%0 Book Section %1 statphys23_0319 %A Tanaka, A. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K condensation electron-pair exact ground long-range off-diagonal order state statphys23 superconductivity topic-8 %T A simple model with electron pairing ground states %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=319 %X Many materials, such as high-$T_c$ cuprates, ruthenates, and organic superconductors, are now known to exhibit superconductivity which does not conform to the Bardeen-Cooper-Schrieffer theory, and mechanisms for this unconventional superconductivity have been attracting much interest. It is confirmed experimentally that, as in the BCS superconductors, electron pairs are essential for the phenomenon. We should thus clarify a source of force producing electron pairs, a type of effective electron-electron interaction (which will determine the symmetry of pairing) derived from the source, and occurrence of electron-pair condensation due to this effective interaction. Here, concentrating on the occurrence of electron-pair condensation, we study a simple electronic model which has electron pairing ground states. The model is an extension of models previously studied in Refs. 1,2. Let $Łambda$ and $K$ be a lattice and its reciprocal lattice, respectively. We denote by $c_k,\sigma(c_k,\sigma^\dagger)$ the annihilation(creation) operator of an electron with wave vector $kK$ and spin $\sigma=\uparrow,\downarrow$. Let $\varepsilon(k)0$ be a single-electron energy associated with the electron corresponding to $c_k,\sigma$. We assume that the electrons around the Fermi surface feel electron-electron interactions. Under this assumption we consider the model whose Hamiltonian is given by eqnarray H=\sum_k\mathcalK\sum_\sigma=\uparrow,\downarrow \varepsilon(k) c_k,\sigma^c_k,\sigma \nonumber\\ && +1|Łambda| \sum_k_1,k_2,p_1,p_2K_\delta \sum_\sigma,\tau=\uparrow,\downarrow W_\sigma,\tau(k_1,k_2;p_1,p_2)\nonumber\\ && c_k_1,\sigma^c_k_2,\tau^ c_p_2,\tau c_p_1,\sigma, eqnarray with $K_\delta=\k~|~\delta_1<\varepsilon(k)<\delta_2, kınK\$, where $\delta_1$ and $\delta_2$ are some values of energy between which the Fermi energy of the non-interacting system is located. It is noted that our Hamiltonian conserves the electron number, unlike the ones usually discussed in the framework of mean field approximations. Under a certain condition on $W_\sigma,\tau(k_1,k_2;p_1,p_2)$ we rigorously prove that the model has ground states in which many electrons condense into an electron pairing state. We demonstrate that various kinds of pairing symmetries are realized depending on the types of $W_\sigma,\tau(k_1,k_2;p_1,p_2)$. It is also shown that the model exhibits off-diagonal long-range order in the ground states. 1) A. Tanaka, J. Phys. Soc. Jpn. 73, 1107 (2004).\\ 2) A. Tanaka and M. Yamanaka, Phys. Rev. B 71, 233102 (2005).

@incollection{statphys23_0319, abstract = {Many materials, such as high-$T_\mathrm{c}$ cuprates, ruthenates, and organic superconductors, are now known to exhibit superconductivity which does not conform to the Bardeen-Cooper-Schrieffer theory, and mechanisms for this unconventional superconductivity have been attracting much interest. It is confirmed experimentally that, as in the BCS superconductors, electron pairs are essential for the phenomenon. We should thus clarify a source of force producing electron pairs, a type of effective electron-electron interaction (which will determine the symmetry of pairing) derived from the source, and occurrence of electron-pair condensation due to this effective interaction. Here, concentrating on the occurrence of electron-pair condensation, we study a simple electronic model which has electron pairing ground states. The model is an extension of models previously studied in Refs. [1,2]. Let $\Lambda$ and $\mathcal{K}$ be a lattice and its reciprocal lattice, respectively. We denote by $\hat{c}_{k,\sigma}(\hat{c}_{k,\sigma}^\dagger)$ the annihilation(creation) operator of an electron with wave vector $k\in \mathcal{K}$ and spin $\sigma=\uparrow,\downarrow$. Let $\varepsilon(k)\ge 0$ be a single-electron energy associated with the electron corresponding to $\hat{c}_{k,\sigma}$. We assume that the electrons around the Fermi surface feel electron-electron interactions. Under this assumption we consider the model whose Hamiltonian is given by \begin{eqnarray} \lefteqn{ H=\sum_{k\in \mathcal{K}}\sum_{\sigma=\uparrow,\downarrow} \varepsilon(k) \hat{c}_{k,\sigma}^\dagger \hat{c}_{k,\sigma} }\nonumber\\ && +\frac{1}{|\Lambda|} \sum_{k_1,k_2,p_1,p_2\in \mathcal{K}_\delta} \sum_{\sigma,\tau=\uparrow,\downarrow} W_{\sigma,\tau}(k_1,k_2;p_1,p_2)\nonumber\\ && \hat{c}_{k_1,\sigma}^\dagger \hat{c}_{k_2,\tau}^\dagger \hat{c}_{p_2,\tau} \hat{c}_{p_1,\sigma}, \end{eqnarray} with $\mathcal{K}_\delta=\{k~|~\delta_1<\varepsilon(k)<\delta_2, k\in\mathcal{K}\}$, where $\delta_1$ and $\delta_2$ are some values of energy between which the Fermi energy of the non-interacting system is located. It is noted that our Hamiltonian conserves the electron number, unlike the ones usually discussed in the framework of mean field approximations. Under a certain condition on $W_{\sigma,\tau}(k_1,k_2;p_1,p_2)$ we rigorously prove that the model has ground states in which many electrons condense into an electron pairing state. We demonstrate that various kinds of pairing symmetries are realized depending on the types of $W_{\sigma,\tau}(k_1,k_2;p_1,p_2)$. It is also shown that the model exhibits off-diagonal long-range order in the ground states. 1) A. Tanaka, J. Phys. Soc. Jpn. 73, 1107 (2004).\\ 2) A. Tanaka and M. Yamanaka, Phys. Rev. B 71, 233102 (2005).}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Tanaka, A.}, biburl = {https://www.bibsonomy.org/bibtex/2195e001174b50fed655139fe63ebb74d/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {4e8cae21cccbaf82f783e2da3759cdb2}, intrahash = {195e001174b50fed655139fe63ebb74d}, keywords = {condensation electron-pair exact ground long-range off-diagonal order state statphys23 superconductivity topic-8}, month = {9-13 July}, timestamp = {2007-06-20T10:16:17.000+0200}, title = {A simple model with electron pairing ground states}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=319}, year = 2007 }