Interplay of pairing correlation and Coulomb correlation in Boson exchange superconductors

Abstract

A theoretical methodology for exploring the conventional Bardeen–Cooper–Schrieffer (BCS) pairing instability for superconductivity from a correlated normal phase for all possible degrees of many-body correlation has been developed. The Gutzwiller projection scheme with a correlation parameter was made use of in generating the BCS pairing state. A variational scheme was thereafter implemented, leading to a self-consistent equation for superconducting gap function. This equation shows explicit dependence of the gap function on the many body correlation parameter. This ‘pairing-gap’ and the corresponding self-consistent gap equation in zero correlation limit, becomes identical in nature with those of the pure (1-well) BCS formalism, as expected and the Coulomb correlation affects the pairing significantly with the strength of correlation. The detailed consequences are being presented here.

Graphic Abstract

Appendices

Appendix A

Normalization of the Coulomb correlated state:

$$\begin{aligned} _{\text {C}}\langle \Psi \vert \Psi \rangle _{\text {C}}&= \Big \langle 0\Big \vert \prod _{m} (u_{m}^{*} + v_{m}^{*}c_{-m,\downarrow }c_{m,\uparrow })\prod _{s} \Big (1-\alpha \sum _{k^{'},m^{'}}c_{m^{'},\downarrow }^{+}c_{m^{'}, \downarrow }c_{-k^{'},\uparrow }^{+}c_{-k^{'},\uparrow } e^{i(m^{'}-k^{'}) \dot{r}_{s}}\Big )\nonumber \\&\quad \times \prod _{s^{'}}\Big (1-\alpha \sum _{k^{''},m^{''}}c_{k^{''},\uparrow }^{+} c_{k^{''},\uparrow }c_{-m^{''},\downarrow }^{+}c_{m^{^{''},\downarrow }} e^{i(k^{''}-m^{''}) \dot{r}_{s^{'}}}\Big )\prod _{l} (u_{l} + v_{l} c_{l,\uparrow }^{+}c_{-l,\downarrow }^{+})\Big \vert 0\Big \rangle \nonumber \\&= \Big \langle 0\Big \vert \prod _{s,m}\prod _{s^{'},l}[u_{m}^{*}u_{l} + u_{m}^{*}v_{l} c_{l,\uparrow }^{+}c_{-l,\downarrow }^{+} - \alpha ( u_{m}^{*}u_{l}\sum _{k^{''},m^{''}} c_{k^{''},\uparrow }^{+} c_{k^{''},\uparrow } c_{-m^{''},\downarrow }^{+} c_{-m^{''},\downarrow } \nonumber \\&\quad \times u_{m}^{*}v_{l} \sum _{k^{''},m^{''}} c_{k^{''},\uparrow }^{+} c_{k^{''},\uparrow } c_{-m^{''},\downarrow }^{+} c_{-m^{''},\downarrow } c_{l,\uparrow }^{+}c_{-l,\downarrow }^{+}) e^{i(k^{''}-m^{''}) \dot{r}_{s^{'}}} + v_{m}^{*} u_{l} c_{-m,\downarrow }c_{m,\uparrow } \nonumber \\&\quad + v_{m}^{*} v_{l} c_{-m,\downarrow }c_{m,\uparrow } c_{l,\uparrow }^{+}c_{-l,\downarrow }^{+} - \alpha \Big (v_{m}^{*}u_{l}\sum _{k^{''},m{''}}c_{m,\downarrow }c_{m,\uparrow }c_{k^{''},\uparrow }^{+} c_{k^{''},\uparrow } c_{-m^{''},\downarrow }^{+} c_{-m^{''},\downarrow } \nonumber \\&\quad + v_{m}^{*} v_{l} \sum _{k^{''},m{''}}c_{m,\downarrow }c_{m,\uparrow }c_{k^{''}, \uparrow }^{+}c_{k^{''},\uparrow } c_{-m^{''},\downarrow }^{+} c_{-m^{''}, \downarrow } c_{l,\uparrow }^{+} c_{-l,\downarrow }\Big ) e^{i(k^{''}-m^{''})\dot{r}_{s^{'}}} \nonumber \\&\quad -\alpha \Big (u_{m}^{*}u_{l} \sum _{k^{'},m^{'}} c_{m^{'},\downarrow }^{+}c_{m^{'},\uparrow } c_{-k^{'},\uparrow } c_{-k^{'},\uparrow } + u_{m}^{*}v_{l}\sum _{k^{'},m^{'}} c_{m^{'},\downarrow }^{+}c_{m^{'},\uparrow } c_{-k^{'},\uparrow }c_{-k^{'},\uparrow } c_{l,\uparrow }^{+}c_{-l'\downarrow }^{+})e^{-i(k^{'}-m^{'})\dot{r}_{s}} \nonumber \\&\quad + \alpha ^{2} (u_{m}^{*}u_{l}\sum _{k^{'},m^{'},k^{''},m{''}} c_{m^{'},\downarrow }^{+}c_{m^{'},\downarrow } c_{-k^{'}, \uparrow }^{+}c_{-k^{'},\uparrow } c_{k^{''},\uparrow }^{+}c_{k^{''},\uparrow } c_{-m^{''},\downarrow }^{+}c_{-m^{''}, \downarrow } e^{i(k^{''}-m^{''})\dot{r}_{s^{'}} -(k^{'}-m^{'}) \dot{r}_{s}} + \nonumber \\&\quad \times u_{m}^{*}v_{l}\sum _{k^{'},m^{'},k^{''},m{''}} c_{m^{'},\downarrow }^{+} c_{m^{'},\downarrow } c_{-k^{'},\uparrow }^{+}c_{-k^{'},\uparrow } c_{k^{''}, \uparrow }^{+}c_{k^{''},\uparrow } c_{-m^{''},\downarrow }^{+}c_{-m^{''}, \downarrow } c_{l,\uparrow }^{+} c_{-l,\downarrow }^{+} e^{i(k^{''}-m^{''}) \dot{r}_{s^{'}} -(k^{'}-m^{'}) \dot{r}_{s}}) \nonumber \\&\quad - \alpha \Big ( v_{m}^{*}u_{l}\sum _{k^{'},m^{'}} c_{-m,\downarrow } c_{m,\uparrow } c_{m^{'},\downarrow }^{+}c_{m^{'},\downarrow }c_{-k^{'},\uparrow }^{+} c_{-k^{'},\uparrow } e^{i(k^{'}-m^{'}) \dot{r}_{s}} \nonumber \\&\quad +v_{m}^{*}v_{l} \sum _{k^{'},m^{'}}c_{-m,\downarrow }c_{-m,\uparrow } c_{m^{'},\downarrow }^{+}c_{m^{'},\downarrow }c_{-k^{''},\uparrow }^{+} c_{-k^{''},\uparrow }c_{l,\uparrow }^{+}c_{-l,\downarrow }^{+} e^{-i(k^{'}-m^{'}) \dot{r}_{s}}) \nonumber \\&\quad +\alpha ^{2}( v_{m}^{*}u_{l} \sum _{k^{'},m^{'},k^{''},m^{''}} c_{-m,\downarrow }c_{-m,\uparrow }c_{m^{'},\downarrow }^{+} c_{m^{'},\downarrow }c_{-k^{'},\uparrow }^{+}c_{-k^{'},\uparrow } c_{k^{''},\uparrow }^{+}c_{k^{''},\uparrow } c_{-m^{''},\downarrow } c_{-m^{''},\downarrow } e^{i\Big ((k^{''}-m^{''})r_{s^{'}} -(k^{'}-m^{'})r_{s})} \nonumber \\&\quad + v_{m}^{*}v_{l} \sum _{k^{'},m^{'},k^{''},m^{''}} c_{-m,\downarrow }c_{-m,\uparrow }c_{m^{'},\downarrow }^{+} c_{m^{'},\downarrow }c_{-k^{'},\uparrow }^{+}c_{-k^{'},\uparrow } c_{k^{''},\uparrow }^{+}c_{k^{''},\uparrow } c_{-m^{''},\downarrow }^{+} c_{-m^{''},\downarrow } c_{l,\uparrow }^{+} c_{-l,\downarrow }^{+} \nonumber \\&\quad e^{i((k^{''}-m^{''})r_{s^{'}}-(k^{'}-m^{'})r_{s})}\Big )\Big ]\vert 0 \Big \rangle . \end{aligned}$$

