arXiv:1001.1343v1 [cond-mat.supr-con] 8 Jan 2010 A new basis set for the description of electrons in superconductors J. E. Hirsch Department of Physics, University of California, San Diego La Jolla, CA 92093-0319 In the usual description of electrons in metals and superconductors, the single electron states are assumed to satisfy Bloch’s theorem. This is because the electron-ion interaction is privileged over the electron-electron interaction. However the theory of hole superconductivity proposes that in the transition to superconductivity carriers ‘undress’ from the electron-ion interaction. I propose here a new basis set to describe electrons in the superconducting state that does not satisfy Bloch’s theorem but instead is designed to optimize the electron-electron interaction. The new basis set favors a superconducting state where a spin current exists and where states near the bottom of the band become partially occupied, as predicted by the theory of hole superconductivity. PACS numbers: It is generally assumed that electron states in a peri- odic solid obey Bloch’s theorem[1]. That is, a wavevector k characterizes the single electron orbital ψ$_{n}$$_{k}$ ( r ) which obeys the relation where the index n (band index) differentiates different states with the same Bloch wavevector k . In Eq. (1), R is any lattice translation vector that leaves the electron- ion potential invariant: Electron-electron interactions are assumed to convert the single-particle electron-ion potential to a new ‘effective’ potential U$_{eff}$ that includes electron-electron interaction direct, exchange and correlation effects (e.g. the Kohn- Sham potential in density-functional theory)[2]. The ef- fective potential still obeys with the same lattice vectors R , and the corresponding single particle orbitals (e.g. Kohn-Sham orbitals) still obey Bloch’s theorem Eq. (1). Similarly, in the generally accepted conventional the- ory of superconductivity (BCS theory)[3] the many-body wavefunction is written as where the index k is the Bloch wavevector of the state where the single electron of spin σ is created by the opera- tor c † k $_{σ}$. The normal state of the system can be expressed in the form Eq. (4) with u$_{k}$v$_{k}$ = 0 and v$_{k}$ = 0 or u$_{k}$ = 0 depending on whether ϵ$_{k}$ < ϵ$_{F}$ or ϵ$_{k}$ > ϵ$_{F}$ with ϵ$_{F}$ the Fermi energy. The superconducting state instead is char- acterized by u$_{k}$v$_{k}$ = 0 for some values of k . For the case where ϕ$_{k}$ are the usual Bloch states for a given band. Note that { ˜ ϕ$_{k}$ } is a complete orthonormal set if and only The new single-particle basis states that we propose here are defined by the relation Furthermore, we have recently proposed that the su- perconducting state should be understood as a (dis- torted) ‘mirror image’ of the normal state, where the ‘mirror’ switches the role of the electron-ion and electron- electron interactions[6]. We argued that in the supercon- ducting state, the electron-electron interaction is priv- ileged over the electron-ion interaction and the many- body wavefunction optimizes the electron-electron in- teraction, with the electron-ion interaction being non- optimized. It is natural that such a state should be de- scribed by a single electron basis that does not satisfy Bloch’s theorem. But what is the optimal basis to de- scribe such state? We have recently argued that experimental evidence clearly indicates that in the transition to superconduc- tivity the carriers undergo a wavelength expansion and end up completely ‘undressed’ from the electron-ion in- teraction, so that they no longer ‘see’ the discrete nature of the electron-ion potential[4, 5]. It is natural to con- clude that the electron states describing the superfluid will no longer obey Bloch’s theorem. However, it is not necessarily true that electron states in a periodic electron-ion potential have to obey Bloch’s theorem. Consider a case where the electron-electron in- teraction completely overwhelms the electron-ion interac- tion. The electronic spatial arrangement that minimizes the energy may be an electronic lattice (Wigner crys- tal, incommensurate charge density wave) with a lattice structure that has no relation with the underlying ionic structure. For example, the electronic structure may be hexagonal and the underlying ionic lattice cubic. In such situation the single particle states describing the elec- trons will not obey Bloch’s theorem Eq. (1). of nearly free electron metals, the Bloch states are often simply taken to be free electron plane wave states. I. INTRODUCTION ψ$_{n}$$_{k}$ ( r + R ) = e i k · $^{R}$ψ$_{n}$$_{k}$ ( r ) (1) U$_{e}$$_{−}$$_{i}$ ( r + R ) = U$_{e}$$_{−}$$_{i}$ ( r ) . (2) U$_{eff}$ ( r + R ) = U$_{eff}$ ( r ) (3) | Ψ >$_{BCS}$ = ∏ k ( u$_{k}$ + v$_{k}$c † k $_{↑}$c † − k $_{↓}$) | 0 > (4) ˜ ϕ$_{k}$ ( r ) = ϕ$_{k}$ ( r ) + iϕ$_{−}$$_{k}$ ( r ) √$_{2}$ (5a)