Random field effects on the isotropic quantum Heisenberg model with Gaussian random magnetic field distribution
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Random field effects on the isotropic quantum Heisenberg model with Gaussianrandom magnetic field distribution¨Umit Akıncı Department of Physics, Dokuz Eyl¨ul University, TR-35160 Izmir, Turkey
Effect of Gaussian random magnetic field distribution which is centered at zero on the phasetransition properties of isotropic quantum Heisenberg model has been investigated on two (2D)and three dimensional (3D) lattices within the framework of effective field theory (EFT) for a twospin cluster (which is abbreviated as EFT-2). Beside the phase diagrams and the evolution ofthe magnetization versus temperature curves with the Gaussian magnetic field distribution width,critical Gaussian distribution width values, which make the critical temperature zero, have beenobtained for several lattices. Moreover, it has been concluded that all critical temperatures are ofthe second order and reentrant behavior does not exist in the phase diagrams.Keywords:
Quantum isotropic Heisenberg model; Random magnetic field; Gaussianmagnetic distribution
Recently, there has been growing theoretical interest in the random field lattice spin models. Themodel was introduced for the first time by Larkin [1] for superconductors and later generalized byImry and Ma [2]. Ising model which is the most basic lattice spin model with quenched randomfield (RFIM) has been studied over three decades, since this model can be used to describe a widevariety of disordered systems. Diluted antiferromagnets (such as
F e x Zn − x F , Rb Co x M g − x F and Co x Zn − x F ) in a homogenous magnetic field behave like ferromagnetic systems in the pres-ence of random fields [3, 4]. Structural phase transitions in random alloys, commensurate charge-density-wave systems with impurity pinning, binary fluid mixtures in random porous media, andthe melting of intercalates in layered compounds, such as T iS [5] are the examples of the ex-perimentally accessible disordered systems which can be described by RFIM. Besides, RFIM canmimic the phase transitions and interfaces in random media [6, 7], e.g pre-wetting transition on adisordered substrate can be mapped onto a 2D RFIM problem [8]. RFIM has also been applied todescribe critical surface behavior of amorphous semi-infinite systems [9, 10].RFIM has been widely studied in the literature with discrete [11, 12, 13, 14, 15], as well ascontinous [16, 17, 18, 19, 20, 21] distributions. Random distribution of the magnetic field mayproduce drastic effects on the phase diagrams and related magnetic properties of the system.It has been shown that Ising systems under the influence of discrete symmetric distributions,like bimodal [11] and trimodal [12] distributions, exhibit tricritical behavior, while continuoussymmetric distributions like Gaussian distribution [16] lead to only second order transitions.Although the results of the RFIM have been reported for both discrete and continuous distri-butions, there has been less attention paid on the random field effects on the Heisenberg model.Since Heisenberg model is more realistic model than the Ising model for the spin systems, it isimportant to investigate this model in the presence of quenched random fields. Albuquerque andArruda [22] studied the effect of the bimodal random field distribution on the phase transitioncharacteristics of the spin-1/2 isotropic classical Heisenberg model and they found tricritical be-havior within the EFT-2 formulation. Oubelkacem et al., studied the same system with anotherapproach, namely EFT with probability distribution technique and they obtained similar results[23]. Albuquerque et al. [24] treated the same system with amorphization effect, again with theEFT-2 formulation. Recently, Sousa et al. have studied the effect of the bimodal random fielddistribution on phase transition characteristics of the isotropic classical- and quantum- spin-1/2Heisenberg model within the EFT-2 formulation and also they found a tricritical behavior [25]. [email protected] We consider a lattice which consists of N identical spins (spin-1 /
2) such that each of the spins has z nearest neighbors. The Hamiltonian of the system is given by H = − J X s i . s j − X i H i s zi (1)where s i and s zi denote the Pauli spin operator and the z component of the Pauli spin operator at asite i , respectively. J stands for the exchange interactions between the nearest neighbor spins and H i is the longitudinal magnetic field acting on the site i . The first summation is carried over thenearest neighbors of the lattice, while the second one is over all the lattice sites. Magnetic fieldsare distributed on the lattice sites according to a Gaussian distribution function which is given by P ( H i ) = 1 √ πσ exp (cid:18) − H i σ (cid:19) (2)where σ is the width of the distribution. According to Eq. (2), the average magnetic field valueof the overall system is zero. The limit σ → s and s ) and treat the interactionsexactly in this two spin cluster. In order to avoid some mathematical difficulties, we replace theperimeter spins of the two spin cluster by Ising spins (axial approximation) [32]. After all, by usingthe differential operator technique with decoupling approximation [29], we get an expression forthe magnetization per spin as m = (cid:28)
12 ( s z + s z ) (cid:29) = h [ A x + mB x ] z [ A y + mB y ] z [ A xy + mB xy ] z i F ( x, y ) | x =0 ,y =0 (3)2here s and s have z distinct nearest neighbors and both of them have z common nearestneighbors.The coefficients are defined by A x = cosh ( J z ∇ x ) B x = sinh ( J z ∇ x ) A y = cosh ( J z ∇ y ) B y = sinh ( J z ∇ y ) A xy = cosh [ J z ( ∇ x + ∇ y )] B xy = sinh [ J z ( ∇ x + ∇ y )] (4)where ∇ x = ∂/∂x and ∇ y = ∂/∂y are the usual differential operators in the differential operatortechnique. Differential operators act on an arbitrary function viaexp ( a ∇ x + b ∇ y ) G ( x, y ) = G ( x + a, y + b ) (5)with any constant a and b . The function in Eq. (3) is given by F ( x, y ) = