Magnetic quantum phase transitions and entropy in Van Vleck magnet

Author’s Accepted Manuscript Magnetic quantum phase transitions and entropy in van vleck MAGNET G.Yu. Lavanov, I.М. Ivanova, V.M. Kalita, V.М. Loktev www.elsevier.com/locate/jmmm

PII: DOI: Reference:

S0304-8853(16)30568-6 http://dx.doi.org/10.1016/j.jmmm.2016.05.017 MAGMA61430

To appear in: Journal of Magnetism and Magnetic Materials Received date: 9 September 2015 Revised date: 16 March 2016 Accepted date: 6 May 2016 Cite this article as: G.Yu. Lavanov, I.М. Ivanova, V.M. Kalita and V.М. Loktev, Magnetic quantum phase transitions and entropy in van vleck MAGNET, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.05.017 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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MAGNETIC QUANTUM PHASE TRANSITIONS AND ENTROPY IN VAN VLECK MAGNET G. Yu. Lavanov1, I. М. Ivanova2, V. M. Kalita3, V. М. Loktev2,4 1

National Aviation University, 1, Kosmonavt Komarov Ave., Kyiv 03058, Ukraine

2

National Technical University of Ukraine “Kyiv Polytechnic Institute”

37, Peremoha Ave., Kyiv 03056, Ukraine 3

Institute of Physics of the NAS of Ukraine, 46, Nauka Ave., Kyiv 03028, Ukraine

4

Bogolyubov Institute for Theoretical Physics of the NAS of Ukraine,

14b, Metrolohichna Str., Kyiv 03680, Ukraine E-mail: [email protected]

Keywords: Quantum phase transition, entropy, magnetization, hysteresis

Field-induced magnetic quantum phase transitions in the Van Vleck paramagnet with easyplane single-ion anisotropy and competing Ising exchange between ions with the spin S=1 have been studied theoretically. The description was made by minimizing the Lagrange function at zero temperature (Т = 0) and the free energy at T  0 . Stable and unstable solutions of equations corresponding to the case T  0 asymptotically transform into those following from the Lagrange function at Т = 0. First-order phase transitions from the Van Vleck paramagnet state into the ferromagnet one were found to take place at a sufficiently high single-ion anisotropy. The entropy of such a magnet was shown to grow with its magnetization, as it occurs for antiferromagnets. At the point of quantum phase transition, the entropy has a jump, which magnitude depends on the ratio between the Ising exchange and anisotropy constants, as well as on the temperature. The described magnetic phase transition was supposed to be accompanied by the magnetocaloric effect. In the case when the Ising exchange dominates over the single-ion anisotropy, the magnetization reversal of ferromagnetic state by an external field was shown to be a phase transition of the first kind, which does not belong to orientational ones and which should be regarded as a quantum order-order phase transition.

1. Introduction In this paper, quantum phase transitions (QPTs) in the Van Vleck paramagnet (VVPM) are studied theoretically. The VVPM is supposed to be characterized by the Ising exchange (IE) of the ferromagnetic (FM) type between ions with the spin S=1 and by the easy-plane single-ion anisotropy (SIA) that is perpendicular to the Ising exchange direction [1-7]. A model with competing IE and 1

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easy-plane SIA was used, e.g, in [7] to describe a phase separation in the Не3 and Не4 mixture. However, it is better known as one of the simplest ones for the theoretical description of the QPT of the first kind (QPT-I). In the easy-plane SIA case, the ion ground state is a non-polarized singlet with the wave function 0 . If the energy of negative IE (by itself or, if a magnetic field is applied, together with the Zeeman energy) exceeds the easy-plane SIA energy, one can easily show using the mean field approximation that the ground state of ions changes to 1 characterized by the maximum spin polarization. The step-wise change of the ground state induced by an external field at Т=0 is a QPT-I [8-11]. However, this interpretation of QPT-I does not correspond to the description of the first-order phase transitions (PT-Is) at Т0 in the thermodynamic theory. Indeed, the ground state of quantum system at T=0, which is obtained as a solution of Schrödinger equation, is found to be stable in quantum mechanics [12]. Spin states 0 and 1 obtained for the mean-field Hamiltonian have also to be stable [13]. However, the finding of the stability limits for those states goes beyond the scope of determining stable solutions for the Schrödinger equation. Unlike the quantum theory, the thermodynamic one uses the Landau potential, either phenomenological or obtained from theoretical calculations, for the description of phases and the PT-I at Т0. Correspondingly, stable and unstable solutions (the latter are S-like curves of the Van der Waals type) are obtained from the equations of state [14-16]. The appearance of unstable solutions is considered in the thermodynamic theory as one of the origins of PT-I formation at Т0. Despite numerous claims that the thermodynamic approach can be applied only at high temperatures, the consideration on the basis of Landau potential can be extended to the limit T  0 . In this case, unstable solutions obtained for this potential at Т  0 can survive. Thus, the formal quantum-mechanical demand concerning the stability of wave functions of quantum system states at T=0, which follows from the mean-field Hamiltonian consideration, turns out inconsistent with the thermodynamic theory for QPT-Is. This is a reason of why the problem of magnon spectrum calculation using the Hubbard operator technique is often applied to find the spinstate stability limits at Т=0 [17-22]. It should be noted that QPTs associated with the change of the ground-state wave function can occur at finite temperatures as well [23-25]. Some researchers interpret the magnetic QPT as a Bose-Einstein magnon condensation [26-31], although virtual magnons rather than real ones are meant in this case. We made an attempt to overcome the methodological problem of the discrepancy between the descriptions of QPT in the Ising magnet with easy-plane SIA in the quantum-mechanical and thermodynamic approaches. A model of the VVPM with easy-plane SIA was used, which allowed us to make all calculations analytically. The QPT-I from the initial VVPM phase into the FM one 2

