This is established in some materials to better than one part in 1015 of the normal state resistance. Almost half a century ago ginzburg addressed the question of possible super conductivity in ferromagnetic material1, and studied the problems posed by orbital supercurrents within a material with intrinsic magnetic ux. Canonical transformations in quantum field theory 1. Computational methods applied to superconductivity and magnetism. Bogoliubov transformation, group structure any feven unitary matrix in 2m will correspond to some bogoliubov transformation.
Since bogoliugov form of this condition is suggestive of the hyperbolic identity. On bogoliubovs method in superconductivity springerlink. Sep 03, 2008 bogoliubov transformation is one of many such reductions transformations, mainly used for reduction special hamiltonians in superfluid theory of interacting particles helium, electron gas to almost noninteracting quasiparticles normal coordinates. Bogoliubov transformations and entanglement of two fermions the manifold of all states 15 is the settheoretic sum of cartesian products of two balls of radii r 1 and r 2 1 2. Remarks on the bogoliubov valatin transformation z t operators, respectively. Xin, a new class of quantum states generated by mfold application of bogoliubov s transforma. A stronger version of bogoliubovs inequality and the heisenberg model roepstorff, g. Surprisingly, this analysis leads to the idea that the majorana neutrino should be regarded as a bogoliubov quasiparticle that is consistently understood, as it will be explained further, only by use of the bogoliubov transformation. Computational methods applied to superconductivity and. In this work a generalization of the bogoliubov transformation is developed to describe a space compacti ed fermionic eld. Bogoliubov mathematics institute, academy of sciences, u.
Superconductivity an overview sciencedirect topics. Two fluid model for superconductivity and london equations lecture 2. The majorana neutrino could thus become the first bogoliubov quasiparticle observed in particle physics. Fernandes here we will discuss a new ground state of the interacting electron gas. The method is the fermion counterpart of the formalism introduced earlier for bosons j. Mathematical analysis of the bcs bogoliubov theory shuji watanabe division of mathematical sciences, graduate school of engineering gunma university 1 introduction superconductivity is one of the historical landmarks in condensed matter physics.
Matrix elements and their generating functions are new for the general. Gauge invariance in the theory of superconductivity londons theory madelung version. Mathematical analysis of the bcsbogoliubov theory 1. Furthermore, it will turn out shortly that all terms in this expression which are linear to yor vanish.
The value of is determined by selfconsistent condition. The bogoliubov transformation is often used to diagonalize hamiltonians, which yields the stationary solutions of the corresponding schrodinger equation. Bogoliubovvalatin transformation in 1947, bogoliubov 2 introduced a novel linear transformation to diagonalize the quantum quadratic hamiltonian present in super. The terms in 15 which are proportional to y represent the hartree and fock contributions which are degenerated due to the contact interaction, those which. Bogoliubov theory for bose gases in random potentials. Hawking was using well accepted principles, but whose consequences were not well understood. A great deal was known about the phenomenology of superconductivity in the 1950s, and it was already suspected that the electron phonon interaction was responsible, but the microscopic form of the wave function was unknown. Generalized bogoliubov transformation, fermion field and. The boson spectrum consists of the normal modes of coupled phononmagnon oscillations. A superconductor is an insulator of bogoliubov quasiparticles. The use of the bogoliubov transform in hawking radiation is not really related to overcoming the lack of a theory of quantum gravity. We will then calculate and study interesting quantities of the su. Pdf bogoliubov transformations for fermibose systems and. In theoretical physics, the bogoliubov transformation, also known as bogoliubovvalatin transformation, were independently developed in 1958 by nikolay bogolyubov and john george valatin for finding solutions of bcs theory in a homogeneous system.
