Smoothness and Monotone Decreasingness of the Solution to the BCS-Bogoliubov Gap Equation for Superconductivity

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1 Journal of Basic Applied Sciences Smoothness and Monotone ecreasingness of the Solution to the BCS-Bogoliubov Gap Equation for Superconductivity Shuji Watanabe and Ken Kuriyama ivision of Mathematical Sciences Graduate School of Engineering Gunma University 4- Aramaki-machi Maebashi Japan Yamaguchi University -6- Tokiwadai Ube Japan Abstract: We show the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation for superconductivity Here the temperature belongs to the closed interval [ ] with > nearly equal to half of the transition temperature We show that the solution is continuous with respect to both the temperature and the energy and that the solution is Lipschit continuous and monotone decreasing with respect to the temperature Moreover we show that the solution is partially differentiable with respect to the temperature twice and the second-order partial derivative is continuous with respect to both the temperature and the energy or that the solution is approximated by such a smooth function Keywords: Smoothness monotone decreasingness temperature solution to the BCS-Bogoliubov gap equation superconductivity INTROUCTION AN MAIN RESULT In this paper we show the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation [ ] for superconductivity: ut x) = Ux)uT ) + ut ) tanh + ut ) d ) where the solution u is a function of the absolute temperature T and the energy x x ) The solution u corresponds to the energy gap between the superconducting ground state and the superconducting first excited state and so the value of the solution is nonnegative ie ut x) The constant > stands for the ebye angular frequency and the potential U satisfies Ux) > at all x) [ ] In ) we consider the solution u as a function of the absolute temperature T and the energy x Accordingly we deal with the integral with respect to the energy Sometimes one considers the solution u as a function of the absolute temperature and the wave vector of an electron Accordingly instead of the integral with respect to the energy in ) one deals with the integral with respect to the wave vector over Address correspondence to this author at the ivision of Mathematical Sciences Graduate School of Engineering Gunma University 4- Aramakimachi Maebashi Japan; Tel/Fax: ; shuwatanabe@gunma-uacjp Mathematics Subject Classification 45G 47H 47N5 855 the three dimensional Euclidean space 3 The existence and uniqueness of the solution to the BCS- Bogoliubov gap equation were established in previous papers [3-8] for each fixed temperature So the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution is not covered except for the paper [9] In [9] the gap equation in the Hubbard model for a constant potential was studied and its solution was shown to be strictly decreasing with respect to the temperature In this connection for interdisciplinary reviews of the BCS-Bogoliubov model of superconductivity see [ ] As is well known studying the temperature dependence of the solution to the BCS-Bogoliubov gap equation is very important in condensed matter physics This is because studying the temperature dependence of the solution by dealing with the thermodynamic potential leads to a mathematical proof of the statement that the transition to the superconducting state is a second-order phase transition in the BCS-Bogoliubov model of superconductivity In order to give its proof we have to differentiate the thermodynamic potential and hence the solution with respect to the temperature twice and we have to study some properties of the second-order partial derivative of the solution So it is highly desirable to study the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS- Bogoliubov gap equation ) We now define a nonlinear integral operator A by AuT x) = Ux)uT ) + ut ) tanh + ut ) d ) ISSN: / E-ISSN: 97-59/7 7 Lifescience Global

