Is a field-induced ferromagnetic phase transition in the magnetar core actually possible?

Spin polarized states in dense neutron matter with BSk20 Skyrme force are considered in magnetic fields up to 10²⁰ G. It is shown that the appearance of the longitudinal instability in a strong magnetic field prevents the formation of a fully spin polarized state in neutron matter, and only the stat...

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Дата:	2012
Автори:	Isayev, А.А., Yang, J.
Формат:	Стаття
Мова:	English
Опубліковано:	Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2012
Назва видання:	Вопросы атомной науки и техники
Теми:	Section A. Quantum Field Theory
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/106843
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Цитувати:	Is a field-induced ferromagnetic phase transition in the magnetar core actually possible? / А.А. Isayev, J. Yang // Вопросы атомной науки и техники. — 2012. — № 1. — С. 11-15. — Бібліогр.: 13 назв. — англ.

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spelling	irk-123456789-1068432016-10-08T03:01:40Z Is a field-induced ferromagnetic phase transition in the magnetar core actually possible? Isayev, А.А. Yang, J. Section A. Quantum Field Theory Spin polarized states in dense neutron matter with BSk20 Skyrme force are considered in magnetic fields up to 10²⁰ G. It is shown that the appearance of the longitudinal instability in a strong magnetic field prevents the formation of a fully spin polarized state in neutron matter, and only the states with moderate spin polarization can be developed. Рассматриваются спиновоупорядоченые состояния в плотной нейтронной материи с силой Скрима BSk20 в магнитных полях вплоть до 10²⁰ Гс. Показано, что появление продольной неустойчивости в сильном поле препятствует формированию полностью поляризованного состояния и возможны только состояния с умеренной поляризацией. Розглядаються спінововпорядковані стани в густій нейтронній матерії з силою Скірма BSk20 в магнітних полях до 10²⁰ Гс. Показане, що поява поздовжньої нестійкості в сильному магнітному полі перешкоджає формуванню повністю поляризованого стану і можливі тільки стани з помірною поляризацією. 2012 Article Is a field-induced ferromagnetic phase transition in the magnetar core actually possible? / А.А. Isayev, J. Yang // Вопросы атомной науки и техники. — 2012. — № 1. — С. 11-15. — Бібліогр.: 13 назв. — англ. 1562-6016 PACS: 21.65.Cd, 26.60.-c, 97.60.Jd, 21.30.Fe http://dspace.nbuv.gov.ua/handle/123456789/106843 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	Section A. Quantum Field Theory Section A. Quantum Field Theory
spellingShingle	Section A. Quantum Field Theory Section A. Quantum Field Theory Isayev, А.А. Yang, J. Is a field-induced ferromagnetic phase transition in the magnetar core actually possible? Вопросы атомной науки и техники
description	Spin polarized states in dense neutron matter with BSk20 Skyrme force are considered in magnetic fields up to 10²⁰ G. It is shown that the appearance of the longitudinal instability in a strong magnetic field prevents the formation of a fully spin polarized state in neutron matter, and only the states with moderate spin polarization can be developed.
format	Article
author	Isayev, А.А. Yang, J.
author_facet	Isayev, А.А. Yang, J.
author_sort	Isayev, А.А.
title	Is a field-induced ferromagnetic phase transition in the magnetar core actually possible?
title_short	Is a field-induced ferromagnetic phase transition in the magnetar core actually possible?
title_full	Is a field-induced ferromagnetic phase transition in the magnetar core actually possible?
title_fullStr	Is a field-induced ferromagnetic phase transition in the magnetar core actually possible?
title_full_unstemmed	Is a field-induced ferromagnetic phase transition in the magnetar core actually possible?
title_sort	is a field-induced ferromagnetic phase transition in the magnetar core actually possible?
