Appendix
1.1 A.1 Coefficients and Their Fourier Transforms of the Model Hamiltonian
The total Hamiltonian in real space H is as given in (1). The Hamiltonian H is transformed into the canonical form \(H^{\prime }\) via a standard displaced oscillator transformation as per (6). The primed terms \(H^{\prime }_{1}\) and \(H^{\prime }_{3}\) in (6) are as follows:
$$\begin{array}{@{}rcl@{}} H^{\prime}_{1}\!\!&=&\!\!\sum\limits_{i,j,\sigma} {t_{ij}c^{\dagger}_{i,\sigma} c_{j,\sigma}} \\ \!\!&\equiv&\!\! \sum\limits_{i,j,\sigma} {t^{\prime}_{ij}c^{\dagger}_{i,\sigma} c_{j,\sigma} }\exp \left\{\frac{1}{\sqrt{N}}\sum\limits_{{\mathbf{k}}}\frac{g\left( {\mathbf{k}}\right)}{\hslash {\omega}_{{\mathbf{k}}}}\left( a_{{\mathbf{k}},\sigma} \,-\,a^{\dagger}_{{\mathbf{-}}{\mathbf{k}}{\mathbf{,-}}\sigma} \right)\right.\\ &&\quad\quad\quad\quad\quad\quad\quad\quad\!\times\!\left.\left( e^{i{\mathbf{k}}{\mathbf{\cdot} }{{\mathbf{R}}}_{i}}\,-\,e^{i{\mathbf{k}}{\mathbf{\cdot} }{{\mathbf{R}}}_{j}}\right)\vphantom{\frac{1}{\sqrt{N}}\sum\limits_{{\mathbf{k}}}}\right\} , \\ H^{\prime}_{3}\!\!&=&\!\!-\frac{1}{2}\sum\limits_{\left\langle ij\right\rangle ,\sigma {,\sigma}^{\prime}}{V_{ij}n_{i,\sigma} n_{j{,\sigma}^{\prime}}} \\ \!\!&\equiv&\!\! -\frac{1}{N}\sum\limits_{i,j,\sigma {,\sigma} ^{\prime}}{\sum\limits_{{\mathbf{k}}}{\frac{g^{2}\left( {\mathbf{k}}\right)}{\hslash {\omega} _{{\mathbf{k}}}}e^{i{\mathbf{k}}\cdot \left( {{\mathbf{R}}}_{j}\boldsymbol{\!-}{{\mathbf{R}}}_{i}\right)}}c^{\dagger}_{i\sigma} c_{i\sigma} c^{\dagger}_{j{\sigma}^{\prime}}c_{j{\sigma}^{\prime}}}.\end{array} $$
(55)
Referring to \(H^{\prime }_{3}\) in (55), we identify
$$\begin{array}{@{}rcl@{}} V_{ij}=\frac{1}{N}\sum\limits_{{\mathbf{k}}}{\frac{g^{2}\left( {\mathbf{k}}\right)}{\hslash {\omega}_{{\mathbf{k}}}}e^{i{\mathbf{k}}\cdot \left( {{\mathbf{R}}}_{j}\boldsymbol{-}{{\mathbf{R}}}_{i}\right)}}.\end{array} $$
(56)
The coefficients in (6), namely t
i
j
, V
i
j
, c
i,σ
, \(c^{\dagger }_{i,\sigma }\) , and G
i
, are related to their Fourier counterparts via
$$\begin{array}{@{}rcl@{}} t_{ij}&=&\frac{1}{N}\sum\limits_{{\mathbf{k}}}{{\epsilon} _{{\mathbf{k}}}e^{i{\mathbf{k}}\cdot \left( {{\mathbf{R}}}_{j}-{{\mathbf{R}}}_{i}\right)}}, \\ V_{ij}&=&\frac{1}{N}\sum\limits_{{\mathbf{k}}}{V_{{\mathbf{k}}}e^{i{\mathbf{k}}\cdot \left( {{\mathbf{R}}}_{j}-{{\mathbf{R}}}_{i}\right)}}, \\ c_{i,\sigma} &=&\frac{1}{\sqrt{N}}\sum\limits_{{\mathbf{k}}}{c_{{\mathbf{k}},\sigma} e^{-i{\mathbf{k}}\cdot {{\mathbf{R}}}_{i}}}, \\ c^{\dagger}_{i,\sigma} &=&\frac{1}{\sqrt{N}}\sum\limits_{{\mathbf{k}}}{c^{\dagger}_{{\mathbf{k}},\sigma} e^{i{\mathbf{k}}\cdot {{\mathbf{R}}}_{i}}}, \\ G_{i}&=&\frac{1}{\sqrt{N}}\sum\limits_{{\mathbf{k}}}{G_{{\mathbf{k}}}e^{i{\mathbf{k}}\cdot {{\mathbf{R}}}_{i}}}.\end{array} $$
(57)
Comparing the expressions of V
i
j
in (56) and (57),
$$\begin{array}{@{}rcl@{}} V_{ij}&=&\frac{1}{N}\sum\limits_{{\mathbf{k}}}{\frac{g^{2}\left( {\mathbf{k}}\right)}{\hslash {\omega}_{{\mathbf{k}}}}e^{i{\mathbf{k}}\cdot \left( {{\mathbf{R}}}_{j}\boldsymbol{-}{{\mathbf{R}}}_{i}\right)}}=\frac{1}{N}\sum\limits_{{\mathbf{k}}}{V_{{\mathbf{k}}}e^{i{\mathbf{k}}\cdot \left( {{\mathbf{R}}}_{j}\boldsymbol{-}{{\mathbf{R}}}_{i}\right)}} \\ \Rightarrow V_{{\mathbf{k}}}&=&\frac{g^{2}\left( {\mathbf{k}}\right)}{\hslash {\omega}_{{\mathbf{k}}}}.\end{array} $$
(58)
This explains how (24) is arrived at.
1.2 A.2 Derivation of (9) from (6)
Each term in (6) can be cast into Bloch representation with the aid of Fourier transformations from (57).
$$\begin{array}{@{}rcl@{}} H^{\prime}_{1}\!&=&\!\sum\limits_{i,j,\sigma} {t_{ij}c^{\dagger}_{i,\sigma} c_{j,\sigma} }\,=\,\frac{1}{N}\sum\limits_{\sigma} {\sum\limits_{i,j}{\sum\limits_{{\mathbf{k}}}{{\epsilon}_{{\mathbf{k}}}}e^{i{\mathbf{k}}\cdot \left( {{\mathbf{R}}}_{j}-{{\mathbf{R}}}_{i}\right)}c^{\dagger}_{i,\sigma} c_{j,\sigma} }} \\ \!&=&\!\sum\limits_{\sigma} \sum\limits_{{\mathbf{k}}}{{\epsilon}_{{\mathbf{k}}}}\left( \frac{1}{\sqrt{N}}\sum\limits_{i}{c^{\dagger}_{i,\sigma} e^{-i{\mathbf{k}}\cdot {{\mathbf{R}}}_{i}}}\right)\\ &&\times\left( \frac{1}{\sqrt{N}}\sum\limits_{j}{c_{j,\sigma} e^{i{\mathbf{k}}\cdot {{\mathbf{R}}}_{j}}}\right) \\ \!&=&\!\sum\limits_{{\mathbf{k}},\sigma} {{\epsilon}_{{\mathbf{k}}}}c^{\dagger}_{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma} .\ \end{array} $$
(59)