2 8 Journal of Basic Applied Sciences 7 Volume 3 Watanabe and Kuriyama Here the right side of this equality is exactly the right side of the BCS-Bogoliubov gap equation ) Since the solution to the BCS-Bogoliubov gap equation is a fixed point of our operator A we apply fixed point theorems to our operator A and study the temperature dependence such as smoothness and monotone decreasingness with respect to the temperature of the solution to the BCS-Bogoliubov gap equation ) Let U > is a positive constant and set Ux) = U at all x) [ ] Then the solution to the BCS- Bogoliubov gap equation becomes a function of the temperature T only and so we denote the solution by :T T ) Accordingly the BCS-Bogoliubov gap equation ) is reduced to the simple gap equation []: ) = > T ) > T ) > ) = < T < T < Moreover the solution is of class C with respect to the temperature T on the interval [ ) and satisfies ) = ) = and lim T ) = T Remark We set T ) = for T > See Figure = U + T ) tanh + T ) d T 3) Here the temperature > is defined by see []) = U tanh d See also Niwa [] and Ziman [3] As is well known in the BCS-Bogoliubov model physicists and engineers studying superconductivity always assume that there is a unique nonnegative solution to the simple gap equation 3) that the solution is continuous and strictly decreasing with respect to the temperature T and that the solution is of class C with respect to the temperature T and so on But as far as the present authors know there is no mathematical proof for these assumptions of the BCS- Bogoliubov model Then applying the implicit function theorem to the simple gap equation 3) one of the present authors obtained the following proposition that indeed gives a mathematical proof for these assumptions mentioned just above: Proposition [4 Proposition ]) Let Ux) = U > ) at all x) [ ] and set = sinh U Then there is a unique nonnegative solution :[ ] [) to the simple gap equation 3) such that the solution is continuous and strictly decreasing with respect to the temperature T on the closed interval [ ] : Figure : The graphs of the functions and with the energy x fixed We introduce another positive constant U > Let < U < U and set Ux) = U at all x) [ ] Then a similar discussion implies that for U there is a unique nonnegative solution :[ ] [) to the simple gap equation = U T + T ) tanh + T ) Here > is defined by = U tanh d We again set T ) = for T > d 4) Lemma 3 [4 Lemma 5]) a) The inequality < holds b) If T < then T ) < T ) If T then T ) = T ) =

3 Smoothness and Monotone ecreasingness of the Solution Journal of Basic Applied Sciences 7 Volume 3 9 See Figure The function has properties similar to those of the function We now deal with the BCS-Bogoliubov gap equation ) where the potential U is not a constant but a function We assume the following condition on U : U) C[ ] ) U Ux) U at all x) [ ] 5) Let T and fix T We now consider the Banach space C[ ] consisting of continuous functions of the energy x only and deal with the following temperature dependent subset V T : efinition 6 Let u T ) be as in Theorem 5 Then the transition temperature T c is defined by T c = inf{t > :u T x) = at all x [ ]} Note that the transition temperature T c satisfies T c Let u T ) be as in Theorem 5 A straightforward calculation gives that if there is a point x [ ] satisfying u T x ) = then u T x) = at all x [ ] We then set u T x) = at all x [ ] for T T c We thus see that u T x) > at all x [ ] for T < T c and that u T x) = at all x [ ] for T T c See Figure V T = ut ) C[ ] : T ) ut x) T ) ) at x [ ] + Remark 4 The set V T depends on the temperature T See Figure and Applying the Schauder fixed-point theorem to our operator ) defined on V T one of the present authors gave another proof of the existence and uniqueness of the nonnegative solution to the BCS- Bogoliubov gap equation ) which shows how the solution varies with the temperature Theorem 5 [4 Theorem ]) Assume 5) and fix T [ ] Then there is a unique nonnegative solution u T ) V T to the BCS-Bogoliubov gap equation ): u T x) = x [ ] Ux)u T ) + u T ) tanh + u T ) d Consequently the solution u T ) with T fixed is continuous with respect to the energy x and varies with the temperature as follows: T ) u T x) T ) at T x) [ ][ ] See Figure Superconductivity is observed when the temperature T satisfies T < T c Here T c is the transition temperature critical temperature) and divides superconductivity T < T c ) and normal conductivity T > T c ) The existence and uniqueness of the transition temperature T c were pointed out in previous papers [4-6 8] In our case we can define it as follows: Figure : For each fixed T the solution u T x) lies between T ) and T ) Remark 7 Theorem 5 tells us nothing about continuity of the solution u with respect to the temperature T Applying the Banach fixed-point theorem we then showed in [5 Theorem ] that the solution u is indeed continuous both with respect to the temperature T and with respect to the energy x under the restriction that the temperature T is sufficiently small See also [6] When the potential U) is not a constant but a function one of the present authors [7] studied the temperature dependence such as smoothness and monotone decreasingness of the solution to the BCS- Bogoliubov gap equation ) with respect to the temperature near the transition temperature T c and gave the behavior of the solution near the transition temperature T c Then dealing with the thermodynamic potential it was shown that the transition to the superconducting state is a second-order phase

