Superfluid states with finite momentum of Cooper pairs in nuclear matter

Superﬂuid states of symmetric nuclear matter with ﬁnite total momentum of Cooper pairs (nuclear LOFF phase) are studied with the use of Fermi–liquid theory in the model with Skyrme eﬀective forces. It is considered the case of four–fold splitting of the excitation spectrum due to ﬁnite superﬂuid mom...

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Дата:	2001
Автор:	Isayev, A.A.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2001
Назва видання:	Вопросы атомной науки и техники
Теми:	Quantum fluids
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Цитувати:	Superfluid states with finite momentum of Cooper pairs in nuclear matter / A.A. Isayev // Вопросы атомной науки и техники. — 2001. — № 6. — С. 370-374. — Бібліогр.: 14 назв. — англ.

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spelling	irk-123456789-800562015-04-10T03:02:26Z Superfluid states with finite momentum of Cooper pairs in nuclear matter Isayev, A.A. Quantum fluids Superﬂuid states of symmetric nuclear matter with ﬁnite total momentum of Cooper pairs (nuclear LOFF phase) are studied with the use of Fermi–liquid theory in the model with Skyrme eﬀective forces. It is considered the case of four–fold splitting of the excitation spectrum due to ﬁnite superﬂuid momentum and coupling of T = 0 and T = 1 pairing channels. It has been shown that at zero temperature the energy gap in triplet–singlet (TS) pairing channel (in spin and isospin spaces) for the SkM∗ force demonstrates double–valued behavior as a function of superﬂuid momentum. As a consequence, the phase transition at the critical superﬂuid momentum from the LOFF phase to the normal state will be of a ﬁrst order. Behavior of the energy gap as a function of density for TS pairing channel under increase of superﬂuid momentum changes from one–valued to universal two–valued. It is shown that two–gap solutions, describing superposition of states with singlet–triplet (ST) and TS pairing of nucleons appear as a result of branching from one–gap ST solution. Comparison of the free energies shows that the state with TS pairing of nucleons is thermodynamically most preferable. 2001 Article Superfluid states with finite momentum of Cooper pairs in nuclear matter / A.A. Isayev // Вопросы атомной науки и техники. — 2001. — № 6. — С. 370-374. — Бібліогр.: 14 назв. — англ. 1562-6016 PACS numbers: 21.65.+f; 21.30.Fe; 71.10.Ay http://dspace.nbuv.gov.ua/handle/123456789/80056 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Quantum fluids Quantum fluids
spellingShingle	Quantum fluids Quantum fluids Isayev, A.A. Superfluid states with finite momentum of Cooper pairs in nuclear matter Вопросы атомной науки и техники
description	Superﬂuid states of symmetric nuclear matter with ﬁnite total momentum of Cooper pairs (nuclear LOFF phase) are studied with the use of Fermi–liquid theory in the model with Skyrme eﬀective forces. It is considered the case of four–fold splitting of the excitation spectrum due to ﬁnite superﬂuid momentum and coupling of T = 0 and T = 1 pairing channels. It has been shown that at zero temperature the energy gap in triplet–singlet (TS) pairing channel (in spin and isospin spaces) for the SkM∗ force demonstrates double–valued behavior as a function of superﬂuid momentum. As a consequence, the phase transition at the critical superﬂuid momentum from the LOFF phase to the normal state will be of a ﬁrst order. Behavior of the energy gap as a function of density for TS pairing channel under increase of superﬂuid momentum changes from one–valued to universal two–valued. It is shown that two–gap solutions, describing superposition of states with singlet–triplet (ST) and TS pairing of nucleons appear as a result of branching from one–gap ST solution. Comparison of the free energies shows that the state with TS pairing of nucleons is thermodynamically most preferable.
format	Article
author	Isayev, A.A.
author_facet	Isayev, A.A.
author_sort	Isayev, A.A.
