Linear Perturbation Renormalization Group method for Ising-like spin systems

Linear Perturbation Renormalization Group method for Ising-like spin systems

The linear perturbation group transformation (LPRG) is used to study the thermodynamics of the axial next-nearest-neighbor Ising model with four spin interactions (extended ANNNI) in a field. The LPRG for weakly interacting Ising chains is presented. The method is used to study finite field para-fer...

Повний опис

Збережено в:
Бібліографічні деталі
Дата:2013
Автор: Sznajd, J.
Формат: Стаття
Мова:English
Опубліковано: Інститут фізики конденсованих систем НАН України 2013
Назва видання:Condensed Matter Physics
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/121079
Теги: Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Linear Perturbation Renormalization Group method for Ising-like spin systems / J. Sznajd // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13704:1–8. — Бібліогр.: 21 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-121079
record_format dspace
spelling irk-123456789-1210792017-06-14T03:02:55Z Linear Perturbation Renormalization Group method for Ising-like spin systems Sznajd, J. The linear perturbation group transformation (LPRG) is used to study the thermodynamics of the axial next-nearest-neighbor Ising model with four spin interactions (extended ANNNI) in a field. The LPRG for weakly interacting Ising chains is presented. The method is used to study finite field para-ferrimagnetic phase transitions observed in layered uranium compounds, UAs₁-xSex, UPd₂Si₂ or UNi₂Si₂. The above-mentioned systems are made of ferromagnetic layers and the spins from the nearest-neighbor and next-nearest-neighbor layers are coupled by the antiferromagnetic interactions J₁<0 and J₂<0, respectively. Each of these systems exhibits a triple point in which two ordered phases (ferrimagnetic and incommensurate) meet the paramagnetic one, and all undergo the high field phase transition from para- to ferrimagnetic (++-) phase. However, if in UAs₁-xSex the para-ferri phase transition is of the first order as expected from the symmetry reason, in UT₂Si₂ (T=Pd, Ni) this transition seems to be a continuous one, at least in the vicinity of the multicritical point. Within the MFA, the critical character of the finite field para-ferrimagnetic transition at least at one isolated point can be described by the ANNNI model supplemented by an additional, e.g., four-spin interaction. However, in LPRG approximation for the ratio κ = J₂/J₁ around 0.5 there is a critical value of the field for which an isolated critical point also exists in the original ANNNI model. The positive four-spin interaction shifts the critical point towards higher fields and changes the shape of the specific heat curve. In the latter case for the fields small enough, the specific heat exhibits two-peak structure in the paramagnetic phase. Лiнiйне пертурбативне перетворення (LPRG) використовується для вивчення термодинамiки аксiальної моделi Iзинга з наступними до найближчих сусiдами з чотириспiновою взаємодiєю (розширена модель ANNNI) у полi. Представлено LPRG для слабовзаємодiючих ланцюжкiв Iзинга. Метод застосовано до ви-вчення пара-феромагнiтних фазових переходiв у скiнченному полi, що спостерiгаються в шаруватих спо-луках урану UAs₁−xSex, UPd₂Si₂ чи UNi₂Si₂. Вище згаданi системи зробленi з феромагнiтних шарiв i спi-ни з найближчих i наступних до найближчих шарiв є зв’язанi антиферомагнiтними взаємодiями J₁ < 0 i J₂ < 0, вiдповiдно. Кожна з цих систем демонструє потрiйну точку, в якiй двi впорядкованi фази (фе-ромагнiтна i неспiвмiрна) зустрiчаються з парамагнiтною фазою i всi фази зазнають фазового переходу у сильному полi з пара- до феромагнiтної (+ + −) фази. Проте, якщо в UAs₁−xSex є пара-феро фазовий перехiд першого роду, як очiкується з симетрiйних мiркувань, в UT₂Si₂ (T = Pd, Ni) цей перехiд видається