A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov
We survey the unification of the Bardeen, Cooper, Schrieffer (BCS) and the Bose–Einstein condensation (BEC) theories via a generalized BEC (GBEC) formalism. The GBEC describes a ternary boson-fermion gas mixture consisting of fermion-particle- as well as fermion-hole-Cooper-pairs (CPs) that are boso...
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De Llano, M. Tolmachev, V.V. 2010-11-04T10:07:31Z 2010-11-04T10:07:31Z 2010 A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov / M. De Llano, V.V. Tolmachev // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 79-84. — Бібліогр.: 32 назв. — англ. 2071-0194 PACS 74.70.-b; 71.10.-w; 71.10.Hf; 71.10.Li https://nasplib.isofts.kiev.ua/handle/123456789/13288 We survey the unification of the Bardeen, Cooper, Schrieffer (BCS) and the Bose–Einstein condensation (BEC) theories via a generalized BEC (GBEC) formalism. The GBEC describes a ternary boson-fermion gas mixture consisting of fermion-particle- as well as fermion-hole-Cooper-pairs (CPs) that are bosons in thermal and chemical equilibrium with unpaired electrons. One then switches on an interaction Hamiltonian (Hint) that is reminiscent of the single-vertex Fr¨ohlich “two-fermion/one-boson” interaction. In contrast with the well-known BCS “four-fermion” two-vertex Hint, the full GBEC H H0 + Hint is exactly diagonalized with a Bogolyubov–Valatin transformation provided only that one ignores nonzero-total-momenta CPs in the interaction Hint although not in the unperturbed H0 that describes an ideal ternary gas. Nonzerototal-momenta CPs are completely ignored in the full BCS H. Exact diagonalization is possible since the reduced GBEC H becomes bilinear in the fermion creation/annihilation operators on applying the Bogolyubov “recipe” of replacing the remaining zero-totalmomenta boson hole- and particle-CP operators by the square root of their respective temperature- and coupling-dependent boson cnumbers. The resulting GBEC theory subsumes all five statistical theories of superconductors, including the Friedberg–T.D. Lee (1989) BEC theory, and yields hundredfold enhancements in predicted Tcs when compared with BCS predictions with the same two-electron BCS model phonon interaction producing the CPs. Дано огляд об’єднання теорiй Бардiна–Купера–Шрiффера (БКШ) i бозе-ейнштейнiвської конденсацiї (БЕК) в узагальненому формалiзмi БЕК, який описує трикомпонентний бозефермiонний газ, що мiстить куперiвськi пари (КП) частинок-фермiонiв i дiрок-фермiонiв, якi є бозонами в тепловiй i хiмiчнiй рiвновазi з неспареними електронами. Введено гамiльтонiан взаємодiї Hint, що нагадує одновершинну “двофермiонну/однобозонну” взаємодiю Фрелiха. На вiдмiну вiд добре вiдомого БКШ “чотирифермiонного” двовершинного Hint, повний узагальнений гамiльтонiан БЕК H H0 + Hint точно дiагоналiзується перетворенням Боголюбова–Валатина, якщо знехтувати КП з ненульовим повним iмпульсом в Hint, а не в незбурюваному H0, який описує iдеальний трикомпонентний газ. Куперiвськi пари з ненульовим повним iмпульсом повнiстю iгноруються в повному БКШ гамiльтонiанi H. Точна дiагоналiзацiя можлива, оскiльки узагальнений редукований гамiльтонiан БЕК H стає бiлiнiйним у термiнах фермiонних операторiв народження/знищення при застосуваннi “процедури” Боголюбова iз замiною iнших операторiв куперiвських пар iз дiрок-бозонiв та частинок-бозонiв iз нульовим iмпульсом коренем квадратним iз вiдповiдних бозонних c-чисел, що залежать вiд температури i взаємодiї. Результуюча узагальнена теорiя БЕК пiдсумовує всi п’ять статистичних теорiй надпровiдностi, також теорiю Фрiдберга–Т.Д. Лi, та дає на два порядки бiльшу Tc, нiж за теорiєю БКШ з тiєю ж електрон-фононною взаємодiєю, яка породжує КП. We thank M. Fortes, M. Grether, F.J. Sevilla, M.A. Sol´ıs, S. Tapia, O. Rojo, J.J. Valencia, and A.A. Valladares. Also S.K. Adhikari, V.C. Aguilera-Navarro, A.S. Alexandrov, P.W. Anderson, J.F. Annett, J. Batle, M. Casas, J.R. Clem, J.D. Dow, D.M. Eagles, B.L. Gy¨orffy, K. Levin, T.A. Mamedov, P.D. Mannheim, W.C. Stwalley, H. Vucetich, and J.A. Wilson for conversations and/or for providing material prior to its publication. MdeLl thanks UNAM-DGAPA-PAPIIT (Mexico) for the partial support through grant IN106908. He also thanks the Physics Department of the University of Connecticut, Storrs, CT, USA for its gracious hospitality during the sabbatical year, as well as CONACyT (Mexico) for its support. en Відділення фізики і астрономії НАН України Тверде тіло A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov Узагальнена теорія надпровідності з бозе-ейнштейнівською конденсацією, стимульована Боголюбовим Article published earlier |