title	Superfluid states with finite momentum of Cooper pairs in nuclear matter
title_short	Superfluid states with finite momentum of Cooper pairs in nuclear matter
title_full	Superfluid states with finite momentum of Cooper pairs in nuclear matter
title_fullStr	Superfluid states with finite momentum of Cooper pairs in nuclear matter
title_full_unstemmed	Superfluid states with finite momentum of Cooper pairs in nuclear matter
title_sort	superfluid states with finite momentum of cooper pairs in nuclear matter
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2001
topic_facet	Quantum fluids
url	http://dspace.nbuv.gov.ua/handle/123456789/80056
citation_txt	Superfluid states with finite momentum of Cooper pairs in nuclear matter / A.A. Isayev // Вопросы атомной науки и техники. — 2001. — № 6. — С. 370-374. — Бібліогр.: 14 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT isayevaa superfluidstateswithfinitemomentumofcooperpairsinnuclearmatter
first_indexed	2025-07-06T03:59:29Z
last_indexed	2025-07-06T03:59:29Z
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fulltext	ar X iv :n uc l- th /0 10 90 41 v1 1 7 Se p 20 01 Superfluid states with moving condensate in nuclear matter A. A. Isayev Kharkov Institute of Physics and Technology, Kharkov, 61108, Ukraine (Dated: November 12, 2013) Superfluid states of symmetric nuclear matter with finite total momentum of Cooper pairs (nuclear LOFF phase) are studied with the use of Fermi–liquid theory in the model with Skyrme effective forces. It is considered the case of four–fold splitting of the excitation spectrum due to finite superfluid momentum and coupling of T = 0 and T = 1 pairing channels. It has been shown that at zero temperature the energy gap in triplet–singlet (TS) pairing channel (in spin and isospin spaces) for the SkM∗ force demonstrates double–valued behavior as a function of superfluid momentum. As a consequence, the phase transition at the critical superfluid momentum from the LOFF phase to the normal state will be of a first order. Behavior of the energy gap as a function of density for TS pairing channel under increase of superfluid momentum changes from one–valued to universal two–valued. It is shown that two–gap solutions, describing superposition of states with singlet–triplet (ST) and TS pairing of nucleons appear as a result of branching from one–gap ST solution. Comparison of the free energies shows that the state with TS pairing of nucleons is thermodynamically most preferable. PACS numbers: 21.65.+f; 21.30.Fe; 71.10.Ay In this report we shall study superfluid states of nu- clear matter with nonzero total momentum of Cooper pairs. First the states with moving condensate were con- sidered in Refs. [1, 2] with respect to metallic supercon- ductors. In this case the superconducting condensate has a spatially periodic structure. Corresponding phase is called the Larkin–Ovchinnikov–Fulde–Ferrel (LOFF) phase. Recent upsurge of interest to the LOFF phase is related with the possibility of formation of this phase in nuclear [3] and quark matter [4, 5]. In Ref. [3] it was con- sidered in model calculations the case of neutron–proton superfluidity, when the quasiparticle spectrum is two– fold split due to asymmetry of nuclear matter and finite superfluid momentum. It was shown, that the nuclear LOFF phase appears as a result of a first order phase transition in the asymmetry parameter from the spatially uniform BCS superconducting state. Further increase of the asymmetry parameter leads to the second order phase transition from the LOFF phase to the normal state. In the study of Ref. [3] it was assumed that coupling be- tween isospin singlet and isospin triplet pairing channels can be neglected. However, as emphasized