(40)

Here, the orthogonality of the states set a condition, for which only the terms with equal number of fermion Creation and annihilation operators will contribute with a non-zero value to that normalization. Therefore,

$$\begin{aligned} _{C}\langle \Psi \vert \Psi \rangle _{C}= & {} \langle 0\vert \prod _{s,m}\prod _{s^{'},l}[ u_{m}^{*}u_{l} + v_{m}^{*}v_{l} c_{-m,\downarrow }c_{m,\uparrow } c_{l,\uparrow }^{+}c_{-l,\downarrow }^{+}\nonumber \\&-\alpha v_{m}^{*}v_{l} \sum _{k^{''},m^{''}} c_{-m,\downarrow }c_{m,\uparrow } c_{k^{''},\uparrow }^{+}c_{k^{''},\uparrow }c_{-m^{''}, \downarrow }^{+}c_{-m^{''},\downarrow } c_{l,\uparrow }^{+}c_{-l,\downarrow }^{+} e^{i(k^{''}-m^{''}).r_{s^{'}}} \nonumber \\&+ \alpha ^{2} v_{m}^{*}v_{l} \sum _{k^{'},m^{'},k^{''},m^{''}} c_{-m,\downarrow }c_{m,\uparrow } c_{m^{'},\downarrow }^{+}c_{m^{'}, \downarrow }c_{-k^{'},\uparrow }^{+}c_{-k^{'},\uparrow } c_{k^{''}, \uparrow }^{+}c_{k^{''},\uparrow }c_{-m^{''}, \downarrow }^{+}c_{-m^{''},\downarrow } c_{l,\uparrow }^{+} c_{-l,\downarrow }^{+}\nonumber \\&e^{i((k^{''}-m^{''})\dot{r}_{s^{'}}-(k^{'}-m^{'})\dot{r}_{s})} \nonumber \\&-\alpha v_{m}^{*}v_{l} \sum _{k^{'},m^{'}}c_{-m,\downarrow }c_{m,\uparrow } c_{-m^{'},\downarrow }^{+} c_{-m^{'},\downarrow } c_{-k^{'},\uparrow }^{+} c_{-k^{'},\uparrow } c_{l,\uparrow }^{+}c_{-l,\downarrow }^{+} e^{i(k^{'}-m^{'})\dot{r}_{s}}] \vert 0\rangle \nonumber \\= & {} \prod _{s,m}\prod _{s^{'},l} [ u_{m}^{*}u_{l} + v_{m}^{*}v_{l} - \alpha v_m^{*}v_{l}\delta _{-l,-m^{''}} \delta _{l,k^{''}} \delta _{-m^{''},-m} \delta _{k^{''},m} + \nonumber \\&\alpha ^{2} v_{m}^{*}v_{l}\delta _{k^{''},m^{''}} \delta _{k^{'},m^{'}} - \alpha v_m^{*}v_{l} \delta _{-m^{'},-l}\delta _{l,k^{'}} \delta _{-k^{'},m}\delta _{m^{'},m}]\nonumber \\ _{C}\langle \Psi \vert \Psi \rangle _{C}= & {} \prod _{l} [ \vert u_{l}\vert ^{2} + \vert v_{l}\vert ^{2}-2\alpha \vert v_{l} \vert ^{2} + \alpha ^{2} \vert v_{l}\vert ^{2}] \nonumber \\ _{C}\langle \Psi \vert \Psi \rangle _{C}= & {} \prod _{l} [1+(\alpha ^{2}-2\alpha )\vert v_{l}\vert ^{2}]; [\because \vert u_{k}\vert ^{2} + \vert v_{k}\vert ^{2} = 1 ]. \end{aligned}$$

