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Theory of Superconductivity

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Theory of Superconductivity Theory of Superconductivity Kwon Park KIAS-SNU Physics Winter Camp Camp Winter KIAS-SNU Physics Phoenix Park Jan 20 – 27, 2013 Outline • Why care about superconductivity? • BCS theory as a trial wave function method • BCS theory as a mean-field theory • High-temperature superconductivity and strong correlation • Effective field theory for superconductivity: Ginzburg- Landau theory Family tree of strongly correlated electron systems Quantum magnetism Topological Mott insulator FQHE insulator HTSC Quantum Hall effect Superconductivity Wigner crystal Breakdown of the Landau-Fermi liquid Collective behavior of a staring crowd Disordered State Ordered State Superconductivity as an emergent phenomenon Superconducting phase coherence: Josephson effect • Cooper-pair box: An artificial two-level system composed of many superconducting electron pairs in a “box.” reservoir box - - - - + + + + gate Josephson effect in the Cooper-pair box Nakamura, Pashkin, Tsai, Nature 398, 786 (99) • Phase vs. number uncertainty relationship: When the phase gets coherent, the Cooper-pair number becomes uncertain, which is nothing but the Josephson effect. Devoret and Schoelkopf, Nature 406, 1039 (00) Meissner effect • mv-momentum = p-momentum − e/c × vector potential • current density operator = 2e × Cooper pair density × velocity operator • θ=0 for a coherent Cooper-pair condensate in a singly connected region.region London equation • The magnetic field is expelled from the inside of a superconductor: Meissner effect. London equation where Electromagnetic field, or wave are attenuated inside a superconductor, which means that, in quantum limit, photons become massive while Maxwell’s equations remain gauge-invariant! Anderson-Higgs mechanism • Quantum field theories should be renormalizable in order to produce physically meaningful predictions via systematic elimination of inherent divergences. A necessary condition is the gauge invariance. • The gauge invariance requires that all gauge bosons should possess no mass. Actually, the problem is so much more severe that all particles including fermions should be massless. Where do all masses come from? Anderson-Higgs mechanism • Is there a way to generate non-zero mass while keeping the underlying quantum field theory gauge-invariant? Yes, it is called the Anderson-Higgs mechanism. The idea is basically that the ground state breaks the gauge symmetry spontaneously even though the underlying Hamiltonian preserves it. This is essentially how photons become massive in the Meissner effect. Large Hadron Collider (LHC) CERN, Geneva Trial (variational) wave function method Hamiltonian: Trial wave function with a variational atomic number, Z: Energy minimization: in comparison with Cooper’s problem Hamiltonian: Trial wave function with the variational Fourier weight, gk: Schrödinger equation: Cooper’s problem Cooper’s problem Energy gain due to pairing: BCS theory as a trial wave function method • The low-temperature superconductivity has been successfully explained by the Bardeen-Cooper-Schrieffer (BCS) theory, where the ground state wave function is given in real space as follows: which is essentially identical to the usual form of the BCS wave function written in momentum space and second quantization: Second quantization • The second quantization formulism is essentially the creation/ annihilation operator method used in the harmonic oscillator problem except that we now think about the fermion version. Fermion anti-commutation relationship Cooper-pair wave function Therefore, BCS many-body wave function • The BCS wave function is a coherent superposition of many different particle-number eigenstates. which, after normalization, becomes the usual form. BCS wave function as a coherent state wave function Notice the resemblance with the coherent state wave function of the 1D harmonic oscillator: • Eigenstate of the annihilation operator while not of the creation counterpart. • Almost invariant with respect to the particle-number change. • Closest thing that can be called as a “phase” eigenstate as opposed to the number eigenstate. Number fluctuation in the BCS wave function • Let us think about how much the particle number fluctuates in the BCS wave function. • The BCS wave function can be expanded in terms of the number-projected components. BCS theory as a variational wave function method How to determine the unknown parameters, uk and vk? • Wave function: • Hamiltonian: q, σ -q, σ’ k, σ -k, σ’ Energy minimization and the BCS gap equation BCS gap equation Physical meaning of the BCS gap Let us first examine what happens when Δk=0. When Δk disappears, the BCS ground state becomes the usual Fermi sea! Physical meaning of the BCS gap Is the