Superconductivity: An OffDiagonal Long Range Order
Shigeyuki AONO
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1 Introduction
In the preface of the big book "Superconductivity" edited by Parks [1], one of the editors has written, "During the preparation of this treatise one of the authors commented that it would be "the last nail in the coffin [of superconductivity]." Further we have found that Anderson was pessimistic about the further advance of superconductivity. For example, he thought that high temperature or room temperature superconductivity is very unlikely. However, against his expectation, the discovery of high temperature superconductivity by Bednorz and Muller [2] is astonishing. The traditional BCS (BardeenCooperSchrieffer)[3] theory has failed to explain the mechanism of this hightemperature superconductivity. Anderson[4] proposed a new idea, called the resonating valence bond theory (RVB), or the t  J model. The term t refers to the transfer integral and J to the electron correlation. We have never heard of any success of his theory. As for the high temperature superconductors or the room temperature superconductors a recent publication[5] is worth reding.
In addition to the current attempts at investigating the mechanism of superconductivity we want to present our idea, which is fundamentally new, to introduce purely electronical electronelectron attractions.
2 Fundamental Remarks
First we discuss briefly the Noether theorem and the Goldston boson. Consider the global transformation or the Abelian gauge transformation for the electron field, f_{a}(x), where a is the spin index,
Under this transformation, we postulate that the Lagrangian for the system in question is invariant. The 4current densities are kept unchanged: (hereafter the repeated indices imply summation, an Einstein convention)
then we have the continuity relation,
If the system is static, the density must be conserved. _{t}j^{0} = 0. Put
it is found that if
then the expectation value over the ground state does not vanish,
That is to say, if the ground state satisfies relation (2.5), we can expect the appearance of the boson whose mass is zero. This boson is called the Goldstone boson, and in this system symmetry breaking takes place. This is what the Goldstone theorem insists. Details are available in the standard book of field theory[7].
3 Nambu Spinors
The algebra representing electrons is spinor. The Dirac relativistic (special relativity) function well describes this property. However the relativistic character seems not so important for the present problem. The spin here is the internal freedom of the particle and is not related to relativity. We now concentrate on the spinor character of electrons. The field operator has two components in the spin space.
3. 1 Nambu Spinors
We now introduce Nambu spinors[6]:
It should be noted that the charge density in terms of the Nambu spinor is not f^{+}(x)f(x), but
To avoid this surplus one might define
so that
The field operators satisfy the anticommutators,
where (a, b) = (↑, ↓), and x = (r, t). Then for spinors (3.1) the matrix commutator holds:
In the present case, the invariant quantities of the charge density and others are restored by the nonAbelian gauge transformation. That is to say,
since
This is the rotation in the spin space around the s^{3} axis with the angle L. It should be noted that, in the Nambu representation, the electron field behaves under the nonAbelian gauge transformation.
Let us investigate the Goldstone theory in this case. It is to seek for a operator which does not remove the commutator with the invariant charge f^{+}s_{3}f,
We can easily find
The space integral of A(x) is the Goldstone integral, and the scalar operator f^{+}(x)s^{3}f(x) is regarded as the boson operator. We then expect that f^{+}(x)s^{±}f(x) causes a new phase, followed by the spontaneous symmetry breaking. This will be seen later as the superconducting phase. The socalled gap function D which is proportional to this ground state average, is
where g is the electronelectron coupling parameter.
3. 2 Pauli Spin
From the conventional set of operators,
we employ other combinations:
Then one can observe the useful relations:
The ordinary commutators are:
The anticommutators are:
4 Propagators
In the previous section, as seen in eq.(3.9), we have found that the superconducting state is strongly connected with the gap function, or the anomalous Green's function. We want to deal with solid state substances. However, the infinite crystals described by the single band are already fully investigated, and the recent investigations are carried out on objects with multiband structures.[10, 11] The infinite system with multibands is constructed from the unit cell which is really a chemical molecule. The atoms in this molecule give the band index of the real crystal. In this respect we investigate the Green's function of the unit cell. The Green's function is now presented in the site representation. In quantum chemistry, the basis sets are the atomic orbitals, and then the molecular orbitals are built by the linear combinations of atomic orbitals. The representation that refers to the former will be called the level representation from now on.