in Ref. [6], in the region of low densities coupling between T = 0 and T = 1 pairing channels may be of importance, leading to the emergence of multi–gap superfluid states, charac- terized by nonvanishing gaps in both pairing channels. Thus, we shall consider the case of four–fold splitting of the quasiparticle spectrum, caused by account of, first, fi- nite superfluid momentum and, second, coupling of T = 0 and T = 1 pairing channels. Another simplifying mo- ment in Ref. [3] is the use of free single particle spectrum and bare interaction in the gap equation without taking into account the effects of medium polarization. A ”first– principle” derivation of the pairing interaction from the bare NN force still encounters many problems, such as, e.g., treatment of core polarization [7]. Hence, it is quite natural to develop some kind of a phenomenological the- ory, where a phenomenological pairing interaction is em- ployed. As such a theory, we shall use the Fermi–liquid (FL) approach [8]. In the Fermi–liquid model the normal and anomalous FL interaction amplitudes are taken into account on an equal footing [9]. This will allow us to con- sider consistently influence of the FL amplitudes on su- perfluid properties of the nuclear LOFF phase. Besides, as a potential of NN interaction we choose the density de- pendent Skyrme effective forces, used earlier in a number of contexts for description of superfluid properties both finite nuclei [10, 11] and infinite nuclear matter [12, 13]. Superfluid states of nuclear matter are described by the normal fκ1κ2 = Tr ̺a+ κ2 aκ1 and anomalous gκ1κ2 = Tr ̺aκ2 aκ1 distribution functions of nucleons (κ ≡ (p, σ, τ), p is momentum, σ(τ) is the projection of spin (isospin) on the third axis, ̺ is the density matrix of the system). We shall study two–gap superfluid states in symmetric nuclear matter, corresponding to superpo- sition of states with total spin S and isospin T of a pair S = 1, T = 0 (triplet–singlet (TS) pairing) and S = 0, T = 1 (singlet–triplet (ST) pairing) with the projections Sz = Tz = 0 (TS–ST states). In this case, assuming that a condensate moves with the finite momentum q, the nor- mal f and anomalous g distribution functions read [6, 8] fκ1κ2 = f00(p1)(σ0τ0)κ1κ2 δp1,p2 , (1) gκ1κ2 = (g30(p1)σ3σ2τ2 + g03(p1)σ2τ3τ2)κ1κ2 δp1,−p2+q, (2) where σi and τk are the Pauli matrices in spin and isospin spaces, respectively. For the energy functional, being invariant with respect to rotations in spin and isospin spaces, the quasiparticle energy and the order parameter have the similar structure εκ1κ2 = ε00(p1)(σ0τ0)κ1κ2 δp1,p2 , (3) ∆κ1κ2 = (∆30(p1)σ3σ2τ2 + ∆03(p1)σ2τ3τ2)κ1κ2 δp1,−p2+q (4) http://arxiv.org/abs/nucl-th/0109041v1 2 If to take into account the antisymmetry properties ∆T = −∆, gT = −g and to set p1 = p + q/2, p2 = −p + q/2 (q is total momentum of a pair, p is momentum of one of nucleons in the center of mass frame of a pair), then we obtain ∆30(p + q 2 ) = ∆30(−p + q 2 ) ≡ ∆30(p,q), (5) ∆03(p + q 2 ) = ∆03(−p + q 2 ) ≡ ∆03(p,q) and analogous relationships hold for the functions g30, g03. Further we shall write the self–consistent equa- tions for the quantities ∆30(p,q), ∆03(p,q). Using the minimum principle of the thermodynamic potential and procedure of block diagonalization [8], one can express evidently the distribution functions f00, g30, g03 in terms of the quantities ε and ∆: f00 = 1 4 [ (1 + n+ + − n− +) − ξs E+ (1 − n+ + − n− +) (6) + (1 + n+ − − n− −) − ξs E− (1 − n+ − − n− −) ] , g30 = − ∆+ 4E+ (1 − n+ + − n− +) − ∆− 4E− (1 − n+ − − n− −), (7) g03 = − ∆+ 4E+ (1 − n+ + − n− +) + ∆− 4E− (1 − n+ − − n− −). (8) Here f00 = f00(p + q 2 ), n± = n(±p + q 2 ) and E± = √ ξ2 s + \|∆±\|2, ∆± = ∆30 ± ∆03, ξs(p + q 2 ) = 1 2 ( ξ(p + q 2 ) + ξ(−p + q 2 ) ) , ξa(p + q 2 ) = 1 2 ( ξ(p + q 2 ) − ξ(−p + q 2 ) ) , ξ(p) = ε00(p) − µ0, n± = {exp Y0(ξa + E±) + 1}−1, µ0 is chemical potential, which should be determined from the normalization condition 4 V ∑ p f00(p + q 2 ) = ̺, (9) ̺ is density of symmetric nuclear matter. As follows from the structure of the distribution functions f00, g30, g03, the quantity ω±,± = ξa(±p + q 2 ) + E±, being the exponent in Fermi distribution functions n±(±p + q 2 ), plays the role of the quasiparticle exci- tation spectrum. In the considering case the spectrum is four–fold split due to 1) finite superfluid momentum (q 6= 0), 2) coupling of TS and ST pairing channels (∆30 6= 0, ∆03 6= 0). To obtain the closed system of equations for the quan- tities ∆ and ξ, it is necessary to set the energy functional of the system. In the case of symmetric nuclear matter with TS and ST pairings of nucleons the energy func- tional is characterized by one normal U0 and two anoma- lous V1, V2 FL amplitudes [6, 13]. Then one can obtain the self–consistent equations in the form ξ(p) = ε0 0(p) − µ0 + ε̃00(p), ε0 0(p) = p 2 2m0 , (10) ε̃00(p) = 1 2V ∑ p′ U0(k)f00(p ′), k = p − p′ 2 , ∆30(p,q) = 1 V ∑ p′ V1(p,p′)g30(p ′,q), (11) ∆03(p,q) = 1 V ∑ p′ V2(p,p′)g03(p ′,q), (12) where m0 being the mass of a bare nucleon. Further for obtaining numerical results we shall use the Skyrme effective interaction. In the case of Skyrme forces the normal and anomalous FL amplitudes read [13] U0(k) = 6t0 + 6t3̺ β (13) + 1 h̄2 [6t1 + 2t2(5 + 4x2)]k 2 ≡ d0 + e0k 2 V1,2(p,p′) = t0(1 ± x0) + 1 6 t3̺ β(1 ± x3) (14) + 1 2h̄2 t1(1 ± x1)(p 2 + p′2), where ti, xi, β are some phenomenological constants, characterizing the given parametrization of Skyrme forces (we shall use the SkP [10], SkM∗ [14] potentials). With account of Eq. (13) we obtain ξs(p + q 2 ) = p2 2m1 + q2 8m2 − µ, (15) where h̄2 2m1,2 = h̄2 2m0 ± ̺ 16 [3t1 + t2(5 + 4x2)] (16) and the effective chemical potential µ should be deter- mined from the normalization condition (9). Besides, expression for the quantity ξa reads ξa(p + q 2 ) = pq 2m0 − e0 4 ∑ p′ f00(p ′ + q 2 )pp′ (17) The normal distribution function f00 in turn depends on the quantity ξa and, hence, expression (17) represents an equation for determining the quantity ξa. Since the second term in Eq. (17) is proportional to the scalar prod- uct pq, solution of Eq. (17) should be found in the form ξa(p + q 2 ) = pq/2m∗, where m∗ is some effective mass. Using Eqs. (6)–(8), we present equations for the order parameters ∆30, ∆03, effective chemical potential µ and 3 effective mass m∗ as ∆30 = − 1 4V ∑ p′ V1(p,p′) { ∆′ + E′ + ( tanh ω′ ++ 2T (18) + tanh ω′ −+ 2T ) + ∆′ − E′ − ( tanh ω′ +− 2T + tanh ω′ −− 2T )} , ∆03 = − 1 4V ∑ p′ V2(p,p′) { ∆′ + E′ + ( tanh ω′ ++ 2T (19) + tanh ω′ −+ 2T ) − ∆′ − E′ − ( tanh ω′ +− 2T + tanh ω′ −− 2T )} , 1 V ∑ p { 2 − ξs 2E+ ( tanh ω++ 2T + tanh ω−+ 2T ) − ξs 2E− ( tanh ω+− 2T + tanh ω−− 2T )} = ̺, (20) pq m0 + e0 16 ∑ p′ pp′ {( tanh ω′ ++ 2T − tanh ω′ −+ 2T ) + ( tanh ω′ +− 2T − tanh ω′ −− 2T )} = pq m∗ (21) Here ∆′ ± = ∆±(p′,q), ω′ ±,± = ± p′q 2m∗ + E±(p′,q) Eqs. (18)–(21) describe two–gap superfluid states of symmetric nuclear matter with moving condensate and contain one–gap solutions with ∆30 6= 0, ∆03 ≡ 0 (TS pairing) and ∆30 ≡ 0, ∆03 6= 0 (ST pairing) as some particular cases. We shall analyze Eqs. (18)–(21) using the simplifying assumption, that FL amplitudes V1, V2 are not equal to zero only in a narrow layer near the Fermi surface: \|ξs\| ≤ θ, θ ≪ εF (further we set θ = 0.1εF ). First we shall find the dependence of the order pa- rameters ∆30(p = pF), ∆03(p = pF) from the superfluid momentum at zero temperature. We begin our analy- sis with finding one–gap solutions of