Model of antiferromagnetic superconductivity

Abstract

We present a simple model that supports coexistent superconductive and antiferromagnetic ordering. The model consists of a system of electrons on a simple cubic lattice that move by tunnel effect and interact via antiferromagnetic Ising spin couplings and short-range repulsions: these include infinitely strong Hubbard forces that prevent double occupancy of any lattice site. Hence, under the filling condition of one electron per site and at sufficiently low temperature, the system is an antiferromagnetic Mott insulator. However, when holes are created by suitable doping, they are mobile charge carriers. We show that, at low concentration, their interactions induced by the above interelectronic ones lead to Schafroth pairing. Hence, under certain plausible but unproven assumptions, the model exhibits the off-diagonal long-range order that characterises superconductivity, while retaining the antiferromagnetic ordering.

Appendix: Proof of Prop. 3.1.

As a preliminary to the proof of Prop. 3.1, we define

$$\begin{aligned} {\alpha }_{-}(x):=a(x,-s(x)) \quad {\forall } \quad x{\in }X \end{aligned}$$

(8.1)

and establish the following lemma.

Lemma 8.1

The above definitions and assumptions imply that

$$\begin{aligned} {\alpha }_{-}(x){\Phi }=0 \quad {\forall } \quad x{\in }X \end{aligned}$$

(8.2)

Proof

By Eqs. (2.9), (3.1) and (8.1),

$$\begin{aligned} {\alpha }_{-}(x){\alpha }^{\star }(x)+{\alpha }^{\star }(x){\alpha }_{-}(x) ={\alpha }^{\star }(x){\alpha }_{-}(x) \end{aligned}$$

and hence

$$\begin{aligned} {\alpha }_{-}(x){\alpha }^{\star }(x)=0. \quad {\forall } \quad x{\in }X. \end{aligned}$$

(8.3)

Furthermore, by Eq. (2.23) and (3.1),

$$\begin{aligned} {\Phi }=\left[ {\Pi }_{x^{\prime }{\in }X}{\alpha }^{\star }(x^{\prime })\right] {\Omega } \end{aligned}$$

and, in view of the anticommutation relation (2.8) and Eq. (3.1), we may move the factor ${\alpha }^{\star }(x)$ of the r.h.s. of this formula to the left of all the other factors, where, in view of Eq. (8.3), it is annihilated by the action of ${\alpha }_{-}(x)$. Eq. (8.2) follows immediately from this observation. $\square$

Proof of Prop. 3.1

By Eqs. (2.24) and (3.2),

$$\begin{aligned} HC_{n}(f)=[H,D_{n}(f)]{\Phi }. \end{aligned}$$

(8.4)

where

$$\begin{aligned} D_{n}(f):=(n!)^{-1/2}{\sum }_{x_{1},\ldots ,x_{n}{\in }X}f(x_{1},\ldots ,x_{n}){\alpha }(x_{1}).\, .{\alpha }(x_{n}). \end{aligned}$$

(8.5)

Hence, defining

$$\begin{aligned}&H^{(n)T}:=-T \sum \nolimits _{j=1}^{n}{\Delta }_{\mathcal{V},j},\end{aligned}$$

(8.6)

$$\begin{aligned}&H^{(n)J}:=-J \sum \nolimits _{j,k(>j)=1}^{n}{\sum }_{u{\in }\mathcal{U}} \bigl ({\delta }(x_{j},x_{k}+u)+{\delta }(x_{j},x_{k}-u)\bigr ), \end{aligned}$$

(8.7)

and

$$\begin{aligned} H^{(n)K}:=K \sum \nolimits _{j.k(>j)=1}^{n} {\sum }_{u,u^{\prime }({\ne }-u){\in }\mathcal{U}} \bigl ({\delta }(x_{j},x_{k}+u+u^{\prime })+ {\delta }(x_{j},x_{k}-u-u^{\prime })\bigr ), \end{aligned}$$

(8.8)

we see from Eqs. (2.12), (3.5), (3.6) and (8.4)–(8.8) that in order to establish the formula (3.4), it suffices to prove that

$$\begin{aligned}{}[H^{T},D_{n}(f)]{\Phi }=D_{n}(H^{(n)T}f){\Phi }, \end{aligned}$$