4 Journal of Basic Applied Sciences 7 Volume 3 Watanabe and Kuriyama transition from the viewpoint of operator theory [7] Moreover the exact and explicit expression for the gap in the specific heat at constant volume at the transition temperature T c was also obtained in [7] Let us denote by > a unique solution to the equation = tanh > ) Note that is nearly equal to 7 and that tanh for Let > ) satisfy ) = 6) From 6) it follows immediately that <) < Remark 8 Observed values in many experiments by using superconductors imply the temperature is nearly equal to T c / Let < < and fix We then deal with the following subset V of the Banach space C[ ][ ]) : u T x) = T x Ux)u T ) + u T ) tanh + u T ) d Consequently the solution u is continuous on [ ][ ] ie the solution u is continuous with respect to both the temperature T and the energy x Moreover the solution u is Lipschit continuous and monotone decreasing with respect to the temperature T and satisfies T ) u T x) T ) at all T x) [ ][ ] Furthermore if u V then the solution u is partially differentiable with respect to the temperature T twice and the second-order partial derivative is continuous with respect to both the temperature T and the energy x On the other hand if u V \ V then the solution u is approximated by such a smooth element of the subset V with respect to the norm of the Banach space C[ ][ ]) See Figure 3 for the graph of the solution u with the energy x fixed ) + V = u C[ ][ ]) : ut x) u T x) + T T ) T < T ) T ) ut x) T ) u is partially differentiable with respect to T twice u T u T C[ ][ ]) Here > is defined by ) below Let us define our operator ) on the subset V of the Banach space C[ ][ ]) We denote by V the closure of the subset V with respect to the norm of the Banach space C[ ][ ]) Remark 9 The constant > depends neither on u V nor on T [ ] nor on x [ ] See ) The following is our main result Theorem Assume 5) Let and V be as above Then the operator A :V V has a unique fixed point u V and so there is a unique nonnegative solution u V to the BCS-Bogoliubov gap equation ): Figure 3: The solution u belongs to the subset V PROOF OF THEOREM We prove Theorem in a sequence of lemmas Lemma Let < < and fix efine a function F on [ ] by FT ) = T [ ] + T ) tanh + T ) d

5 Smoothness and Monotone ecreasingness of the Solution Journal of Basic Applied Sciences 7 Volume 3 Then the function F is continuous on [ ] Proof Let ) and that tanh FT + h) FT ) ) tanh + T + h) T ) + d T [ ] Note that ) 3/ ) Then + d + + d T + h) T ) T + h) T ) cosh + d cosh tanh -d + d ) tanh + d d 3/ d + d ) Here d is between T + h) d ) it follows that and T ) Since FT + h) FT ) T + h) T ) ) arctan ) set Continuity of the function proves the lemma Let < < and fix In view of Lemma we a = max T + T ) tanh + T ) d ) = U b U a > ) ) As mentioned in Remark 9 the constant > depends neither on u V nor on T [ ] nor on x [ ] Lemma Let T [ ] and let X [ ) ) efine a function G by GT X) = tanh + X + 4XT + X Then G is a monotone increasing function with respect to T [ ] Consequently GT X) G X) Proof A straightforward calculation gives G T = + X 8X + + X T cosh ) + X + X 8X T cosh + X Since 8X cosh + X T cosh + X ) it follows from 6) that 8X 8T + X b = 3 ) arctan ) = 8X + X 8T + X 8 ) 8 + X Then for T [ ] = U > U + T ) tanh + T ) d + T ) tanh + T ) d by 3) Lemma implies > U a where a is that in ) We choose U > U ) such that > U a holds true Set = 8 + X ) Note that X ) and that is nearly equal to 7 The result thus follows A straightforward calculation gives the following Lemma 3 The subset V is bounded closed convex and nonempty