$$\begin{array}{@{}rcl@{}} H^{\prime}_{2} &=& U \sum\limits_{i,\sigma} n_{i,\sigma}n_{i,-\sigma} \\ &=& U \sum\limits_{\sigma} \left\{ \left( \frac{1}{\sqrt{N}}\sum\limits_{{\mathbf{k}}}{c^{\dagger} _{{\mathbf{k}},\sigma} e^{i{\mathbf{k}}\cdot {{\mathbf{R}}}_{i}}}\right)\left( \frac{1}{\sqrt{N}}\sum\limits_{{{\mathbf{k}}}^{\prime}}{c_{{{\mathbf{k}}}^{{\prime}},\sigma} e^{-i{{\mathbf{k}}}^{{\prime}}\cdot {{\mathbf{R}}}_{i}}}\right) \right. \\ &&\left. \left( \frac{1}{\sqrt{N}}\sum\limits_{{{\mathbf{k}}}^{\prime\prime}}{c^{\dagger} _{{\mathbf{k}},-\sigma} e^{i{{\mathbf{k}}}^{\prime\prime}\cdot {{\mathbf{R}}}_{i}}}\right)\left( \frac{1}{\sqrt{N}}\sum\limits_{{{\mathbf{k}}}^{\prime\prime\prime}}{c_{{{\mathbf{k}}}^{\prime\prime\prime},-\sigma} e^{-i{{\mathbf{k}}}^{\prime\prime\prime}\cdot {{\mathbf{R}}}_{i}}}\right) \right\} \\ &=&\ \frac{U}{N^{2}}\sum\limits_{\sigma} \left( \sum\limits_{{\mathbf{k}},{{\mathbf{k}}}^{{\prime}},{{\mathbf{k}}}^{\prime\prime}},{{\mathbf{k}}^{{\prime\prime\prime}}}{c^{\dagger} _{{\mathbf{k}},\sigma} c_{{{\mathbf{k}}}^{{\prime}},\sigma} }c^{\dagger} _{{{\mathbf{k}}}^{\prime\prime},-\sigma} c_{{{\mathbf{k}}}^{\prime\prime\prime},-\sigma} \right)\\ &&\times N\delta \left( {\mathbf{k}}-{{\mathbf{k}}}^{{\prime}}+{{\mathbf{k}}}^{\prime\prime}-{{\mathbf{k}}}^{\prime\prime\prime}\right) \\ &=&\frac{U}{N^{2}}\sum\limits_{\sigma ,{\mathbf{k}}{,{\mathbf{k}}}^{\prime},{\mathbf{q}}}{{c^{\dagger} _{{\mathbf{k}}+{\mathbf{q}},\sigma} c_{{\mathbf{k}},\sigma} c}^{\dagger} _{{{\mathbf{k}}}^{{\prime}}-{\mathbf{q}},-\sigma} c_{{{\mathbf{k}}}^{{\prime}},-\sigma} }.\end{array} $$
(60)
$$\begin{array}{@{}rcl@{}} H^{\prime}_{3}&=&-\frac{1}{2}\sum\limits_{\left\langle ij\right\rangle ,\sigma {,\sigma} ^{\prime}}{V_{ij}c^{\dagger}_{i,\sigma} c_{i,\sigma} c^{\dagger}_{j,{\sigma}^{\prime}}c_{j,{\sigma} ^{\prime}}} \\ &=& -\frac{1}{2}\sum\limits_{\sigma {,\sigma}^{\prime}}{\sum\limits_{\left\langle ij\right\rangle} {V_{ij}}} \left\{ \sum\limits_{{\mathbf{k}}}{\frac{1}{\sqrt{N}}c^{\dagger} _{{\mathbf{k}},\sigma} e^{i{\mathbf{k}}\cdot {{\mathbf{R}}}_{i}}}\right.\\ &&\times\left. \sum\limits_{{{\mathbf{k}}}^{{\prime}}}{\frac{1}{\sqrt{N}}c_{{{\mathbf{k}}}^{{\prime}},\sigma} e^{-i{{\mathbf{k}}}^{{\prime}} {{\mathbf{R}}}_{i}}} \cdot\sum\limits_{{{\mathbf{k}}}^{{\prime\prime}}}{\frac{1}{\sqrt{N}}c^{\dagger} _{{{\mathbf{k}}}^{{\prime\prime}},{\sigma} ^{\prime}}e^{i{{\mathbf{k}}}^{{\prime\prime}}\cdot {{\mathbf{R}}}_{j}}}\right.\\ &&\times\left.\sum\limits_{{{\mathbf{k}}}^{{\prime\prime\prime}}}\frac{1}{\sqrt{N}}c_{{{\mathbf{k}}}^{{\prime\prime\prime}},{\sigma} ^{\prime}}e^{-i{{\mathbf{k}}}^{{\prime\prime\prime}}\cdot {{\mathbf{R}}}_{j}} \right\} \\ &=& -\frac{1}{2N^{2}}\sum\limits_{\sigma {,\sigma}^{\prime}} \sum\limits_{{\mathbf{k}},{{\mathbf{k}}}^{{\prime}},{{\mathbf{k}}}^{{\prime\prime}}{{\mathbf{k}}}^{{\prime\prime\prime}}} \sum\limits_{\left\langle i,j\right\rangle} \sum\limits_{{\mathbf{q}}}\\ &&\times\left\{ \frac{1}{N} V_{{\mathbf{q}}}\times e^{i{\mathbf{q}}\cdot \left( {{\mathbf{R}}}_{j}-{{\mathbf{R}}}_{i}\right)} e^i\left( {\mathbf{k}}-{{\mathbf{k}}}^{{\prime}}\right)\right. \\ &&\left.\cdot {{\mathbf{R}}}_{i}e^{i\left( {{\mathbf{k}}}^{{\prime\prime}}-{{\mathbf{k}}}^{{\prime\prime\prime}}\right)\cdot {{\mathbf{R}}}_{j}} c^{\dagger}_{{\mathbf{k}},\sigma} c_{{{\mathbf{k}}}^{{\prime}},\sigma} c^{\dagger} _{{{\mathbf{k}}}^{{\prime\prime}},{\sigma}^{\prime}} c^{\dagger} _{{{\mathbf{k}}}^{{\prime\prime\prime}},{\sigma}^{\prime}} \vphantom{\frac{1}{N} V_{{\mathbf{q}}}}\right\} \\ &=&-\frac{1}{2N}\sum\limits_{\sigma {,\sigma} ^{\prime}}{\sum\limits_{{\mathbf{q}}}{\sum\limits_{{\mathbf{k}},{{\mathbf{k}}}^{{\prime}}}{V_{{\mathbf{q}}}}}c^{\dagger} _{{\mathbf{k}}+{\mathbf{q}},\sigma} c_{{\mathbf{k}},\sigma} c^{\dagger} _{{{\mathbf{k}}}^{{\prime}}-{\mathbf{q}},{\sigma}^{\prime}}c^{\dagger} _{{{\mathbf{k}}}^{{\prime}},{\sigma}^{\prime}}}. \end{array} $$
(61)
$$\begin{array}{@{}rcl@{}} H^{\prime}_{4}&=&-\sum\limits_{i,\sigma} {G_{i}c^{\dagger}_{i,\sigma} c_{i,\sigma} } \\ &=&-\sum\limits_{i,\sigma} \frac{1}{\sqrt{N}}\sum\limits_{{\mathbf{k}}}{G_{{\mathbf{k}}}e^{i{\mathbf{k}}\cdot {{\mathbf{R}}}_{i}}}\left( \sum\limits_{{{\mathbf{k}}}^{{\prime}}}{\frac{1}{\sqrt{N}}c^{\dagger} _{{{\mathbf{k}}}^{{\prime}},\sigma} e^{i{{\mathbf{k}}}^{{\prime}}\cdot {{\mathbf{R}}}_{i}}}\right)\\ &&\times\left( \sum\limits_{{{\mathbf{k}}}^{{\prime\prime}}}{\frac{1}{\sqrt{N}}c_{{{\mathbf{k}}}^{{\prime\prime}},\sigma} e^{-i{{\mathbf{k}}}^{{\prime\prime}}\cdot {{\mathbf{R}}}_{i}}}\right) \\ &=& -\frac{1}{\sqrt{N}}\sum\limits_{\sigma} {\sum\limits_{{\mathbf{q}},{\mathbf{k}}}{G_{{\mathbf{q}}}c^{\dagger}_{{\mathbf{k}}-{\mathbf{q}},\sigma} c_{{\mathbf{k}},\sigma} }}.\end{array} $$
(62)
$$\begin{array}{@{}rcl@{}} H^{\prime}_{5}\!&=&\!-\mu\! \sum\limits_{i,\sigma} {c^{\dagger}_{i,\sigma} c_{i,\sigma} }\mathrm{=}\,-\,\mu\! \sum\limits_{i,\sigma} \!\left( \!\sum\limits_{\mathbf{k}}\!\frac{1}{\sqrt{N}}c^{\dagger}_{{\mathbf{k}},\sigma} e^{i{\mathbf{k}}\cdot {{\mathbf{R}}}_{i}}\!\right)\\ &&\times\left( \sum\limits_{{{\mathbf{k}}}^{{\prime}}}{\frac{1}{\sqrt{N}}c_{{{\mathbf{k}}}^{{\prime}},\sigma} e^{-i{{\mathbf{k}}}^{{\prime}}\cdot {{\mathbf{R}}}_{i}}}\right) \\ &=& -\mu \sum\limits_{\sigma, {\mathbf{k}}}{c^{\dagger}_{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma} }. \end{array} $$
(63)
Adding up (59)–(63), \(H^{\prime }_{1}+H^{\prime }_{2}+H^{\prime }_{3}+H^{\prime }_{4}+H^{\prime }_{5}\) produces the Bloch representation model Hamiltonian, (9).
