Emergence of macroscopic models from the BCS theory of superconductivity

This thesis aims to study the derivation of a suitable effective macroscopic theory from the quantum microscopic description of a low-temperature fermionic gas confined in a trap and possibly in the presence of a strong external magnetic field. More precisely, our goal is to study the quantum macroscopic counterpart of the microscopic pairing mechanism modeled by the BCS theory of superconductivity. The most spectacular phenomenon observable for gas of particles at low temperatures is quantum condensation. When this phenomenon occurs in a degenerate system of bosons (e.g., bosonic atoms), it is known as \textit{Bose–Einstein condensation} \cite{Af: paper, AFBN: paper, AH: paper, DSY: paper, HSc: paper, HS1: paper, LSY: paper, LSSY}; in a system of degenerate fermions (e.g., electrons), we refer to it as \textit{Cooper pairing} \cite{AL: paper, L: paper}. Depending on the bosonic/fermionic nature of the gas, a macroscopic fraction of particles or pairs of particles, which from now on we refer to as the \textit{condensate}, behave in the same way. For example, in a gas of interacting bosons in a trap, one observes that the momentum distribution becomes very peaked around a point (typically zero momentum) at extremely low temperature as a consequence of the condensation of a large portion of the particles in the same (zero-momentum) one particle state. Fermions give another typical example in a metal: at room temperature, electrons move independently, while when the sample is cooled down below a specific critical temperature, the particles get bound in Cooper pairs by an attractive interaction due to the presence of the underlying ionic structure. A very striking consequence of quantum condensation is \textit{superconductivity} (in a charged fermionic system, as electrons in metals \cite{dG: paper}), a quantum mechanical phenomenon consisting of a sudden drop of electrical resistance and expulsion of magnetic fields occurring in certain materials when cooled below a critical temperature \cite{Bardeen: paper, B: paper, CHO: paper, DR: paper, Evetts: paper, Ly: paper, Parks: paper, R: paper, R-IR: paper, Schieffer: paper}. This phenomenon is related to the formation of electronic Cooper pairs: a small attraction between electrons with opposite spin may generate pair-bound states that behave like bosons and, at low temperatures, show a collective behavior as in a Bose-Einstein condensate. This behavior is responsible for the phenomenon of superconductivity via the generations or charge carriers (Cooper pairs) flowing in the material without friction. This phenomenon was studied widely also for material with high-critical temperature \cite{Bedell: paper, CP: paper, Ginzberg: paper, Hall: paper, Phillips: paper}. The \textit{BCS theory of superconductivity} was formulated in 1957 by the American physicists Bardeen, Cooper, and Schrieffer \cite{BCS: paper}. In this model, all the information about the state of the system is encoded in two variables: the reduced one-particle density matrix $\gamma$ and the pairing density matrix $\alpha$. The free energy of the system is given by the BCS functional, which depends on $\gamma$ and $\alpha$ so that any stationary state is a critical point of the functional and thus a solution of the corresponding Euler-Lagrange equations (the BCS gap equation). The BCS gap equation must have a solution with $\alpha$ different from $0$ for the system to be superconducting. If this is the case, the system is said to be in a \textit{superconducting state}; otherwise, it is in a normal state. This condition is, in turn, equivalent to the existence of a negative eigenvalue of an operator, which can be explicitly determined from the gap equation so that the variational problem may be reduced to a spectral one \cite{HHSS: paper, HS2: paper, HS4: paper}. The BCS theory is already an effective description of the system, although still microscopic, and can be derived at least heuristically from first principles description of a gas of electrons inside a metal under the assumption that the underlying ionic array mediates the quantum state in quasi-free and the interaction between the electrons \cite{BLS: paper, Yang: paper}. It was realized in the eighties \cite{L: paper} that BCS theory applies to both BECs of tightly bound fermions and cases where the pairing mechanism is very weak. The only difference is in the nature of the pairing mechanism, but the consequent condensation phenomenon is the same. The intermediate regime between weak and strong binding is called the BEC-BCS crossover regime \cite{L: book}. A few years before the BCS description of superconductivity appeared, a phenomenological macroscopic explanation was provided in \cite{GL: paper, GL: collected papers} by V.L. Ginzburg and L.D. Landau. In the \textit{Ginzburg-Landau} (GL) theory, the superconducting features of the sample are encoded in an order parameter $ \psi $, i.e., a complex wave function minimizing a suitable energy functional, which is supposed to approximate the free energy of the system. In the presence of an applied magnetic field, the response of the system is encoded in another variable besides the order parameter which is the induced magnetic field: outside the sample and at its boundary, the magnetic field is given by the applied one, while inside the sample it is provided by a self-generated field which minimizes the free energy functional \cite{AT: paper, BH: paper, BJP: paper, FH: paper, Ginz: paper, GS: paper, JZ: paper, K: paper, KP: paper, SS2: paper, BHS1: paper, Deu: paper, DGHL: paper}. The connection between the two models was heuristically investigated in \cite{G: paper}. For a dilute bosonic system, the counterpart of the GL theory is the \textit{Gross-Pitaevskii} (GP) one: when BEC takes place, all the particles of the gas behave in the same way, and their one-particle wave function is shown to minimize a suitable nonlinear functional - the GP functional - which provides an effective macroscopic description of the phenomenon. For a fermionic gas, if the system favors the formation of stable bosonic molecules, the Gross-Pitaevskii theory description is more appropriate. Indeed, if the physical regime under investigation is a BEC one, the mechanism behind the emergence of collective behavior in the low-temperature Fermi gas is not the usual BCS pairing phenomenon but rather a condensation of fermionic pairs playing the role of bosonic molecules. The effective representation of the condensate is slightly different depending on the pairing, even if the two functionals used to describe it may appear mathematically similar. As anticipated, the GP theory describes a low-density regime where tightly bound bosonic molecules are far apart \cite{BHS: paper, CC: paper, FLS: paper, Sei: paper}. Conversely, when weakly bound Cooper pairs are formed, the GL theory is the appropriate limiting theory \cite{DHS: paper, DHS1: paper, FHSS: paper}. Note that both models describe the phase transition of a quantum system between a normal and a condensed state. They have been widely studied in the mathematical literature thanks to their formulation as variational problems. The BCS model offers a significant simplification of the full many-body problem. In the many-body theory, a state is characterized by a complex wave function involving a large number of variables. In contrast, the BCS model encodes all information in quantities depending on only two variables: the reduced one-particle density matrix $\gamma$, a positive trace class operator in the one-particle space, and the Cooper pair wave function $\alpha$, a two-particle wave function which is non identically zero only below the critical temperature. The GL model describes the system through a single function $\psi$ depending only on one variable, which satisfies a nonlinear second-order PDE, known as the GL equation. This variable is the center of mass of the fermionic pair. While $\psi$ captures macroscopic variations in the system, BCS states $\gamma$ and $\alpha$ contain both microscopic and macroscopic information on the system. Consequently, GL represents a substantial simplification compared to BCS theory. To better understand the relationship between these theories, one can think of atomic physics, where quantum mechanics, Hartree-Fock theory, and Thomas-Fermi theory embody increasing levels of simplicity. However, unlike the atomic case, where the Hartree-Fock approximation is well understood mathematically, establishing the rigorous connection between the BCS model and the complete many-body quantum-mechanical description remains an open problem \cite{HS: review}. From a mathematical point of view, the rigorous derivation of GL from BCS theory was first addressed by C. Hainzl, R. Frank, R. Seiringer, and J.P. Solovej \cite{FHSS: paper, HHSS: paper}, who provided the first solution to the problem in the translation-invariant case. To concretely realize the BCS model, they introduce a scale parameter $h>0$, yielding the ratio between the microscopic and the macroscopic length scales to make a scale separation apparent. They consider a temperature range where $T$ is very close to the critical temperature $|T - T_c| < h^2$ in the presence of periodic external magnetic and electric fields (so that the translation invariance is not broken). Then, they show that the GL theory arises as an effective theory from the microscopic BCS theory. The fermion pairing function $\alpha$ of any approximate BCS minimizer lives on two different scales: on the microscopic scale $h$, one observes the pair binding, while the energy leading order lives on the macroscopic scale $1$. In that framework, the external fields lead to a variation of the pair wave function in its center of mass coordinate, which on the macroscopic scale is described by the GL theory. In contrast, the behavior in the relative coordinate remains unaffected. Since the derivation of the effective GP and GL theories has primarily addressed in translationally invariant systems or systems with weak external potentials that do not substantially influence the physics of the problem, a natural question arises: \textit{is it possible to derive an effective theory in the case of physically relevant domains, with external potentials and/or in the presence of external magnetic fields that break translational invariance?