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

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Data Availability Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: Based on our analytical results, we added all the data to this manuscript and there are no more data to deposit.]

References

  1. 1.

    D. K. Thapa, S. Islam, S. K. Saha, P. S. Mahapatra, B. Bhattacharyya, T. P. Sai, R. Mahadevu, S. Patil, A. Ghosh, A. Pandey, arXiv:1807.08572, (2019)

  2. 2.

    D.J. Scalapino, R.D. Parks, Superconductivity (Marcel Dekker, New York, 1969), p. 449

    Google Scholar 

  3. 3.

    P.B. Allen, B. Mitrovic, Solid State Physies, edited by H. Ehrenreich, F. Seitz, D., vol. 37 (Turnbull Academic, New York, 1982), p. 1

    Google Scholar 

  4. 4.

    V.L. Ginzburg, D.A. Kirzhnits, High-Temperature Superconductivity, Chapter-I (Consultants Bureau, New York, 1982)

    Google Scholar 

  5. 5.

    N.W. Ashcroft, Phys. Rev. Lett. 21, 1748–1750 (1968)

    ADS  Article  Google Scholar 

  6. 6.

    N.W. Ashcroft, Phys. Rev. Lett. 92, 187002 (2004)

    ADS  Article  Google Scholar 

  7. 7.

    A.P. Drozdrov, M.I. Eremets, I.A. Troyan, V. Ksenofontov, S.I. Shylin, Nature 525, 73–76 (2015)

    ADS  Article  Google Scholar 

  8. 8.

    M. Somayazulu, M. Ahart, A. Mishra, Z.M. Geballe, M. Baldini, Y. Meng, V.V. Struzhkin, R.J. Memley, Phys. Rev. Lett. 122, 027001 (2019)

    ADS  Article  Google Scholar 

  9. 9.

    E. Snider, N. Dasenbrock-Gammon, R. McBride, M. Debessai, H. Vindana, K. Vencatasamy, K.V. Lawler, A. Salamat, R.P. Dias, Nature 588(7837), E18 (2020)

    ADS  Article  Google Scholar 

  10. 10.

    D. Rybicki, M. Jurkutat, S. Reichardt, C. Kapusta, J. Haase, Nat. Commun. 7, 11413 (2016)

    ADS  Article  Google Scholar 

  11. 11.

    D. Chakraborty, C. Morice, C. Pépin, Phys. Rev. B 97, 214501 (2018)

    ADS  Article  Google Scholar 

  12. 12.

    J. Bardeen, L.N. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957)

    ADS  MathSciNet  Article  Google Scholar 

  13. 13.

    M. Tinkham, Introduction to Superconductivity (Dover Publications, Mineola, 1996)

    Google Scholar 

  14. 14.

    T.C. Paulick, C.E. Campbell, Phys. Rev. B 16, 2000 (1977)

    ADS  Article  Google Scholar 

  15. 15.

    Z. Wang, J.R. Engelbrecht, S. Wang, H. Ding, S.H. Pan, Phys. Rev. B 65, 064509 (2002)

    ADS  Article  Google Scholar 

  16. 16.

    E. Krotscheck, R.A. Smith, A.D. Jackson, Phys. Lett. B 104, 421 (1981)

    ADS  Article  Google Scholar 

  17. 17.

    M.C. Gutzwiller, Phys. Rev. Lett. 10, 159 (1963)

    ADS  Article  Google Scholar 

  18. 18.

    M. Causula, S. Sorella, J. Chem. Phys. 119, 6500 (2003)

    ADS  Article  Google Scholar 

  19. 19.

    J.R. Schrieffer, Theory of Superconductivity (CRR Press, New York, 2018), pp. 36–39

    Google Scholar 

  20. 20.

    S. Sorella, Phys. Rev. B 64, 024512 (2001)

    ADS  Article  Google Scholar 

  21. 21.

    E. Krotschek, J. Low. Temp. Phys. 119, 103 (2000)

    ADS  Article  Google Scholar 

  22. 22.

    K. Mandal, Study of electron-electron correlation in Cooper pair problem in a low dimensional system and modified gap equation Post M.Sc. project at S N Bose Centre (done under the supervision of R. Chaudhury) (2017)

  23. 23.

    K. Chatterejee, Two-well model with Cooper pairing mechanism and isotope exponent from BCS theory M.Sc. project at S N Bose Centre (done under the supervision of R. Chaudhury) (2018)

  24. 24.

    S. Bhattacharjee, R. Chaudhury, Phys. B 500, 133 (2016)

    ADS  Article  Google Scholar 

  25. 25.

    S. Bhattacharjee, R. Chaudhury, J. Low. Temp. Phys. 193, 21 (2018)

    ADS  Article  Google Scholar 

  26. 26.

    M. Saarela, E. Krotscheck, J. Low. Temp. Phys. 90, 415 (1993)

    ADS  Article  Google Scholar 

  27. 27.

    H.H. Fan, E. Krotscheck, T. Lichtenegger, D. Mateo, R.E. Zillich, Phys. Rev. A 92, 023640 (2015)

    ADS  Article  Google Scholar 

  28. 28.

    R.B. Wiringa, S.C. Pieper, Phys. Rev. Lett. 89, 182501 (2002)

    ADS  Article  Google Scholar 

  29. 29.

    G.R. Stewart, Rev. Sci. Instrum. 54(1), 1–11 (1983)

    ADS  Article  Google Scholar 

  30. 30.

    T.C. Chi, J. Phys. Chem. Ref. Data 8(2), 339–438 (1979)

    ADS  Article  Google Scholar 

  31. 31.

    R.A. Matula, J. Phys. Chem. Ref. Data 8(4), 1147–1298 (1979)

    ADS  Article  Google Scholar 

  32. 32.

    J.P. Carbotte, Rev. Mod. Phys. 62, 1027 (1990)

    ADS  Article  Google Scholar 

  33. 33.

    S. C. Ganguly, V. Vano, S. Kezilebieke, J. L. Lado, P. Liljeroth, arxiv: 2009.12422v1 [con-matt.mess-hall] (2020)

Download references

Acknowledgements

KM and RC would like to thank the Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for Basic Sciences, for their kind hospitality. KM thanks UGC for the financial support [ref. no. 522495(2016)].

Author information

Affiliations

Authors

Contributions

KM and RC defined this problem. KM did calculations. KM and RC wrote the manuscript together.

Corresponding author

Correspondence to Koushik Mandal.

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)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mandal, K., Chaudhury, R. Interplay of pairing correlation and Coulomb correlation in Boson exchange superconductors. Eur. Phys. J. B 94, 46 (2021). https://doi.org/10.1140/epjb/s10051-021-00051-9

Download citation