Editing Bogoliubov transformation (section)

=== Fermionic condensates ===
Bogoliubov transformations are a crucial mathematical tool for understanding and describing [[Fermionic condensate|fermionic condensates]]. They provide a way to diagonalize the Hamiltonian of an interacting fermion system in the presence of a condensate, allowing us to identify the elementary excitations, or quasiparticles, of the system.

In a system where fermions can form pairs, the standard approach of filling single-particle energy levels (the Fermi sea) is insufficient. The presence of a condensate implies a coherent superposition of states with different particle numbers, making the usual creation and annihilation operators inadequate. The Hamiltonian of such a system typically contains terms that create or annihilate pairs of fermions, such as:<blockquote><math>H \sim \sum_k \epsilon_k c_k^\dagger c_k + \sum_k \Delta_k c_k^\dagger c_{-k}^\dagger + \Delta_k^* c_{-k} c_k</math></blockquote>where <math>c_k^\dagger</math> and <math>c_k</math> are the creation and annihilation operators for a fermion with momentum <math>k</math>, <math>\epsilon_k</math> is the single-particle energy, and <math>\Delta_k</math> is the pairing amplitude, which characterizes the strength of the condensate. This Hamiltonian is not diagonal in terms of the original fermion operators, making it difficult to directly interpret the physical properties of the system.

Bogoliubov transformations provide a solution by introducing a new set of quasiparticle operators, <math>\gamma_k^\dagger</math> and <math>\gamma_k</math>, which are linear combinations of the original fermion operators:<blockquote><math>\begin{aligned}
\gamma_k &= u_k c_k - v_k c_{-k}^\dagger \\
\gamma_k^\dagger &= u_k^* c_k^\dagger - v_k^* c_{-k}
\end{aligned}</math></blockquote>where <math>u_k</math> and <math>v_k</math> are complex coefficients that satisfy the normalization condition <math>|u_k|^2 + |v_k|^2 = 1</math>. This transformation mixes particle and hole creation operators, reflecting the fact that the quasiparticles are a superposition of particles and holes due to the pairing interaction. This transformation was first introduced by  N. N. Bogoliubov in his seminal work on superfluidity.<ref name=":0" />

The coefficients <math>u_k</math> and <math>v_k</math> are chosen such that the Hamiltonian, when expressed in terms of the quasiparticle operators, becomes diagonal:<blockquote><math>H = E_0 + \sum_k E_k \gamma_k^\dagger \gamma_k</math></blockquote>where <math>E_0</math> is the ground state energy and <math>E_k</math> is the energy of the quasiparticle with momentum <math>k</math>. The diagonalization process involves solving the Bogoliubov-de Gennes equations, which are a set of self-consistent equations for the coefficients <math>u_k</math>, <math>v_k</math>, and the pairing amplitude <math>\Delta_k</math>. A detailed discussion of the Bogoliubov-de Gennes equations can be found in de Gennes' book on superconductivity.<ref>{{cite book |last1=de Gennes |first1=P. G. |title=Superconductivity of metals and alloys |date=1999 |publisher=Westview press}}</ref>

==== Physical interpretation ====

The Bogoliubov transformation reveals several key features of fermion condensates:

*   Quasiparticles''':''' The elementary excitations of the system are not individual fermions but quasiparticles, which are coherent superpositions of particles and [[Electron hole|holes]]. These quasiparticles have a modified energy spectrum <math>E_k = \sqrt{\epsilon_k^2 + |\Delta_k|^2}</math>, which includes a gap of size <math>|\Delta_k|</math> at zero momentum. This gap represents the energy required to break a [[Cooper pair]] and is a hallmark of superconductivity and other fermionic condensate phenomena.
*   Ground state''':''' The ground state of the system is not simply an empty Fermi sea but a state where all quasiparticle levels are unoccupied, i.e., <math>\gamma_k |\mathrm{BCS}\rangle = 0</math> for all <math>k</math>. This state, often called the BCS state in the context of superconductivity, is a coherent superposition of states with different particle numbers and represents the macroscopic condensate.
*   Broken symmetry''':''' The formation of a fermion condensate is often associated with the spontaneous breaking of a symmetry, such as the [[Circle group|U(1)]] gauge symmetry in superconductors. The Bogoliubov transformation provides a way to describe the system in the broken symmetry phase. The connection between broken symmetry and Bogoliubov transformations is explored in Anderson's work on pseudo-spin and gauge invariance.<ref>{{cite journal |last1=Anderson |first1=P. W. |date=1958 |title=Random-phase approximation in the theory of superconductivity |journal=Physical Review |volume=112 |issue=6 |page=1900}}</ref>