} The aim of this thesis is to investigate non-translation-invariant systems, e.g., because of an external trap or the presence of a magnetic field. That is, we want to describe not only systems which are approximately translation-invariant, such as liquid $^4{He}$, but also, for example, the magnetically trapped atomic alkali gases and electrons in amorphous (noncrystalline) metals, see \cite{L: book}. To do this, it is convenient to allow our particles to move in an arbitrarily bounded one-particle external potential that is a function of the coordinates. In this way we will try to expand the possibility of studying systems on different relevant physical domains. After presenting the physical models in which superconductivity can occur, in \cref{chapter: 2 - supercond e BCS theory} we present a heuristic derivation of the BCS theory starting from the many-body quantum mechanics. Moreover, we review the main mathematical properties of the BCS functional and some results related to the critical temperature \cite{FHL: paper, FHNS: paper, FHSS1: paper, HS3: paper, FH1: paper}. In \cref{chapter: 3 derivation op GP and GL theories}, we present the GL and GP theories, focusing on the central question of their derivation from the microscopic BCS theory. In this chapter, we review some results from the literature addressing non-translationally invariant cases. In \cref{section: derivation of GP low density}, the starting point is the Bogoliubov-Hartree-Fock functional \cite{BHS: paper, BFJ: paper}, containing also additional exchange and direct term, and still the leading order of the energy is given by the energy of a repulsive Bose gas. The setting is a limiting regime where the particles are far away from one another, and the small parameter $h$ plays the role of the inverse of the particle number. The choice of this parameter and its relation to the external potential, representing the size of the box in which the particles are confined, implies that the system is in a low-density regime (\cref{proposition: mu and low density}): in this regime, particles pair up, giving rise to the condensate. In \cref{section: Derivation of the GP Functional in a Bounded Domain}, the setting is similar: $N \gg 1$ fermions are confined in a bounded domain with Dirichlet boundary conditions. It has been proven that \cite{FLS: paper}, for fermions in a low-density regime, from the BCS theory with Dirichlet boundary conditions arises GP theory again with Dirichlet boundary conditions. This is the first attempt to derive GP theory from a system of particles in a hard walls trap, but, as the authors also point out, the result is in contrast with De Gennes' predictions in \cite{dG: paper}. Finally, in \cite{DHS: paper, DHS1: paper}, it is shown a possible extension of the work of \cite{FHSS: paper} to the case of external magnetic potentials with a nonzero magnetic flux through the unit cell, using phase approximation methods instead of semiclassical analysis techniques. In \cref{chapter: 4 - derivation of GP trap}, we study a system of electrons trapped by an external potential $W$ with a polynomial asymptotic behavior in the presence of a two-body interaction $V$ strong enough to bind two particles together. Our work presents the first proof of the occurrence of pairing in the ground state of a non-translation invariant BCS system, where the particles are contained in a soft trapping external potential. Let us briefly present the setting: we fixed the length scale of the trap to be 1, while the interaction varies on a microscopic scale of order $h$. The parameter $h$ thus describes the ratio between the microscopic and macroscopic scales, and we study this system in a limit of small $h$. We do not fix the number of particles a priori. Indeed, we study the grand canonical problem in the presence of a chemical potential $\mu$. We also provide an expansion of the Cooper pair wave function $\alpha$: in the limit of small $h$, to leading order, one gets \begin{equation} \label{alpha intro} \alpha(x,y) = h^{-2}\psi\left(\tfrac{x+y}{2}\right)\alpha_{0}\left( \tfrac{x-y}{h} \right), \end{equation} where $\psi$ is an approximate minimizer of the GP functional and describes the center-of-mass motion of the Cooper pair wave function $\alpha$. Thus, the two-particle wave function really separates into relative and center-of-mass coordinates, and from this behavior, one observes the emergence of superconductivity. The expansion \eqref{alpha intro} together with the heuristics $ \gamma \simeq \alpha \overline{\alpha}$ suggests that the density of the gas in our setting is proportional to $ h^{-1} \lf| \psi \ri|^2 $, i.e., the total number of particles is of order $ h^{-1} $. Then, we get to our main result, stated in \cref{thm: main thm capitolo 4}, providing a bound from above and a bound from below of the BCS energy through the GP energy, up to an error of order $h^2$: \begin{equation} E_{\mu}^{\BCS} = h E_{D}^{\GP} + O(h^{2}). \end{equation} An upper bound quickly follows from a variational principle. The lower bound, instead, requires more work; as anticipated above, its proof is based on the combination of a gap condition for the interaction problem and a priori estimates on some error terms. In \cref{chapter: toward a derivation of GP}, we present an explanatory scheme of a model going toward the first semiclassical derivation of the GP theory in the presence of a strong self-consistent magnetic field from the microscopic BCS model, except for some aspects still in the form of conjectures, susceptible to further investigation. This means that by considering a small parameter in the system, which will play the role of an effective Planck constant, we can derive an expansion of the BCS energy depending on that parameter. As in the previous chapter, our goal is to obtain, at leading order, an expression of the BCS energy in terms of the GP one. This is the first step towards studying the system in the presence of more physically relevant fields (i.e., completing the case of strong magnetic fields or anyonic particles carrying Aharonov-Bohm fluxes), along with physical boundary conditions. We study the self-adjoint realization of a magnetic Schrödinger operator in a bounded domain with Neumann boundary conditions. Thus, to get a more physically coherent understanding of the topic, we consider a system consisting of $N$ interacting fermions in the presence of an external magnetic field: this describes an effective theory in which we expect to observe how the spins of the electrons generate a magnetic field within the material that compensates for the external one. The added complexities arise from the fact that we consider an intense induced magnetic potential. In this setting, the behavior of the sample can be read off from the properties of the minimizer of the GP functional, which now is a pair: the order parameter of the theory and an induced or self-consistent magnetic potential. We consider a low-density fermionic system in a regime in which the parameter describing the intensity of the applied magnetic field is significantly large, and we fix the temperature at $T=0$. In contrast with the case of the previous chapter, we consider how the induced (or self-consistent) magnetic potential $\textbf{A}$, and in turn, the induced field $hb\curl{\textbf{A}}$ responds to the influence of the external magnetic field. The length of microscopic interactions is order $h$, while, in this setting, we also consider a large parameter $b$ related to the strength of the external magnetic potential. In this setting, we also consider a low-density regime, in which the particles undergo a superconductivity transition. Again, the idea of the proof is based on matching an upper and a lower bound for the BCS energy in terms of the GP energy \begin{equation} E^{\BCS}_{\mu,b} = h^2 \left(E^{\GP}_{D,b} + O\left(h^2 b^{1+\alpha}\right)\right). \end{equation} Here, to control some errors and minimize the BCS functional, it is not sufficient to rely on a priori estimates based solely on the knowledge of the energy of the system. Instead, we require a more refined technique that considers a priori estimates one can derive from the Euler-Lagrange equations for the BCS functional. This approach allows us to conclude that the magnetic potential minimizing the BCS functional is close to the one describing a uniform magnetic field. Subsequently, considering a relation between the scale parameter $h$ and the parameter $b$ measuring the magnetic field strength, i.e. $hb \gg 1$, $h^2 b^{1+\alpha} \ll 1$, we get an expression for the BCS energy in terms of the energy minimizing the GP functional, up to small corrections. To better understand the method, one adds to the BCS energy a term that shows how an induced magnetic field responds to the influence of an external magnetic field. Here we simply want to describe an effective theory in which you eventually see that electron spins produce a magnetic field inside the material that compensates in some way for the external field. This regime can be expressed through in a condition on alpha, which results in the fact that the external magnetic potential contribution is dominant in the system. The magnetic field is strong, but still, chosen a chemical potential that ensures the system is close to the phase transition, the condition $\alpha < 1$ allows the system to be in a regime of superconductivity.

Emergence of Macroscopic Models from the BCS Theory of Superconductivity