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induced by a magnetic field (MF) was consistently studied at all temperatures, including Т=0, and assuming that ions have the spin S=1. The results showed that, in the limit T  0 , all stable and unstable solutions, irrespective of whether they are equilibrium or not, which are obtained by minimizing the model non-equilibrium free energy, transform onto the solutions that are obtained by minimizing the Lagrange function with respect to the wave function parameters at Т=0. In this work, we consider the QPT-I in the range of parameters when the easy-plane SIA and IE constants are close by magnitude. In this case, the MF-induced QPT-I does not require high fields, which is important, first of all, from the experimental point of view. Besides, the study of QPT-I as an anomalous effect because of its unusual properties is also interesting. The entropy change, which is an indicator of magnetocaloric effect and, generally speaking, has an anomalous character in the PT-I region [32-35], is also analyzed. In particular, the field dependence of entropy at magnetization of the VVPM with IE is similar to its counterpart for antiferromagnets; namely, with the growth of MF, the entropy first increases and, after having a jump at the phase transition point, decreases. Hence, in this work, we consistently describe the QPT-I induced by an external MF in the VVPM with IE and easy-plane SIA at any temperature, T ≥ 0, and analyze its specific features at magnetization. 2. Quantum phase transitions at Т=0 The Hamiltonian of model system is written as follows [1-3]: ˆ   1  I s z s z  D  ( s z ) 2  h s z H nm n m n n 2 n ,m n n

(1)

where the first sum is the IE ( I n,m  0 ) with the easy magnetization axis Z, the second one describes the easy-plane SIA with the constant D>0 , the third one is the Zeeman contribution (expressed in terms of energy units) at magnetization by the longitudinal MF h ( h Z ), and snz is the z-projection of operator for the spin S = 1, which position in the lattice is given by the vector n. The wave functions of the n-th ion in the ground state can be written in the general form as [36]

 n  C 1  C0 0  C 1

(2)

where Сj are constants that satisfy the normalization condition

C2  C02  C2  1.

(3)

Assuming the magnet to be homogeneous, the ion states can be considered identical in the mean-field approximation. Then, the energy per ion for Hamiltonian (1) takes rather a simple form

1 E   Isz2  Dqzz  h sz 2 3

(4)

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where s z is the average spin projection of ion in state (2), qzz   n ( snz )2  n is the average spinquadrupole moment, and the exchange constant I

is determined in the nearest-neighbor

approximation as I   I nm . Expression (4) can easily be rewritten using formula (2): m

1 E   I (C2  C2 )2  D(C2  C2 )  h (C2  C2 ) . 2

(5)

The magnetic order in the ground state at Т=0 can be found by minimizing energy (5) as a function of coefficients Сj providing condition (3). For this purpose, let us introduce the Lagrange function

1 L   I (C2  C2 )2  D(C2  C2 )  h (C2  C2 )   (C2  C02  C2  1) , 2

(6)

where  is a multiplier. Its differentiation with respect to Сj brings us to the equalities

L  2 IC (C2  C2 )  2 DC  2hC  2C  0 ; C

(7)

L  2 IC (C2  C2 )  2 DC  2hC  2C  0 ; C

(8)

L  2C0  0 , C

(9)

which are regarded as equations of state for QPTs. The solutions of the system of equations (7)--(9) are as follows. (i) C0  0 , C  0 , and C2  ( D  h   ) / I . From the normalization condition   I  D  h we obtain C  1 . This solution describes a FM phase. The corresponding single-ion wave function is

1  n  1 , and the energy equals E   I  D  h . 2 (ii) C0  0 , C  0 and C2  ( D  h   ) / I . Now,   I  D  h , so that C  1 . This

1 solution describes a FM phase with the single-ion state  n  1 , the energy E   I  D  h , 2 and the magnetization directed oppositely to that in case (i). It is easy to see that solution (ii) transforms into solution (i) if the field changes its sign. Bboth solutions can be related to real situations occurring at high enough fields exceeding the SIA. The equality E  E for the energies of FM states with sz  1 and sz  1 is satisfied in the absence of MF ( h  0 ). Therefore, the PT between those states takes places in the zero field, hI (T  0)  0 . The value and the sign of initial exchange field change step-wise at the QPT point.

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(iii) C0  1 , C  0 , C  0 and 0  0 . This is the VVPM state with  n  0 , E0  0 , and no magnetization. If I  2D , the initial state at h  0 is a non-magnetized VVPM state. The QPT-I from the VVPM phase to the FM one occurs in the field. The transition field for this magnetic QPT-I is found from the condition of energy equality between the initial VVPM ( E0  0 ) and FM

1 1 ( E   I  D  h ) phases: h  hI (T  0)  D  I . The transition into the phase with sz  1 2 2 takes place at h  hI . In the limit h  0 , the FM phase is preferable in comparison with the VVPM

1 one if E   I  D  E0  0 , i.e. if I  2D . 2 The stability of the solutions of system (7)--(9) can be determined using the Hessian matrix G. For function (6), it looks like 0  2C G    2C  0  2C  

2C  2 I (3C2  C2 )  2 D  2h  2

2C0 4 IC C 2

0 4 IC C

0

  0  . (10)  0   2 I (3C2  C2 )  2 D  2h  2  2C

Actual are the third, G3 , and fourth, G4 , corner minors of this matrix. In particular, the solutions of the equations of state correspond to the minimum if G3  0 and G4  0 . For solution (i), a conditional minimum occurs if the inequalities G3  8( I  D  h)  0 and G4  32( I  D  h)( I  h)  0 are satisfied. Thus, this solution is stable if h  hcrFM (T  0) , where hcrFM (T  0)  D  I . Provided this condition, the other one, h   I , is satisfied automatically.