неперервним, принаймнi в околi мультикритичної точки. В рамках наближення середнього поля, критич-ний характер пара-феромагнiтного переходу в скiнченному полi, принаймнi в однiй iзольованiй точцi, може бути описаний за допомогою моделi ANNNI, яка доповнена додатковою, наприклад, чотириспiно-вою взаємодiєю. Проте, в наближеннi LPRG для коефiцiєнта κ = J₂/J₁ поблизу 0.5 є критичне значення поля, для якого iзольована критична точка також iснує в оригiнальнiй моделi ANNNI. Позитивна чотири-спiнова взаємодiя зсуває критичну точку до вищих полiв i змiнює форму кривої питомої теплоємностi. В останньому випадку для достатньо малих полiв питома теплоємнiсть демонструє двопiкову структуру в парамагнiтнiй фазi. 2013 Article Linear Perturbation Renormalization Group method for Ising-like spin systems / J. Sznajd // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13704:1–8. — Бібліогр.: 21 назв. — англ. 1607-324X PACS: 75.10.Hk, 75.40.Cx DOI:10.5488/CMP.16.13704 arXiv:1303.5585 http://dspace.nbuv.gov.ua/handle/123456789/121079 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description The linear perturbation group transformation (LPRG) is used to study the thermodynamics of the axial next-nearest-neighbor Ising model with four spin interactions (extended ANNNI) in a field. The LPRG for weakly interacting Ising chains is presented. The method is used to study finite field para-ferrimagnetic phase transitions observed in layered uranium compounds, UAs₁-xSex, UPd₂Si₂ or UNi₂Si₂. The above-mentioned systems are made of ferromagnetic layers and the spins from the nearest-neighbor and next-nearest-neighbor layers are coupled by the antiferromagnetic interactions J₁<0 and J₂<0, respectively. Each of these systems exhibits a triple point in which two ordered phases (ferrimagnetic and incommensurate) meet the paramagnetic one, and all undergo the high field phase transition from para- to ferrimagnetic (++-) phase. However, if in UAs₁-xSex the para-ferri phase transition is of the first order as expected from the symmetry reason, in UT₂Si₂ (T=Pd, Ni) this transition seems to be a continuous one, at least in the vicinity of the multicritical point. Within the MFA, the critical character of the finite field para-ferrimagnetic transition at least at one isolated point can be described by the ANNNI model supplemented by an additional, e.g., four-spin interaction. However, in LPRG approximation for the ratio κ = J₂/J₁ around 0.5 there is a critical value of the field for which an isolated critical point also exists in the original ANNNI model. The positive four-spin interaction shifts the critical point towards higher fields and changes the shape of the specific heat curve. In the latter case for the fields small enough, the specific heat exhibits two-peak structure in the paramagnetic phase.
format Article
author Sznajd, J.
spellingShingle Sznajd, J.
Linear Perturbation Renormalization Group method for Ising-like spin systems
Condensed Matter Physics
author_facet Sznajd, J.
author_sort Sznajd, J.
title Linear Perturbation Renormalization Group method for Ising-like spin systems
title_short Linear Perturbation Renormalization Group method for Ising-like spin systems
title_full Linear Perturbation Renormalization Group method for Ising-like spin systems
title_fullStr Linear Perturbation Renormalization Group method for Ising-like spin systems
title_full_unstemmed Linear Perturbation Renormalization Group method for Ising-like spin systems
title_sort linear perturbation renormalization group method for ising-like spin systems
publisher Інститут фізики конденсованих систем НАН України
publishDate 2013
url http://dspace.nbuv.gov.ua/handle/123456789/121079
citation_txt Linear Perturbation Renormalization Group method for Ising-like spin systems / J. Sznajd // Condensed Matter Physics. — 2013. — Т. 16, № 1. — С. 13704:1–8. — Бібліогр.: 21 назв. — англ.