1.3 A.3 Diagonalization of Trial Hamiltonian of (10) into the Form (11)
The trial Hamiltonian of (10) is diagonalized by following a textbook approach, such as that discussed in p. 164 in [19]. To this end, we also require the anti-commutative relations for the operators \(c_{{\mathbf {k}},\sigma } ,c^{\dagger }_{{\mathbf {k}},\sigma } \). We write down the Heisenberg equation of motion for c
k,σ
in terms of the trial Hamiltonian of (10),
$$\begin{array}{@{}rcl@{}} i\frac{dc_{{\mathbf{k}},\sigma} }{dt}\equiv i{\dot{c}}_{{\mathbf{k}},\sigma} =\left[c_{{\mathbf{k}},\sigma} ,H_{t}\right].\end{array} $$
(64)
Expanding and simplifying the commutator in Eq. (64),
$$\begin{array}{@{}rcl@{}} \left[c_{{\mathbf{k}},\sigma} ,H_{t}\right]&\,=\,&\sum\limits_{{{\mathbf{k}}}^{\prime},{\sigma}^{\prime}}{\left( {\epsilon} _{{{\mathbf{k}}}^{\prime}}\,-\,\mu \,-\,A_{{{\mathbf{k}}}^{\prime}}\right)}\underbrace{\left( c_{{\mathbf{k}},\sigma} c^{\dagger} _{{{\mathbf{k}}}^{\prime},{\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{\prime},{\sigma}^{\prime}}-c^{\dagger} _{{{\mathbf{k}}}^{{\prime}},{\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{{\prime}},{\sigma} ^{\prime}}c_{{\mathbf{k}},\sigma} \right)}_{I} \\ &&-\frac{1}{2}\sum\limits_{{{\mathbf{k}}}^{\prime},{\sigma}^{\prime}} \left[ B_{{{\mathbf{k}}}^{\prime}}\underbrace{\left( {c^{\dagger} _{{{\mathbf{k}}}^{\prime},{\sigma}^{\prime}}c^{\dagger} _{-{{\mathbf{k}}}^{\prime},-{\sigma}^{\prime}}c_{{\mathbf{k}},\sigma} \,-\,c}_{{\mathbf{k}},\sigma} c^{\dagger}_{{{\mathbf{k}}}^{\prime},{\sigma} ^{\prime}}c^{\dagger}_{-{{\mathbf{k}}}^{\prime},-{\sigma}^{\prime}}\right)}_{II} \right. \\ && \left. +B^{*}_{{{\mathbf{k}}}^{{\prime}}}\underbrace{\left( {c_{-{{\mathbf{k}}}^{{\prime}},-{\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{{\prime}},{\sigma} ^{\prime}}c_{{\mathbf{k}},\sigma} \,-\,c}_{{\mathbf{k}},\sigma} c_{-{{\mathbf{k}}}^{{\prime}},-{\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{{\prime}},{\sigma}^{\prime}}\right)}_{III} \right]\!. \end{array} $$
(65)
We now look at the I, II, and III terms in (65) in turn. The term I in (65) can be simplified to
$$\begin{array}{@{}rcl@{}} I=\delta \left( {\mathbf{k}}-{{\mathbf{k}}}^{\prime}\right){\delta}_{\sigma ,{\sigma}^{\prime}}c_{{{\mathbf{k}}}^{\prime},{\sigma}^{\prime}}\end{array} $$
by using the following anti-commutative relations
$$\begin{array}{@{}rcl@{}} \left\{c_{{\mathbf{k}},\sigma} ,c_{{{\mathbf{k}}}^{{\prime}},{\sigma} ^{\prime}}\right\}=0;\ \left\{c_{{\mathbf{k}},\sigma} ,c^{\dagger} _{{{\mathbf{k}}}^{{\prime}},{\sigma}^{\prime}}\right\}=\delta \left( {\mathbf{k}}-{{\mathbf{k}}}^{\prime}\right){\delta}_{\sigma ,{\sigma} ^{\prime}}.\end{array} $$
(66)
The term II in (65) can be simplified to
$$\begin{array}{@{}rcl@{}} II=\delta \left( {\mathbf{k}}+{{\mathbf{k}}}^{\prime}\right){\delta} _{\sigma ,-{\sigma}^{\prime}}c^{\dagger}_{{{\mathbf{k}}}^{\prime},\sigma} -\delta \left( {\mathbf{k}}-{{\mathbf{k}}}^{\prime}\right){\delta}_{\sigma ,{\sigma} ^{\prime}}c^{\dagger}_{-{{\mathbf{k}}}^{\prime},-{\sigma}^{\prime}}\end{array} $$
by using the following anti-commutative relation
$$\begin{array}{@{}rcl@{}}c^{\dagger}_{{-{\mathbf{k}}}^{{\prime}},-{\sigma}^{\prime}}c_{{\mathbf{k}},\sigma} =\delta \left( {\mathbf{k}}+{{\mathbf{k}}}^{\prime}\right){\delta}_{\sigma ,-{\sigma}^{\prime}}-c_{{\mathbf{k}},\sigma} c^{\dagger}_{-{{\mathbf{k}}}^{{\prime}},-{\sigma}^{\prime}}.\end{array} $$
(67)
The term III in (65) can be simplified to
$$\begin{array}{@{}rcl@{}} III={c_{{\mathbf{k}},\sigma} c_{-{{\mathbf{k}}}^{{\prime}},-{\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{{\prime}},\sigma} -c}_{{\mathbf{k}},\sigma} c_{-{{\mathbf{k}}}^{{\prime}},-{\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{{\prime}},{\sigma}^{\prime}}\end{array} $$
by using the following anti-commutative relation,
$$\begin{array}{@{}rcl@{}} \left\{c_{{\mathbf{k}},\sigma} ,c_{-{{\mathbf{k}}}^{{\prime}},-{\sigma}^{\prime}}\right\}=0.\end{array} $$
(68)
Putting I, II, and III into (65),
$$\begin{array}{@{}rcl@{}} i{\dot{c}}_{{\mathbf{k}},\sigma} \!\!&=&\!\sum\limits_{{{\mathbf{k}}}^{\prime},{\sigma} ^{\prime}}{\left( {\epsilon}_{{{\mathbf{k}}}^{\prime}}-\mu -A_{{{\mathbf{k}}}^{\prime}}\right)}{\delta \left( {\mathbf{k}}-{{\mathbf{k}}}^{\prime}\right){\delta}_{\sigma ,{\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{\prime},{\sigma}^{\prime}}}_{I} \\ &&-\frac{1}{2}\sum\limits_{{{\mathbf{k}}}^{\prime},{\sigma} ^{\prime}}{B_{{{\mathbf{k}}}^{\prime}}\left[\delta \left( {\mathbf{k}}+{{\mathbf{k}}}^{\prime}\right){\delta}_{\sigma ,-{\sigma} ^{\prime}}c^{\dagger}_{{{\mathbf{k}}}^{\prime},\sigma} \,-\,\delta \left( {\mathbf{k}}-{{\mathbf{k}}}^{\prime}\right){\delta}_{\sigma ,{\sigma} ^{\prime}}c^{\dagger}_{-{{\mathbf{k}}}^{\prime},-{\sigma}^{\prime}}\right]} \\&=& E_{{\mathbf{k}}}c_{{\mathbf{k}},\sigma} +{\frac{1}{2}B}_{{\mathbf{k}}}c^{\dagger} _{-{\mathbf{k}}{\mathbf{,-}}\sigma} .\end{array} $$
(69)
Similarly,
$$\begin{array}{@{}rcl@{}} i{\dot{c}}^{\dagger}_{{\mathbf{k}},\sigma}={-E}_{{\mathbf{k}}}c^{\dagger}_{{\mathbf{k}},\sigma} -\frac{1}{2}B_{{\mathbf{k}}}c_{-{\mathbf{k}},-\sigma} .\end{array} $$
(70)
In deriving (69) and (70), we have defined
$$\begin{array}{@{}rcl@{}} E_{{\mathbf{k}}}\equiv {\epsilon}_{{\mathbf{k}}}-\mu -A_{{\mathbf{k}}},\end{array} $$
(71)
and assumed
$$\begin{array}{@{}rcl@{}} B_{{\mathbf{k}}}&=&B^{*}_{{\mathbf{k}}}\ \left( \mathrm{i.e.,} B_{{\mathbf{k}}}\mathrm{a\ real\ number}\right); \\ E_{{\mathbf{k}}}&=&E_{-{\mathbf{k}}},B_{{\mathbf{k}}}=B_{-{\mathbf{k}}}.\end{array} $$
(72)
For the next step, we shall perform Bogolyubov-Valatin transformation (see e.g., p. 307 of Haken, [20]) on \(c^{\dagger }_{-{\mathbf {k}}{\mathbf {,-}}\sigma } ,c_{{\mathbf {-}}{\mathbf {k}}{\mathbf {,-}}\sigma } ,c_{{\mathbf {k}},\sigma } {,c}^{\dagger }_{{\mathbf {k}},\sigma } \) to obtain
$$\begin{array}{@{}rcl@{}} {\alpha}^{\dagger}_{{\mathbf{k}},\sigma}=u_{{\mathbf{k}}}c^{\dagger} _{{\mathbf{k}},\sigma} -v_{{\mathbf{k}}}c_{-{\mathbf{k}}{\mathbf{,-}}\sigma} ,\end{array} $$
$$\begin{array}{@{}rcl@{}} {\alpha}_{{\mathbf{k}},\sigma} =u_{{\mathbf{k}}}c_{{\mathbf{k}},\sigma} -v_{{\mathbf{k}}}c^{\dagger}_{-{\mathbf{k}}{\mathbf{,-}}\sigma} ,\end{array} $$