Solution (ii) is «inverse» to solution (i) in the sense that only the sign of critical field changes. For this solution, the critical field hcrFM (T  0)  I  D , and the hysteresis loop for the FM with

I  2D at Т=0 lies in the interval  hcrFM (T  0)  h  hcrFM (T  0) . For solution (iii), the corner minors are G3  (h  D) and G4  16(h2  D2 ) . The stability of VVPM state is given by the condition h  hcrSF (T  0) , where hcrSF (T  0)  D . This solution corresponds to the equilibrium state at h  0 if I / 2D  1 . It should be noted that, for the given parameter ratio I /( 2D)  1 , the critical field for the FM phase induced by an external field can change its sign and become negative for the phase with sz=1. It is easy to see that system (7)-(9) has two more solutions: (iv) C  0 , C2  ( D  h) / I , C02  1  ( D  h) / I ; and (v) C  0 , C2  ( D  h) / I , C02  1  ( D  h) / I .

5

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Solution (iv) gives rise to the value sz  C2  ( D  h) / I , which is actual in the field interval [  hcrFM , hcrSF ] and varies there from 1 to 0. At the same time, solution (v) gives rise to

sz  C2  ( D  h) / I , which varies from 0 to –1 in the interval [  hcrSF , hcrFM ]. Both solutions correspond to continuous transitions from the FM state into the VVPM one. Since the corresponding derivative sz / h  0 at that, those states turn out unstable. It is of interest that the spin x-projection for solutions (iv) and (v) is not equal to zero. The exchange field acts only along the z-axis, so that all directions in the easy-magnetization plane are identical. Therefore, the sign of the average spin at the site is not fixed, and its magnitude can be arbitrary, ranging from 0 to 1. Within the field interval of [  hcrFM , hcrSF ], the average spin projection

sx  2CC0   2

Dh Dh 1 , so that it vanishes at the critical points. When averaging over I I

all ions, we find that only the z-projection differs from zero for the solutions concerned. System of equations (7)-(9) has no other solutions with the average spin deviating from the zaxis. This conclusion is consistent with the assumption that there is no coherent, i.e. identical, spin rotations in the Ising system with easy-plane SIA and undergoing a QPT-I in the MF h Z . Stable solutions (i)--(iii) obtained above satisfy the single-ion Hamiltonian [1-3] ˆ   Is s z  D ( s z )2  h s z , H n z n n n

(11) z n

Where Isz is the mean exchange field that affects the z-projection of the n-th spin, s . Notice that the energy spectrum and the eigen functions of Hamiltonian (11) correspond to stable solutions only:

 n(1)  1 ,

 n(0)  0 ,

 n(  )  1 ,

(12)

1   Isz  D  h ,

0  0 ,

 1   Isz  D  h ,

(13)

here,  j are the energies of ions in the exchange field. They differ from energy (4) of system (1) in the ground state. One can see that the conditions of FM state stability, which were found by analyzing the Lagrange function, correspond to the inequalities 1   0 ,  1 (or  1   0 , 1 ), whereas the conditions of VVPM state stability to the inequality  0  0 > 1  D  h . The corresponding Lagrange multiplier equals     for the FM state with sz  1 , and     for the FM state with sz  1 . If the VVPM is the initial (at h  0 ) state of the system, the regions of FM phase stability may not overlap. In this case, the curves of non-equilibrium magnetization pass through the point

sz  0 at h  0 [2]. However, a situation is also possible when the non-equilibrium magnetization of

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the VVPM has a hysteresis loop like the FM. Indeed, the FM magnetization reversal is possible for

1 the VVPM in the case D  I  I  D , i.e. when the inequality I  4D / 3 is satisfied. 2 Figure 1 illustrates the magnetization of the system described by Hamiltonian (1) at T=0 and for various ratios between the IE and easy-plane SIA constants. Bold solid lines mark stable equilibrium states that correspond to equilibrium magnetization, and thin solid lines mark unstable equilibrium solutions. The inclined (from left up to right down) sections of the latter correspond to non-equilibrium solutions. The arrows show the hysteresis at magnetization. For D / I  0.45 (Fig. 1а), the ground state in the field h  0 is the FM phase with the average spin sz  1 or sz  1 . The value hI  0 is the point of PT between those states, and the critical fields of stability for those states equal hcrFM  0.55I . The hysteresis loop has a rectangular form. The magnetization of the system with D / I  0.8 , for which the ground state at h  0 is the non-magnetized VVPM, is shown in Fig. 1b. All notations are the same as in Fig. 1a. At hI  0.3I , a QPT-I takes place from the VVPM phase into the FM one. Taking into account that hcrFM  0.2I is lower by magnitude than hI , the hysteresis loop turns out to be composed of two rectangles shifted with respect to each other. At D / I  0.55 , the ground state of ions is the non-magnetized VVPM, but the hysteresis loop is also а rectangle (see Fig. 1с). In this case, the critical field hcrFM  0.45I is higher by magnitude than the fields of QPT-I from the VVPM state into the FM one ( hI  0.05I ). Thus, the system under consideration has two PTs of the first kind from the VVPM phase into the FM one in the case of its equilibrium magnetization. If the magnetization is non-equilibrium, the system demonstrates a hysteresis loop of the FM type. The critical field of the VVPM phase stability, hcrSP  0.5I , is shown by dotted lines. Its value is higher than that for the FM phase. As a result, at the beginning of magnetization process, the curve sz (h) corresponding to the first field sweep is located outside the hysteresis loop. In Figs. 1a and 1b, the inclined thin straight lines show the dependences sz (h) for solutions (iii). As was said above, those solutions are unstable and non-equilibrium. However, as will be shown below, they are limits (at T  0 ) for non-equilibrium solutions obtained at finite temperatures by the free energy minimization. Those solutions will be found in the next section. 3. Free energy and equations of state at Т0 The Gibbs free energy is written in the form F  E  T , where

   p j ln p j

(14)

j

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is the entropy of the system in self-consistent approximation (2), and p j ( j = 1, 0) are the probabilities of states (12), which are connected with thermodynamic means by the relations

p1  p1  sz and p1  p1  qzz , provided that

p

j

 1.