Corresponding to the spinors (3.1) (and (3.3)), we define the spinor in the site representation
The commutator is
The matrix propagator is defined as
where t is the imaginary time so that the propagator is the temperature Green's function or the Matubara function. In the following, we put
and the system depends on (t_{1} t_{2}). The main problem in this article is to investigate the gap function responsible for superconductivity,
Various terms in the Hamiltonian can be written by using the charge density matrix:
where, for example,
4. 1 Hamiltonian
The electronic Hamiltonian can be written in the spinor notation[12]. The Hamiltonian consists of two parts. The noninteracting one implies that the Hamiltonian is bilinear which is diagonalized in principle.
where
Since we now assume that we are dealing with the pure electronic procedure so that the quantum mechanical Hamiltonian has nothing to do with spin operator, or the matrix {h_{rs}} is spin diagonal:
Then s^{μ} in eq.(4.9) is restricted to s^{↑} and s^{↓}.
Other twoparticle Hamiltonians are given in a similar way. However, in the previous case, the type of the Hamiltonian specifies the type of the Pauli spin, while in the present case, as will be seen, the type of the operator specifies the element of the density matrix.
where
The way to build Hamiltonians above is noted for the direct interaction. This quantum mechanical one is
The corresponding field theoretical one is written as: First the above one is put into the matrix elements and the wave functions are replaced by the operators a^{+}_{r↑}, a_{r↑} and so on. Then these are written in the spinor notation, for example:
Note that H^{ex} is obtained by reversing indices as (u « t) in v_{rs;tu}. This arises when the Wick's theorem is applied in ordering operators.
4. 2 Noninteracting
'Noninteracting' implies that the Hamiltonian is bilinear with respect to the operators so that diagonalization is possible. It should be instructive to begin with the singleparticle case, since even if we manipulate the complicated twoparticle case, the procedures are almost the same when the mean field approximation is employed.
The energy for the Hamiltonian (4.8) is
where
with the normalization factor W. We now define the temperature Green's function, in which t is the imaginary time, t = it [8, 9],
where
and it is assumed that the system is only dependent of the relative time t . Then we can written E^{0} in terms of the temperature Green's function,
The equation of motion for the noninteracting Hamiltonian (4.7) can be read as
The equation of motion can be solved by the Fourier transform[8]:
with
since we deal with fermions. Then (4.6) becomes
In this step, the matrix structure of the above is carefully investigated. Let us assume that the singleparticle Hamiltonian is spindiagonal,
It is preferable to introduce the flame diagonalization in each element,
Then we have
where
Another sophisticated way starts from the decomposed Hamiltonian:
The commutator is evaluated as
The equation of motion becomes
In the matrix notation, we have
or
Note that in these representations h is diagonal as
eq.(4.27) becomes
The first term gives
and the second term gives
Finally
Carrying out the w_{n} sum gives
This relation holds for both ↑ and ↓, which is consistent with result (4.22).
By the way, we can give the energy expression for eq.(4.7).
It may be needless to present another illustration:
The result is meaningful if s^{μ}s^{ν} = s^{0}, which leads to, in the present case,
The above manipulations will be used in the later investigation.
5 Interacting
In this section, the electronelectron interactions are considered, and we will discuss how these interactions lead to the superconducting state. The Hamiltonians are given in eq.(4.11). Up to now, the superconductivity has been dealt with the mean field approximation. In this approximation, the Hamiltonians are written as
where
Here < ... > implies the ground state average, which is actually obtained by the wave functions in the previous step during the SCF calculations.
