PHYSICS

I. A. KVASNIKOV

APPLICATION OF THE VARIATIONAL PRINCIPLE IN A NEW METHOD OF THE THEORY OF SUPERCONDUCTIVITY

(Presented by Academician N. N. Bogolyubov on 18 XI 1957)

In the work on a new method in the theory of superconductivity ((^{1})), N. N. Bogolyubov, considering Bardeen’s model ((^{2})), in order to sum a special class of diagrams constructs the Hamiltonian of an equivalent model dynamical system, which after certain transformations is written in the form (see ((^{1}), (13)))

[
\begin{aligned}
H={}&\sum_k 2\xi(k)v_k^2-\frac{J}{V}\sum_{k_1,k_2}\theta(k_1)\theta(k_2)u_{k_1}u_{k_2}v_{k_1}v_{k_2}
+\sum_k 2E_e(k)\beta_k^+\beta_k
\
&-\frac{J}{V}\sum_{(k_1\ne k_2)}
\left(u_{k_1}^2\beta_{k_1}^+-v_{k_1}^2\beta_{k_1}-2u_{k_1}v_{k_1}\beta_{k_1}^+\beta_{k_1}\right)\times
\
&\times
\left(u_{k_2}^2\beta_{k_2}-v_{k_2}^2\beta_{k_2}^+-2u_{k_2}v_{k_2}\beta_{k_2}^+\beta_{k_2}\right)\theta(k_1)\theta(k_2),
\end{aligned}
\tag{1}
]

where (u_k) and (v_k) are determined by the expressions ((^{3}))

[
u_k^2=1-v_k^2=\frac{1}{2}\left{1+\frac{\xi(k)}{E_e(k)}\right},
\tag{2}
]

where

[
\xi(k)=E(k)-E(k_F),\qquad
E_e(k)=\sqrt{\xi^2(k)+\theta(k)c^2},
]

[
c=2\omega\exp\left{-\frac{1}{\rho}\right},\qquad
\rho=\frac{J}{2\pi^2}\left(k^2\frac{dk}{dE(k)}\right)_{k=k_F},
]

[
\theta(k)=
\begin{cases}
1, & |\xi(k)|<\omega,\
0, & |\xi(k)|>\omega.
\end{cases}
]

(J) and (\omega) are parameters of the Bardeen model, which, according to ((^{3})), correspond to the quantities (g^2) and (\tilde{\omega}/2) of the Fröhlich system.

As was shown earlier ((^{4})), proceeding from the variational principle of N. N. Bogolyubov, one can obtain the basic equations of the phenomenological theories of Weiss and Bragg—Williams; moreover, it turned out that the mathematical apparatus in both cases is extremely similar. Let us apply the method developed in ((^{4})) to the theory of the phase transition in a superconductor. For this, in (1) we pass from the Pauli amplitudes (\beta_k^+) and (\beta_k) to new operators (S_k^x, S_k^y), and (S_k^z), possessing the properties of ordinary spin operators:

[
\beta_k^+=S_k^x-iS_k^y;\qquad
\beta_k=S_k^x+iS_k^y;\qquad
\beta_k^+\beta_k=\frac{1}{2}-S_k^z.
\tag{3}
]

When applying the variational principle to such a system, as (H_0) we choose a sum of the form

[
H_0=-\sum_k \mathcal{E}(k)\sigma_k,
\tag{4}
]

where (\mathcal E(k)) are as yet undetermined functions, and (\sigma_k=2S_k^z=\pm 1) are the usual Ising symbols. Since the contributions from the terms of the Hamiltonian that include (S^x) or (S^y), when the variational principle is applied with such an (H_0), are equal to zero, then, discarding them at once, we shall have an equivalent Hamiltonian of the system, similar to the Ising Hamiltonian:

[
H' = U-\sum_k E'(k)\sigma_k-\frac{1}{V}\sum_{(k_1\ne k_2)} F(k_1,k_2)\sigma_{k_1}\sigma_{k_2},
\tag{5}
]

where

[
U=\sum_k \xi(k),\qquad
E'(k)=E_e(k)-\frac{2J}{V}u_k v_k\theta(k)\sum_{\substack{k'\(k'\ne k)}} u_{k'}v_{k'}\theta(k'),
\tag{6}
]

[
F(k_1,k_2)=Ju_{k_1}u_{k_2}v_{k_1}v_{k_2}\theta(k_1)\theta(k_2).
]

According to the variational principle, the upper bound of the free energy of the system (F) has the form

[
F\leq F_{\sup}=-\theta\ln \operatorname{Sp}{e^{-H_0/\theta}}
+\operatorname{Sp}{(H'-H_0)e^{-H_0/\theta}}/\operatorname{Sp}{e^{-H_0/\theta}}.
\tag{7}
]

As in (4), we shall regard the upper bound of its free energy as its approximate value. In calculating the free energy (7) it is necessary to take into account that the operators (\beta^+) and (\beta) in (1) correspond to the creation and annihilation of pairs of particles; therefore we shall have

[
F\leq U-2\theta\sum_k \ln 2\,\operatorname{ch}\frac{\mathcal E(k)}{2}
-\sum_k\bigl(E'(k)-\mathcal E(k)\bigr)\operatorname{th}\frac{\mathcal E(k)}{2\theta}
-
]