$$\begin{array}{@{}rcl@{}} {\alpha}_{-{\mathbf{k}}{\mathbf{,-}}\sigma} =u_{{\mathbf{k}}}v_{{\mathbf{k}}}c_{-{\mathbf{k}}{\mathbf{,-}}\sigma} +v_{{\mathbf{k}}}c^{\dagger}_{{\mathbf{k}},\sigma} ,\end{array} $$
$$\begin{array}{@{}rcl@{}}{\alpha}^{\dagger}_{-{\mathbf{k}}{\mathbf{,-}}\sigma} =u_{{\mathbf{k}}}c^{\dagger}_{-{\mathbf{k}}{\mathbf{,-}}\sigma} +v_{{\mathbf{k}}}c_{{\mathbf{k}},\sigma} ,\end{array} $$
(73)
with the condition \(u^{2}_{{\mathbf {k}}}+v^{2}_{{\mathbf {k}}}=1\) and u
k
,v
k
are real numbers. In addition, we shall also assume
$$\begin{array}{@{}rcl@{}} u_{-{\mathbf{k}}}=u_{{\mathbf{k}}},v_{-{\mathbf{k}}}=v_{{\mathbf{k}}}.\end{array} $$
(74)
Unless otherwise specify, as a short-hand notation, we shall suppress the spin subscript in the c
k
,c
‡
k
, and \({\alpha }^{\dagger } _{{\mathbf {k}}},{\alpha }_{{\mathbf {k}}}\) operators with the understanding that they are implicitly σ-bearing, i.e.,
$$\begin{array}{@{}rcl@{}} {\alpha}^{\dagger}_{{\mathbf{k}}}\equiv {\alpha}^{\dagger}_{{\mathbf{k}},\sigma};{\alpha}_{{\mathbf{k}}}\equiv {\alpha}_{{\mathbf{k}},\sigma} ;{{\alpha} _{{\mathbf{-}}{\mathbf{k}}}\equiv \alpha} _{{\mathbf{-}}{\mathbf{k}}{\mathbf{,-}}\sigma} ;{\alpha}^{\dagger} _{-{\mathbf{k}}}\equiv {\alpha}^{\dagger}_{-{\mathbf{k}}{\mathbf{,-}}\sigma} ;\end{array} $$
$$\begin{array}{@{}rcl@{}} c^{\dagger}_{{\mathbf{k}}}\!\equiv \!c^{\dagger}_{{\mathbf{k}},\sigma} ;\ c_{{\mathbf{-}}{\mathbf{k}}}\!\equiv \!c_{-{\mathbf{k}}{\mathbf{,-}}\sigma} ;c_{-{\mathbf{k}}}\!\equiv \!c_{-{\mathbf{k}}{\mathbf{,-}}\sigma};c^{\dagger}_{-{\mathbf{k}}}\!\equiv \!c^{\dagger}_{-{\mathbf{k}}{\mathbf{,-}}\sigma .}\end{array} $$
(75)
The operators in (75) fulfill the following anti-commutative relations:
$$\begin{array}{@{}rcl@{}} \left\{{\alpha}_{{\mathbf{k}}},{\alpha}^{\dagger} _{{{\mathbf{k}}}^{{\prime}}}\right\}={\delta} _{{{\mathbf{k}},{\mathbf{k}}}^{\prime}}{\delta}_{\sigma ,{\sigma}^{\prime}};\ \left\{{\alpha}_{{\mathbf{k}}},{\alpha}_{{{\mathbf{k}}}^{{\prime}}}\right\}=0;\ \left\{{\alpha}^{\dagger}_{{\mathbf{k}}},{\alpha}^{\dagger} _{{{\mathbf{k}}}^{{\prime}}}\right\}=0.\end{array} $$
(76)
Inverting (76) we have
$$\begin{array}{@{}rcl@{}} c_{{\mathbf{k}}}=u_{{\mathbf{k}}}{\alpha} _{{\mathbf{k}}}+v_{{\mathbf{k}}}{\alpha}^{\dagger}_{-{\mathbf{k}}},\end{array} $$
$$\begin{array}{@{}rcl@{}} c^{\dagger}_{{\mathbf{k}}}=u_{{\mathbf{k}}}{\alpha}^{\dagger} _{{\mathbf{k}}}+v_{{\mathbf{k}}}{\alpha}_{-{\mathbf{k}}},\end{array} $$
$$\begin{array}{@{}rcl@{}} c_{-{\mathbf{k}}}=u_{{\mathbf{k}}}{\alpha} _{-{\mathbf{k}}}-v_{{\mathbf{k}}}{\alpha}^{\dagger}_{{\mathbf{k}}},\end{array} $$
$$\begin{array}{@{}rcl@{}}c^{\dagger}_{-{\mathbf{k}}}=u_{{\mathbf{k}}}{\alpha}^{\dagger}_{-{\mathbf{k}}}-v_{{\mathbf{k}}}{\alpha}_{{\mathbf{k}}}.\end{array} $$
(77)
Insert (77) into the trial Hamiltonian H
t
of (10),
$$\begin{array}{@{}rcl@{}} H_{t}&=&\underbrace{\sum\limits_{{\mathbf{k}},\sigma} {{\alpha}^{\dagger} _{{\mathbf{k}}}{\alpha} _{{\mathbf{k}}}\left( u^{2}_{{\mathbf{k}}}E_{{\mathbf{k}}}-\frac{1}{2}u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}-\frac{1}{2}u_{{\mathbf{k}}}v_{{\mathbf{k}}}B^{*}_{{\mathbf{k}}}\right)}}_{H_{t,I}} \\ && +\underbrace{\sum\limits_{{\mathbf{k}},\sigma} {{{\alpha}_{-{\mathbf{k}}}\alpha} ^{\dagger} _{-{\mathbf{k}}}\left( v^{2}_{{\mathbf{k}}}E_{{\mathbf{k}}}+\frac{1}{2}u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}+\frac{1}{2}u_{{\mathbf{k}}}v_{{\mathbf{k}}}B^{*}_{{\mathbf{k}}}\right)}}_{H_{t,II}} \\ && +\underbrace{\sum\limits_{{\mathbf{k}},\sigma} {{\alpha}^{\dagger} _{{\mathbf{k}}}{\alpha}^{\dagger} _{-{\mathbf{k}}}\left( {u_{{\mathbf{k}}}v_{{\mathbf{k}}}E}_{{\mathbf{k}}}+ \frac{1}{2}u^{2}_{{\mathbf{k}}}B_{{\mathbf{k}}}-\frac{1}{2}v^{2}_{{\mathbf{k}}}B^{*}_{{\mathbf{k}}}\right)}}_{H_{t,III}} \\ && +\underbrace{\sum\limits_{{\mathbf{k}},\sigma} {{\alpha}_{-{\mathbf{k}}}{\alpha} _{{\mathbf{k}}}\left( {u_{{\mathbf{k}}}v_{{\mathbf{k}}}E}_{{\mathbf{k}}}+\frac{1}{2}u^{2}_{{\mathbf{k}}}B^{*}_{{\mathbf{k}}}-\frac{1}{2}v^{2}_{{\mathbf{k}}}B_{{\mathbf{k}}}\right)}}_{H_{t,IV}} \end{array} $$
(78)
If H
t
were diagonal in the \(\left \{{{\alpha }^{\dagger }_{{\mathbf {k}}},\alpha } _{{\mathbf {k}}}\right \}\) representation, it can be expressed in the form
$$\begin{array}{@{}rcl@{}} H_{t}=\sum\limits_{{\mathbf{k}},\sigma} {{\lambda}_{{\mathbf{k}}}{{\alpha}^{\dagger} _{{\mathbf{k}}}\alpha}_{{\mathbf{k}}}}+\text{constant},\end{array} $$
(79)
where λ
k
the eigenvalue of H
t
. The constant term, which does not contain any α operator, can be neglected when applying variation calculation on H
t
. The equations of motion for α
k
and \({\ \alpha }^{\dagger }_{{\mathbf {k}}}\) are respectively given by
$$\begin{array}{@{}rcl@{}} i\frac{d{\alpha}^{\dagger}_{{\mathbf{k}}}}{dt}&=&\left[{\alpha}^{\dagger} _{{\mathbf{k}}},H_{t}\right]=-{\lambda}_{{\mathbf{k}}}{\alpha}^{\dagger} _{{\mathbf{k}}}, \\ i\frac{d{\alpha}_{{\mathbf{k}}}}{dt}&=&\left[{\alpha} _{{\mathbf{k}}},H_{t}\right]={\lambda}_{{\mathbf{k}}}{\alpha} _{{\mathbf{k}}}.\end{array} $$
(80)
Putting H
t
in (78) together with α
k‡ in a commutator yields
$$\begin{array}{@{}rcl@{}} \left[{\alpha}^{\dagger}_{{\mathbf{k}}},H_{t}\right]=\left[{\alpha}^{\dagger} _{{\mathbf{k}}},H_{t,I}\right]+\left[{\alpha}^{\dagger} _{{\mathbf{k}}},H_{t,II}\right].\end{array} $$
(81)
The terms \(\left [{\alpha }^{\dagger } _{{\mathbf {k}}},H_{t,III}\right ]\ \text {and} \left [{\alpha }^{\dagger } _{{\mathbf {k}}},H_{t,IV}\right ]\) that should otherwise appear in the RHS of (81) vanish due to the following anti-commutative relations:
$$\begin{array}{@{}rcl@{}} \left\{{\alpha}^{\dagger}_{{\mathbf{k}}}{,\alpha}^{\dagger} _{{{\mathbf{k}}}^{{\prime}}}\right\}&=&0; \ \left\{{\alpha} _{{{\mathbf{k}}}^{\prime}},{\alpha}^{\dagger}_{{\mathbf{k}}}\right\}=\delta \left( {\mathbf{k}}{\mathbf{-}}{\mathbf{k}}{\prime}\right){\delta} _{\sigma ,{\sigma}^{\prime}}; \\ \left\{{\alpha}_{-{{\mathbf{k}}}^{\prime}},{\alpha}^{\dagger} _{{\mathbf{k}}}\right\}&=&\delta \left( {\mathbf{k}}{\mathbf{+}}{\mathbf{k}}{\prime}\right){\delta} _{\sigma ,{-\sigma}^{\prime}}. \end{array} $$