These Hamiltonians are classified into two kinds called modes, one for the normal many electron problem and the other for the superconductivity:
where
The main difference between the above two is that in the former we have the single particle part, while in the latter we do not. Various D_{rs}'s are complicated, but merely in the cnumbers at this stage. The propagator in question of eq.(4.13) is presented here again:
The equation of motion can be read as
Making the Fourier transform on t = t_{1}  t_{2} gives
or in the matrix form
Note that D^{a} consists of the single electron part and the two electron interaction, which involves another mate r^{a}_{tu} combined with the propagator s^{a} < a_{t}a_{u} >:
However, the mean field approximation makes this look like the single electron interaction. A few comments are required about the matrix character of G. This is a big matrix with site indices, and each element is the (2, 2)matrix in the spin space. The index a characterizes the mode of the mean field potential. Each mode is independent of all others, and is individually diagonalized. We now introduce a flame in which the (r, s)part is diagonalized,
Look at eq. (5.7) and remember the Einstein convention where repeated indices implies summation.
Note the second line in a simple notation. Then we have
Taking the (r, s)matrix elements and selecting the mode c, which is achieved by the operation,
while in the right hand side, the terms with the mode c satisfying
are selected. Otherwise Tr operation leads to the vanishing result.
By carrying out the w_{n} sum, we have
where
In estimations of matrix elements, the chemical potential which is disregarded up to now is required to be taken into count. Namely the Hamiltonian has an additive term ma_{r}a_{r} which causes
The mean field potentials are carefully treated. In modes with ↑ and ↓ we have the nonvanishing single particle parts, h^{↑}_{rs}d_{rs} and h^{↓}_{rs}d_{rs}, which are usually negative. However for superconducting modes, h^{+}_{rs} = h^{–}_{rs} = 0.
5. 1 Unrestricted HF
Let us review the SCF(selfconsistent field) procedure. We now discuss, as an example, the ordinary manyelectron system, the propagator with the up spin. Equation (5.13) is
where < r^{↑}s^{↑} > is the overlap integral between sites r and s and then vanishes if we use the orthogonalized atomic orbitals. We thus have
Note that
Look at the potential of the mode a,
In this case there is h^{↑}(x) whose matrix element should be negative, so that even if the matrix elements of the second terms are positive, the (r, s)matrix element of D^{↑}(x) is probably negative. This usually arises in atoms, molecules and the solid state. In evaluating D^{↑}(x), propagators (wave functions) of all other modes are required.
5. 2 Gap equation for superconductivity
In the case of superconductivity, since s^{+}(s^{}) is traceless, the first term of eq.(5.14) vanishes and also h^{+} = 0. Now eq.(5.14) can be read as
This complicated equation gives the relation between G^{+}_{rs} and G^{}_{ut} both referring to superconductivity, and is called the gap equation. A few points should be presented. In the procedure to select the superconducting mode, one can use the fact that
so that
The relation (5.19) tells that the following relationship must hold:
Therefore for eq. (5.23) to hold it is required that
This is really possible. Let us perform an example by using a pilot calculation by successive approximations.
We adopt a chain polymer whose unit cell is the butadiene molecule, a foursite molecule each of which sites has a p electron. The Huckel theory offers the quantum chemical data: orbital energies and eigen functions (LCAO).
level  LCAO coeff. 
e1  1.618  0.3718  0.6015  0.6015  0.3718 
e2  0.618  0.6015  0.3718  0.3718  0.6015 
e3  0.618  0.6015  0.3718  0.3718  0.6015 
e4  1.618  0.3718  0.6015  0.6015  0.3718 
t : transfer integral (5.25)
a. If band energies in question h_{i} are restricted to those just below the Fermi level, we may be allowed to consider only the highest occupied level, denoted by H >, whose orbital energy is e_{2};
b. while, if we are in the ground state, both e_{1} and e_{2} should be considered.
We then carry on the approximation as follows:
Here (r^{+}_{rs}; r^{–}_{tu}) is the electronelectron interaction between two electron densities and is certainly positive (repulsive). The bond orders are defined as
where the latter is the partial bond order corresponding to the highest occupied orbital. These are tabulated in the following:
and
Look at the negative value of q^{H}_{14} and q_{14}. If we are concerned with the interaction (r_{11}; r_{14}), this interaction becomes effectively negative due to q^{H}_{14} or q_{14}. We have now arrived at a new possibility of the onset of superconductivity, which is not mediated by phonons, but is purely electronic. Further discussion will appear in the next section.