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| title |
A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov |
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A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov De Llano, M. Tolmachev, V.V. Тверде тіло |
| title_short |
A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov |
| title_full |
A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov |
| title_fullStr |
A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov |
| title_full_unstemmed |
A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov |
| title_sort |
generalized bose–einstein condensation theory of superconductivity inspired by bogolyubov |
| author |
De Llano, M. Tolmachev, V.V. |
| author_facet |
De Llano, M. Tolmachev, V.V. |
| topic |
Тверде тіло |
| topic_facet |
Тверде тіло |
| publishDate |
2010 |
| language |
English |
| publisher |
Відділення фізики і астрономії НАН України |
| format |
Article |
| title_alt |
Узагальнена теорія надпровідності з бозе-ейнштейнівською конденсацією, стимульована Боголюбовим |
| description |
We survey the unification of the Bardeen, Cooper, Schrieffer (BCS) and the Bose–Einstein condensation (BEC) theories via a generalized BEC (GBEC) formalism. The GBEC describes a ternary boson-fermion gas mixture consisting of fermion-particle- as well as fermion-hole-Cooper-pairs (CPs) that are bosons in thermal and chemical equilibrium with unpaired electrons. One then switches on an interaction Hamiltonian (Hint) that is reminiscent of the single-vertex Fr¨ohlich “two-fermion/one-boson” interaction. In contrast with the well-known BCS “four-fermion” two-vertex Hint, the full GBEC H H0 + Hint is exactly diagonalized with a Bogolyubov–Valatin transformation provided only that one ignores nonzero-total-momenta CPs in the interaction Hint although not in the unperturbed H0 that describes an ideal ternary gas. Nonzerototal-momenta CPs are completely ignored in the full BCS H. Exact diagonalization is possible since the reduced GBEC H becomes bilinear in the fermion creation/annihilation operators on applying the Bogolyubov “recipe” of replacing the remaining zero-totalmomenta boson hole- and particle-CP operators by the square root of their respective temperature- and coupling-dependent boson cnumbers. The resulting GBEC theory subsumes all five statistical theories of superconductors, including the Friedberg–T.D. Lee (1989) BEC theory, and yields hundredfold enhancements in predicted Tcs when compared with BCS predictions with the same two-electron BCS model phonon interaction producing the CPs.
Дано огляд об’єднання теорiй Бардiна–Купера–Шрiффера (БКШ) i бозе-ейнштейнiвської конденсацiї (БЕК) в узагальненому формалiзмi БЕК, який описує трикомпонентний бозефермiонний газ, що мiстить куперiвськi пари (КП) частинок-фермiонiв i дiрок-фермiонiв, якi є бозонами в тепловiй i хiмiчнiй рiвновазi з неспареними електронами. Введено гамiльтонiан взаємодiї Hint, що нагадує одновершинну “двофермiонну/однобозонну” взаємодiю Фрелiха. На вiдмiну вiд добре вiдомого БКШ “чотирифермiонного” двовершинного Hint, повний узагальнений гамiльтонiан БЕК H H0 + Hint точно дiагоналiзується перетворенням Боголюбова–Валатина, якщо знехтувати КП з ненульовим повним iмпульсом в Hint, а не в незбурюваному H0, який описує iдеальний трикомпонентний газ. Куперiвськi пари з ненульовим повним iмпульсом повнiстю iгноруються в повному БКШ гамiльтонiанi H. Точна дiагоналiзацiя можлива, оскiльки узагальнений редукований гамiльтонiан БЕК H стає бiлiнiйним у термiнах фермiонних операторiв народження/знищення при застосуваннi “процедури” Боголюбова iз замiною iнших операторiв куперiвських пар iз дiрок-бозонiв та частинок-бозонiв iз нульовим iмпульсом коренем квадратним iз вiдповiдних бозонних c-чисел, що залежать вiд температури i взаємодiї. Результуюча узагальнена теорiя БЕК пiдсумовує всi п’ять статистичних теорiй надпровiдностi, також теорiю Фрiдберга–Т.Д. Лi, та дає на два порядки бiльшу Tc, нiж за теорiєю БКШ з тiєю ж електрон-фононною взаємодiєю, яка породжує КП.