[
-\frac{1}{V}\sum_{(k_1\ne k_2)}
F(k_1,k_2)\operatorname{th}\frac{\mathcal E(k_1)}{2\theta}
\operatorname{th}\frac{\mathcal E(k_2)}{2\theta}.
\tag{8}
]

Determining the functions (\mathcal E(k)) from the condition of a minimum of the upper bound of the free energy, we obtain a system of transcendental equations:

[
\mathcal E(k)=E'(k)+\frac{2}{V}\sum_{\substack{k'\(k'\ne k)}}F(k,k')\operatorname{th}\frac{\mathcal E(k')}{2\theta}.
\tag{9}
]

The solution of these equations, taking into account the concrete form of (F(k,k')) (6), can be written at once:

[
\mathcal E(k)=
\frac{\xi^2(k)+\theta(k)c\,\mathcal E(k_F)}
{\sqrt{\xi^2(k)+\theta(k)c^2}},
\tag{10}
]

where the quantity (\mathcal E(k_F)) is determined by the equation:

[
\frac{\mathcal E(k_F)}{c}
=
\frac{J}{4\pi^2}
\int_{k_F-\Delta}^{k_F+\Delta}
\frac{k^2\,dk}{\sqrt{\xi^2(k)+c^2}}\,
\operatorname{th}
\left{
\frac{\xi^2(k)+c\mathcal E(k_F)}
{2\theta\sqrt{\xi^2(k)+c^2}}
\right}.
\tag{11}
]

Let us note that at (\theta=0), (\mathcal E(k_F)=c), and we obtain the natural result (\mathcal E(k)=E_e(k)).

Noting that the first factor in the integrand has, for small (c), a sharp maximum at the point (k=k_F), and taking the value of the tangent at this point outside the integral sign, we obtain the approximate equation for (\mathcal E(k_F)):

[
\frac{\mathcal E(k_F)}{c}
=
\operatorname{th}\frac{\mathcal E(k_F)}{2\theta}.
\tag{12}
]

From it, in particular, it follows that the temperature (\theta_0 = {}^1/_2 c) is critical, since for (\theta > \theta_0) we have (\mathcal E(k_F)=0). However, just in the region (\theta \sim \theta_0), equation (12) gives only a qualitative approximation to (11), and therefore the values of (\theta_0) and of the heat capacity in this temperature region are only indicative.

Let us investigate the behavior of the specific heat of the system:

[
C=\frac{1}{2N}\sum_k \mathcal E(k)\operatorname{ch}^{-2}\frac{\mathcal E(k)}{2\theta}\cdot
\left{\frac{\mathcal E(k)}{\theta^2}-\frac{1}{\theta}\frac{\partial \mathcal E(k)}{\partial \theta}\right}.
\tag{13}
]

At temperatures above the critical temperature, but still sufficiently low, after straightforward calculations we obtain

[
C \simeq \frac{Vmk_F\theta}{3N},
\tag{14}
]

which, naturally, coincides with the heat capacity of a free electron gas at low temperatures.

In the range of temperatures below the critical one, but such that (\theta_0-\theta \ll \theta_0), the heat capacity, according to (12) and (13), has a singularity. Indeed, near (\theta_0)

[
\frac{\partial \mathcal E(k_F)}{\partial \theta}\simeq
-\sqrt{\frac{3\theta_0}{\theta_0-\theta}},
]

and in the immediate vicinity of the critical point, to within constant terms, we shall have

[
C\simeq \sqrt{\frac{3\theta_0}{\theta_0-\theta}}\,
\frac{Vmk_F\theta_0}{2\pi^2 N}.
\tag{15}
]

At very low temperatures, such that (\theta \ll \theta_0), the heat capacity falls according to an exponential law:

[
C\simeq
\frac{Vmk_Fc^2\sqrt{2\pi c\theta}}{N\pi^2\theta^2}
e^{-c/\theta}.
\tag{16}
]

To study the dependence of the entropy on temperature, it is convenient to use the relation

[
\frac{\partial S}{\partial \theta}=\frac{1}{\theta}C.
\tag{17}
]

At temperatures above the critical temperature, the entropy increases with increasing temperature according to the law (14). As the temperature is lowered, the entropy curve has a sharp downward kink at the point (\theta_0) and tends exponentially to zero as (\theta \to 0).

While writing the present note, the author became aware of the work ((^5)), in which the same problem is solved in a more correct manner. The results given above are quite close to the results of ((^5)). In particular, the basic equation (11), upon replacing the quantities (c) entering it by (\mathcal E(k_F)), becomes equation (12) of work ((^5)). Therefore, in contrast to ((^5)), near the critical temperature our (\mathcal E(k_F)) has, according to (11), a certain smearing.

In conclusion, I express my deep gratitude to Acad. N. N. Bogolyubov and V. V. Tolmachev for discussion and advice in carrying out this work.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
18 XI 1957

CITED LITERATURE

(^{1}) N. N. Bogolyubov, ZhETF, 34, issue 1 (1958).
(^{2}) J. Bardeen, L. N. Cooper, J. R. Schrieffer, Phys. Rev., 106, 162 (1957).
(^{3}) V. V. Tolmachev, S. V. Tyablikov, ZhETF, 34, issue 1 (1958).
(^{4}) I. A. Kvasnikov, DAN, 110, 755 (1956); 113, 544, 777 (1957).
(^{5}) N. N. Bogolyubov, D. N. Zubarev, Yu. A. Tserkovnikov, DAN, 117, No. 5 (1957).