publisher	Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate	2012
topic_facet	Section A. Quantum Field Theory
url	http://dspace.nbuv.gov.ua/handle/123456789/106843
citation_txt	Is a field-induced ferromagnetic phase transition in the magnetar core actually possible? / А.А. Isayev, J. Yang // Вопросы атомной науки и техники. — 2012. — № 1. — С. 11-15. — Бібліогр.: 13 назв. — англ.
series	Вопросы атомной науки и техники
work_keys_str_mv	AT isayevaa isafieldinducedferromagneticphasetransitioninthemagnetarcoreactuallypossible AT yangj isafieldinducedferromagneticphasetransitioninthemagnetarcoreactuallypossible
first_indexed	2025-07-07T19:06:41Z
last_indexed	2025-07-07T19:06:41Z
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fulltext	IS A FIELD-INDUCED FERROMAGNETIC PHASE TRANSITION IN THE MAGNETAR CORE ACTUALLY POSSIBLE? A.A. Isayev 1, 2∗and J. Yang 3† 1National Science Center “Kharkov Institute of Physics and Technology”, 61108, Kharkov, Ukraine 2Kharkov National University, 61077, Kharkov, Ukraine 3Department of Physics and the Institute for the Early Universe, Ewha Womans University, Seoul 120-750, Korea (Received October 20, 2011) Spin polarized states in dense neutron matter with BSk20 Skyrme force are considered in magnetic fields up to 1020 G. It is shown that the appearance of the longitudinal instability in a strong magnetic field prevents the formation of a fully spin polarized state in neutron matter, and only the states with moderate spin polarization can be developed. PACS: 21.65.Cd, 26.60.-c, 97.60.Jd, 21.30.Fe 1. INTRODUCTION. BASIC EQUATIONS Magnetars are strongly magnetized neutron stars with emissions powered by the dissipation of mag- netic energy. The magnetic field strength at the sur- face of a magnetar is of about 1014-1015 G. Such huge magnetic fields can be inferred from observations of magnetar periods and spin-down rates, or from hy- drogen spectral lines. In the interior of a magnetar the magnetic field strength could reach values up to 1020 G [1]. Then the issue of interest is the behavior of neutron star matter, which further will be approx- imated by pure neutron matter, in a strong magnetic field [2]. In particular, a scenario is possible in which a field-induced ferromagnetic phase transition occurs in the magnetar core. This idea was explored in the recent research [3], where it was shown that a fully spin polarized state in neutron matter could be formed in the magnetic field larger than 1019 G. Note, however, that the breaking of the O(3) rota- tional symmetry in such ultrastrong magnetic fields results in the anisotropy of the total pressure, hav- ing a smaller value along than perpendicular to the field direction [1, 4]. The possible outcome could be the gravitational collapse of a magnetar along the magnetic field, if the magnetic field strength is large enough. Thus, exploring the possibility of a field-induced ferromagnetic phase transition in neu- tron matter in a strong magnetic field, the effect of the pressure anisotropy has to be taken into ac- count because this kind of instability could prevent the formation of a fully polarized state in neutron matter. In the present study, we determine thermo- dynamic quantities of strongly magnetized neutron matter taking into account this effect. Let us stop on the basic equations of the theory. The normal (nonsuperfluid) states of neutron matter are described by the normal distribution function of neutrons fκ1κ2 = Tr �a+ κ2 aκ1 , where κ ≡ (p, σ), p is momentum, σ is the projection of spin on the third axis, and � is the density matrix of the system [5, 6]. The energy of the system is specified as a functional of the distribution function f , E = E(f), and deter- mines the single particle energy [7, 8] εκ1κ2(f) = ∂E(f) ∂fκ2κ1 . (1) The self-consistent matrix equation for determining the distribution function f follows from