|
| issn |
2071-0194 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/13288 |
| citation_txt |
A Generalized Bose–Einstein Condensation Theory of Superconductivity Inspired by Bogolyubov / M. De Llano, V.V. Tolmachev // Укр. фіз. журн. — 2010. — Т. 55, № 1. — С. 79-84. — Бібліогр.: 32 назв. — англ. |
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| fulltext |
A GENERALIZED BOSE–EINSTEIN CONDENSATION THEORY
A GENERALIZED BOSE–EINSTEIN CONDENSATION
THEORY OF SUPERCONDUCTIVITY INSPIRED
BY BOGOLYUBOV
M. DE LLANO,1, 2 V.V. TOLMACHEV3
1Physics Department, University of Connecticut, Storrs
(CT 06269 USA)
2Instituto de Investigaciones en Materiales, Universidad Nacional Autónoma de México
(Apdo. Postal 70-360 04510 México, DF, Mexico)
3N.E. Baumann State Technical University
(5, 2-ya Baumanskaya Str., Moscow 107005, Russia)
PACS 74.70.-b; 71.10.-w;
71.10.Hf; 71.10.Li
c©2010
We survey the unification of the Bardeen, Cooper, Schrieffer (BCS)
and the Bose–Einstein condensation (BEC) theories via a gener-
alized BEC (GBEC) formalism. The GBEC describes a ternary
boson-fermion gas mixture consisting of fermion-particle- as well
as fermion-hole-Cooper-pairs (CPs) that are bosons in thermal and
chemical equilibrium with unpaired electrons. One then switches
on an interaction Hamiltonian (Hint) that is reminiscent of the
single-vertex Fröhlich “two-fermion/one-boson” interaction. In con-
trast with the well-known BCS “four-fermion” two-vertex Hint,
the full GBEC H ≡ H0 + Hint is exactly diagonalized with a
Bogolyubov–Valatin transformation provided only that one ignores
nonzero-total-momenta CPs in the interaction Hint although not
in the unperturbed H0 that describes an ideal ternary gas. Nonzero-
total-momenta CPs are completely ignored in the full BCS H. Ex-
act diagonalization is possible since the reduced GBEC H becomes
bilinear in the fermion creation/annihilation operators on apply-
ing the Bogolyubov “recipe” of replacing the remaining zero-total-
momenta boson hole- and particle-CP operators by the square root
of their respective temperature- and coupling-dependent boson c-
numbers. The resulting GBEC theory subsumes all five statisti-
cal theories of superconductors, including the Friedberg–T.D. Lee
(1989) BEC theory, and yields hundredfold enhancements in pre-
dicted Tcs when compared with BCS predictions with the same
two-electron BCS model phonon interaction producing the CPs.
1. Introduction
Boson-fermion (BF) models of superconductivity (SC) as
a Bose–Einstein condensation (BEC) go back to the mid-
1950s [1–4], pre-dating even the BCS–Bogoliubov theory
[5–7]. Although BCS theory only envisions the presence
of “Cooper correlations” of single-electron states, BF
models [1–4, 8–19] posit the existence of actual bosonic
Cooper pairs (CPs). With two [18, 19] exceptions, how-
ever, all BF models neglect the effect of hole CPs in-
cluded on an equal footing with electron CPs to give the
“complete” BF model (CBFM) that constitutes the gen-
eralized Bose–Einstein condensation (GBEC) formalism
to be surveyed.