(41)

Appendix B

Total energy expectation W:

$$\begin{aligned} W = \dfrac{1}{_{c}\Big \langle \Psi \Big \vert \Psi \rangle _{c}} \left[ { _{c}\langle \Psi \vert \sum _{k} 2\varepsilon _{k} b^{+}_{k}b_{k} + \sum _{k,l} V_{k,l}b^{+}_{k} b_{l} \vert \Psi \rangle _{c} }\right] . \end{aligned}$$

(42)

Kinetic energy operator expectation value:

$$\begin{aligned} T= & {} \dfrac{1}{\prod _{l} [1+(\alpha ^{2}-2\alpha ) \vert v_{l}\vert ^{2}]} \sum _{k,\sigma } \varepsilon _{k} \Big \langle 0\Big \vert \prod _{m} (u_{m}^{*} + v_{m}^{*}c_{-m,\downarrow }c_{m,\uparrow }\Big )\prod _{s}\Big (1-\alpha \sum _{k^{'},m^{'}}c_{m^{'},\downarrow }^{+}c_{m^{'},\downarrow }c_{-k^{'},\uparrow }^{+}c_{-k^{'},\uparrow }\\&e^{i(m^{'}-k^{'}).r_{s}}) c_{k,\sigma }^{+}c_{k,\sigma }\prod _{s^{'}}\Big (1-\alpha \sum _{k^{''},m^{''}}c_{k^{''},\uparrow }^{+}c_{k^{''},\uparrow }c_{-m^{''},\downarrow }^{+}c_{m^{^{''},\downarrow }} e^{i(k^{''}-m^{''}) \dot{r}_{s^{'}}}\Big )\prod _{l} (u_{l} + v_{l}c_{l,\uparrow }^{+}c_{-l,\downarrow }^{+}) \Big \vert 0\Big \rangle . \end{aligned}$$

This calculation is done by classifying the contribution coming out from the states which are specified with the power of $\alpha$, and they are listed as below:

$$\begin{aligned}&\alpha ^{0} : 2 \sum _{k} \dfrac{\varepsilon _{k} \vert v_{k}\vert ^{2}}{1+\alpha (\alpha -2)\vert v_{k}\vert ^{2}}. \end{aligned}$$

(43)

$$\begin{aligned}&\alpha ^{1} : 2 \sum _{k} \dfrac{-2\alpha \varepsilon _{k} \vert v_{k}\vert ^{2}}{1+\alpha (\alpha -2)\vert v_{k}\vert ^{2}}. \end{aligned}$$

(44)

$$\begin{aligned}&\alpha ^{2} : 2 \sum _{k} \dfrac{\alpha ^{2}\varepsilon _{k} \vert v_{k}\vert ^{2}}{1+\alpha (\alpha -2)\vert v_{k}\vert ^{2}}. \end{aligned}$$

(45)

The contribution of potential energy expectation value is deduced in a similar manner and these are added in this section.

$$\begin{aligned}&\alpha ^{0} : 0 \end{aligned}$$

(46)

$$\begin{aligned}&\alpha ^{1} : \sum _{k,l} V_{k,l}\dfrac{-2\alpha u_{l}^{*}u_{k}v_{k}^{*}v_{l}}{[1+\alpha (\alpha -2)\vert v_{k}\vert ^{2}][1+\alpha (\alpha -2)\vert v_{l}\vert ^{2}]}\nonumber \\ \end{aligned}$$

(47)

$$\begin{aligned}&\alpha ^{2} : \sum _{k,l} V_{k,l}\dfrac{(1+\alpha ^{2}) u_{l}^{*}u_{k}v_{k}^{*}v_{l}}{[1+\alpha (\alpha -2)\vert v_{k}\vert ^{2}][1+\alpha (\alpha -2)\vert v_{l}\vert ^{2}]}.\nonumber \\ \end{aligned}$$

(48)