j

The expression for the non-equilibrium free energy per ion has a form similar to that used in work [37]: q s q s q s 1 q  s  F  Isz2  Dqzz  T  zz z ln zz z  zz z ln zz z  (1  qzz ) ln(1  qzz )   hsz . 2 2 2 2  2 

(15)

Unlike the case Т=0, the average values sz (T , h) and qzz (T , h) in (15) depend on the field and the temperature. By differentiating (15), one can obtain two equations of state:

F T  q s 1 q s   I sz   ln zz z  ln zz z  sz 2 2 2 2

   h 0 , 

(16)

F 1 q s 1 q s   D  T  ln zz z  ln zz z  ln(1  qzz )   0 .  qzz 2 2 2 2 

(17)

Equation (17) allows the quantity qzz to be expressed in terms of s z : qzz (sz ) 

where   4 e



2D T

 k ,  1

(18)

and k  sz2 (1  )  . Then, the equation of state can be written in the form  Isz 

2sz   1  T  ln 1  h  0. 2    k  sz   1 

(19)

The latter makes it possible to find solutions sz (h) , which enable one to calculate the field dependencies for the free energy, F (h) . The PT-I points can be calculated by equating energies obtained for different solutions. The critical fields of phase stability are found by simultaneously solving the equations of state (8) and (9), and zeroing the determinant of the Hessian matrix for energy (15). This determinant consists of the second-order derivatives of the free energy with respect to the order parameters s z and qzz . Its zeroing allows a relation between s z and qzz to be obtained. The corresponding result looks like

sz2 

4T / I (  1)  1 1 1 ( (1  T / I )   T / I  )  1 2 2

8

,

(20)

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By substituting it into (19), we obtain an equation for the critical fields of phase stability. The latter can also be determined from the condition (sz / h h )1  0 for the derivative at the phase stability cr

boundaries. 4. Phase diagram of Van Vleck paramagnet At I  2D (or D / I  1/ 2 ), the ground state of ions is singlet in the absence of external field. The most interesting situation arises when the easy-plane SIA constant is only slightly exceeds this limit. In this case, the competition between interactions makes the field hI much lower than the easyplane SIA constant, i.e. the PT field becomes not large, which is actual from the experimental point of view. The field dependences of the solutions for equations of state (16)-(17) obtained at

D / I  0.55

and for various temperatures are shown in Fig. 2. At T / I  0.5, the magnetization

has a paramagnetic character, and sz (h) changes continuously. The lowest state in the ion spectrum at h  0 is a non-magnetized singlet, and two others with opposite spin projections are degenerate. The magnetic field eliminates the degeneracy. The IE enhances the field effect. As a result, a crossover inevitably takes place, which makes the ground ion state to be polarized (magnetized). This transition is continuous, and no peculiar features in the magnetization are observed. At the lower temperature, T / I  0.2 , the curve sz (h) has two S-like sections: one at positive and the other at negative fields. Those intervals do not overlap. In this case, a PT-I between the VVPM and FM states takes place. It is associated with the step-wise change of the ground state function of ions, therefore being a magnetic QPT at finite temperatures. Figure 2b shows the field dependence sz (h) at an even lower temperature, T / I  0.1 . It is easy to see that the critical field of the FM phase stability is higher than the equilibrium field hI . In the case of equilibrium magnetization at this temperature, there are two PT-Is as well. However, non-equilibrium magnetization occurs as a hysteresis loop between two FM states with the oppositely directed magnetization, although the initial equilibrium state in the field absence is the VVPM. The field hcrFM is equal to the coercive force. The non-equilibrium transition between the VVPM and FM states occurs at the field hcrSF . This fact means that the magnetization curve lies outside the hysteresis loop at the first field sweep. Note that the unstable sections of the dependence

sz (h) at T  0 correspond to non-equilibrium solutions (iv) and (v) at Т=0. Figure 3 illustrates the field dependences of free energy for the solutions plotted in Fig. 2. At the temperature T / I  0.5 , the free energy has a parabolic dependence, as it has to be for the paramagnet. At T / I  0.2 , the non-equilibrium and unstable sections of the solutions correspond to higher free energies, and the field dependence character changes. In the case T / I  0.1 , the 9

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behavior of free energy corresponds to a magnetization with the FM hysteresis loop with hcrFM  hI . Thin vertical lines in Fig. 3 show the free energy jump at the field hcrFM of the transition into the FM state with the opposite direction of average magnetization. In this state, the energy has the lowest value. Figure 4 shows the h-T phase diagram for the VVPM with

D / I  0.55 . The diagram

illustrates the temperature dependences for the field of QPT-I between the VVPM and FM phases, hI (T ) , and the critical fields of phase stability, hcrFM (T ) and hcrSF (T ) . When crossing the curve hI (T ) ,

the ion ground state changes in a step-wise manner from the non-magnetized singlet to the state with maximum spin polarization. There is no QPT-I above the tricritical point temperature Ttrk . The temperature dependence of the field hI (T ) in Fig. 4 is satisfactorily approximated by the formula ln

hI (T )  hI (T  0) I



D . Thus, single-site heat fluctuations caused by easy-plane SIA T

increase the temperature of magnetic QPT-I between the VVPM and FM states. At a certain temperature Tk , the equation hcrFM  hI (T )

is satisfied (see Fig. 4). In the

temperature interval 0  T  Tk , the magnetization of VVPM is similar to that of FM with a rectangular hysteresis loop. At the same time, in the interval Tk  T  Ttkr , the hysteresis consists of two loops, as in Figs. 1b and 2a. At the QPT-I, the ground state changes in a step-like manner. At D / I  0.55 , if h = 0, the singlet  0  0

is the ground state, the states  1  1 are polarized, and  0   1  0 . After the

QPT into the FM phase, the polarized state becomes the ground one. For the FM phase in the field

h  hI , we have 1   0  0 . This inequality, as is shown in the lower inset in Fig. 4, is satisfied at