2. The GBEC Hamiltonian
The GBEC [18,19] formalism is described by the Hamil-
tonian H = H0 +Hint, where
H0 =
∑
k1,s1
εk1a
+
k1,s1
ak1,s1
+
∑
K
E+(K)b+KbK−
−
∑
K
E−(K)c+KcK, (1)
and K ≡ k1 + k2 is the CP total or center-of-mass-
momentum (CMM) wavevector, while εk1 ≡ ~2k2
1/2m
are the single-electron and E±(K) the 2e-/2h-CP phe-
nomenological, energies. Here, a+
k1,s1
(ak1,s1) are the cre-
ation (annihilation) operators for electrons and similarly
b+K (bK ) and c+K (cK) for 2e- and 2h-CP bosons, respec-
tively. As originally suggested by the work of Cooper
[20], the b and c CP operators depend only on K and
so are distinct from the BCS pair operators depending
on both K and the relative wavevector k ≡ 1
2 (k1 − k2)
discussed in [5] [Eqs. (2.9) to (2.13)] for the particular
case of K = 0 and shown there not to satisfy the ordi-
nary Bose commutation relations. Nonetheless, CPs are
objects easily seen to obey Bose–Einstein statistics as, in
the thermodynamic limit, an indefinitely large number
of distinct k values correspond to a given K value defin-
ing the energy levels E+(K) or E−(K). This is all that
is needed to ensure a BEC (or the macroscopic occupa-
tion of a given state that appears below a certain fixed
T = Tc). This was found [18, 19] numerically a posteri-
ori in the GBEC theory. Also, the BCS gap equation is
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 79
M. de LLANO, V.V. TOLMACHEV
recovered for equal numbers of both kinds of pairs, both
in the K = 0 state and in all K 6= 0 states taken collec-
tively, and in weak coupling, regardless of CP overlaps.
The precise familiar BEC Tc formula emerges [18] when
i) 2h-CPs are ignored (whereupon the Friedberg–T.D.
Lee model [13]-[16] equations are recovered) and ii) one
switches off the BF interaction but under a strong inter-
electron coupling, whereby no unpaired electrons survive
in the remaining binary mixture. The interaction Hamil-
tonian Hint consists of four distinct BF interaction single
vertices each with two-fermion/one-boson creation or an-
nihilation operators. Each vertex is reminiscent of the
Fröhlich electron-phonon interaction with CPs replac-
ing phonons. Here, Hint depicts how unpaired electrons
(or holes) combine to form the 2e- (and 2h-CPs), and
vice versa, assumed in a d-dimensional system of size L,
namely
Hint = L−d/2
∑
k,K
f+(k)×
×[a+
k+ 1
2K,↑a
+
−k+ 1
2K,↓bK + a−k +
1
2
K, ↓ ak+ 1
2K, ↑ b+K]+
+L−d/2
∑
k,K
f−(k)×
×[a+
k+ 1
2K,↑a
+
−k+ 1
2K,↓c
+
K + a−k+ 1
2K,↓ak+ 1
2K,↑cK]. (2)
The energy form factors f±(k) in (2) are taken as those
in [18,19], where the associated quantities Ef and δε are
new phenomenological dynamical energy parameters (in
addition to the positive BF vertex coupling parameter
f introduced in [18, 19]) that replace the previous such
E±(0), through the relations Ef ≡ 1
4 [E+(0) + E−(0)]
and δε ≡ 1
2 [E+(0) − E−(0)] ≥ 0, where E±(0) are the
(empirically unknown) zero-CMM energies of the 2e- and
2h-CPs, respectively.
We refer to Ef as the “pseudo-Fermi” energy. It serves
as a convenient energy scale and is not to be confused
with the usual Fermi energy EF = 1
2mv
2
F ≡ kBTF,
where TF is the Fermi temperature. If n is the total
number-density of charge-carrier electrons of effective
mass m, the Fermi energy EF equals π~2n/m in 2D and
(~2/2m)(3π2n)2/3 in 3D, while Ef is similarly related
to another density nf which serves to scale the ordinary
density n. The two quantities Ef and EF , and conse-
quently also n and nf , coincide only when the perfect
2e/2h-CP symmetry holds as in the BCS instance.