the self–consistent equations. Results of numerical determination of the en- ergy gap are presented in the Fig. 1a. It is seen, that a general tendency is quite clear: at low superfluid mo- menta the energy gap retains its constant value and then rapidly decreases and vanishes at some critical point. This means, that one–gap superfluid states with moving condensate will disappear at large enough superfluid mo- menta. However, for TS pairing of nucleons, interacting via SkM∗ effective potential, the phase curve has an inter- esting peculiarity, namely, the energy gap demonstrates nontrivial double–valued behavior in the region, where it sharply falls. Such behavior differs from the ordinary one–valued behavior of the energy gap, as, e.g., in the case of TS pairing and SkP effective interaction. To un- derstand this difference, we have determined the effective mass m∗ as a function of superfluid momentum, Fig. 1b. For the SkP potential the mass m∗ is equal to the bare mass m0 of a nucleon practically for all superfluid mo- menta. Unlike to this behavior, the effective mass m∗ 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 (a) ts (SkP) ts (SkM * ) st (SkM * ) ∆ ( M e V ) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 890 900 910 920 930 940 (b) ts (SkP) ts (SkM * )st (SkM * ) m * ( M e V ) q/p f FIG. 1: Energy gap (top) and effective mass m∗ (bottom) vs. superfluid momentum for different types of pairing and Skyrme forces at ̺ = 0.03 fm−3 and zero temperature. for the SkM∗ potential rapidly decreases close to the re- gion of phase transition to the normal state. Since the mass m∗ enters into Eqs. (18)–(21) only through the ratio q/m∗, descent of the effective mass leads to the increase of the effective shift between the centers of proton and neutron Fermi spheres, on which the paired proton and neutron lie. This gives the possibility to the appearance of the second solution for the energy gap. According to Eq. (13), the parameter e0 of the normal FL amplitude U0 determines the sign before the sum in the l.h.s. of Eq. (21), and, hence, determines whether the mass m∗ will be greater (if e0 < 0) or less (if e0 > 0), than the bare mass m0 (if corresponding sum is nonzero). For the SkM∗ potential e0 > 0 while for the SkP potential e0 < 0, that explains the difference in the behavior of the effective mass for these two forces. However, diminishing behavior of the effective mass m∗ does not guarantee that the energy gap will have two–valued behavior as a func- 4 tion of superfluid momentum. In Fig. 1a we have plotted also the dependence of the energy gap for ST pairing of nucleons and SkM∗ potential. As follows from here, in spite of correctness of the inequality e0 > 0, the energy gap demonstrates the usual one–valued behavior. Since ST coupling constant in the Skyrme model is always less then TS one [6], one can conclude, that the pairing inter- action should be strong enough for the second solution to be developed. The possible two–valued behavior of the energy gap has important consequence. If to go from the region of large enough superfluid momenta in the di- rection of low momenta, then at some critical point the energy gap will arise by a jump, and, hence, the phase transition to the superfluid phase will be of a first order. Now we go to the study of two–gap superfluid states when both order parameters, ∆30 and ∆03, are not equal to zero. Results of numerical determination of the or- der parameters ∆30(q), ∆03(q) on the base of Eqs. (18)– (21) are presented in Fig. 2. Here tsst(ts) and tsst(st) 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.0 0.2 0.4 0.6 0.8 1.0 1.2 SkM * tsst (st) tsst (ts) st ts ∆ ( M e V ) q/p f FIG. 2: Order parameters ∆30, ∆03 vs. superfluid