series Condensed Matter Physics
work_keys_str_mv AT sznajdj linearperturbationrenormalizationgroupmethodforisinglikespinsystems
first_indexed 2025-07-08T19:09:36Z
last_indexed 2025-07-08T19:09:36Z
_version_ 1837107016657534976
fulltext Condensed Matter Physics, 2013, Vol. 16, No 1, 13704: 1–8 DOI: 10.5488/CMP.16.13704 http://www.icmp.lviv.ua/journal Linear perturbation renormalization group method for Ising-like spin systems J. Sznajd Institute for Low Temperature and Structure Research, Polish Academy of Sciences, Wroclaw Received July 4, 2012, in final form January 3, 2013 The linear perturbation group transformation (LPRG) is used to study the thermodynamics of the axial next- nearest-neighbor Ising model with four spin interactions (extended ANNNI) in a field. The LPRG for weakly in- teracting Ising chains is presented. The method is used to study finite field para-ferrimagnetic phase transitions observed in layered uranium compounds, UAs1−x Sex , UPd2Si2 or UNi2Si2. The above-mentioned systems are made of ferromagnetic layers and the spins from the nearest-neighbor and next-nearest-neighbor layers are coupled by the antiferromagnetic interactions J1 < 0 and J2 < 0, respectively. Each of these systems exhibits a triple point in which two ordered phases (ferrimagnetic and incommensurate) meet the paramagnetic one, and all undergo the high field phase transition from para- to ferrimagnetic (++−) phase. However, if in UAs1−x Sex the para-ferri phase transition is of the first order as expected from the symmetry reason, in UT2Si2 (T = Pd, Ni) this transition seems to be a continuous one, at least in the vicinity of the multicritical point. Within the MFA, the critical character of the finite field para-ferrimagnetic transition at least at one isolated point can be described by the ANNNI model supplemented by an additional, e.g., four-spin interaction. However, in LPRG approxima- tion for the ratio κ = J2/J1 around 0.5 there is a critical value of the field for which an isolated critical point also exists in the original ANNNI model. The positive four-spin interaction shifts the critical point towards higher fields and changes the shape of the specific heat curve. In the latter case for the fields small enough, the specific heat exhibits two-peak structure in the paramagnetic phase. Key words: ANNNI model, renormalization group, isolated critical point PACS: 75.10.Hk, 75.40.Cx 1. Introduction The Linear Perturbation Renormalization Group (LPRG) [1] method uses a simple one-dimensional decimation to study universal (critical) and non-universal such as a location of the critical temperature and temperature or field dependence of the properties of thermodynamic quantities of several classical and quantum higher-dimensional models. For the first time, this kind of method was proposed by Suzuki and Takano (ST) [2]. In the ST approach the one-dimensional decimation is combined with the Migdal- Kadanoff (MK) bond moving approximation. The disadvantages of the latter method especially for the quantum systems were discussed by Barma et al. [3] and Castellani et al. [4]. Here, we wish only to remind that the MK approach gives rather poor quantitative results even for the two-dimensional Ising model and there is no possibility to construct any systematic approximation procedure within this method. For example, the MK procedure gives for the Ising model on the square lattice the values of the inverse critical temperature kc ≈ 0.61 whereas the exact value is kc ≈ 0.44 and for the s = 1 2 XY model kc ≈ 1.2 [2] much larger than the value kc ≈ 0.64 estimated from the high-temperature series expansion [5], kc ≈ 0.71(67) found from Monte Carlo simulations by fitting to the exponential law [6, 7] and power law [8], respectively or rotationally invariant non-linear (block) transformation kc ≈ 0.62 [9, 10]. The LPRG method has been proposed for the so-called quasi-one-dimensional magnets made of spin chains with the intrachain coupling k and much weaker interchain coupling k1 < k. However, even for the standard Ising model k1 = k , the LPRG approximation gives the results which are in very good agreement with the © J. Sznajd, 2013 13704-1 http://dx.doi.org/10.5488/CMP.16.13704 http://www.icmp.lviv.ua/journal J. Sznajd exact ones kc ≈ 0.45 . For k > k1 > 0.15k , the deviation from the critical temperature exact values is less than 2% and for 0.5k > k1 > 0.15k even less than 1% [11]. In this paper, the LPRG is used to study the thermodynamics and the existence of a critical point in two-dimensional axial next-nearest-neighbour Ising model with four spin interactions (extended ANNNI) in a field. 