the minimum condition of the thermodynamic potential [7] and is f = {exp(Y0ε + Yi · μnσi + Y4) + 1}−1 (2) ≡ {exp(Y0ξ) + 1}−1 . Here the quantities ε, Yi and Y4 are matrices in the space of κ variables, with ( Yi,4 ) κ1κ2 = Yi,4δκ1κ2 , Y0 = 1/T , Yi = −Hi/T and Y4 = −μ0/T being the Lagrange multipliers, μ0 being the chemical poten- tial of neutrons, and T the temperature. In Eq. (2), μn = −1.9130427(5)μN is the neutron magnetic mo- ment (μN being the nuclear magneton), σi are the Pauli matrices. Further it will be assumed that the third axis is directed along the external magnetic field H. Given the possibility for alignment of neutron spins along or opposite to the magnetic field H, the normal distrib- ution function of neutrons and the matrix quantity ξ ∗E-mail address: isayev@kipt.kharkov.ua †E-mail address: jyang@ewha.ac.kr PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY, 2012, N 1. Series: Nuclear Physics Investigations (57), p. 11-15. 11 (which we will also call a single particle energy) can be expanded in the Pauli matrices σi in spin space f(p) = f0(p)σ0 + f3(p)σ3, (3) ξ(p) = ξ0(p)σ0 + ξ3(p)σ3. (4) The distribution functions f0, f3 satisfy the nor- malization conditions: 2 V ∑ p f0(p) = �, (5) 2 V ∑ p f3(p) = �↑ − �↓ ≡ Δ�. (6) Here � = �↑ + �↓ is the total density of neutron mat- ter, �↑ and �↓ are the neutron number densities with spin up and spin down, respectively. The quantity Δ� may be regarded as the neutron spin order pa- rameter which determines the magnetization of the system M = μnΔ�. The magnetization may con- tribute to the internal magnetic field B = H + 4πM. However, we will assume, analogously to the previ- ous studies [2], that, because of the tiny value of the neutron magnetic moment, the contribution of the magnetization to the inner magnetic field B re- mains small for all relevant densities and magnetic field strengths, and, hence, B ≈ H. In order to get the self–consistent equations for the components of the single particle energy, one has to set the energy functional of the system. It represents the sum of the matter and field energy contributions E(f, H) = Em(f) + H2 8π V . (7) The matter energy is the sum of the kinetic and Fermi-liquid interaction energy terms [5, 6] Em(f) = E0(f) + Eint(f), (8) E0(f) = 2 ∑ p ε 0(p)f0(p), Eint(f) = ∑ p {ε̃0(p)f0(p) + ε̃3(p)f3(p)}, where ε̃0(p) = 1 2V ∑ q Un 0 (k)f0(q), k = p− q 2 , (9) ε̃3(p) = 1 2V ∑ q Un 1 (k)f3(q). (10) Here ε 0(p) = p 2 2m0 is the free single particle spectrum, m0 is the bare mass of a neutron, Un 0 (k), Un 1 (k) are the normal Fermi liquid (FL) amplitudes, and ε̃0, ε̃3 are the FL corrections to the free single particle spec- trum. Taking into account Eqs. (1),(2) and (8), ex- pressions for the components of the single particle energy read ξ0(p) = ε 0(p) + ε̃0(p) − μ0, ξ3(p) = −μnH + ε̃3(p). (11) In Eqs. (11), the quantities ε̃0, ε̃3 are the function- als of the distribution functions f0, f3 which, using Eqs. (2) and (3), can be expressed, in turn, through the quantities ξ: f0 = 1 2 {n(ω+) + n(ω−)}, (12) f3 = 1 2 {n(ω+) − n(ω−)}, (13) where n(ω±) = {exp(Y0ω±) + 1}−1, ω± = ξ0 ± ξ3. Thus, Eqs. (11)–(13) form the self-consistency equations for the components of the single particle energy, which should be solved jointly with the nor- malization conditions (5), (6). The pressures (longitudinal and transverse with respect to the direction of the magnetic field) in the system are related to the diagonal elements of the stress tensor whose explicit expression reads [9] σik = [̃ f − � ( ∂ f̃ ∂� ) H,T ] δik + HiBk 4π . (14) Here f̃ = fH − H2 4π , (15) fH = 1 V (E −TS)−HM is the Helmholtz free energy density. For the isotropic medium, the stress ten- sor (14) is symmetric. The transverse pt and longitu- dinal pl pressures are determined from the formulas pt = −σ11 = −σ22, pl = −σ33. At zero temperature, using Eqs. (7), (14), one can get the approximate expressions pt = � (∂em ∂� ) H − em + H2 8π , (16) pl = � (∂em ∂� ) H − em − H2 8π , (17) where em is the matter energy density, and we disre- garded the terms proportional to M . In ultrastrong magnetic fields, the quadratic on the magnetic field term (the Maxwell term) will be dominating, leading to increasing the transverse pressure and to decreas- ing the longitudinal pressure. Hence, at some critical magnetic field, the longitudinal pressure vanishes, re- sulting in the longitudinal instability of neutron mat- ter. The question then is: What is the magnitude of the critical field and the corresponding maximum de- gree of spin polarization in neutron matter? 2. EOS OF DENSE NEUTRON MATTER IN A STRONG MAGNETIC FIELD In numerical calculations, we utilize the BSk20 Skyrme force [10] constrained such as to avoid the spontaneous spin instability of neutron matter at densities beyond the nuclear saturation density and 12 to reproduce a microscopic EoS of nonpolarized neu- tron matter. Expressions for the normal FL ampli- tudes in Eqs. (9),(10) in terms of the parameters of the Skyrme interaction are given in Ref. [11]. Now we present the results of the numerical solution of the self-consistency equations. Fig. 1 shows the spin polarization parameter Π = Δ� � of neutron matter as a function of the magnetic field H at two differ- ent values of the neutron matter density, � = 3�0 and � = 4�0, which can be relevant for the magnetar core. It is seen that the impact of the magnetic field remains small up to the field strength 1017 G. The larger the density is, the smaller the effect produced by the magnetic field on spin polarization of neutron matter. Fig. 1. Neutron spin polarization parameter as a function of the magnetic field H for the Skyrme force BSk20 at zero temperature and fixed values of the density, � = 3�0 and � = 4�0. The vertical arrows indicate the maximum magnitude of spin polarization attainable at the given density, see further details in the text At the magnetic field H = 1018 G, usually con- sidered as the maximum magnetic field strength in the core of a magnetar (according to a scalar vir- ial theorem, see Ref. [1] and references therein), the magnitude of the spin polarization parameter doesn’t exceed 33% at � = 3�0 and 18% at � = 4�0. However, the situation changes if the larger magnetic fields are allowable: With further increasing the magnetic field strength, the magnitude of the spin polarization pa- rameter increases till it reaches the limiting value Π = −1, corresponding to a fully spin polarized state. For example, this happens at H ≈ 1.25 · 1019 G for � = 3�0 and at H ≈ 1.98 · 1019 G for � = 4�0, i.e., certainly, for magnetic fields larger than 1019 G. Nev- ertheless, we should check whether the formation of a fully spin polarized state in a strong magnetic field is actually possible by calculating the anisotropic pres- sure in dense neutron matter. The meaning of the vertical arrows in Fig. 1 is explained later in the text. Fig. 2. Pressures, longitudinal (descending bran- ches) and transverse (ascending branches), as functions of the magnetic field H for the Skyrme force BSk20 at zero temperature and fixed values of the density, � = 3�0 and � = 4�0 (a). Same as in the top panel but for the normalized difference between the transverse and longitudinal pressures (b) Fig. 2, a shows the pressures (longitudinal and transverse) in neutron matter as functions of the magnetic field H at the same densities, � = 3�0 and � = 4�0. First, it is clearly seen that up to some threshold magnetic field the difference between transverse and longitudinal pressures is unessential that corresponds to the isotropic regime. Beyond this threshold magnetic field strength, the anisotropic regime holds for which the transverse pressure in- creases with H while the longitudinal pressure de- creases. The