BCS gap really the gap for excitation? Δ is the minimum energy that is required for excitation! Solution of the BCS gap equation • With an assumption that V is momentum-independent and negative (= -V < 0) and Δ is non-zero only within a narrow energy window (±ħωD ) near the Fermi surface, the BCS gap equation is further simplified. Solution of the BCS gap equation Therefore, BCS wave function for the excited states How about creating two excitations? The above wave function does not always create the excited state that is independent of the ground state. To see this, consider the following situation: BCS wave function for the orthogonal excited states: Bogoliubov quasi-particle creation: annihilation: BCS ground state as the vacuum of the Bogoliubov quasi-particle Creation operator for Annihilation operator for the Bogoliubov quasi-particle the Bogoliubov quasi-particle The BCS ground state should be the vacuum of the Bogoliubov quasi-particle. BCS theory as a mean-field theory BCS theory as a mean-field theory BCS gap equation from the self-consistency condition Solution of the BCS gap equation • With the same assumption that V is momentum-independent and negative (= -V < 0) and Δ is non-zero only within a narrow energy window (±ħωD ) near the Fermi surface, Solution of the BCS gap equation: limits 1. What is the gap at zero temperature? Solution of the BCS gap equation: limits 2. What is the critical temperature, TC, where the gap closes? How does the attraction arise?: pairing mechanism 1. Electrons form a Landau-Fermi liquid where electron charge is completely screened by the depletion of surrounding electrons. As a result, the original strong Coulomb interaction becomes replaced by a much weaker effective interaction between dressed quasi-particles. 2. These dressed quasi-particles are attracted to each other via exchange of phonon. Time line of superconductivity FeAs Feb 2008 High-temperature superconductivity (HTSC) What is so special about HTSC? • Higher is different (as more is different). • Nearby phases are unusual. • Pairing is probably not mediated by phonon. Is there any common thread weaving all of the unusual properties of HTSC? A natural point of view is that all is due to the fact that HTSC is strongly correlated superconductivity under the influence of large repulsive interaction between electrons. Minimal theoretical model for HTSC La2CuO4 2D copper oxide plane La weak interlayer coupling O 2D copper oxide plane Cu (1) Mott insulator (due to strong Coulomb repulsion) (2) Antiferromagnetic spin order (due to super-exchange) Strongly correlated electrons in lattice: the Hubbard model Antiferromagnetism in weak-coupling viewpoint: Nesting instability Nesting instability of the Hubbard Model Ek φ (0,0) (π,π) (kx,ky) Antiferromagnetism in strong-coupling viewpoint: Super-exchange • In the large-U limit, the Hubbard model has exactly the same matrix elements as the antiferromagnetic Heisenberg model at half filling. Antiferromagnetism versus pairing Both the antiferromagnetic spin exchange and the pairing interaction tend to promote the formation of spin singlet pairs! • Anderson’s conjecture (87): Electrons are already Cooper-paired inside the antiferromagnetic phase, but cannot move due to strong repulsive interaction. Electrons will become superconducting once they get mobile upon doping. Short history of the RVB wave function • In 1987, Anderson proposed an ansatz wave function for doped antiferromagnets called the resonating valence bond (RVB) state, which is actually nothing but the Gutzwiller-projected BCS wave function. • However, it was realized that the RVB state could not be the ground state of the Heisenberg model at half filling since it is not Néel ordered. • Is it a good ansatz function for the ground state at non-zero doping? G. Baskaran et al. (87), C. Gros (88), Y. Hasegawa et al.(89), E. Dagotto (94), A. Paramekanti et al. (01), S. Sorella et al. (02) and many more The t-J versus Gutzwiller-projected BCS model KP, PRL 95, 027001 (05); PRB 72, 245116 (05) • It is shown that the ground state wave function of the t-J and the Gutzwiller- projected BCS model is closely related to each other at half filling as well as at moderate doping. Numerical method • Our numerical method is exact diagonalization. • We need a trick to deal with the coherent particle-number fluctuation present in superconductivity where the particle number is not exactly conserved. • The solution is to first enlarge the Hilbert space by combining different particle-number sectors, then to diagonalize the BCS Hamiltonian in the enlarged Hilbert space, and finally to project the ground state into a relevant number sector. Numerical evidence: exact equivalence at half filling where n and m are either both even or both odd integers Simple example Consider a simple
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