Using the following results,
$$\begin{array}{@{}rcl@{}} \left[{\alpha}^{\dagger}_{{\mathbf{k}}},{\alpha}_{{{\mathbf{-}}{\mathbf{k}}}^{{\prime}}}{\alpha}^{\dagger}_{{-{\mathbf{k}}}^{{\prime}}}\right]&=&{{{\delta}_{{{\mathbf{k}}{\mathbf{,-}}{\mathbf{k}}}^{\prime}}\delta}_{\sigma ,-{\sigma}^{\prime}}\alpha}^{\dagger}_{{\mathbf{-}}{{\mathbf{k}}}^{{\prime}}}; \\ \left[{\alpha}^{\dagger}_{{\mathbf{k}}},{\alpha}^{\dagger} _{{{\mathbf{k}}}^{{\prime}}}{\alpha}^{\dagger} _{{{\mathbf{k}}}^{{\prime}}}\right] &=& -{{\delta} _{{{\mathbf{k}},{\mathbf{k}}}^{\prime}}\delta}_{\sigma ,{\sigma}^{\prime}}{\alpha} ^{\dagger}_{{{\mathbf{k}}}^{{\prime}}},\end{array} $$
(82)
the commutators in the RHS of (81) are reduced to
$$\begin{array}{@{}rcl@{}} \left[{\alpha}^{\dagger} _{{\mathbf{k}}},H_{t,I}\right]=-\left( u^{2}_{{\mathbf{k}}}E_{{\mathbf{k}}}-u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}\right){\alpha} ^{\dagger}_{{\mathbf{k}}}\end{array} $$
(83)
and
$$\begin{array}{@{}rcl@{}} \left[{\alpha}^{\dagger}_{{\mathbf{k}}},H_{t,II}\right]={\alpha}^{\dagger} _{{\mathbf{k}}}\left( v^{2}_{{\mathbf{k}}}E_{{\mathbf{k}}}+u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}\right).\end{array} $$
(84)
By (81), (83), and (84), the commutator \(\left [{\alpha }^{\dagger } _{{\mathbf {k}}},H_{t}\right ]\) in (80) becomes
$$\begin{array}{@{}rcl@{}} \left[{\alpha}^{\dagger} _{{\mathbf{k}}},H_{t}\right]=-\left[\left( u^{2}_{{\mathbf{k}}}-v^{2}_{{\mathbf{k}}}\right)E_{{\mathbf{k}}}-2u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}\right]{\alpha} ^{\dagger}_{{\mathbf{k}}}.\end{array} $$
(85)
Comparing (80) and (85), the eigenvalue λ
k
is identified, namely,
$$\begin{array}{@{}rcl@{}} {\lambda} _{{\mathbf{k}}}\mathrm{=}\left( u^{2}_{{\mathbf{k}}}-v^{2}_{{\mathbf{k}}}\right)E_{{\mathbf{k}}}-2u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}.\end{array} $$
(86)
We next look at \(u^{2}_{{\mathbf {k}}},v^{2}_{{\mathbf {k}}}\) in (86). Upon diagonalization, the coefficients for the off-diagonal terms of H
t
in (78) (i.e., H
t,I
I
I
, H
t,I
V
), which involve \({\alpha }^{\dagger }_{{\mathbf {k}}}{\alpha }^{\dagger } _{{-}{\mathbf {k}}},\ {\alpha }_{-{\mathbf {k}}}{\alpha } _{{\mathbf {k}}}\), should vanish, i.e.,
$$\begin{array}{@{}rcl@{}} {u_{{\mathbf{k}}}v_{{\mathbf{k}}}E}_{{\mathbf{k}}}+\frac{1}{2}\left( u^{2}_{{\mathbf{k}}}-v^{2}_{{\mathbf{k}}}\right)B_{{\mathbf{k}}}=0.\end{array} $$
(87)
Since \(u^{2}_{{\mathbf {k}}}+v^{2}_{{\mathbf {k}}}=1\), we parametrise
$$\begin{array}{@{}rcl@{}} u_{{\mathbf{k}}}={\text{cos} \frac{{\theta}_{{\mathbf{k}}}}{2}} ,\ v_{{\mathbf{k}}}={\sin \frac{{\theta}_{{\mathbf{k}}}}{2}} .\end{array} $$
(88)
It is also assumed that the parameter 𝜃
k
= 𝜃
−k
so that (88) is consistent with (74). The will cast (87) into
$$\begin{array}{@{}rcl@{}} {{\text{cos} \frac{{\theta}_{{\mathbf{k}}}}{2}} {\sin \frac{{\theta} _{{\mathbf{k}}}}{2}} E}_{{\mathbf{k}}}+\frac{1}{2}\!\left( {{\text{cos}}^{2} \frac{{\theta} _{{\mathbf{k}}}}{2}} -{{\sin}^{2} \frac{{\theta}_{{\mathbf{k}}}}{2}} \right)B_{{\mathbf{k}}}&=&0 \\ \Rightarrow {\tan {\theta}_{{\mathbf{k}}}} &=&-\frac{B_{{\mathbf{k}}}}{E_{{\mathbf{k}}}}.\end{array} $$
(89)
Hence, from (87), (88), and (89), the eigenvalue λ
k
in (86) is now simplified to
$$\begin{array}{@{}rcl@{}} {\lambda}_{{\mathbf{k}}}={\text{cos} {\theta}_{{\mathbf{k}}}} E_{{\mathbf{k}}}-{\sin {\theta}_{{\mathbf{k}}}} B_{{\mathbf{k}}}=\sqrt{E^{2}_{{\mathbf{k}}}+B^{2}_{{\mathbf{k}}}}.\end{array} $$
(90)
This completes the derivation of (12). We note that the result (90) does not have four branches as obtained in [17] because no charge-ordered states are considered in the present model. The term H
t,I
I
in (78) can be cast into the form
$$\begin{array}{@{}rcl@{}} H_{t,II}&=&\sum\limits_{{\mathbf{k}},\sigma} {{{\alpha}_{-{\mathbf{k}}}\alpha}^{\dagger} _{-{\mathbf{k}}}\left( v^{2}_{{\mathbf{k}}}E_{{\mathbf{k}}}+u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}\right)} \\ &=& \sum\limits_{{\mathbf{k}},\sigma} {{\alpha}^{\dagger}_{{\mathbf{k}}}{\alpha} _{{\mathbf{k}}}\left( v^{2}_{{\mathbf{k}}}E_{{\mathbf{k}}}+u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}\right)}+\underbrace{\sum\limits_{{\mathbf{k}},\sigma} {v^{2}_{{\mathbf{k}}}E_{{\mathbf{k}}}+u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}}}_{\text{constant}}\ \end{array} $$
(91)
where we have used the relation \({{\alpha }_{-{\mathbf {k}}}\alpha }^{\dagger } _{-{\mathbf {k}}}=1-{\alpha }^{\dagger }_{-{\mathbf {k}}}{\alpha }_{-{\mathbf {k}}}\) and replaced −k →k. The constant term in (91) is evaluated by summing over the spin degree of freedom,
$$\begin{array}{@{}rcl@{}} \text{constant} \!&=&\!\sum\limits_{{\mathbf{k}},\sigma} {\left( v^{2}_{{\mathbf{k}}}E_{{\mathbf{k}}}+u_{{\mathbf{k}}}v_{{\mathbf{k}}}B_{{\mathbf{k}}}\right)} \\ &=&\! \sum\limits_{{\mathbf{k}},\sigma}{\left( \frac{E_{{\mathbf{k}}}}{2}-\frac{1}{2}{\lambda} _{{\mathbf{k}}}\right)\,=\,\sum\limits_{{\mathbf{k}}}{E_{{\mathbf{k}}}}\,-\,\sum\limits_{{\mathbf{k}}}{{\lambda} _{{\mathbf{k}}}}}\equiv C.\end{array} $$
(92)
In arriving at (92), the following identities, which can be derived from (88) and (89), have been used, namely,
$$\begin{array}{@{}rcl@{}} u_{{\mathbf{k}}}v_{{\mathbf{k}}}&=&-\frac{1}{2}\frac{B_{{\mathbf{k}}}}{\sqrt{B^{2}_{{\mathbf{k}}}+E^{2}_{{\mathbf{k}}}}}, \\ v^{2}_{{\mathbf{k}}}&=&\frac{1}{2}-\frac{1}{2}\frac{E_{{\mathbf{k}}}}{\sqrt{B^{2}_{{\mathbf{k}}}+E^{2}_{{\mathbf{k}}}}}, \\ {u^{2}_{{\mathbf{k}}}-v}^{2}_{{\mathbf{k}}}&=&\frac{E_{{\mathbf{k}}}}{\sqrt{B^{2}_{{\mathbf{k}}}+E^{2}_{{\mathbf{k}}}}}.\end{array} $$
(93)
Be noted that the expression for the constant in (92), \(C={\sum }_{{\mathbf {k}}}{E_{{\mathbf {k}}}}-{\sum }_{{\mathbf {k}}}{{\lambda } _{{\mathbf {k}}}}\), is defined with summation over the variable k only. Upon diagonalization, H
t,I
I
I
, H
t,I
V
in H
t
in (78) vanish. Combining H
t,I
I
(from (91)) with H
t,I
in (78) gives (11),
$$\begin{array}{@{}rcl@{}} H_{t}=\sum\limits_{{\mathbf{k}},\sigma} {{\lambda}_{{\mathbf{k}}}{\alpha}^{\dagger} _{{\mathbf{k}}}{\alpha}_{{\mathbf{k}}}}+C.\end{array} $$
(94)
This completes the diagonalization of the trial Hamiltonian.