T  TQPT . At a certain temperature within the interval [TQPT , Ttkr ] , an ordinary PT-I takes place, after which the non-magnetized singlet remains to be the ground state of the ion in the FM phase. Figure 5 shows the field dependences of entropy at the magnetization of the VVPM with

D / I  0.55 . The field dependence of the entropy,  (h) , at T / I  0.5 is standard for the paramagnetic state: its value decreases with the field growth, since the MF-induced ordering of ionic magnetic moments becomes stronger. At the QPT, the behavior of entropy changes.

As the field grows, the entropy of

paramagnetic state increases, which should change the sign of magnetocaloric effect in comparison with the high-temperature magnetization. This entropy change is similar to that observed at antiferromagnet magnetization [38].

10

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At the magnetization of PM, the energy 1  D  h of the excited state  1  1 decreases, and its population increases. On the contrary, the energy  1  D  h / 2 of the excited state  1  1 increases, and its population decreases. One can see from Fig. 5 that, at high-temperature

(T

D ) magnetization, both processes take place, being enhanced by the exchange field. For the

VVPM in the field interval [0, hI ] , when the population of those state substantially decreases, the decrease in the energy 1  D  h of the state  1  1 prevails, which is followed by the decrease in the ground singlet state population. The effect of exchange strengthening at the MF growth diminishes significantly at low temperatures (see Fig. 2). As a result, the magnetic disorder in the system increases, and the entropy grows. At the field of magnetic QPT, h  hI , the entropy change is negative,  (hI )  0 , which corresponds to the establishment of FM order. In higher fields, h  hI , the entropy of the ordered FM phase decreases further. Such a tendency is especially pronounced for non-equilibrium stable states in the vicinity of critical fields. At the QPT-I field, h  hI , we may consider that the adiabaticity conditions are satisfied and heat is exchanged only between the magnetic subsystem and the lattice. The heat change in the magnetic subsystem equals

QhI (T )  T  (hI ) ,

(21)

where  (hI ) is the entropy jump at the QPT point [39]. The temperature dependence of this quantity calculated for the VVPM with D / I  0.55 and normalized by the exchange constant is shown in Fig. 6. At the point of magnetic QPT, heat (21) is negative, QhI  0 , and the heat transferred to the lattice is positive. The dependence QhI (T ) has a minimum at the point where the heat transferred to the lattice at the QPT is maximum. The appearance of the maximum in the dependence QhI (T ) corresponds to the ClapeyronClausius relation [39, 40]. In our case, the latter looks like

dhI sz (hI ) , (22) dT is the entropy jump at h  hI , and sz (hI ) is the order parameter jump at the QPT  mag  

where  mag

point. The value of

dhI / dT increases as the temperature grows. At the same time, sz (hI )

decreases at that and vanishes at the tricritical point. As a result, the dependence  mag (T ) has to reveal an extremum. Such a behavior is confirmed by the dependence shown in Fig. 6.

11

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The entropy jump depends on the model parameters. Figure 7 illustrates the dependence of the maximum of the entropy jump magnitude on the ratio D / I. One can see that, in the case of m magnetically induced QPT,  mag

is expected to be maximum at D  I / 2 .

5. Magnetization of Ising ferromagnet with easy-plane single-ion anisotropy The PT between the FM and PM states in the VVPM was studied by a number of methods [41-43]. Let us recall that in the zero field ( h  0 ), if D  I / 2 , both the PT-I and the PT-II are possible as the temperature changes. If D / I  0.463 , the second-order phase transition between the FM and PM phases is of the order-disorder type [42]. At this PT, the spin doublet of single-ion Hamiltonian becomes split at the Curie temperature by a spontaneous exchange field. If 0.463  D / I  0.466 , the doublet splitting occurs in a step-wise manner, which corresponds to the

aforementioned PT-I. Another situation takes place if 0.466  D / I  0.5 : after the doublet becomes split, the spin-polarized state becomes ground at the Curie point. Therefore, this PT between the FM and PM phases at h  0 is a magnetic order-order QPT [42]. Below we analyze only the features of the FM state magnetization. Figure 8 shows the solutions for the equation of state with D / I  0.45 obtained at various temperatures. We consider only the case when the SIA constant is slightly below the critical value D  I / 2 . The equation of state was unexpectedly found to possess a solution at a temperature lower

than the Curie one (this solution corresponds to the VVPM state), as well as non-equilibrium states that tend to solutions (iv) and (v) at T  0 (see Fig. 1). At temperatures higher than the Curie point, the magnetization curves have an almost PM behavior (see the dependence sz (h) for T / I  0.5 in Fig. 8а). Below the Curie point (at T / I  0.3 , Fig. 8a), the curve sz (h) has an S-like form. The points  hcrFM are the critical fields of the FM phase stability. In the case T / I  0.1 (Fig. 8b), the curve sz (h) consists of two S-like sections. The magnetization sz (h) in the range of fields [ hcrSF , hcrSF ] corresponds to the VVPM state, which also tends at T  0 to the solutions obtained from the Lagrange function minimization. Figure 9 shows the free-energy curves F (h) for the dependences sz (h) depicted in Fig. 8. It is evident that the PT between the phases with sz  0 and sz  0 occurs at the point hI  0 . This order-order PT is accompanied by the change of spin projection in the ion ground state. It is also easy to see from Fig. 9b that the solution corresponding to the VVPM state is unstable, and its energy is higher than that of the FM phase. At non-equilibrium magnetization, a hysteresis with the coercive field hcrFM is observed. The phase diagram for the Ising FM with easy-plane SIA ( D / I  0.45 ) is shown in Fig. 10. The solid line corresponds to the zero-field ( hI (T )  0 ) PT-I, which exists at T  TC . The dashed 12