(8.9)

$$\begin{aligned}{}[H^{J},D_{n}(f)]{\Phi }=D_{n}(H^{(n)J}f){\Phi }, \end{aligned}$$

(8.10)

and

$$\begin{aligned}{}[H^{K},D_{n}(f)]{\Phi }=D_{n}(H^{(n)K}f){\Phi }. \end{aligned}$$

(8.11)

Proof of Eq. (8.9)

By Eq. (8.5),

$$\begin{aligned}{}[H^{T},D_{n}(f)]{\Phi }= (n!)^{-1/2}{\sum }_{x_{1},\ldots ,x_{n}{\in }X}{\sum }_{j=1}^{n} f(x_{1},\ldots ,x_{n}){\alpha }(x_{1}).. \, {\alpha }(x_{j-1}) [H^{T},{\alpha }(x_{j})]_{-}{\alpha }(x_{j+1})..{\alpha }(x_{n}){\Phi }\nonumber \\ \end{aligned}$$

(8.12)

and, by Eqs. (2.13) and (3.1),

$$\begin{aligned}{}[H^{T},{\alpha }(x_{j})]_{-}&= -T{\sum }_{x{\in }X,{\lambda }={\pm }1,v{\in }\mathcal{V}} \left[ a^{\star }(x,{\lambda })a(x+v,{\lambda })+a^{\star }(x+v,{\lambda }) a(x,{\lambda }),a(x_{j},s(x_{j})\right] _{-} \end{aligned}$$

In view of the anticommutation relations (2.8) and (2.9), together with Eqs. (3.1) and (8.1), this formula reduces, after some manipulation, to the following equation:

$$\begin{aligned}{}[H^{T},{\alpha }(x_{j})]_{-}= & {} -T{\sum }_{v{\in }\mathcal{V}}\left[ \left( {\alpha }(x_{j}+v)+ {\alpha }(x_{j}-v)\right) \left( I-{\alpha }_{-}^{\star }(x_{j}){\alpha }_{-}(x_{j})\right) \right. \nonumber \\&+ \left. {\alpha }_{-}^{\star }(x_{j}){\alpha }(x_{j})\left( {\alpha }_{-}(x_{j}+v)+ {\alpha }_{-}(x_{j}-v)\right) \right] . \end{aligned}$$

(8.13)

Furthermore, by Eqs. (2.8), (3.1) and (8.1), the terms ${\alpha }_{-}(x_{j})$ and $({\alpha }_{-}(x_{j}+v)+{\alpha }_{-}(x_{j}-v))$ commute or anticommute with ${\alpha }(x_{j+1}).\ldots {\alpha }(x_{n})$, according to whether $(n-j)$ is odd or even; and, by Lemma (8.1), they annihilate ${\Phi }$. Consequently, by Eq. (8.13), the commutator $[H^{T},{\alpha }(x_{j})]_{-}$ may be replaced by $-T{\sum }_{v{\in }\mathcal{V}}\bigl ({\alpha }(x_{j}+v)+{\alpha }(x_{j}-v)\bigr )$ in the formula (8.12). On transferring the action of this operator to the function f, we arrive at Eq. (8.9), with $H^{(n)T}$ given by Eq. (8.6), up to a harmless additive constant.

Proof of Eq. (8.10)

By Eqs. (2.14) and (8.5),

$$\begin{aligned}{}[H^{J},D_{n}(f)]_{-}=({n!})^{-1/2}{\sum }_{j=1}^{n} f(x_{1},\ldots ,x_{n}){\alpha }(x_{1})\ldots {\alpha }(x_{j-1}) [H^{J},{\alpha }(x_{j})]_{-}{\alpha }(x_{j+1})\ldots {\alpha }(x_{n}) \end{aligned}$$

(8.14)

and, by Eqs. (2.7)–(2.9), (2.14) and (3.1),

$$\begin{aligned}{}[H^{J},{\alpha }(x_{j})]_{-}= & {} J{\sum }_{x{\in }X,u{\in } \mathcal{U}}\left( {\sigma }(x)[{\sigma }(x+u),{\alpha }(x_{j})]_{-}+ [{\sigma }(x),{\alpha }(x_{j})]_{-}{\sigma }(x+u)\right) \\= & {} Js(x_{j}){\sum }_{x{\in }X,u{\in } \mathcal{U}}\left( {\delta }(x+u,x_{j}){\sigma }(x) {\alpha }(x_{j})+{\delta }(x,x_{j}){\alpha }(x_{j}){\sigma }(x+u)\right) \\= & {} Js(x_{j}){\sum }_{u{\in }\mathcal{U}}\left( {\sigma }(x_{j}-u){\alpha }(x_{j})+ {\alpha }(x_{j}){\sigma }(x_{j}+u)\right) . \end{aligned}$$

