework of the Ginzburg-Landau theory. We prove the existence of an interval of values of the applied field, where superconductivity is spread uniformly along the boundary of the sample. More precisely the energy is not affected to leading order by the presence of corners and the modulus of the Ginzburg-Landau minimizer is approximately constant along the transversal direction. The critical fields delimiting this surface superconductivity regime coincide with the ones in absence of boundary singularities.

math-ph↗

arXiv2016

Low-Temperature Spin-Wave Approximation for the Heisenberg Ferromagnet

We study the low temperature thermodynamics of the quantum Heisenberg ferromagnet in three dimension for any value of the spin $ S \geq 1/2 $. We report on a rigorous proof of the validity of the spin-wave approximation for the excitation spectrum, at the level of the first non-trivial contribution to the free energy as the inverse temperature $ β\to \infty $.

math-ph↗

arXiv2016

Two-dimensional Time-dependent Point Interactions

We study the time-evolution of a quantum particle subjected to time-dependent zero-range forces in two dimensions. After establishing a conceivable ansatz for the solution to the Schrödinger equation, we prove that the wave packet time-evolution is completely specified by the solutions of a system of Volterra-type equations -- the {\it charge equations} -- involving the coefficients of the singular part of the wave function, thus extending to the two-dimensional case known results in one and three dimensions.

math-ph↗

arXiv2015

A Class of Hamiltonians for a Three-Particle Fermionic System at Unitarity

We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass $ m $, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for $ m $ larger than a critical value $ m^* \simeq (13.607)^{-1} $ a self-adjoint and lower bounded Hamiltonian $ H_0 $ can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for $ m \in( m^*,m^{**}) $, where $ m^{**} \simeq (8.62)^{-1} $, there is a further family of self-adjoint and lower bounded Hamiltonians $ H_{0,β} $, $ β\in \mathbb{R} $, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.

math-ph↗

arXiv2015

Boundary Behavior of the Ginzburg-Landau Order Parameter in the Surface Superconductivity Regime

We study the 2D Ginzburg-Landau theory for a type-II superconductor in an applied magnetic field varying between the second and third critical value. In this regime the order parameter minimizing the GL energy is concentrated along the boundary of the sample and is well approximated to leading order by a simplified 1D profile in the direction perpendicular to the boundary. Motivated by a conjecture of Xing-Bin Pan, we address the question of whether this approximation can hold uniformly in the boundary region. We prove that this is indeed the case as a corollary of a refined, second order energy expansion including contributions due to the curvature of the sample. Local variations of the GL order parameter are controlled by the second order term of this energy expansion, which allows us to prove the desired uniformity of the surface superconductivity layer.

math-ph↗

arXiv2014

On the Ginzburg--Landau Functional in the Surface Superconductivity Regime

We present new estimates on the two-dimensional Ginzburg-Landau energy of a type-II superconductor in an applied magnetic field varying between the second and third critical fields. In this regime, superconductivity is restricted to a thin layer along the boundary of the sample. We provide new energy lower bounds, proving that the Ginzburg-Landau energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. Estimates relating the density of the Ginzburg-Landau order parameter to that of the 1D problem follow. In the particular case of a disc sample, a refinement of our method leads to a pointwise estimate on the Ginzburg-Landau order parameter, thereby proving a strong form of uniformity of the surface superconductivity layer, related to a conjecture by Xing-Bin Pan.

math-ph↗

arXiv2012

Vortex Phases of Rotating Superfluids

We report on the first mathematically rigorous proofs of a transition to a giant vortex state of a superfluid in rotating anharmonic traps. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and large rotational velocity and is based on precise asymptotic estimates on the ground state energy. An interesting aspect is a significant difference between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary. In the former case vortices persist in the bulk until the width of the annulus becomes comparable to the size of the vortex cores. In the second case the transition already takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus. Moreover, the density profiles in the annulus are different in the two cases. In both cases rotational symmetry of the density in a true ground state is broken, even though a symmetric variational ansatz gives an excellent approximation to the energy.

cond-mat.quant-gas↗

arXiv2012

The Free Energy of the Quantum Heisenberg Ferromagnet at Large Spin

We consider the spin-S ferromagnetic Heisenberg model in three dimensions, in the absence of an external field. Spin wave theory suggests that in a suitable temperature regime the system behaves effectively as a system of non-interacting bosons (magnons). We prove this fact at the level of the specific free energy: if $ S \to \infty $ and the inverse temperature $ β\to 0 $ in such a way that $ βS $ stays constant, we rigorously show that the free energy per unit volume converges to the one suggested by spin wave theory. The proof is based on the localization of the system in small boxes and on upper and lower bounds on the local free energy, and it also provides explicit error bounds on the remainder.

math-ph↗

arXiv2012

Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions

We study the stability problem for a non-relativistic quantum system in dimension three composed by $ N \geq 2 $ identical fermions, with unit mass, interacting with a different particle, with mass $ m $, via a zero-range interaction of strength $ α\in \R $. We construct the corresponding renormalised quadratic (or energy) form $ \form $ and the so-called Skornyakov-Ter-Martirosyan symmetric extension $ H_α $, which is the natural candidate as Hamiltonian of the system. We find a value of the mass $ m^*(N) $ such that for $ m > m^*(N)$ the form $ \form $ is closed and bounded from below. As a consequence, $ \form $ defines a unique self-adjoint and bounded from below extension of $ H_α$ and therefore the system is stable. On the other hand, we also show that the form $ \form $ is unbounded from below for $ m < m^*(2)$. In analogy with the well-known bosonic case, this suggests that the system is unstable for $ m < m^*(2)$ and the so-called Thomas effect occurs.

math-ph↗