Using a bakercampbellhausdorff formula for the disentangling of exponential operators, a simple straightforward derivation of bogoliubov transformation matrix elements could be given by elementary algebra. We could have, instead, assumed the expectation was nonzero for pairs with a nonzero total momentum. Matrix elements and their generating functions are new for the general linear. The resultant smallsize hamiltonian contains the information of the original bdg hamiltonian. On the general linear bogoliubov transformation for. Cooper rst quanti ed this instability as due to a small attractive interaction between two electrons above the fermi surface. We will look at features of superconductors before the discovery of the bcs theory, and examine the assumptions and methods used to develop the theory. To solve it, we employ the socalled bogoliubov transformation. First, we reformulate the bcsbogoliubov theory of superconductivity. The bogoliubov transformation of bosons correspond to the linear canonical transformation of oscillators in classical mechanics because bosons are quanta of oscillators, and we know the linear canonical transformations are symplectic due to the symplectic structure of the classical phase space. Convergence of bogoliubov s method of renormalization in momentum space zimmermann, w. This canonical transformation is called the bogoliubov transformation. The set of such matrices forms a group a subgroup of the unitary group of degree 2m let us now make a unitary rotation of all matrices a and define.
Until now, superconductivity at high temperatures remains to some ex tend unresolved. London theory in a nutshell reinterpretation of as macroscopic pseudo bosonic condensate wave function, postulate of macroscopic phase coherence associatedwith a macroscopic numberof i bosons k 1, bose condensateiiii fermionfermion pairspairs k 2 pair 2, pair condensatecondensate reinterpretation of as the macroscopic condensate density. The frolich hamiltonian, with the addition of the electronboson interaction of the fourth degree in fermion operators, is studied. We will mostly focus on conventional superconductors, where the cooper pairs originate from. U vafu v avf, if and only if kvk2 hs trv v on the bogoliubovde gennes equations. This chapter covers some of the basics of superfluidity. Bogoliubov transformations for fermibose systems 84 w. A phase transition is accompanied by an instability of the normal state. A symplectic transformation vis unitarily implementable i.
He derived this requirement from the observation that h. Abovethe criticaltemperaturetc, thedc resistivity ath 0is. The bogoliubov transformation is an isomorphism of either the canonical commutation relation algebra or canonical anticommutation relation algebra. Superconductivity is one of the historical landmarks in condensed matter physics. Lets start on the details of making the bogoliubov transformation work. Mathematical analysis of the bcsbogoliubov theory shuji watanabe division of mathematical sciences, graduate school of engineering gunma university 1 introduction superconductivity is one of the historical landmarks in condensed matter physics. P h ysi cal revi ew volum e 117, number 3 february 1, 1960 quasiparticles and gauge invariance in the theory of superconductivity yoichmo namsu the enrico fermi institute for nuclear studies and the department of physics, the university of chicago, chicago, illinois received july 23, 1959 ideas and techniques known in quantum electrodynamics have been applied to the bardeencooper. Convergence of bogoliubovs method of renormalization in momentum space zimmermann, w.
Mathematical analysis of the bcsbogoliubov theory 1 introduction. This canonical transformation is called the bogoliubov transformation, and is useful for studying the spectrum of the system of the electrons. Since onnes found out the fact that the electrical resistivity of mercury drops to zero below the. The second essential observation was that superconductors not only prevent a magnetic. Bogoliubov transformation d 2, 1 equivalent circuit for josephson junction and analysis lecture 2. Superconductivity occurs in many metallic elements, their alloys, and in intermetallic compounds. An introduction to superfluidity and superconductivity. Jun 01, 2018 the use of the bogoliubov transform in hawking radiation is not really related to overcoming the lack of a theory of quantum gravity. The most prominent application is again by nikolai bogoliubov himself, this time for the bcs theory of superconductivity. Unlike an ordinary metallic conductor, whose resistance decreases gradually as its temperature is lowered even down to near absolute zero, a superconductor has a. Bogoliubovvalatin canonical transformation and the model. Bosonic linear unitary bogoliubov transformation reduction. A great deal was known about the phenomenology of super conductivity in the 1950s, and it was. Bcs ansatz, bogoliubov approach to superconductivity and richardsongaudin exact wa ve function m.
Bogoliubov transformation is one of many such reductions transformations, mainly used for reduction special hamiltonians in superfluid theory of interacting particles helium, electron gas to almost noninteracting quasiparticles normal coordinates. A great deal was known about the phenomenology of super conductivity in the 1950s, and it was already suspected that the electron phonon interaction was responsible, but the microscopic form of the wave function was unknown. Bosonic linear unitary bogoliubov transformation reduction theorem samuel l. It discusses boseeinstein condensation in an ideal gas. Numerical construction of a lowenergy effective hamiltonian. This method was later extended by bogoliubov himself 3, 4, 5 and also by. A stronger version of bogoliubov s inequality and the heisenberg model roepstorff, g. Experiments will be mentioned if they motivate certain theoretical ideas or support or contradict theoretical predictions, but a systematic discussion of experimental results will not be given. This mapping is called bogoliubov or bogoliubovvalatin transformation. The hamiltonian takes into account the exchange interactions of the magnetic moments of the conduction electrons.
Joginder singh galsin, in solid state physics, 2019. Superconductivity is the set of physical properties observed in certain materials, wherein electrical resistance vanishes and from which magnetic flux fields are expelled. The bogoliubov transformation is also important for understanding the unruh effect, hawking radiation, pairing effects in nuclear physics, and many other topics. We want to understand superconductivity using methods of theoretical physics. Recent much more complicated derivations by means of hypergeometric functions are simplified considerably. Pdf bogoliubov transformations for fermibose systems. The microscopical theory of superconductivity was developed by bardeen, cooper. Theoretical study of the interplay of superconductivity and. A clue was provided by leon cooper, who showed that the noninter. What is the physical meaning of bogoliubov transformation. About a decade later, and armed with bcs theory2, fulde and ferrell ff3 and separately larkin and ovchin. Theoretical study of the interplay of superconductivity. Pdf bcs ansatz, bogoliubov approach to superconductivity.
Dzhioev joint institute for nuclear research, dubna thermal bogoliubov transformation is an essential ingredient of the thermo. Bogoliubov, a new method in the theory of superconductivity. Application of the bogoliubov transformation to superconductivity and field theory paul reeves auvil department of physics, stanford university. Xin, a new class of quantum states generated by mfold application of bogoliubovs transforma. Quasiparticles and gauge invariance in the theory of. The transformation is clearly a rotation in the space of anihilation and creation operators. Majorana neutrino as bogoliubov quasiparticle sciencedirect. Introduction superconductivity discovered in 1911 by onnes 9, is the. The role of gauge invariance in the theory of superconductivity. The ground state of the bcs hamiltonian is discussed by several. Nov 21, 2011 bcs ansatz, bogoliubov approach to superconductivity and richardsongaudin exact wa ve function m.
Bcs theory finite temperatures outline 1 cooperpairs formation of pairs origin of attractive interaction 2 bcs theory the model hamiltonian bogoliubovvalatintransformation calculation of the condensation energy 3 finite temperatures excitation energies and the energy gap determination of t c temperature dependence of the energy gap. Using a bakercampbellhausdorff formula for the disentangling of exponential operators, a simple straightforward derivation of bogoliubovtransformation matrix elements could be given by elementary algebra. Lecture notes on superconductivity a work in progress. A 66 2002 052101, and is based on the thermo eld dynamics. Any material exhibiting these properties is a superconductor. Let us now perform the following transformation on a, called bogoliubov translation for coherent states or boson translation. Bcs theory finite temperatures outline 1 cooperpairs formation of pairs origin of attractive interaction 2 bcs theory the model hamiltonian bogoliubov valatin transformation calculation of the condensation energy 3 finite temperatures excitation energies and the energy gap determination of t c temperature dependence of the energy gap. At present, the range of transition temperature extends from about 23 k for the alloy nb 3 ge to 0. For comparison, the resistivity of copper at room temperature is. Thus, the transition from the normal metal state to the super conducting state must be a phase transition. Using a canonical transformation bogoliubov 2 obtained the results similar to those in the bcs theory in 1958. A simple analogy for why the bogoliubov transformation is useful is given by the fourier transform or the laplace. On the general linear bogoliubov transformation for bosons.
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