longitudinal pressure vanishes at some critical magnetic field Hc marking the onset of the longitudinal collapse of a neutron star. For example, Hc ≈ 1.56 · 1018 G at � = 3�0 and Hc ≈ 2.42 · 1018 G at � = 4�0. In all cases under consideration, this critical value doesn’t exceed 1019 G. The magnitude of the spin polarization parameter Π cannot also exceed some limiting value correspond- ing to the critical field Hc. These maximum values of the Π’s magnitude are shown in Fig. 1 by the ver- tical arrows. In particular, Πc ≈ −0.46 at � = 3�0 and Πc ≈ −0.38 at � = 4�0. As can be inferred from these values, the appearance of the negative longitu- dinal pressure in an ultrastrong magnetic field pre- vents the formation of a fully spin polarized state in the core of a magnetar. Therefore, only the onset 13 of a field-induced ferromagnetic phase transition, or its near vicinity, can be catched under increasing the magnetic field strength in dense neutron matter. A complete spin polarization in the magnetar core is not allowed by the appearance of the negative pres- sure along the direction of the magnetic field, con- trary to the conclusion of Ref. [3] where the pressure anisotropy in a strong magnetic field was disregarded. Fig. 2, b shows the difference between the trans- verse and longitudinal pressures normalized to the value of the pressure p0 in the isotropic regime (which corresponds to the weak field limit with pl = pt = p0) being δ = pt−pl p0 . Applying for the transition from the isotropic regime to the anisotropic one the criterion δ � 1, the transition occurs at the threshold field Hth ≈ 1.15·1018 G for � = 3�0 and Hth ≈ 1.83·1018 G for � = 4�0. In all cases under consideration, the threshold field Hth is larger than 1018 G, and, hence, the isotropic regime holds for the fields up to 1018 G. The vertical arrows in Fig. 2, b indicate the points corresponding to the onset of the longitudinal insta- bility in neutron matter. The maximum allowable normalized splitting of the pressures corresponding to the critical field Hc is δ ∼ 2. Fig. 3. Same as in Fig. 2 but for: the Helmholtz free energy density of the system (a); the ratio of the magnetic field energy density to the Helmholtz free energy density of the system (b) Fig. 3, a shows the Helmholtz free energy density of the system as a function of the magnetic field H . It is seen that the magnetic fields up to H ∼ 1018 G have practically small effect on the Helmholtz free energy density fH , but beyond this field strength the contribution of the magnetic field energy to the free energy fH rapidly increases with H . However, this in- crease is limited by the values of the critical magnetic field corresponding to the onset of the longitudinal in- stability in neutron matter. The respective points on the curves are indicated by the vertical arrows. Fig. 4. The Helmholtz free energy density of the system as a function of: the transverse pressure pt (a), the longitudinal pressure pl for the Skyrme force BSk20 at zero temperature and fixed values of the density, � = 3�0 and � = 4�0 (b) Fig. 3, b shows the ratio of the magnetic field en- ergy density ef = H2 8π to the Helmholtz free energy density at the same assumptions as in Fig. 2. The in- tersection points of the respective curves in this panel with the line ef/fH = 0.5 correspond to the magnetic fields at which the matter and field contributions to the Helmholtz free energy density are equal. This happens at H ≈ 1.18 · 1018 G for � = 3�0, and at H ≈ 1.81 · 1018 G for � = 4�0. These values are quite close to the respective values of the threshold field Hth, and, hence, the transition to the anisotropic regime occurs at the magnetic field strength at which the field and matter contributions to the Helmholtz free energy density become equally important. It is also seen from Fig. 3, b that in all cases when the lon- gitudinal instability occurs in the magnetic field Hc the contribution of the magnetic field energy density to the Helmholtz free energy density of the system dominates over the matter contribution. 