`

curves show the temperature dependences of the critical fields of the FM state stability, ± hcrFM (T ) . The figure also exhibits the temperature dependences of the critical field hcrSF (T ) of the VVPM stability below the temperature TSF . The VVPM phase is non-equilibrium at that, and the transition fields are absent. The field dependence of entropy  (h) for the Ising FM with easy-plane SIA is shown for D / I  0.45 in Fig. 11. At high temperatures, T  TC , this dependence has a form typical of the

paramagnetic state, and the entropy value diminishes with the field growth. In the FM state, the entropy jump is equal to zero at the PT-I point,  (hI )  0 , so that no magnetocaloric effect, which is usually registered at the equilibrium transition, should be expected at this PT. Nevetheless, the entropy jump at the phase stability boundary is not zero at non-equilibrium magnetization. 6. Conclusions In this paper, the magnetically induced QPT into the VVPM state with the IE of the FM type has been studied consistently. The procedure of Lagrange function minimization was used in the case Т=0, and the procedure of non-equilibrium free energy minimization following the Landau theory in the case T  0 . All the solutions, both equilibrium and non-equilibrium (stable and unstable), were found to transform at T  0 into the solutions obtained for Т=0. Our results showed that, like the case of FM, magnetization of the initially non-magnetic VVPM with S=1 and IE can have a hysteresis if the critical field of the FM phase stability exceeds the field of equilibrium PT-I from the singlet PM phase into the FM phase. An unusual character of entropy field dependence was also established for the VVPM state at equilibrium magnetization in the fields lower than the QPT-I one. In this field interval, the entropy was found to increase with the field growth. If T  0 , the entropy jump accompanied by the magnetocaloric effect is observed at the QPT point. If the constants of Ising FM exchange are larger than the SIA one, and the initial equilibrium state below the Curie point is the FM phase, then the orientational PT is impossible at magnetization. In the FM state, the initial state decays as the field sign changes, and a new FM state with the change of the ion ground state is formed. The equilibrium field for this PT equals zero, and the corresponding entropy has no jump. This work was partially supported by the Special program of fundamental researches of the Branch of Physics and Astronomy of the NAS of Ukraine.

References

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`

1. H.W. Capel, On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting, Physica 32 (1966) 966-988. 2. H.W. Capel, On the possibility of first-order transitions in Ising systems of triplet ions with zerofield splitting II, Physica 33 (1967) 295-331. 3. B.S. Dumesh, Influence of quantum fluctuations on the magnetic properties of quasi-onedimensional triangular antiferromagnets, Phys.-Uspehi 43 (2000) 365-380. 4. E.V. Rosenfel’d, Quantum magnetization jumps in magnets with easy-plane anisotropy, JETP Letters 24 (1976) 50-53. 5. M.F. Collins, O.A. Petrenko, Triangular antiferromagnets, Can. J. Phys 75 (1997) 605-655. 6. R.S. Gekht, Magnetic states and phase transitions in frustrated triangular-lattice antiferromagnets, Phys.-Uspehi 32 (1989) 871-890. 7. M. Blume, V.J. Emery, R.B. Griffiths, Ising model for the λ transition and phase separation in He3 -He4 mixtures, Phys. Rev. A 4 (1971) 1071. 8. S. Sachdev, Quantum Phase Transitions, Cambridge: Cambridge Univ. Press (2000) 517. 9. M. Vojta, Quantum phase transitions, Rep. Prog. Phys. 66 (2003) 2069. 10. V.F. Gantmaher, V.T. Dolgopolov, Localized–delocalized electron quantum phase transitions, Phys.-Uspehi 51 (2008) 3-22. 11. S.M. Stishov, Quantum phase transitions, Phys –Uspehi 47 (2004) 789-795. 12. A. Davydov, Quantum Mechanics, Oxford Pergamon Press (1965) 680. 13. J.H. Van Vleck, Theory of electric and magnetic susceptibilities, Oxford, Oxford Univ. (1932) 384 14. L.D. Landau, E.M. Lifshitz, Statistical Physics. Vol.5 (3rd ed.), Oxford Butterworth-Heinemann (1980) 544. 15 J. C. Toledano, P. Toledano, The Landau theory of phase transitions, Singapore World Scientific (1987) 451. 16 E.I. Kut’in, V.L. Lorman, S.V. Pavlov, Methods of the theory of singularities in the phenomenology of phase transitions, Phys.-Uspehi 34 (1991) 497-514. 17. V.M. Loktev, V.S. Ostrovski, Peculiarities of the statics and dynamics of magnetic insulators with single-ion anisotropy, Low. Temp. Phys. 20 (1994) 775. 18. V.V. Val’kov, T.A. Val’kova, S.G. Ovchinnikov, Quantum spin-wave theory of ferromagnets with arbitrary single-ion anisotropy, JETP 61 (1985) 323-330. 19. V.S. Ostrovskii, Nonlinear dynamics of highly anisotropic spin-1 magnetic materials, JETP 64 (1986) 999-1006. 20. Yu.N. Mitsai, Yu.A. Fridman, Application of hubbard operators in the theory of magnetoelastc waves, Theor. Math. Phys. 81 (1989) 1194-1199. 14