6 Critical Temperature
The polymer whose unit cell is butadiene is a polyacene, say benzene, naphthalene, anthracene and so on. The band structure of these series is already investigated[13, 14]. The results are given in the following:
The band structure of the levels and the LCAO (linear combination of atomic orbitals) are
with
The matrix of LCAO coefficients is
where for example
The numerical data of the previous section are those for the bottom of the band (k = 0). The investigation of superconductivity of this example seems possible qualitatively from eqs.(5.29) and (5.30).
In the first instance, the single band theory is usually adopted [3]; this is not adequate in this case since two bands are required through the electronelectron coupling (r_{11}; r_{14}). At the same time, the theory neglecting the above sort of coupling cannot give a desirable result. That is to say, that the treatment with the deltafunctiontype coupling or neglecting the overlap charge effect will be useless.
Based on the above consideration, we seek for the new possibility of superconductivity:

Each level with h_{i} has really the band structure and then is written as h_{i;k} stressing the band structure by k with the band index i. Then i is put to be the highest occupied level, and the integration over k is carried out. The bond orders thus obtained are approximated by the partial bond orders q^{H}_{rs}.

That electronelectron interaction has been effectively negative due to the chemical structure of species is simply written as g < 0, in the present case,

The energy interval hw_{D} (Debye frequency) to establish superconductivity in the BCS theory is, in the present theory, replaced by the band width being nearly equal to g.

Assuming
in eq.(5.20) and absorbing the minus sign in g, we have
where x = h^{normal} and as usual,
The critical temperature T_{C} is determined by the condition that at this temperature h^{super} vanishes. The integration in eq.(6.7) is carried out as usual,
where in the second integration, the upper bound is replaced by ¥ which makes it integrable:[8]
A simple rearrangement of the result gives
The result is entirely the same as the current one. However, since it is probable that
the critical temperature is, at most, enhanced by this value, even though it is considerably reduced by the factor e^{1/N(0)g}.
7 Conclusion
As has been presented, superconductivity is not such a complicated phenomenon. If we employ the spinor representation, superconductivity can be described in parallel with the normal electronic process. If we find, in the copper oxide complex, the four site unit as a butadiene molecule, it might be the origin of this superconducting material. We think it is not such a difficult problem for the quantum chemists.
References
[ 1] Superconductivity, edited by R. D. Parks, Marcel Dekker, New York (1969).
[ 2] J. G. Bednorz and K. A. Muller, Z. Phys. B, 64, 189 (1986).
[ 3] J. R. Schrieffer, Theory of Superconductivity, Perseus Books, Reading (1983).
[ 4] P. W. Anderson, Science, 235, 1196 (1987).
[ 5] A. Marouchkine, RoomTemperature Superconductivity, Cambridge International Science Publishing (2004).
[ 6] K. Takahashi, Field Theory for Solid State Physics, (in Japanese), Baifukan (1974).
[ 7] L. H. Ryder, Quantum Field Theory, second edition, Cambridge University Press, Cambridge (1996).
[ 8] A. L. Fetter and J. D. Walecka, Quantum Theory of ManyParticle Systems, McGrawHill (1971).
[ 9] A. A. Abrikosov, L. P. Gorkov and L. E. Dyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Dover (1963).
[10] H. Nagao, H. Nishino, Y. Shigeta and K. Yamaguchi, J. Chem. Phys., 113, 11237 (2000).
[11] J. Kondo, J. Phys. Soc. Jpn., 71, 1353 (2002).
[12] S. Aono, J. Phys. Soc. Jpn., 72, 3097 (2003).
[13] M. Kimura, H. Kawabe, K. Nishikawa, and S. Aono, J. Chem. Phys., 85, 1 (1986).
[14] M. Kimura, H. Kawabe, A. Nakajima, K. Nishikawa, and S. Aono, Bull. Chem. Soc. Jpn., 61, 4239 (1988).
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