6 Journal of Basic Applied Sciences 7 Volume 3 Watanabe and Kuriyama Lemma 4 If u V then T ) AuT x) T ) at all T x) [ ][ ] Proof Since ut x) T ) it follows that ut ) + ut ) Therefore 4) gives T ) + T ) K = tanh + c ) 3/ { } ) ut ) u T + ) + c + c ) GT 3/ c ) T T ) + c + c cosh + c AuT x) U = T ) T ) + T ) tanh + T ) Similarly we can show the rest d + c ) G c ) T T ) 3/ where c satisfies ut ) > c > u T ) and depends on T T and u Note that Lemma 5 For T T [ ] let T < T If u V then AuT x) Au T x) T T ) x [ ] Proof Step We first show AuT x) Au T x) AuT x) Au T x) = Ux) K + K )d + c c > ) = by 6) The substitution = tanh ) into + c therefore turns where K = ut ) + ut ) tanh u T ) + u T ) tanh + ut ) + u T ) 4 + c tanh K Hence + c tanh + c + c T T ) K = u T ) + u T ) tanh tanh + u T ) + u T ) T Since ut ) u T ) it follows that ) + T ) Since tanh cosh + T ) T T ) ) it follows that ut ) + ut ) u T ) + u T ) Hence K Clearly K Thus AuT x) Au T x) Step We next show AuT x) Au T x) T T ) Since cosh ) it follows from Lemma that K = u T ) T T ) + u T ) + u T ) T u ) 6 T ) + u T ) + u T ) T T )3 ) + ) cosh + u T ) T )

7 Smoothness and Monotone ecreasingness of the Solution Journal of Basic Applied Sciences 7 Volume 3 3 where T < T < T Thus by ) AuT x) Au T x) T T )U ) T T )U a + b + 3) + ) ) + T ) tanh + ) ) + T ) ) = T T ) - / / / d / / Lemma 6 If u V then Au C[ ][ ]) Proof Let T < T Then AuT x) Au T x ) AuT x) Au T x) + Au T x) Au T x ) 3) The preceding lemmas imply the following Lemma 8 AV V Lemma 9 The set AV is relatively compact Proof Let u V Lemma 4 then implies AuT x) ) = sinh U So the set AV is uniformly bounded As mentioned in the proof of Lemma 6 the does not depend on u V Hence the set AV is equicontinuous The result thus follows from the Ascoli Arelà theorem Lemma The operator A :V V is continuous Proof Let uv V Then combining a similar discussion to that in the proof of Lemma 5 with 3) gives Since U) is uniformly continuous for an arbitrary > there is a > such that x x < implies Ux) U x ) < Note that the > depends neither on x nor on x nor on nor on u V Hence the second term on the right of 3) becomes Au T x) Au T x ) Ux) U x ) d < On the other hand the first term on the right of 3) becomes AuT x) Au T x) T T ) < AuT x) AvT x) U tanh + d ) 3/ ut ) - vt ) d U U U = U U u v + d + d tanh + d d u v U + T ) tanh + T ) + d + d cosh d u v + d ) + by the preceding lemma Here T T < / ) Thus AuT x) Au T x ) < T T )+ x x < = min ) + Note that the > depends neither on x nor on x nor on nor on u V nor on T nor on T A straightforward calculation gives the following Lemma 7 Let u V Then Au is partially differentiable with respect to T twice T ) and Au T Au T C[ ][ ]) Here d is between ut ) and vt ) and denotes the norm of the Banach space C[ ][ ]) The result thus follows We now extend the domain V of our operator A to the closure V Let u V Then there is a sequence {u n } n= V satisfying u u n as n A similar discussion to that in the proof of Lemma gives {Au n } n= V is a Cauchy sequence and hence there is an Au V satisfying Au Au n as n Note that Au V does not depend on how to choose the sequence {u n } n= V We thus have the following