3. Diagonalization of GBEC Hamiltonian
The interaction Hamiltonian (2) can be further reduced
by keeping only the K = 0 terms, so that
Hint ' L−d/2
∑
k
f+(k)[a+
k↑a
+
−k↓b0 + a−k↓ak↑b
+
0 ]+
+L−d/2
∑
k
f−(k)[a+
k↑a
+
−k↓c
+
0 + a−k↓ak↑c0] (3)
which allows the exact diagonalization as follows. One
applies the Bogoliubov “recipe” [21] (see also [22]
p. 199 ff.) of replacing all zero-CMM 2e- and 2h-CP bo-
son creation and annihilation operators in the full Hamil-
tonian Ĥ = H0 + Hint by their respective c-numbers,
namely b0, b+0 →
√
N0(T ) and c0, c+0 →
√
M0(T ), where
N0(T ) and M0(T ) are the as yet to be determined T -
dependent thermodynamically equilibrated number of
zero-CMM 2e- and 2h-CPs, respectively. One eventu-
ally seeks, numerically at worst, the highest tempera-
ture, say Tc, above which N0(T ) or M0(Tc) vanishes and
below which one or the other is nonzero. Note that Tc
calculated thusly can, in principle, turn out to be zero,
in which case there is no BEC, but this will not turn
out to be for the BCS model interaction to be employed
here. If the number operator is
N̂ ≡
∑
k1,s1
a+
k1,s1
ak1,s1
+ 2
∑
K
b+KbK − 2
∑
K
c+KcK, (4)
the reduced Ĥ − µN̂ with (1) plus (3) is now entirely
bilinear in the a+ and a operators. It can thus be diago-
nalized exactly via a Bogoliubov–Valatin transformation
[23, 24]
ak,s ≡ ukαk,s + 2svkα
†
−k,−s, (5)
where s = ± 1
2 . Transformation (5) simplifies (1) plus
(3) to the fully bilinear form
Ĥ − µN̂ '
∑
k,s
[ ξk
(
u2
k − v2
k
)
+ 2Δkukvk]︸ ︷︷ ︸
≡Ek
α†k,sαk,s+
+
∑
k,s
2s[
≡0︷ ︸︸ ︷
ξkukvk −Δk
(
u2
k − v2
k
)
]×
×
(
α†k,sα
†
−k,−s + αk,sα−k,−s
)
+
∑
k,s
2
[
ξkv
2
k + Δkukvk
]
+
80 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
A GENERALIZED BOSE–EINSTEIN CONDENSATION THEORY
+ [E+(0)− 2µ]N0 +
∑
K 6=0
[E+(K)− 2µ] b†KbK+
+ [2µ− E−(0)]M0 +
∑
K6=0
[2µ− E−(K)] c†KcK GBEC
(6)
with ξk ≡ εk − µ. A little algebra shows that v2
k ≡
1
2 [1 − ξk/Ek] and Ek =
√
ξ2k + Δ2
k, precisely as in the
BCS theory [5] as reformulated [6] by Bogoliubov. The
term set equal to zero in (6) is justified, as this merely
fixes the coefficient, say vk, that was restricted only by
u2
k + v2
k = 1 which follows, in turn, from the require-
ment that both the a and α operators obey Fermi anti-
commutation relations. There are no products such as
α†k,sα
†
−k,−s remaining, nor any other nonbilinear terms,
as with [25] the BCS two-vertex, four -fermion Hamilto-
nian [5] that neglects other than K = 0 pairings
H ≡ H0 +Hint =
∑
k,s
εka
+
k,sak,s−
−V
∑
k,k′,s
a+
k′,sa
+
−k′,−sa−k,−sak,s BCS, (7)
where −V 6 0, and the last summation is restricted by
EF − ~ωD 6 ~2k2/2m ≡ εk, εk′ 6 EF − ~ωD.
Eigenstates of the now fully diagonalized reduced
GBEC Ĥ − µN̂ (6) are
| ...nk,s...NK...MK...〉=
∏
k,s
(
α†k,s
)nk,s∏
K6=0
1√
NK
(
b†K
)NK
×
×
∏
K 6=0
1√
MK !
(
c†K
)MK
|O〉,
where the three exponents nk,s = 0, 1 and NK and
MK = 0, 1, 2 · · · are occupation numbers. Here, | O〉
is the vacuum state for a fermionic “bogolon” quasipar-
ticle with the gapped dispersion energy Ek appearing
in (6) and rewritten below in (10) as E(ε). It is simul-
taneously a vacuum state for 2e-CP and 2h-CP boson
creation and annihilation operators which is to say that
| O〉 is defined by αk,s | O〉 ≡ bK | O〉 ≡ cK | O〉 ≡ 0.