2. LPRG We first describe in detail the LPRG approach in the simplest possible case, i.e., the Ising chains with intrachain interaction J coupled by the weak interchain interactions J1 defined by the Hamiltonian H = k ∑ 〈i j 〉 Si , j Si , j+1 +k1 ∑ 〈i j 〉 Si , j Si+1, j , (2.1) where the label i refers to rows and j refers to columns, the factor −1/kBT has already been absorbed in the Hamiltonian (k ≡ J/kBT , k1 ≡ J1/kBT ), and k1 < k. The LPRG approach starts with an exact deci- mation for one-dimensional system. Dividing the spins of the chain into two groups, in the simplest case, s2i+1 (odd spins — survived) and s2i+2 (even spin — decimated) one can write the decimation transfor- mation in the form e H (s2i+1) = Trs2i+2 e H (s2i+1,s2i+2) . (2.2) In each step of the transformation (2.2), every other spin is decimated. The same transformation can be written by using a linear weight operator P (σi , si ) e H ′(σ) = TrsP (σ, s)e H (s) , (2.3) where the weight operator P (σ, s) which couples the original (s) to the new spins (σ) is chosen in a linear form as follows: P (σ, s) = 1 2N N ∏ i=1 (1+σi+1s2i+1) . (2.4) The weight operator P (σ, s) projects the original spin s space onto the space of the effective spins σ. For the one dimensional Ising model with interchain interaction k1 = 0, the transformation (2.2) or equivalently (2.3) can be carried out exactly. So, we can separate the Hamiltonian (2.1) in a manageably exactly unperturbated part H0 containing intrachain interaction k [first term of the Hamiltonian (2.1)], and a remainder H I containing interchain interaction k1 [second term in equation (2.1)]. With the nota- tion z0 = TrsP (σ, s)e H0(s) (2.5) and 〈A〉0 = 1 z0 Trs AP (σ, s)e H0(s) (2.6) the transformation (2.3) can be written as H ′ (σ) = ln z0 + ln〈e H I (s) 〉, (2.7) with the following cumulant expansion for ln〈eH I (s)〉 [12] ln〈e H I (s) 〉 = 〈H I 〉0 + 1 2! ( 〈H I 2 〉0 −〈H I 〉 2 0 ) . (2.8) The idea of LPRG in two dimensions is presented in figure 1. The chains are divided into two black and white groups. In each renormalization step, every other spin from a black chain is decimated and in a white chain all spins are removed. As a result, one gets the system of effective spins σ. The single chain “partial” partition function z0 can be easily found as [13] z0 = (cosh2k +1)+ (cosh 2k −1)σi , j σi , j+1 , (2.9) 13704-2 LPRG for Ising-like systems Figure 1. The LPRG procedure for weakly interacting chains in two dimensions. and ln z0 = 1 2 (ln2+ ln cosh2k)+ 1 2 σi , j σi , j+1 ln cosh2k. (2.10) To evaluate the cumulants (2.8), one has to know the averages 〈si , j 〉 and 〈si , j . . . si , j+n 〉 from the black (decimated) and white (removed) rows. For the spins from the black rows (figure 1) 〈s1,1〉 =σ1,1 , 〈s1,2〉 = 1 2 (σ1,1 +σ1,2) tanh 2k, 〈s1,3〉 =σ1,2 , (2.11) and 〈s1,1s1,2〉 = 1 2 (1+σ1,1σ1,2) tanh 2k, 〈s1,1s1,3〉 =σ1,1σ1,2 . (2.12) For the spins from the white rows 〈s2, j 〉 = 0, 〈s2, j s2, j+n〉 = tanh n k. (2.13) Now it is relatively easy to find the renormalized Hamiltonian H ′(σ) (2.7) in the form H ′ (σ) =G(ki )+k ′ ∑ σi , j σi , j+1 +k ′ 1 ∑ σi , j σi+1, j +Ω(σ), (2.14) where G(ki ) is a constant (independent of effective spins σ) term which can be used to calculate the free energy per site according to the formula f = ∞ ∑ n=1 G(k(n) i ) 3n , (2.15) and “n” numbers the LPRG steps.