Hence since, by Eqs. (2.5) , (2.7)–(2.9) and (3.1), ${\alpha }(x_{j})$ commutes with ${\sigma }(x_{j}-u)$,

$$\begin{aligned}{}[H^{J},{\alpha }(x_{j})]_{-}=Js(x_{j}) {\sum }_{u{\in }\mathcal{U}}{\alpha }(x_{j})\left( {\sigma }(x-u)+{\sigma }(x+u)\right) . \end{aligned}$$

(8.15)

Further, since

$$\begin{aligned}{}[{\sigma }(x_{j}{\pm }u),{\alpha }(x_{j+1}).\ldots {\alpha }(x_{n})]_{-}{\Phi }= {\sum }_{k=j+1}^{n}{\alpha }(x_{j+1})\ldots ,{\alpha }(x_{k}) [{\sigma }(x_{j}{\pm }u),{\alpha }(x_{k})]_{-}{\alpha }(x_{k+1})\ldots {\alpha }(x_{n}){\Phi } \end{aligned}$$

(8.16)

and since, by Eqs. (2.5), (2.7)–(2.9) and (3.1),

$$\begin{aligned}{}[{\sigma }(x_{j}{\pm }u),{\alpha }(x_{k})]_{-}= {\delta }(x_{j}{\pm }u,x_{k})s(x_{j}{\pm }u){\alpha }(x_{k}), \end{aligned}$$

(8.17)

it follows from Eqs. (2.21), (8.16) and (8.17) that

$$\begin{aligned}&{\sigma }(x_{j}{\pm }u){\alpha }(x_{j+1})\ldots {\alpha }(x_{n}){\Phi }= s(x_{j}{\pm }u){\alpha }(x_{j+1})\ldots {\alpha }(x_{n}){\Phi }\nonumber \\&\quad - {\sum }_{k=j+1}^{n}{\delta }(x_{j}{\pm }u,x_{k})s(x_{j}{\pm }u) {\alpha }(x_{j+1})\ldots {\alpha }(x_{n}){\Phi }. \end{aligned}$$

(8.18)

Since, for $u{\in }\mathcal{U}$, the sites $x_{j}$ and $(x_{j}+u)$ are nearest neighbours and therefore carry opposite spins in the antiferromagnetic configuration,

$$\begin{aligned} s(x_{j})s(x_{j}-u)=-1. \end{aligned}$$

(8.19)

Hence, by Eqs. (8.15), (8.18) and (8.19),

$$\begin{aligned}&[H^{J},{\alpha }(x_{j})]_{-}{\alpha }(x_{j+1})\ldots {\alpha }(x_{n}){\Phi }= 4dj{\alpha }(x_{j})\ldots {\alpha }(x_{n}){\Phi }-\\&\quad J {\sum }_{u{\in }\mathcal{U}}{\sum }_{k=j+1}^{n}\left( {\delta }(x_{j},x_{k}+u)+ {\delta }(x_{j},x_{k}-u)\right) {\alpha }(x_{j})\ldots {\alpha }(x_{n}){\Phi }. \end{aligned}$$

It follows from this equation and Eq. (8.14) that Eq. (8.10) is satisfied, with $H^{(n)J}$ given by Eq. (8.4), up to an irrelevant additive constant 2J.

Proof of Eq. (8.11)

Noting that, by Eqs. (2.5), (2.6) and (3.1),

$$\begin{aligned}{}[{\nu }(x),{\alpha }(x^{\prime })]_{-}=-{\alpha }(x){\delta }(x,x^{\prime }), \end{aligned}$$

(8.20)

we see that it is a simple matter to prove (8.11) by replacing $s(x), \, {\sigma }(x), \, u$ and J by ${\nu }, \, 1, \, u+u^{\prime }$ and K, respectively, in the proof of (8.10). This completes the proof of the proposition.