momentum for SkM∗ force at ̺ = 0.03 fm−3 and zero temperature. are notations for the dependencies of TS and ST order parameters in the TS–ST solution of the self–consistent equations. As seen from Fig. 2, TS–ST solutions appear as a result of branching from one–gap ST solution (in the branching point ∆30 = 0, ∆03 = ∆st 03, ∆st 03 being one– gap ST solution). Note that the self–consistent equations have two–gap solutions in the case of SkM∗ potential, but have no such solutions for the SkP potential. As clarified in the Ref. [6], for the existence of TS–ST solutions it is necessary that TS and ST coupling constants must be of the same order of magnitude. However, this condition is broken for the SkP potential, where TS coupling constant is much larger than ST one. Since we use density dependent effective interaction, it allows us to study also the dependence of the order pa- rameters from density of nuclear matter. The results of numerical determination of one–gap and two–gap solu- tions of self–consistent equations at fixed superfluid mo- mentum are presented in Fig. 3. As one can see, super- 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 SkM * st (tsst) ts (tsst) ts st ρ (fm -3 ) ∆ ( M e V ) FIG. 3: Order parameters ∆30, ∆03 vs. density for SkM∗ force at q/pF = 0.04 and zero temperature. fluidity with finite superfluid momentum exists in finite density region (̺1(q), ̺2(q)), excluding some vicinity of the point ̺ = 0 (for TS pairing the left point ̺1(q) is very close to ̺ = 0). The most important peculiarity, i.e., the double–valued behavior of the energy gap for TS pairing in the case of SkM∗ potential is preserved for the given ratio q/pF. For other types of pairing and Skyrme forces, considered earlier (including the SkP potential, which is not shown in Fig. 3), the order parameter for one–gap so- lutions has one–valued behavior. In the case of two–gap solutions the mechanism of their appearance is similar to the considered above, i.e., it is branching from one–gap ST solution. However, in general case behavior of the energy gap as a function of density is more complicated. In Fig. 4 we plot the dependence of the energy gap in the case of TS pairing and SkM∗ interaction for the set of fixed val- ues of the ratio q/pF . It is seen, that behavior of the phase curves may be one–valued or two–valued, that de- pends on the value of the ratio q/pF . At small enough superfluid momentum (e.g., for q = 0.009pf) the gap be- haves as a one–valued function of density; for the ratios q/pF , larger than some critical value q1/pF , the gap has two–valued behavior in the region close to the right crit- ical point ̺2(q). Further increase of the ratio q/pF leads to formation of the second part of the phase curve with double–valued behavior in the region close to the left crit- ical point ̺1(q) (e.g., for q = 0.1pf). When q increases, the regions with double–valued behavior of the gap begin to approach and at q = qc (qc ≈ 0.106pF ) it takes place contiguity of the regions with two solutions. For q > qc the phase curves are separated from the density axis and turn into the closed oval curves. Under further increase of q the dimension of the oval curves is reduced and at some 5 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 SkM * 0.1065 0.1 0.06 0.04 0.02 q/p f =0.009 ∆ ( M e V ) ρ (fm -3 ) FIG. 4: Energy gap vs. density in the case of TS pairing and SkM∗ force for different values of the ratio q/pF . q = qm the oval curves shrink to a point (qm ≈ 0.108pF ). For the values q > qm TS superfluidity of nuclear mat- ter vanishes. Thus, in the range 0 < q < q1 the gap is one–valued function of density, in the range q1 < q < q2 the phase curve ∆30 = ∆30(̺) has one part with double– valued