1.4 A.4 Derivation of \({{F}}_{{t}},\ {\left \langle {H}\right \rangle }_{{t}},{\left \langle {{H}}_{{t}}\right \rangle }_{{t}}\) as Appear in (15)
Derivation of
F
t
:
Partition function Z
t
for trial Hamiltonian H
t
is defined as
$$\begin{array}{@{}rcl@{}} Z_{t}&=&\text{Tr}\ e^{-\beta H_{t}}\mathrm{=}e^{-\beta C}\mathrm{\cdot} \text{Tr}\ {\mathrm{e}}^{-\beta \left( \sum\limits_{{\mathbf{k}},\sigma} {{\lambda}_{{\mathbf{k}}}{\alpha}^{\dagger} _{{\mathbf{k}}}{\alpha}_{{\mathbf{k}}}}\right)} \\ &=& e^{-\beta C}\mathrm{\cdot} \prod\limits_{{\mathbf{k}},\sigma} {\left[2{\mathrm{e}}^{-\frac{\beta {\lambda} _{{\mathbf{k}}}}{2}}\cdot \frac{1}{2}\left( {\mathrm{e}}^{\frac{\beta {\lambda} _{{\mathbf{k}}}}{2}}+{\mathrm{e}}^{-\frac{\beta {\lambda}_{{\mathbf{k}}}}{2}}\right)\right]} \\ &=& e^{-\beta C}\mathrm{\cdot} \prod\limits_{{\mathbf{k}},\sigma} {2{\mathrm{e}}^{-\frac{\beta {\lambda}_{{\mathbf{k}}}}{2}}{\cosh \left( \frac{\beta {\lambda} _{{\mathbf{k}}}}{2}\right)} },\end{array} $$
(95)
where the constant C is as defined in (92). The trial free energy is
$$\begin{array}{@{}rcl@{}} F_{t}&=&-k_{B}T{\ln Z_{t}} =C-k_{B}T{\ln \left\{\prod\limits_{{\mathbf{k}},\sigma} {\left[{\mathrm{e}}^{-\frac{\beta {\lambda}_{{\mathbf{k}}}}{2}}\cdot 2{\cosh \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)} \right]}\right\}} \\ &=&-k_{B}T\sum\limits_{{\mathbf{k}},\sigma} {\left( -\frac{\beta {\lambda} _{{\mathbf{k}}}}{2}\right)}-\frac{1}{\beta} \sum\limits_{{\mathbf{k}},\sigma} {{\ln \left[\mathrm{2cosh}\left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)\right]} }+C \\ &=&\frac{1}{2}\sum\limits_{{\mathbf{k}},\sigma} {{\lambda}_{{\mathbf{k}}}}-\frac{1}{\beta} \sum\limits_{{\mathbf{k}},\sigma} {{\ln \left[\mathrm{2cosh}\left( \frac{\beta {\lambda} _{{\mathbf{k}}}}{2}\right)\right]} }+\left( \sum\limits_{{\mathbf{k}}}{E_{{\mathbf{k}}}}-\sum\limits_{{\mathbf{k}}}{{\lambda} _{{\mathbf{k}}}}\right) \\ &=& \sum\limits_{{\mathbf{k}}}{{\lambda} _{{\mathbf{k}}}}-\frac{1}{\beta} \sum\limits_{{\mathbf{k}},\sigma} {{\ln \left[\mathrm{2cosh}\left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)\right]} }+\sum\limits_{{\mathbf{k}}}{E_{{\mathbf{k}}}}-\sum\limits_{{\mathbf{k}}}{{\lambda} _{{\mathbf{k}}}} \\ &=& -\frac{1}{\beta} \sum\limits_{{\mathbf{k}},\sigma} {{\ln \left[\mathrm{2cosh}\left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)\right]} }+\sum\limits_{{\mathbf{k}}}{{(\epsilon}_{{\mathbf{k}}}-\mu -A_{{\mathbf{k}}})}.\end{array} $$
(96)
Derivation of
\({\left \langle \boldsymbol {H}\right \rangle }_{\boldsymbol {t}}\):
The derivation \({\left \langle H\right \rangle }_{t}\) proceeds by beginning with the definition of the model Hamiltonian H, (9), which is separated into three parts,
$$\begin{array}{@{}rcl@{}} {\left\langle H\right\rangle}_{t}={\left\langle h_{1}\right\rangle}_{t}+{\left\langle h_{2}\right\rangle} _{t}+{\left\langle h_{3}\right\rangle}_{t},\end{array} $$
$$\begin{array}{@{}rcl@{}} h_{1}&=&\sum\limits_{{\mathbf{k}},\sigma} {\left( {\epsilon}_{\mathrm{k}}-\mu \right)c^{\dagger} _{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma} }, \\ h_{2}&=&-\frac{1}{2N}\sum\limits_{\sigma ,{\sigma} ^{\prime}}{\sum\limits_{{\mathbf{k}},{{\mathbf{k}}}^{\prime},{\mathbf{q}}}{\left( V_{{\mathbf{q}}}-{2U\delta} _{\sigma ,{-\sigma}^{\prime}}\right)c^{\dagger}_{{\mathbf{k}}+{\mathbf{q}},\sigma} c_{{\mathbf{k}},\sigma} }c^{\dagger} _{{{\mathbf{k}}}^{\prime}-{\mathbf{q}},{\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{\prime},{\sigma}^{\prime}}}, \\ h_{3}&=&-\frac{1}{\sqrt{N}}\sum\limits_{\sigma} {\sum\limits_{{\mathbf{k}},{\mathbf{q}}}{G_{{\mathbf{q}}}c^{\dagger} _{{\mathbf{k}}-{\mathbf{q}},\sigma} c_{{\mathbf{k}},\sigma} }}.\end{array} $$
(97)
In terms of the operators \({\alpha }_{{\mathbf {k}},\sigma } ,\ {\alpha }^{\dagger }_{{\mathbf {k}},\sigma } \), \({\left \langle h_{1}\right \rangle }_{t}\) reads
$$\begin{array}{@{}rcl@{}} {\left\langle h_{1}\right\rangle}_{t}\!&=&\!\!{\left\langle \sum\limits_{{\mathbf{k}},\sigma} {\left( {\epsilon}_{{\mathbf{k}}}\,-\,\mu \right)\left( u_{{\mathbf{k}}}{\alpha}^{\dagger} _{{\mathbf{k}},\sigma} +v_{{\mathbf{k}}}{\alpha}_{-{\mathbf{k}},-\sigma} \right)\left( u_{{\mathbf{k}}}{\alpha}_{{\mathbf{k}},\sigma} \,+\,v_{{\mathbf{k}}}{\alpha} ^{\dagger}_{-{\mathbf{k}},-\sigma} \right)}\right\rangle}_{t} \\ &=&\!\! {\left\langle \sum\limits_{{\mathbf{k}},\sigma}{\kern-.5pt} {\left( {\epsilon}_{{\mathbf{k}}}\,-\,\mu \right){}\left[u^{2}_{{\mathbf{k}}}{\alpha}^{\dagger}_{{\mathbf{k}},\sigma} {\alpha} _{{\mathbf{k}},\sigma} \!+v^{2}_{{\mathbf{k}}}{}\left( {}1\!-{\!\alpha}^{\dagger} _{{\mathbf{-}}{\mathbf{k}},-\sigma} {\alpha} _{{\mathbf{-}}{\mathbf{k}},-\sigma} {}\right){}\right]}\right\rangle}_{t}.\end{array} $$
(98)
The expectation values \({\left \langle {\alpha }^{\dagger }_{{\mathbf {k}},\sigma } {\alpha }^{\dagger } _{{\mathbf {-}}{\mathbf {k}},-\sigma } \right \rangle }_{t},\ {\left \langle {\alpha } _{{\mathbf {-}}{\mathbf {k}},-\sigma } {\alpha } _{{\mathbf {-}}{\mathbf {k}},-\sigma } \right \rangle }_{t}\) in (98) vanish in representation in which \({\alpha }^{\dagger }_{{\mathbf {k}},\sigma } {,\alpha } _{{\mathbf {k}},\sigma } \) are diagonal. There are two expectation values left to be evaluated in (98), i.e., \({\left \langle {\alpha }^{\dagger }_{{\mathbf {k}},\sigma } {\alpha } _{{\mathbf {k}},\sigma } \right \rangle }_{t}\) and \({\left \langle {\alpha }^{\dagger } _{{\mathbf {-}}{\mathbf {k}},-\sigma } {\alpha } _{{\mathbf {-}}{\mathbf {k}},-\sigma } \right \rangle }_{t}\). \({\left \langle {\alpha } _{{\mathbf {k}},\sigma } {\alpha }^{\dagger }_{{\mathbf {k}},\sigma } \right \rangle }_{t}\) is evaluated via
$$\begin{array}{@{}rcl@{}} {\left\langle {\alpha}^{\dagger}_{{\mathbf{k}},\sigma} {\alpha} _{{\mathbf{k}},\sigma} \right\rangle}_{t}&=&\frac{\text{Tr}\ \left( {\alpha}^{\dagger} _{{\mathbf{k}},\sigma} {\alpha}_{{\mathbf{k}},\sigma} \right)e^{-\beta H_{t}}}{\text{Tr}\ \left( e^{-\beta H_{t}}\right)} \\ &=&-\frac{\partial} {\partial \left( \beta {\lambda} _{{\mathbf{k}}}\right)}{\ln Z_{t}} =\frac{1}{2}-\frac{1}{2}{\tanh \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)} .\end{array} $$
(99)
In arriving at A 45, we have ignored the contribution from constant term C in H
t
of A 41.
The subscripts in \({\left \langle {\alpha }^{\dagger }_{{\mathbf {-}}{\mathbf {k}},-\sigma } {\alpha } _{{\mathbf {-}}{\mathbf {k}},-\sigma } \right \rangle }_{t}\) can be manipulated so that it can be expressed in terms of \({\left \langle {{\alpha }^{\dagger }_{{\mathbf {k}},\sigma } \alpha } _{{\mathbf {k}},\sigma } \right \rangle }_{t}\),
$$\begin{array}{@{}rcl@{}} &&-\sum\limits_{{\mathbf{k}},\sigma} {\left( {\epsilon}_{{\mathbf{k}}}-\mu \right)v^{2}_{{\mathbf{k}}}{\left\langle {\alpha}^{\dagger} _{{-}{\mathbf{k}}{,-}\sigma} {\alpha} _{{-}{\mathbf{k}}{,-}\sigma} \right\rangle} _{t}}\\&=&-\sum\limits_{{\mathbf{k}},\sigma} {\left( {\epsilon}_{{\mathbf{k}}}-\mu \right)v^{2}_{{\mathbf{k}},\sigma} {\left\langle {\alpha}^{\dagger}_{{\mathbf{k}},\sigma} {\alpha}_{{\mathbf{k}},\sigma} \right\rangle}_{t}},\end{array} $$
(100)
where we have made use of 𝜖
−k
= 𝜖
k
. Using (93), (99), and (100), the square bracket term in (98) can be simplified to