`

21. Yu.A. Fridman, O.A. Kosmachev, Ph.N. Klevets, Phase states of S=1 magnetic with anisotropic exchange interactions, JMMM 320 (2008) 435-449. 22. Yu.A. Fridman, D.A. Matyunin, Phase transitions in 2D and 3D non-Heisenberg ferromagnets with temperature-dependent anisotropy, Physics of the Solid State 50 (2008) 696-702. 23. V.M. Kalita,V.M. Loktev, Displacive magnetic phase transitions upon spin ordering in magnets with strong single-ion anisotropy, Physics of the Solid State 45 (2003) 1523-1529. 24. V.M. Kalita, V. M. Loktev, A multisublattice magnetic phase induced by external field in a singlet magnet, JETP 98 (2004) 1006-1014. 25. V.M. Kalita, V.M. Loktev. Temperature-induced magnetic phase transitions in crystals with competing single-ion and interionic magnetic anisotropies, Physics of the Solid State 47 (2005) 690696 26. V.S. Zapf, D. Zocco, B.R. Hansen, M. Jaime, N. Harrison, C.D. Batista, M. Kenzelmann, C. Niedermayer, A. Lacerda, A. Paduan-Filho, Bose-Einstein condensation of S = 1 nickel spin degrees of freedom in NiCl2-4SC(NH2)2,. Phys. Rev. Lett. 96 (2006) 077204. 27. S.A. Zvyagin, J. Wosnitza, C.D. Batista, M. Tsukamoto, N. Kawashima, J. Krzystek, V.S. Zapf, M. Jaime, N.F. Oliveira Jr., A. Paduan-Filho, Magnetic Excitations in the Spin-1 Anisotropic Heisenberg Antiferromagnetic Chain System NiCl2−4SC(NH2)2, Phys. Rev. Lett. 98 (2007) 047205. 28. S.A. Zvyagin, J. Wosnitza, A.K. Kolezhuk, V.S. Zapf, M. Jaime, A. Paduan-Filho, V.N. Glazkov, S.S. Sosin, A.I. Smirnov, Spin dynamics of NiCl2−4SC(NH2)2 in the field-induced ordered phase, Phys. Rev. B 77 (2008) 092413. 29. L. Yin, J.S. Xia, V.S. Zapf, N.S. Sullivan, A. Paduan-Filho, Direct measurement of the BoseEinstein condensation universality class in NiCl2-4SC(NH2)2 at ultralow temperatures, Phys. Rev. Lett. 101 (2008) 187205. 30. X.F. Sun, W. Tao, X.M. Wang, C. Fan, Low-Temperature Heat Transport in the LowDimensional Quantum Magnet NiCl2−4SC(NH2)2, Phys. Rev. Lett. 102 (2009) 167202. 31. A. Paduan-Filho, K.A. Al-Hassanieh, P. Sengupta, M. Jaime, Critical Properties at the FieldInduced Bose-Einstein Condensation in NiCl2−4SC(NH2)2, Phys. Rev. Lett. 102 (2009) 077204. 32. L. Li, K. Nishimura, H. Yamane, Giant reversible magnetocaloric effect in antiferromagnetic GdCo2B2 compound, Appl. Phys. Lett. 94 (2009) 102509. 33. J. Chen, B.G. Shen, Q.Y. Dong, J.R. Sun, Giant magnetic entropy change in antiferromagnetic DyCuSi compound, Solid State Communications 150 (2010) 1429-1431. 34. J. Chen, B.G. Shen, Q.Y. Dong, F.X. Hu, J.R. Sun, Giant reversible magnetocaloric effect in metamagnetic HoCuSi compound, Appl. Phys. Lett. 96 (2010) 152501.

15

`

35. Z.Y. Xu, J. Shen, X.Q. Zheng, H. Zhang, Magnetocaloric effect in ErSi compound, EEE Transactions on Magnetics 47 (2011) 2470-2473. 36. V.M. Kalita, I.M. Ivanova, V. M. Loktev, Quantum effects of magnetization of an easy-axis ferromagnet with S = 1, Theoretical and Mathematical Physics 173 (2012) 1620-1635. 37. J. Barr´e, D. Mukamel, S. Ruffo, Inequivalence of ensembles in a system with long range interactions, Phys. Rev. Lett. 87 (2001) 030601. 38. G.Yu. Lavanov, V.M. Kalita, V.M. Loktev, Isostructural magnetic phase transitions and the magnetocaloric effect in Ising ferromagnets, Low Temp. Phys. 40 (2014) 823-829. 39. S.A. Nikitin, A.S, Andreenko, G.E. Chuprikov, V.P. Posyado, Magnetic phase transformations and the magnetocaloric effect in single crystals of Tb-Y alloys, JETP 46 (1977) 118-122. 40. D. Kim, B. Revaz, B. L. Zink, F. Hellman, J.J. Rhyne, J.F. Mitchell, Tricritical point and the doping dependence of the order of the ferromagnetic phase transition of La1−xCaxMnO3 , Phys. Rev. Lett. 89 (2002) 227202. 41. J.A. Plascak, J.G. Moreira, F.C. saBarreto, Mean field solution of the general spin Blume-Capel model, Physics Letters A 173 (1993) 360-364. 42. V.M. Kalita, G.Yu. Lavanov, V. M. Loktev, Specific features of spin ordering in an Ising antiferromagnet with single-ion easy-plane anisotropy, Physics of the Solid State 50 (2008) 295-301. 43. A.A. Khamzin, R.R. Nigmatullin, Thermodynamics of Ising rare-earth magnet in the static fluctuation approximation, Optics and Spectroscopy 116 (2014) 842-848.