With the Hamiltonian explicitly diagonalized, one can
now straightforwardly construct the thermodynamic po-
tential Ω ≡ −PLd for the GBEC, with Ld the system
“volume” and P its pressure, which is defined as ([22],
p. 228)
Ω(T, Ld, µ,N0,M0) =
= −kBT ln
[
Tr exp{−β(H − µN̂)}
]
, (8)
where “Tr” stands for “trace.” Inserting (1) plus (2) into
(8) [18], one obtains, after some algebra, an explicit ex-
pression for Ω(T, Ld, µ, N0, M0)/Ld (see [26], Eq. 10).
In d = 3, one usually has
N(ε) ≡ m3/2
21/2π2~3
√
ε and M(ε) ≡ 2m3/2
π2~3
√
ε (9)
for the (one-spin) fermion density-of-states (DOS) at en-
ergies ε = ~2k2/2m and the boson DOS for an assumed
quadratic [1] boson dispersion ε = ~2K2/2(2m), respec-
tively. The latter assumption is to be lifted later so as to
include Fermi-sea effects which change the boson disper-
sion relation from quadratic to linear, as mentioned be-
fore. Finally, the relation between the resulting fermion
spectrum E(ε), which is as before, and the fermion en-
ergy gap Δ(ε), are of the form
E(ε) =
√
(ε− µ)2 + Δ2(ε), (10)
Δ(ε) ≡
√
n0f+(ε) +
√
m0f−(ε). (11)
This last expression for the gap Δ(ε) implies a simple
T -dependence rooted in the 2e-CP n0(T ) ≡ N0(T )/Ld
and 2h-CP m0(T ) ≡ M0(T )/Ld number densities of
BE-condensed bosons, i.e., Δ(T ) =
√
n0(T )f+(ε) +√
m0(T )f−(ε).
4. Minimizing the Helmholtz Free Energy
By definition, the Helmholtz free energy is
F (T, Ld, µ,N0,M0) ≡ Ω(T, Ld, µ,N0,M0) + µN.
Minimizing it with respect to N0 and M0, and simul-
taneously fixing the total number N of electrons by in-
troducing the electron chemical potential µ in the usual
way, specifies an equilibrium state of the system at fixed
volume Ld and temperature T . The necessary conditions
for an equilibrium thermodynamic state are thus
∂F/∂N0 = 0, ∂F/∂M0 = 0, and ∂Ω/∂µ = −N, (12)
whereN evidently includes both paired and unpaired CP
fermions. The second partial derivatives of F have been
examined in [27]. After some algebra, Eqs. (12) then
lead to the three coupled transcendental Eqs. (7)–(9) of
[18]. These can be rewritten somewhat more transpar-
ently as: a) two “gap-like equations”
[2Ef + δε− 2µ(T )] =
1
2
f2
Ef +δε∫
Ef
dεN(ε)×
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 81
M. de LLANO, V.V. TOLMACHEV
×
tanh 1
2β
√
[ε− µ(T )]2 + f2n0(T )√
[ε− µ(T )]2 + f2n0(T )
(13)
and
[2µ(T )− 2Ef + δε] =
1
2
f2
Ef∫
Ef−δε
dεN(ε)×
×
tanh 1
2β
√
[ε− µ(T )]2 + f2m0(T )√
[ε− µ(T )]2 + f2m0(T )
(14)
with β ≡ 1/kBT, as well as b) a single “number equation”
2nB(T )− 2mB(T ) + nf (T ) = n. (15)
This last relation ensures the charge conservation in a
ternary mixture. In general, n ≡ N/Ld is the total num-
ber density of electrons, nf (T ) that of the unpaired elec-
trons, while nB(T ) and mB(T ) are, respectively, those
of 2e- and 2h-CPs in all bosonic states, ground plus ex-
cited, i.e., condensed and noncondensed. These turn out
to be
nB(T ) ≡ n0(T ) +
∞∫
0+
dεM(ε)×
× (expβ[2Ef + δε− 2µ+ ε]− 1)−1
, (16)
mB(T ) ≡ m0(T ) +
∞∫
0+
dεM(ε)×
× (expβ[2µ+ ε− 2Ef + δε]− 1)−1
, (17)
which are clear manifestations of the bosonic nature of
both kinds of CPs. For the number density of unpaired
electrons at any T , one also obtains
nf (T ) ≡
∞∫
0
dεN(ε)[1−ε− µ
E(ε)
tanh
1
2
βE(ε)] = 2
∑
k
v2
k(T ),
(18)
where v2
k(T ) ≡ 1
2 [1 − (εk − µ)/Ek] −→
T →0
v2
k with Ek be-
ing given by (10) is precisely the BCS–Bogoliubov T -
dependent coefficient that is linked with uk(0) ≡ uk
through v2
k + u2
k = 1 of the normalized BCS trial wave-
function
| O〉 ≡
∏
k
(uk + vka
+
k↑a
+
−k↓) | O〉 with 〈O | O〉 = 1,
(19)
where | O〉 is the ordinary vacuum. The zero-T version
of the two amplitude coefficients vk and uk originally ap-
peared in (19) and shortly afterwards in the Bogoliubov–
Valatin canonical transformation. Next, one picks δε =
~ωD and identifies [18, 19] nonzero f+(ε) and nonzero
f−(ε) with f ≡
√
2~ωDV but such that f+(ε)f−(ε) ≡ 0.