Ω(σ) denotes the additional interactions generated eventually by LPRG transformation. The approach presented above can be also used to consider a quantum spin model or an interacting electron model. However, in these cases the LPRG does not start with an exact but with an approximate decimation for one-dimensional systems [2]. We should also emphasize that generally, the approach is relevant for higher temperatures. Using LPRG one can show the existence of the critical point, and can obtain the location of a transition point, if any, and the temperature or field dependences of the thermo- dynamic quantities. 3. Extended ANNNI model in a field There exists a class of compounds with a layered structure and strong c-axial anisotropy which can be described by the Ising-like ANNNI model or its extensions. For example, models of this kind have been proposed to describe the phase diagrams of the uranium compounds: UAs1−xSex with x < 0.1 [14] or UT2Si2 (T= Pd, Ni) [15, 16] made of ferromagnetic layers which comprise three ordered phases: anti- ferromagnetic (+−+−), ferrimagnetic (++−) and incommensurate. Each of this system exhibits a triple 13704-3 J. Sznajd point in which two ordered phases (ferrimagnetic and incommensurate) meet the paramagnetic one, and all undergo the high field phase transition from paramagnetic to ferrimagnetic (++−) phase. However, if in UAs1−xSex the para-ferri phase transition is of the first order, in UT2Si2 this transition, at least in the vicinity of the multicritical point, seems to be a continuous one. For the symmetry reason, the para- ferrimagnetic (++−) phase transition in the presence of an external field should be discontinuous and a question has arised if the ANNNI model can exhibit in such a case a critical transition. Within the MFA it has been shown that the ANNNI model should be supplemented by some additional, e.g., four-spin inter- action to exhibit an isolated critical point [17]. In order to answer this question, we have used the LPRG to study the extended ANNNI model in a field H = k ∑ 〈i j 〉 Si , j Si , j+1 +k1 ∑ 〈i j 〉 Si , j Si+1, j +k2 ∑ 〈i j 〉 Si , j Si+2, j (3.1) +k4 ∑ 〈i j 〉 Si , j Si+1, j Si , j+1Si+1, j+1 +h ∑ 〈i〉 Si . In the ground state in zero field, the standard ANNNI model with k4 = 0, k1,k2 < 0 and κ≡ k2/k1 = 0.5 exhibits a multicritical point where antiferromagnetic phases (+−+−) and (++−−) meet a ferrimagnetic phase with the spins of the ferromagnetic planes ordered in the sequence (++−) (〈2,1〉 phase in the notation of the paper [16]). The external magnetic field extends the (++−) phase in the (κ, H) plane [19]. All magnetic structures of UPd2Si2 as well as UNi2Si2 and UAs1−xSex (0 É x É 0.1) in the fields along c axis exhibit ferromagnetic layers with moments perpendicular to the layers. So, only the magnetic order between the ferromagnetic layers is of interest. Consequently, it seems that to understand the main feature of the paramagnetic-ferrimagnetic (++−) phase transition one can confine oneself to consider the two dimensional (2D) problem. Thus, in the present paper we use the LPRGmethod to study the extended ANNNI model in two dimensions. The idea of the LPRG for ANNNI-type model in 2D with the following projector for one decimated row P (σ, s) = 1 4 (1+σ1,1S1,3)(1+σ1,2S1,6) (3.2) is presented in figure 2. The full circles represent the spins which survive in the decimation procedure. According to figure 2, in each step of the RG transformation every other row (“even row”) is removed, and from odd rows every third spin survives. The cluster presented in figure 2 can be used to study sys- tems made of ferromagnetic or antiferromagnetic chains but it does not preserve the antiferromagnetic and much less modulated order between the chains. However, this cluster preserves the ferrimagnetic (++−) ordering which is a matter of interest in this paper. In order to take into account both possibilities (antiferromagnetic and ferrimagnetic orders) one should consider a cluster of 11 chains and for three Figure 2. Cluster (8-10-8-10-8) used to get renormalized Hamiltonian of the extended ANNNI model in the LPRG procedure. 