$$\begin{array}{@{}rcl@{}} \left[\cdots \right]\,=\,\left( u^{2}_{{\mathbf{k}}}\,-\,v^{2}_{{\mathbf{k}}}\right){\!\left\langle {\alpha}^{\dagger} _{{\mathbf{k}},\sigma} {\alpha}^{\dagger}_{{\mathbf{k}},\sigma} \right\rangle} _{t}\,+\,v^{2}_{{\mathbf{k}}}\,=\,\frac{1}{2}\,-\,\frac{E_{{\mathbf{k}}}}{2{\lambda} _{{\mathbf{k}}}} {\tanh \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)} .\end{array} $$
(101)
$$\begin{array}{@{}rcl@{}} \Rightarrow {\left\langle h_{1}\right\rangle}_{t}=\frac{1}{2}\sum\limits_{{\mathbf{k}},\sigma} {\left( {\epsilon}_{{\mathbf{k}}}-\mu \right)\left[1-\frac{E_{{\mathbf{k}}}}{{\lambda} _{{\mathbf{k}}}} {\tanh \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)} \right]}. \end{array} $$
(102)
To evaluate \({\left \langle h_{2}\right \rangle }_{t}\), we rearrange \({\left \langle c^{\dagger } _{{\mathbf {k}}+{\mathbf {q}},\sigma } c_{{\mathbf {k}},\sigma } c^{\dagger } _{{{\mathbf {k}}}^{\prime }-{\mathbf {q}}{,\sigma } ^{\prime }}c_{{{\mathbf {k}}}^{\prime },{\sigma }^{\prime }}\right \rangle }_{t}\) by reverting the position of the c
‡operator twice and using anti-commutative relations for c
k‡,c
k
to arrive at
$$\begin{array}{@{}rcl@{}} {\left\langle c^{\dagger}_{{\mathbf{k}}+{\mathbf{q}},\sigma} c_{{\mathbf{k}},\sigma} c^{\dagger} _{{{\mathbf{k}}}^{\prime}-{\mathbf{q}}{,\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{\prime},{\sigma}^{\prime}}\right\rangle}_{t}\!&=&{\!-\left\langle c^{\dagger}_{{\mathbf{k}}+{\mathbf{q}},\sigma} {c^{\dagger} _{{{\mathbf{k}}}^{\prime}-{\mathbf{q}}{,\sigma} ^{\prime}}c}_{{\mathbf{k}},\sigma} c_{{{\mathbf{k}}}^{\prime},{\sigma} ^{\prime}}\right\rangle}_{t} \\ &=&{\!\left\langle c^{\dagger} _{{\mathbf{k}}+{\mathbf{q}},\sigma} {c^{\dagger} _{{{\mathbf{k}}}^{\prime}-{\mathbf{q}}{,\sigma} ^{\prime}}c_{{{\mathbf{k}}}^{\prime},{\sigma}^{\prime}}c}_{{\mathbf{k}},\sigma} \right\rangle}_{t}.\end{array} $$
(103)
In view of (103), h
2 reads (after a few manipulative steps in the subscript symbols)
$$\begin{array}{@{}rcl@{}} h_{2}\,=\,-\frac{1}{2N}\!\!\sum\limits_{\sigma} {\!\sum\limits_{{\mathbf{k}},{\mathbf{q}}}{\!\left( \!V_{{\mathbf{q}}\boldsymbol{-}{\mathbf{k}}}\!\,-\,2U\right){\!\left\langle \!c^{\dagger}_{{\mathbf{q}},\sigma} {c^{\dagger} _{\mathrm{-}{\mathbf{q}}{\mathbf{,-}}\sigma} c_{\mathrm{-}{\mathbf{k}} ,-\sigma} c}_{{\mathbf{k}},\sigma} \!\right\rangle}_{\!t}}}.\\ \end{array} $$
(104)
It can be shown that \({\left \langle c^{\dagger }_{{\mathbf {q}},\sigma } {c^{\dagger } _{\mathrm {-}{\mathbf {q}}{\mathbf {,-}}\sigma } c_{\mathrm {-}{\mathbf {k}},-\sigma } c}_{{\mathbf {k}},\sigma } \right \rangle }_{t}={\left \langle c^{\dagger } _{{\mathbf {q}},\sigma } c^{\dagger }_{\mathrm {-}{\mathbf {q}}{\mathbf {,-}}\sigma } \right \rangle }_{t}{\left \langle {c_{\mathrm {-}{\mathbf {k}},-\sigma } c}_{{\mathbf {k}},\sigma } \right \rangle }_{t}\) provided q≠k, or equivalently, V
0 is excluded in the summation in (104). \({\left \langle c^{\dagger }_{{\mathbf {q}},\sigma } c^{\dagger } _{\mathrm {-}{\mathbf {q}}{\mathbf {,-}}\sigma } \right \rangle }_{t}\) is given by the following expression (after some algebra, and making use of (93), (99), (100), and (105)),
$$\begin{array}{@{}rcl@{}} {\left\langle c^{\dagger}_{{\mathbf{q}},\sigma} c^{\dagger} _{\mathrm{-}{\mathbf{q}}{\mathbf{,-}}\sigma} \right\rangle} _{t}\!&=&\!u_{{\mathbf{q}}}v_{{\mathbf{q}}}\left( 1\,-\,{\left\langle {{\alpha}^{\dagger} _{{\mathbf{-}}{\mathbf{q}},-\sigma} \alpha} _{{\mathbf{-}}{\mathbf{q}},-\sigma} \right\rangle}_{t}\,-\,{\left\langle {\alpha}^{\dagger} _{{\mathbf{q}},\sigma} {\alpha}_{{\mathbf{q}},\sigma} \!\right\rangle}_{\!t}\right) \\ &=&\!-\frac{1}{2}\frac{B_{{\mathbf{q}}}}{{\lambda}_{{\mathbf{q}}}}{\tanh \left( \frac{\beta {\lambda}_{{\mathbf{q}}}}{2}\right)} \end{array} $$
(105)
As for the derivation of (105), we can similarly obtain
$$\begin{array}{@{}rcl@{}} {\left\langle {c_{\mathrm{-}{\mathbf{k}},-\sigma} c}_{{\mathbf{k}},\sigma} \right\rangle}_{t}=-\frac{1}{2}\frac{B_{{\mathbf{k}}}}{{\lambda}_{{\mathbf{k}}}}{\tanh \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)} . \end{array} $$
(106)
Using (105) and (106), the expectation value of \({\left \langle h_{2}\right \rangle }_{t}\) is now obtained,
$$\begin{array}{@{}rcl@{}} {\left\langle h_{2}\right\rangle}_{t}\,=\,-\frac{1}{8N}\!\sum\limits_{\sigma ,{\mathbf{k}}}{\!\sum\limits_{{\mathbf{q}}}{\left( V_{{\mathbf{q}}\boldsymbol{-}\mathrm{k}}\,-\,2U\right)\!\frac{B_{{\mathbf{q}}}B_{{\mathbf{k}}}}{{\lambda}_{{\mathbf{q}}}{\lambda}_{{\mathbf{k}}}}{\mathrm{tanh\!} \left( \frac{\beta {\lambda}_{{\mathbf{q}}}}{2}\right)} {\mathrm{tanh\!} \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)} }}. \\ \end{array} $$
(107)
The G
q
term that enters \({\left \langle h_{3}\right \rangle }_{t}\) has d-wave symmetry and is physically significant pertaining to the pseudo-gap [1]. However, as \({\left \langle h_{3}\right \rangle }_{t}\) does not contribute to the process of minimization of F
v
with respect to B
k
for deriving superconducting gap equation, we leave it as it is without explicitly evaluating it. It is relabeled as \({\left \langle G\ \text {term}\right \rangle }_{t}\),
$$\begin{array}{@{}rcl@{}} {\left\langle h_{3}\right\rangle}_{t}\equiv {\left\langle G\ \text{term}\right\rangle} _{t}. \end{array} $$
(108)
Putting the final expression of \({\left \langle h_{1}\right \rangle }_{t},{\left \langle h_{2}\right \rangle }_{t}\), and \({\left \langle h_{3}\right \rangle }_{t}\) together,
$$\begin{array}{@{}rcl@{}} {\left\langle H\right\rangle}_{t}\!&=&\!\!\sum\limits_{{\mathbf{k}},\sigma} {\left( {\epsilon} _{{\mathbf{k}}}-\mu \right){\left\langle c^{\dagger}_{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma} \right\rangle}_{t}} \\ && -\frac{1}{8N}\sum\limits_{\sigma ,{\mathbf{k}}}{\sum\limits_{{\mathbf{q}}}{\left( V_{{\mathbf{q}}-\mathrm{k}}\,-\,2U\right)\!\frac{B_{{\mathbf{q}}}B_{{\mathbf{k}}}}{{\lambda} _{{\mathbf{q}}}{\lambda}_{{\mathbf{k}}}}{\mathrm{tanh\!} \left( \frac{\beta {\lambda} _{{\mathbf{q}}}}{2}\right)} {\mathrm{tanh\!} \left( \frac{\beta {\lambda} _{{\mathbf{k}}}}{2}\right)} }} \\ && +{\left\langle G\ \text{term}\right\rangle} _{t}\end{array} $$
(109)
Derivation of
\({\left \langle {\boldsymbol {H}}_{\boldsymbol {t}}\right \rangle }_{\boldsymbol {t}}\):
Referring to (64),
$$\begin{array}{@{}rcl@{}} {\left\langle H_{t}\right\rangle}_{t}\!\!&=&\!\sum\limits_{{\mathbf{k}},\sigma} {\left( {\epsilon} _{{\mathbf{k}}}-\mu -A_{{\mathbf{k}}}\right){\left\langle c^{\dagger} _{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma} \right\rangle}_{t}} \\ && \!+ \frac{1}{2}\sum\limits_{{\mathbf{k}},\sigma} \!\left( B_{{\mathbf{k}}}{\left\langle \!c^{\dagger} _{{\mathbf{k}},\sigma} c^{\dagger}_{-{\mathbf{k}},-\sigma} \!\right\rangle}_{t} \,+\,B^{*}_{{\mathbf{k}}}{\left\langle c_{-{\mathbf{k}},-\sigma} c_{{\mathbf{k}},\sigma} \right\rangle}_{t}\right) \!.\end{array} $$
(110)
Using (72), (105), and (106), we have
$$\begin{array}{@{}rcl@{}} {\left\langle H_{t}\right\rangle}_{t}&=&\sum\limits_{{\mathbf{k}},\sigma} {\left( {\epsilon} _{{\mathbf{k}}}-\mu -A_{{\mathbf{k}}}\right){\left\langle c^{\dagger} _{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma} \right\rangle} _{t}}\\&&-\frac{1}{2}\sum\limits_{{\mathbf{k}},\sigma} {\frac{B^{2}_{{\mathbf{k}}}}{{\lambda} _{{\mathbf{k}}}}{\tanh \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right).} }\end{array} $$