behavior, for q2 < q < qc it contains two distinct parts with double–valued behavior and for qc < q < qm the gap has a universal double–valued behavior. Since we have a few solutions of self–consistent equa- tions, it is necessary to check, which solution is thermo- dynamically favorable. Calculations show that due to the large size of the gap in TS pairing channel the free energy of the corresponding state as a function of momen- tum or density much smaller then for the case of ST and TS–ST pairing. Hence, TS superfluid state wins compe- tition for the thermodynamic stability. If to compare the free energies of two different branches, corresponding to double–valued behavior of the energy gap in TS pairing channel, then the branch with larger size of the gap will be thermodynamically favorable. In summary, we studied superfluidity of symmet- ric nuclear matter with moving condensate in the FL model with density dependent Skyrme effective interac- tion (SkM∗, SkP forces). It has been considered the case, when the quasiparticle excitation spectrum is four–fold split due to finite superfluid momentum and coupling of T = 0 and T = 1 pairing channels. Apart from the renormalization of the chemical potential and bare mass of a nucleon, taking into account of the normal FL am- plitude leads to appearance of additional effective mass m∗ in the linear on superfluid momentum term in sin- gle particle energy. It is shown that at zero temperature the energy gap in TS pairing channel for SkM∗ poten- tial demonstrates two–valued behavior as a function of superfluid momentum. This is caused by the decreasing behavior of the effective mass m∗ close to the region of the phase transition to the normal state and strong enough interaction in TS pairing channel. The behavior of the energy gap as a function of density in TS pairing channel in general case is more complicated and under increase of superfluid momentum it changes from one–valued to universal two–valued character (until superfluidity disap- pears at some critical momentum). Two–gap solutions of self–consistent equations, corresponding to the case when both TS and ST order parameters are not equal to zero, appear as a result of branching from one–gap ST solution. Calculation of the free energy shows that TS superfluid state is thermodynamically most preferable state. In the case of double–valued behavior the gap changes in the critical point by a jump and, hence, the phase transition from the LOFF phase to the normal state will be of a first order (in superfluid momentum or density). Since the possible two–valued behavior of the gap will be preserved for small asymmetry, this will be also true for weakly asymmetric nuclear matter, that differs from the picture of a second order phase transition, considered in [3]. The author is grateful for discussions and useful com- ments to S. Peletminsky, G. Roepke, H.-J. Schulze and A. Yatsenko. The financial support of STCU (grant No. 1480) is acknowledged. [1] A.I. Larkin, and Yu. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964) [Sov. Phys. JETP 20, 762 (1965)]. [2] P. Fulde and R. A. Ferrell, Phys. Rev. 135, 550 (1964). [3] A. Sedrakian, Phys. Rev. C 63, 025801 (2001). [4] M. Alford, J. Bowers, and K. Rajagopal, Phys. Rev. D 63, 074016 (2001). [5] R. Rapp, E. V. Shuryak, and I. Zahed, Phys. Rev. D 63, 034008 (2001). [6] A.I. Akhiezer, A.A. Isayev, S.V. Peletminsky, and A.A. Yatsenko, Phys. Lett. B 451, 430 (1999). [7] H. Kucharek and P. Ring, Z. Phys. A 339, 23 (1991). [8] A.I. Akhiezer, V.V. Krasil’nikov, S.V. 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Superfluid states with finite momentum of Cooper pairs in nuclear matter

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