In the very special case where n0(T ) = m0(T ), adding
together (13) and (14) gives the precise BCS gap pro-
vided one identifies the pseudo-Fermi energy Ef with µ.
This is guaranteed, in turn, if nB(T ) = mB(T ), namely,
if (16) and (17) are set equal to each other so that the
arguments of the two exponentials become identical.
The self-consistent (at worst, numerical) solution of
the three coupled equations (13) to (15) yields the three
thermodynamic variables of the GBEC formalism
n0(T, n, µ), m0(T, n, µ) and µ(T, n). (20)
The existence of a nonzero Tc associated with these ex-
pressions vindicates the GBEC theory. The numeri-
cal elimination of µ(T, n) shows [19] that, in addition
to the normal phase at high temperatures defined by
n0(T, n) = m0(T, n) = 0, three condensed phases appear
at lower temperatures: two pure phases of 2e-CP- and
2h-CP-BE-condensed states and one mixed phase with
arbitrary proportions of both kinds of BE-condensed
states.
If hole pairs are ignored, the relation Δ(T ) =
f
√
n0(T ) resulting from (11) has recently been general-
ized [28] to include nonzero-K pairs beyond expression
(3) with the help of two-time Green functions [29, 30].
This leads to a generalized gap Eg(λ, T ) defined as
Eg(λ, T ) =
√
2~ωDV nB(λ, T ) ≡ f
√
nB(λ, T ), (21)
where nB(λ, T ) is the net number density of CPs, both in
and above the BE-condensate, in the BF mixture which
was taken in [28] for simplicity as a binary mixture in-
stead of a ternary one. The generalized gap Eg(λ, T )
accommodates recently discovered pseudogap phenom-
ena [31], whereby the so-called “depairing” or pseudo-
gap critical temperature T ∗ > Tc arises. The pseudogap
T ∗ is the solution of Eg(λ, T ∗) = 0, whereas the super-
conducting Tc is that of Δ(Tc) = 0.
82 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
A GENERALIZED BOSE–EINSTEIN CONDENSATION THEORY
Flowchart outlining conditions, under which the GBEC formalism reduces to all five statistical theories of superconductivity (ovals).
The GBEC formalism has alternately been called the “complete boson-fermion model” (CBFM) in that it does not neglect hole CPs
5. Five Statistical Theories Subsumed
All told, the three GBEC equations (13) to (15) subsume
five different theories as special cases, see a flowchart in
Figure. The vastly more general GBEC formalism has
been applied and gives sizeable enhancements in Tcs over
the BCS theory that emerge [32] by admitting, appar-
ently for the first time, departures from the very special
case of the perfect 2e/2h-pair symmetry in the mixed
phase.