13704-4 LPRG for Ising-like systems possibilities (antiferromagnetic +−+−, antiferromagnetic ++−− , and ferrimagnetic orders) 16 chains. Although calculation with such clusters is straightforward it becomes quite involved. It should be em- phasized once more that the LPRG with the cluster presented in figure 2 is suitable to describe, except for para-ferro, only para-ferrimagnetic (++−) phase transitions. Such phase transitions can occur in both 2D and 3D systems. Thus, we have confined ourselves to consider 2D problem although the incommensurate phases observed in the mentioned uranium compounds can be described by 3D ANNNI model. As usual, the RG procedure leads from the set of original parameters of the Hamiltonian (2.3) (h, k , k1, k2, k4) to the set of renormalized parameters (h′, k ′, k ′ 1 , k ′ 2 , k ′ 4 ) h′ σ i j , k ′ σ ( j ) 1 σ ( j ) 2 , k ′ 1σ j 1 σ ( j+1) 1 , k ′ 2σ ( j ) 1 σ ( j+2) 1 , k ′ 4σ ( j ) 1 σ ( j+1) 1 σ ( j ) 2 σ ( j+1) 2 . (3.3) However, in the lowest nontrivial order of the cumulant expansion, which in our case is the second order, several new odd and even interactions come into play. For the sake of simplicity, we will take into account the contributions up to the second order only from the one- and two-spin interactions, contributions to the first order from three- and four-spin interactions, and neglect any contributions from the higher order spin terms [21]. Accordingly, we consider 17 parameters, i.e., five original (3.3) and 12 generated by the RG transformation. To evaluate the transformation (2.8) one has to know the averages of spins 〈Si , j 〉 and two spins prod- ucts 〈Si , j Si , j+n〉 from the decimated (“odd”) and removed (“even”) rows. The chain averages for the spins from decimated rows of the Ising model in a field with the weight operator (3.2) have been presented in paper [9], and for the spins from the removed rows, these averages are known exactly (see for example [18] and references therein). Now, we are able to numerically evaluate the renormalization transforma- tion (2.8) which in our approximation has a form of 17 recursion relations. As usual, in order to deter- mine the critical temperature, one has to find a critical surface which separates in the parameter space the region of attraction of the two stable fixed points, zero temperature, kα = ∞ and infinite tempera- ture kα = 0. The existence of such a surface means that the system can undergo a second order phase transition. Moreover, we can also numerically calculate the free energy per spin collecting the constant terms generated in each step of the iteration process (2.15). In a disordered phase we obtain a rapidly convergent infinite series for the free energy which can be used to find the specific heat as a function of the reduced temperature t = T /J . Let us start with the standard ANNNI model with k = 1,k1 =−0.6,k2 =−0.3 and k4 = 0 [21]. The value of the critical temperature has been determined on the ground of the recursion relation analysis. Using the formula (2.15), the free energy per site and the specific heat as functions of temperature and a field have been found. In figure 3, the temperature dependence of the specific heat for several values of the field is presented. The specific heat divergence corresponding to a critical point is visible for h = 0 and for h around 0.11. For 0 < h < hc ≈ 0.11, the specific heat exhibits a maximum. Unfortunately, the LPRG fails 1.55 1.60 1.65 1.70 1.75 1.80 1.85 t0 500 1000 1500 chHtL h=0 h=0.01 h=0.06 h=0.08 h=0.1 h=0.108 h=0.15 Figure 3. (Color online) The specific heat temperature dependences of the ANNNI model with k1 =−0.6, k2 =−0.3, k4 = 0 for several values of the field. 13704-5 J. Sznajd 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74 t0 5000 10 000 15 000 chHtL k4=-0.2 k4=-0.1 k4=0 k4=0.1 k4=0.2 Figure 4. (Color online) The specific heat temperature dependences of the extended ANNNI model with k1 =−0.6, k2 =−0.3 at zero field for several values of the four spin coupling k4. 1.55 1.60 1.65 1.70 1.75 1.80 t0 500 1000 1500 2000 chHtL h=0 h=0.03h=0.08 h=0.1 h=0.112 h=0.1214 h=0.13 Figure 5. (Color online) The specific heat temperature dependences of the extended ANNNI model with k1 =−0.6,k2 =−0.3,k4 = 0.2 for several values of the field. 