14 Because of the pressure anisotropy, the EoS of neutron matter in a strong magnetic field is also anisotropic. Fig. 4 shows the dependence of the Helmholtz free energy density fH of the system on the transverse pressure (top panel) and on the longi- tudinal pressure (bottom panel) at the same densities considered above. Since in an ultrastrong magnetic field the dominant Maxwell term enters the pressure pt and free energy density fH with positive sign and the pressure pl with negative sign, the free energy density fH is the increasing function of pt and de- creasing function of pl. In the bottom panel, the physical region corresponds to the positive values of the longitudinal pressure. The obtained results can be of importance in the structure studies of magnetars. It would be also of interest to extend this research to finite tempera- tures relevant for proto-neutron stars which can lead to a number of interesting effects, such as, e.g., an unusual behavior of the entropy of a spin polarized state [12, 13]. J.Y. was supported by grant 2010-0011378 from Basic Science Research Program through NRF of Ko- rea funded by MEST and by grant R32-10130 from WCU project of MEST and NRF. References 1. E.J. Ferrer, V. de la Incera, J.P. Keith, I. Por- tillo, and P.L. Springsteen. Equation of state of a dense and magnetized fermion system // Phys. Rev. C. 2010, v. 82, 065802, 15 p. 2. A.A. Isayev and J. Yang. Spin-polarized states in neutron matter in a strong magnetic field // Phys. Rev. C. 2009, v. 80, 065801, 7 p. 3. G.H. Bordbar, Z. Rezaei, and A. Montakhab. Investigation of the field-induced ferromagnetic phase transition in spin-polarized neutron mat- ter: A lowest order constrained variational ap- proach // Phys. Rev. C. 2011, v. 83, 044310, 7 p. 4. V.R. Khalilov. Macroscopic effects in cold mag- netized nucleons and electrons with anomalous magnetic moments // Phys. Rev. D. 2002, v. 65, 056001, 6 p. 5. A.A. Isayev and J. Yang. Spin polarized states in strongly asymmetric nuclear matter // Phys. Rev. C. 2004, v. 69, 025801, 8 p. 6. A.A. Isayev. Spin ordered phase transitions in isospin asymmetric nuclear matter // Phys. Rev. C. 2006, v. 74, 057301, 4 p. 7. A.I. Akhiezer, A.A. Isayev, S.V. Peletminsky, A.P. Rekalo, and A.A. Yatsenko. On a theory of superfluidity of nuclear matter based on the Fer- mi-liquid approach // JETP. 1997, v. 85, p. 1-12. 8. A.I. Akhiezer, A.A. Isayev, S.V. Peletminsky, and A.A. Yatsenko. Multi-gap superfluidity in nuclear matter // Phys. Lett. B. 1999, v. 451, p. 430–436. 9. L.D. Landau, E.M. Lifshitz, and L.P. Pitaevskii. Electrodynamics of Continuous Media. New York: Pergamon, 1984, 2nd ed. 10. S. Goriely, N. Chamel, and J.M. Pearson. Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. XII. Stiffness and stability of neutron-star matter // Phys. Rev. C. 2010, v. 82, 035804, 18 p. 11. A.A. Isayev and J. Yang. Phase transition to the state with nonzero average helicity in dense neu- tron matter // JETP Lett. 2010, v. 92, p. 783-787. 12. A.A. Isayev. Finite temperature effects in anti- ferromagnetism of nuclear matter // Phys. Rev. C. 2005, v. 72, 014313, 4 p. 13. A.A. Isayev. Unusual temperature behavior of the entropy of the antiferromagnetic spin state in nuclear matter with an effective finite range inter- action //Phys. Rev.C. 2007, v. 76, 047305, 4 p. �� 1020 �� ! ��"�� # �� $ �� " �� "% � &�� "�'�� "�� (� � �� ( �� (� �� (� ��() " �� (�� (�% �� 1020 �� ! ��"��# * � �� "� �&�� ) ��(�� ( � �� (�� ( ��+% � �&�, $ �� (�� " �� ( � &��( �(�� " � �(�� "�'(, -.

Is a field-induced ferromagnetic phase transition in the magnetar core actually possible?

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