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Figure captions Fig. 1. Dependences sz (h) at Т=0 for various normalized anisotropy values D / I  0.45 (a), 0.8 (b), and 0.55 (c). Bold solid lines correspond to equilibrium magnetization, thin solid lines to nonequilibrium magnetization in the intervals of stability fields for the ferromagnetic,  hcrFM , and singlet,  hcrSP , phases. Arrows mark the boundaries of hysteresis loop at magnetization. Fig. 2. Field dependences sz (h) for the VVPM with D / I  0.55 at various normalized temperatures. Magnetization hysteresis is marked by arrows. Fig. 3. Free energy vs field dependences, F (h) , for the VVPM with D / I  0.55 at various normalized temperatures. Fig. 4. h-T phase diagram for the VVPM with D / I  0.55 . hI (T ) is the temperature dependence for the field of the first-order magnetic PT between the VVPM and FM phases (shown scaled-up in the upper inset). The lower inset demonstrates the temperature dependence of the difference between the one-ion energies in the FM phase in the field h  hI (T ) . Fig. 5. Field dependences of entropy,  (h) , for the VVPM with D / I  0.55 at various normalized temperatures. Solid curves correspond to the equilibrium entropy dependences, dashed ones to the non-equilibrium states. Double-headed arrows show the entropy jumps at critical fields. Fig. 6. Temperature dependence for the normalized heat, QhI (T ) / I , at the QPT point h  hI for D/I=0.55. Fig. 7. Dependence of the magnitude of the maximum entropy jump  m mag on the normalized easy-plane anisotropy constant D / I. Fig. 8. Field dependences of magnetization, sz (h) , for the Ising FM with D / I  0.45 at various normalized temperatures. The dashed lines mark the hysteresis. Fig. 9. Field dependencies of free energy, F (h) , for the Ising FM with D / I  0.45 at various normalized temperatures. The dashed lines mark the end points of hysteresis. 17

`

Fig. 10. Phase diagram for the Ising FM with easy-plane anisotropy ( D / I  0.45 ) and S=1. Solid line corresponds to the zero-field ( hI (T )  0 ) first-order PT; dashed ones, hcrFM (T ) , describe the critical fields of FM phase stability; and dotted ones, hcrSF (T ) , the critical fields of VVPM phase stability. Fig. 11. Field dependences of entropy,  (h) , for the Ising FM with easy-plane anisotropy ( D / I  0.45 ) at various normalized temperatures.

18

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Fig. 1. Dependences sz (h) at Т=0 for various normalized anisotropy values D / I  0.45 (a), 0.8 (b), and 0.55 (c). Bold solid lines correspond to equilibrium magnetization, thin solid lines to nonequilibrium magnetization in the intervals of stability fields for the ferromagnetic,  hcrFM , and singlet,  hcrSP , phases. Arrows mark the boundaries of hysteresis loop at magnetization.

19

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Fig. 2. Field dependences sz (h) for the VVPM with D / I  0.55 at various normalized temperatures. Magnetization hysteresis is marked by arrows.

20

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Fig. 3. Free energy vs field dependences, F (h) , for the VVPM with D / I  0.55 at various normalized temperatures.

21

`

0,5

D/I=0.55

0,06

hI

hI / I

0,05

SF

hcr (T)

0,0

hI(T)

0,1

0,2

T/I

0,3

0,0 0,1

T/I

0,2

Tk

 

0,0

FM

hcr (T)

0,2 0,0 0,0

TQPT 0,1

Ttkr

0,2

Ttkr

T/I 0,3

-0,2 -0,4

-0,5

-0,6

Fig. 4. h-T phase diagram for the VVPM with D / I  0.55 . hI (T ) is the temperature dependence for the field of the first-order magnetic PT between the VVPM and FM phases (shown scaled-up in the upper inset). The lower inset demonstrates the temperature dependence of the difference between the one-ion energies in the FM phase in the field h  hI (T ) .

22

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Fig. 5. Field dependences of entropy,  (h) , for the VVPM with D / I  0.55 at various normalized temperatures. Solid curves correspond to the equilibrium entropy dependences, dashed ones to the non-equilibrium states. Double-headed arrows show the entropy jumps at critical fields.

23

`

0,00 0,1

T/I

0,2

Ttkr

Q

hI(T)/I

0,0

D/I=0.55 -0,01

m

 Qh

I

Fig. 6. Temperature dependence for the normalized heat, QhI (T ) / I , at the QPT point h  hI for D/I=0.55.

24

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m

mag(D)|

0,10

0,05

0,00 0,5

0,6

0,7

D/I

0,8

0,9

Fig. 7. Dependence of the magnitude of the maximum entropy jump  m mag on the normalized easy-plane anisotropy constant D / I.

25

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Fig. 8. Field dependences of magnetization, sz (h) , for the Ising FM with D / I  0.45 at various normalized temperatures. The dashed lines mark the hysteresis.

26

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Fig. 9. Field dependencies of free energy, F (h) , for the Ising FM with D / I  0.45 at various normalized temperatures. The dashed lines mark the end points of hysteresis.

27

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Fig. 10. Phase diagram for the Ising FM with easy-plane anisotropy ( D / I  0.45 ) and S=1. Solid line corresponds to the zero-field ( hI (T )  0 ) first-order PT; dashed ones, hcrFM (T ) , describe the critical fields of FM phase stability; and dotted ones, hcrSF (T ) , the critical fields of VVPM phase stability.

28

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Fig. 11. Field dependences of entropy,  (h) , for the Ising FM with easy-plane anisotropy ( D / I  0.45 ) at various normalized temperatures.

Highlights Consistent theory is proposed for the phase transition description at T=0 and T≠0. We receive phase H-T diagram of van Vleck paramagnet with quantum phase transitions. Van Vleck paramagnet with S=1 and Ising exchange has a hystresis loop as ferromagnet. The entropy change in van Vleck system is similar to that at antiferromagnets magnetization. We show that quantum phase transition is followed by magnetocaloric effect.

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