1.56 1.58 1.60 1.62 1.64 1.66 1.68 1.70 t0 2000 4000 6000 8000 10 000 chHtL k4=0.0, h=0.0k4=0.0, h=0.11 k4=0.2, h=0.121 Figure 6. (Color online) Comparison of the specific heat curves of the standard ANNNI model at fields h = 0 and h = 0.11 around the critical field with such a curve for the model with k4 = 0.2 around the critical field h = 0.121. 13704-6 LPRG for Ising-like systems 1.580 1.585 1.590 1.595 t -1.0 -0.5 0.0 0.5 1.0 <sisi+1>,<sisi+2> Figure 7. (Color online) The temperature dependences of the nearest-neighbor 〈si si+1〉 (bottom curve) and next-nearest-neighbor 〈si si+2〉 chains correlation functions for extended ANNNI (k4 = 0.2) model at h = 0.121. to properly describe the specific heat behavior below themaximum because it is not capable of describing the ordered phase which is expected below the maximum, if the system undergoes a discontinuous phase transition. For some value of the field h > hc (h = 0.15 in figure 3), specific heat has only a broad hump. This means that the considered ANNNI in the presence of the external field can exhibit a continuous phase transition to the ferrimagnetic phase (++−) at h = 0 and h = hc ≈ 0.11. Thus, within the LPRG unlike in MFA it is not necessary to supplement the original ANNNI model by an additional four-spin interaction to reach an isolated critical point for a finite field around h = 0.11 [21]. Now, we proceed to the extended ANNNI model with k = 1,k1 =−0.6,k2 =−0.3 and k4 , 0. In figure 4 we present the temperature dependences of the specific heat in zero field for several values of negative and positive values of k4. As seen, the four-spin interaction simply shifts the critical point towards higher temperature for k4 > 0 and towards lower temperature for k4 < 0. Figure 5 shows the temperature de- pendence of the specific heat for the model with k4 = 0.2 in finite fields. Similarly to the standard ANNNI model case, the application of the external field changes the divergence of the specific heat into a maxi- mum for 0 < h < hc. For h ≈ 0.1214, the divergence appears again indicating a critical point. In figure 6 we compare the shapes of the specific heat curves for the field strengths close to the critical values for standard and extended ANNNI models. In the case of the k4 = 0.2 model, a small peak in specific heat appears above the critical temperature. In figure 7, the two-site correlation functions for the spin from adjacent (G1 = 〈si si+1〉) and next nearest neighbor (G2 = 〈si si+2〉) chains are presented. As seen, a kink of G2 around the temperature of the specific maximum is observed. It can suggest some preliminary or- dering between the spins from the next nearest neighbour chains. To summarize, it has been shown that within the LPRG approximation, both the original ANNNI model and the one supplemented by the four- spin interaction with k2/k1 = 0.5 and k1,k2 < 0 undergo the continuous phase transition at h = 0 and can undergo such a transition for some finite value of the field h = hc. In the calculations presented in this paper, a sharp anomaly in specific heat is evident at fields around hc which could explain the experimen- tally observed anomalies in UPd2Si2. The four-spin interaction shifts both the critical temperature and field, and changes the shape of the specific heat curves. References 1. Sznajd J., Phys. Rev. B, 2001, 63, 184404; doi:10.1103/PhysRevB.63.184404. 2. Suzuki M., Takano H., Phys. Lett. A, 1979, 69, 426; doi:10.1016/0375-9601(79)90397-9. 3. Barma M., Kumar D., Pandey R.B., J. Phys. C: Solid State Phys., 1979, 12, L909; doi:10.1088/0022-3719/12/23/008. 4. Castellani C., Di Castro C., Ranninger J., Nucl. Phys. B, 1982, 200, 45; doi:10.1016/0550-3213(82)90057-8. 5. Rogiers J., Grundke W., Betts D.D., Can. J. Phys., 1979, 57, 1719; doi:10.1139/p79-237. 13704-7 http://dx.doi.org/10.1103/PhysRevB.63.184404 http://dx.doi.org/10.1016/0375-9601(79)90397-9 http://dx.doi.org/10.1088/0022-3719/12/23/008 http://dx.doi.org/10.1016/0550-3213(82)90057-8 http://dx.doi.org/10.1139/p79-237 J. Sznajd 6. Ding H.-Q., Makivić M.S., Phys. Rev. B, 1990, 42, 6827; doi:10.1103/PhysRevB.42.6827. 7. Ding H.-Q., Makivić M.S., Phys. Rev. B, 1992, 45, 230; doi:10.1103/PhysRevB.45.230. 8. Jonsson A., Minnhagen P., Nylén M., Phys. Rev. Lett., 1993, 70, 1327; doi:10.1103/PhysRevLett.70.1327. 