8 4 Journal of Basic Applied Sciences 7 Volume 3 Watanabe and Kuriyama Lemma A :V V It is not obvious that Au u V ) is expressed as ) The next lemma shows this is the case A similar discussion to that in the proof of Lemma gives the following Lemma Let u V Then AuT x) = Proof For u V set IT x) = T x) [ ][ ] Ux)uT ) + ut ) tanh + ut ) Ux)uT ) + ut ) tanh + ut ) d d and let {u n } n= V be a sequence satisfying u u n as n Note that the function T x) IT x) just above is well-defined and continuous Then AuT x) IT x) AuT x) Au n T x) + Au n T x) IT x) Since Au n Au in the Banach space C[ ][ ]) the first term on the right becomes AuT x) Au n T x) Au Au n n ) A similar discussion to that in the proof of Lemma gives the second term becomes Au n T x) IT x) U U u n u n ) The result thus follows Similar discussions to those in Lemmas 4 and 5 give the following Lemma 3 Let u V and let be as in ) Then T ) AuT x) T ) Moreover if T < T then AuT x) Au T x) T T ) Lemma 3 implies AuT x) ) for u V since the function is strictly decreasing with respect to the temperature T Hence the set AV is uniformly bounded Similar discussions to those in the proofs of Lemmas 6 and 9 give the following Lemma 4 Let u V Then Au C[ ][ ]) Moreover the set AV is equicontinuous and hence the set AV is relatively compact By Lemma a similar discussion to that in the proof of Lemma gives the following Lemma 5 The operator A :V V is continuous Lemmas 4 and 5 imply the following Lemma 6 The operator A :V V is compact Combining Lemma 6 with Lemma 3 and then applying the Schauder fixed-point theorem give the following Lemma 7 The operator A :V V has at least one fixed point u V ie u = Au The uniqueness of the nonero fixed point of A :V V was pointed out in Theorem 5 Our proof of Theorem is now complete ACKNOWLEGEMENTS S Watanabe is supported in part by the JSPS Grant-in-Aid for Scientific Research C) 454 K Kuriyama is supported by the Yamaguchi University Foundation REFERENCES [] Bardeen J Cooper LN Schrieffer JR Theory of superconductivity Phys Rev 957; 8: [] Bogoliubov NN A new method in the theory of superconductivity I Soviet Phys JETP 958; 34: [3] Billard P Fano G An existence proof for the gap equation in the superconductivity theory Commun Math Phys 968; : [4] Frank RL Hainl C Naboko S Seiringer R The critical temperature for the BCS equation at weak coupling J Geom Anal 7; 7: [5] Hainl C Hama E Seiringer R Solovej JP The BCS functional for general pair interactions Commun Math Phys 8; 8: [6] Hainl C Seiringer R Critical temperature and energy gap for the BCS equation Phys Rev B 8; 77: [7] Odeh F An existence theorem for the BCS integral equation IBM J Res evelop 964; 8: [8] Vansevenant A The gap equation in the superconductivity theory Physica 985; 7: [9] Bach V Lieb EH Solovej JP Generalied Hartree-Fock theory and the Hubbard model J Stat Phys 994; 76:

9 Smoothness and Monotone ecreasingness of the Solution Journal of Basic Applied Sciences 7 Volume 3 5 [] Kuemsky AL Bogoliubov s vision: quasiaverages and broken symmetry to quantum protectorate and emergence Internat J Mod Phys B ; 4: [] Kuemsky AL Variational principle of Bogoliubov and generalied mean fields in many-particle interacting systems Internat J Mod Phys B 5; 9: pages) [] Niwa M Fundamentals of Superconductivity Tokyo enki University Press Tokyo ; in Japanese) [3] Ziman JM Principles of the Theory of Solids Cambridge University Press Cambridge 97 [4] Watanabe S The solution to the BCS gap equation and the second-order phase transition in superconductivity J Math Anal Appl ; 383: [5] Watanabe S Addendum to The solution to the BCS gap equation and the second-order phase transition in superconductivity J Math Anal Appl 3; 45: [6] Watanabe S An operator-theoretical treatment of the Maskawa-Nakajima equation in the massless abelian gluon model J Math Anal Appl 4; 48: [7] Watanabe S An operator theoretical proof for the secondorder phase transition in the BCS-Bogoliubov model of superconductivity arxiv: 679v Received on 3--6 Accepted on 8--7 Published on Watanabe and Kuriyama; Licensee Lifescience Global This is an open access article licensed under the terms of the Creative Commons Attribution Non-Commercial License which permits unrestricted non-commercial use distribution and reproduction in any medium provided the work is properly cited