6. Conclusions
In conclusion, five statistical continuum theories of su-
perconductivity, including both the BCS and BEC the-
ories, are contained as special limiting cases within a
single generalized Bose–Einstein condensation (GBEC)
model. This model includes, for the first time, along with
unpaired electrons, both two-electron and two-hole pair-
condensates in freely variable proportions. The BCS
and BEC theories are thus completely unified within
the GBEC. The BCS condensate emerges directly from
the GBEC as a BE condensate through the condi-
tion for phase equilibria when both total 2e- and 2h-
pair number, as well as their condensate, densities are
equal at the given T and coupling provided the cou-
pling is weak enough so that the electron chemical po-
tential µ roughly equals the Fermi energy EF . The or-
dinary BEC Tc-formula, on the other hand, is recov-
ered from the GBEC when hole pairs are completely
neglected, the BF coupling f is made to vanish, and
the limit of zero unpaired electrons is taken, this im-
plying a very strong interelectron coupling. The practi-
cal outcome of the BCS-BEC unification via the GBEC
is an enhancement in Tc by more than two orders-of-
magnitude in 3D. This enhancement in Tc falls within
empirical ranges for 2D and 3D “exotic” SCs, whereas
BCS Tc values remain low and within the empirical
ranges for conventional, elemental SCs using standard
interaction-parameter values. Lastly, room tempera-
ture superconductivity is possible for a material with
a Fermi temperature TF . 103K, with the same inter-
action parameters used in BCS theory for conventional
SCs.
ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1 83
M. de LLANO, V.V. TOLMACHEV
We thank M. Fortes, M. Grether, F.J. Sevilla,
M.A. Soĺıs, S. Tapia, O. Rojo, J.J. Valencia, and
A.A. Valladares. Also S.K. Adhikari, V.C. Aguilera-
Navarro, A.S. Alexandrov, P.W. Anderson, J.F. Annett,
J. Batle, M. Casas, J.R. Clem, J.D. Dow, D.M. Eagles,
B.L. Györffy, K. Levin, T.A. Mamedov, P.D. Mannheim,
W.C. Stwalley, H. Vucetich, and J.A. Wilson for conver-
sations and/or for providing material prior to its pub-
lication. MdeLl thanks UNAM-DGAPA-PAPIIT (Mex-
ico) for the partial support through grant IN106908. He
also thanks the Physics Department of the University
of Connecticut, Storrs, CT, USA for its gracious hospi-
tality during the sabbatical year, as well as CONACyT
(Mexico) for its support.
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Received 28.09.09
УЗАГАЛЬНЕНА ТЕОРIЯ
НАДПРОВIДНОСТI З БОЗЕ-ЕЙНШТЕЙНIВСЬКОЮ
КОНДЕНСАЦIЄЮ, СТИМУЛЬОВАНА БОГОЛЮБОВИМ
М. де Ллано, В.В. Толмачев
Р е з ю м е
Дано огляд об’єднання теорiй Бардiна–Купера–Шрiффера
(БКШ) i бозе-ейнштейнiвської конденсацiї (БЕК) в узагаль-
неному формалiзмi БЕК, який описує трикомпонентний бозе-
фермiонний газ, що мiстить куперiвськi пари (КП) частинок-
фермiонiв i дiрок-фермiонiв, якi є бозонами в тепловiй i хi-
мiчнiй рiвновазi з неспареними електронами. Введено гамiль-
тонiан взаємодiї Hint, що нагадує одновершинну “двофермiон-
ну/однобозонну” взаємодiю Фрелiха. На вiдмiну вiд добре вiдо-
мого БКШ “чотирифермiонного” двовершинного Hint, повний
узагальнений гамiльтонiан БЕК H ≡ H0 + Hint точно дiаго-
налiзується перетворенням Боголюбова–Валатина, якщо зне-
хтувати КП з ненульовим повним iмпульсом в Hint, а не в
незбурюваному H0, який описує iдеальний трикомпонентний
газ. Куперiвськi пари з ненульовим повним iмпульсом повнi-
стю iгноруються в повному БКШ гамiльтонiанi H. Точна дiа-
гоналiзацiя можлива, оскiльки узагальнений редукований га-
мiльтонiан БЕК H стає бiлiнiйним у термiнах фермiонних опе-
раторiв народження/знищення при застосуваннi “процедури”
Боголюбова iз замiною iнших операторiв куперiвських пар iз
дiрок-бозонiв та частинок-бозонiв iз нульовим iмпульсом коре-
нем квадратним iз вiдповiдних бозонних c-чисел, що залежать
вiд температури i взаємодiї. Результуюча узагальнена теорiя
БЕК пiдсумовує всi п’ять статистичних теорiй надпровiдностi,
також теорiю Фрiдберга–Т.Д. Лi, та дає на два порядки бiльшу
Tc, нiж за теорiєю БКШ з тiєю ж електрон-фононною взаємо-
дiєю, яка породжує КП.
84 ISSN 2071-0194. Ukr. J. Phys. 2010. Vol. 55, No. 1
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