(111)
This completes the derivation of \({\left \langle H_{t}\right \rangle }_{t}.\)
Derivation of
F
v
:
Finally, putting everything together, i.e., (96) (for F
t
), (109) (for \({\left \langle H\right \rangle }_{t}\)) (111) (for \({\left \langle H_{t}\right \rangle }_{t}\)), the variational free energy is written as
$$\begin{array}{@{}rcl@{}} F_{v}\!\!&=&F_{t}+{\left\langle H\right\rangle}_{t}-{\left\langle H_{t}\right\rangle}_{t}+\mu N_{e} \\ &=& \underbrace{-\frac{1}{\beta} \sum\limits_{{\mathbf{k}},\sigma} {{\ln \left[\mathrm{2cosh}\left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)\right]} }}_{F_{v,I}} \\ && \underbrace{-\frac{1}{8N}\sum\limits_{\sigma ,{\mathbf{k}}}{\sum\limits_{{\mathbf{q}}}{\left( V_{{\mathbf{q}}\boldsymbol{-}\mathrm{k}}-2U\right)\frac{B_{{\mathbf{q}}}B_{{\mathbf{k}}}}{{\lambda} _{{\mathbf{q}}}{\lambda}_{{\mathbf{k}}}}{\tanh \left( \frac{\beta {\lambda} _{{\mathbf{q}}}}{2}\right)} {\tanh \left( \frac{\beta {\lambda} _{{\mathbf{k}}}}{2}\right)} }}}_{F_{v,II}} \\ &&+\underbrace{\frac{1}{2}\sum\limits_{{\mathbf{k}},\sigma} {\frac{B^{2}_{{\mathbf{k}}}}{{\lambda} _{{\mathbf{k}}}}{\tanh \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)} }}_{F_{v,III}} \\ && {+\underbrace{{\left\langle G\ \text{term}\right\rangle} _{t}\,+\,\sum\limits_{{\mathbf{k}}}{{\epsilon}_{{\mathbf{k}}}}+\mu N\left( 1-n\right)\,+\,\sum\limits_{{\mathbf{k}},\sigma} {A_{{\mathbf{k\!}}}\left( {\left\langle c^{\dagger} _{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma} \!\right\rangle} _{t}\,-\,\frac{1}{2}\right)}}}_{\mathrm{independent~of}~{\mathrm{B}}_{{\mathbf{k}}}}. \\ \end{array} $$
(112)
Note that the last three terms in (112) are independent of B
k
. Also, we have used the relation N
e
= N
n, where n denotes concentration of charge carries, defined as
$$\begin{array}{@{}rcl@{}} n\equiv \frac{1}{N}\sum\limits_{{\mathbf{k}},\sigma} {n_{{\mathbf{k}},\sigma} }=\frac{1}{N}\sum\limits_{{\mathbf{k}},\sigma} {{\left\langle c^{\dagger}_{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma} \right\rangle}_{t}}.\end{array} $$
(113)
From (102), we can identify \(n_{{\mathbf {k}},\sigma } \equiv {\left \langle c^{\dagger }_{{\mathbf {k}},\sigma } c_{{\mathbf {k}},\sigma } \right \rangle }_{t}\) in terms of the variables E
k
and λ
k
, namely,
$$\begin{array}{@{}rcl@{}} n_{{\mathbf{k}},\sigma} \mathrm{=}\frac{1}{2}\left[1-\frac{E_{{\mathbf{k}}}}{{\lambda}_{{\mathbf{k}}}} {\tanh \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)\ }\right].\end{array} $$
(114)
In view of (113) and (114), we have
$$\begin{array}{@{}rcl@{}} nN=\sum\limits_{{\mathbf{k}},\sigma} {{\left\langle c^{\dagger}_{{\mathbf{k}},\sigma} c_{{\mathbf{k}},\sigma} \right\rangle} _{t}}=N-\sum\limits_{{\mathbf{k}}}{\frac{E_{{\mathbf{k}}}}{{\lambda}_{{\mathbf{k}}}} {\tanh \left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)}}.\end{array} $$
(115)
Note that in the RHS of (115), the spin degree of freedom has already been summed over.
1.5 A.5 Minimization of F
v
with Respect to B
k
to Derive the Recurrent Gap Equation
F
v
in (112) can be minimized term by term. Differentiating the first term in F
v
by making use of the relation
$$\begin{array}{@{}rcl@{}} \frac{\partial {\lambda}_{{\mathbf{k}}}}{\partial B_{{\mathbf{k}}}}=\frac{B_{{\mathbf{k}}}}{{\lambda}_{{\mathbf{k}}}},\end{array} $$
(116)
we obtain
$$\begin{array}{@{}rcl@{}} \frac{\partial F_{v,I}}{\partial B_{{\mathbf{k}}}}=-\frac{1}{2}\sum\limits_{{\mathbf{k}},\sigma} {B_{{\mathbf{k}}}T_{{\mathbf{k}}}},\end{array} $$
(117)
where
$$\begin{array}{@{}rcl@{}} T_{{\mathbf{k}}}\equiv \frac{\tanh\left( \frac{\beta {\lambda} _{{\mathbf{k}}}}{2}\right)}{{\lambda}_{{\mathbf{k}}}}.\end{array} $$
(118)
Differentiating the second term in F
v
results in
$$\begin{array}{@{}rcl@{}} \frac{\partial F_{v,II}}{\partial B_{{\mathbf{k}}}}\,=\,-\frac{1}{8N}\!\sum\limits_{{\mathbf{k}},\sigma} {\!\sum\limits_{{\mathbf{q}}}{\left( V_{{\mathbf{q}}-{\mathbf{k}}}\,-\,2U\right)\!B_{{\mathbf{q}}}{\mathrm{T}}_{{\mathbf{q}}}\!\left[{\mathrm{\!T}}_{{\mathbf{k}}}\,+\,B_{{\mathbf{k}}}\frac{\partial {\mathrm{T}}_{{\mathbf{k}}}}{\partial B_{{\mathbf{k}}}}\right]}}.\end{array} $$
(119)
Differentiating the third term in F
v
results in
$$\begin{array}{@{}rcl@{}} \frac{\partial F_{v,III}}{\partial B_{{\mathbf{k}}}}=\sum\limits_{{\mathbf{k}},\sigma} {B_{{\mathbf{k}}}\left( {\mathrm{T}}_{{\mathbf{k}}}+\frac{1}{2}B_{{\mathbf{k}}}\frac{\partial {\mathrm{T}}_{{\mathbf{k}}}}{\partial B_{{\mathbf{k}}}}\right)}. \end{array} $$
(120)
Putting (117), (118), (119), and (120) together into (112),
$$\begin{array}{@{}rcl@{}} \frac{\partial F_{v}}{\partial B_{{\mathbf{k}}}}&=&\left[\sum\limits_{{\mathbf{k}},\sigma} {\frac{1}{2}B_{{\mathbf{k}}}}-\frac{1}{8N}\sum\limits_{{\mathbf{k}},\sigma} {\sum\limits_{{\mathbf{q}}}{\left( V_{{\mathbf{q}}{\mathbf{-}}{\mathbf{k}}}-2U\right)B_{{\mathbf{q}}}{\mathrm{T}}_{{\mathbf{q}}}}}\right]\\&&\times\left( {\mathrm{T}}_{{\mathbf{k}}}+B_{{\mathbf{k}}}\frac{\partial {\mathrm{T}}_{{\mathbf{k}}}}{\partial B_{{\mathbf{k}}}}\right)=0 \\ \Rightarrow B_{{\mathbf{k}}}&=&\frac{1}{4N}\sum\limits_{{\mathbf{q}}}{\left( V_{{\mathbf{k}}{\mathbf{-}}{\mathbf{q}}}-2U\right)B_{{\mathbf{q}}}\frac{\tanh\left( \frac{\beta {\lambda}_{{\mathbf{k}}}}{2}\right)}{{\lambda}_{{\mathbf{k}}}}.}\end{array} $$
(121)
1.6 A.6 Derivation of (53) and (54)
In linear algebra, a homogeneous system of (47) has infinitely many non-trivial solutions if the determinant vanishes, or equivalently, the matrix M in (47) is a rank 1 matrix. The non-trivial solutions \(\left (\begin {array}{c} {{\Delta } }_{0} \\ {{\Delta } }_{\eta } \end {array} \right )\) has only one linearly independent arbitrary variable, which we will choose as Δ0. Δ
η
is parametrized in terms of Δ0 via \(r_{\eta } =\frac {{{\Delta } }_{\eta } }{{{\Delta } }_{0}}\). If the root for \(D_{{\mathbf {k}}}\left (\beta \right )=\left |M\right |=T_{1}T_{2{\mathbf {k}}}+T_{3}T_{4{\mathbf {k}}}\ =0\) [i.e., (49)] exists, the matrix M is reduced into row-reduced echelon form (which is always possible because it is a rank 1 matrix with vanishing determinant),
$$\begin{array}{@{}rcl@{}}M\sim\left( \begin{array}{cc} 1 & \frac{T_{3}}{T_{1}} \\ 0 & 0 \end{array} \right). \end{array} $$
(122)
Thus,
$$\begin{array}{@{}rcl@{}}\left( \! \begin{array}{cc} 1 & \frac{T_{3}}{T_{1}} \\ 0 & 0 \end{array} \!\right)\!\left( \! \begin{array}{c} {{\Delta} }_{0} \\ {{\Delta} }_{\eta} \end{array} \!\right)\,=\,\left( \!\begin{array}{c} 0 \\ 0 \end{array} \!\right)\!\Rightarrow\! r_{\eta} \,=\,\frac{{{\Delta}}_{\eta} }{{{\Delta} }_{0}}\,=\,-\frac{T_{1}}{T_{3}}\,=\,-\frac{1+2U{\sum}_{{\mathbf{q}}}{F_{{\mathbf{q}}}}}{2U{\sum}_{{\mathbf{q}}}{{\eta} _{{\mathbf{q}}}F_{{\mathbf{q}}}}}.\end{array} $$
(123)
The numerical value of the gap Δ
k
at k-point k is given as per
$$\begin{array}{@{}rcl@{}}{{\Delta} }_{{\mathbf{k}}}={{\Delta} }_{0}+{{\Delta}}_{\eta} {\eta}_{{\mathbf{k}}}={{\Delta} }_{0}\left( 1+r_{\eta} {\eta}_{{\mathbf{k}}}\right).\end{array} $$
(124)
This completes the derivation of (53) and (54).