9. Sznajd J., Z. Phys. B: Condens. Matter, 1986, 62, 349; doi:10.1007/BF01313458. 10. Drzewiński A., Sznajd J., Physica A, 1991, 170, 415; doi:10.1016/0378-4371(91)90052-E. 11. Sznajd J., Phys. Rev. B, 2008, 78, 214411; doi:10.1103/PhysRevB.78.214411. 12. Niemeijer Th., Van Leeuwen J.M.J., Physica, 1974, 71, 17; doi:10.1016/0031-8914(74)90044-5. 13. Nauenberg M., J. Math. Phys., 1975, 16, 703; doi:10.1063/1.522584. 14. Rossat-Mignod J., Lander G.H., Burlet P., In: Handbook on the Physics and Chemistry of the Actinides, Vol. 1, Freeman A.J., Lander G.H. (Eds.), North-Holland, Amsterdam, 1984, p. 415. 15. Shemirani B., Lin H., Collins M.F., Stager C.V., Garrett J.D., Buyers W.J.L., Phys. Rev. B, 1993, 47, 8672; doi:10.1103/PhysRevB.47.8672. 16. Honma T., Amitsuka H., Yasunami S., Tenya K., Sakakibara T., Mitamura H., Goto T., Kido G., Kawarazaki D., Miyako Y., Sugiyama K., Date M., J. Phys. Soc. Jpn., 1998, 67, 1017; doi:10.1143/JPSJ.67.1017. 17. Plackowski T., Kaczorowski D., Sznajd J., Phys. Rev. B, 2011, 83, 174443; doi:10.1103/PhysRevB.83.174443. 18. Selke W., Phys. Rep., 1988, 170, 213; doi:10.1016/0370-1573(88)90140-8. 19. Rujan P., Selke W., Uimin G.V., Z. Phys. B: Condens. Matter, 1983, 53, 221; doi:10.1007/BF01388543. 20. Thompson C.J., In: Phase Transition and Critical Phenomena, Domb C., Green M.S. (Eds.), Academic Press, 1972, p. 177. 21. Sznajd J., J. Phys.: Condens. Matter, 2012, 24 , 436006; doi:10.1088/0953-8984/24/43/436006. Метод лiнiйної пертурбативної ренормалiзацiйної групи для iзингоподiбних спiнових систем Й. Шнайд Iнститут низької температури i структурних дослiджень, Польська академiя наук, Вроцлав, Польща Лiнiйне пертурбативне перетворення (LPRG) використовується для вивчення термодинамiки аксiальної моделi Iзинга з наступними до найближчих сусiдами з чотириспiновою взаємодiєю (розширена модель ANNNI) у полi. Представлено LPRG для слабовзаємодiючих ланцюжкiв Iзинга. Метод застосовано до ви- вчення пара-феромагнiтних фазових переходiв у скiнченному полi, що спостерiгаються в шаруватих спо- луках урану UAs1−x Sex , UPd2Si2 чи UNi2Si2. Вище згаданi системи зробленi з феромагнiтних шарiв i спi- ни з найближчих i наступних до найближчих шарiв є зв’язанi антиферомагнiтними взаємодiями J1 < 0 i J2 < 0, вiдповiдно. Кожна з цих систем демонструє потрiйну точку, в якiй двi впорядкованi фази (фе- ромагнiтна i неспiвмiрна) зустрiчаються з парамагнiтною фазою i всi фази зазнають фазового переходу у сильному полi з пара- до феромагнiтної (++−) фази. Проте, якщо в UAs1−x Sex є пара-феро фазовий перехiд першого роду, як очiкується з симетрiйних мiркувань, в UT2Si2 (T = Pd, Ni) цей перехiд видається неперервним, принаймнi в околi мультикритичної точки. В рамках наближення середнього поля, критич- ний характер пара-феромагнiтного переходу в скiнченному полi, принаймнi в однiй iзольованiй точцi, може бути описаний за допомогою моделi ANNNI, яка доповнена додатковою, наприклад, чотириспiно- вою взаємодiєю. Проте, в наближеннi LPRG для коефiцiєнта κ = J2/J1 поблизу 0.5 є критичне значення поля, для якого iзольована критична точка також iснує в оригiнальнiй моделi ANNNI. Позитивна чотири- спiнова взаємодiя зсуває критичну точку до вищих полiв i змiнює форму кривої питомої теплоємностi. В останньому випадку для достатньо малих полiв питома теплоємнiсть демонструє двопiкову структуру в парамагнiтнiй фазi. Ключовi слова: модель ANNNI, ренормалiзацiйна група, iзольована критична точка 13704-8 http://dx.doi.org/10.1103/PhysRevB.42.6827 http://dx.doi.org/10.1103/PhysRevB.45.230 http://dx.doi.org/10.1103/PhysRevLett.70.1327 http://dx.doi.org/10.1007/BF01313458 http://dx.doi.org/10.1016/0378-4371(91)90052-E http://dx.doi.org/10.1103/PhysRevB.78.214411 http://dx.doi.org/10.1016/0031-8914(74)90044-5 http://dx.doi.org/10.1063/1.522584 http://dx.doi.org/10.1103/PhysRevB.47.8672 http://dx.doi.org/10.1143/JPSJ.67.1017 http://dx.doi.org/10.1103/PhysRevB.83.174443 http://dx.doi.org/10.1016/0370-1573(88)90140-8 http://dx.doi.org/10.1007/BF01388543 http://dx.doi.org/10.1088/0953-8984/24/43/436006 Introduction LPRG Extended ANNNI model in a field

Схожі ресурси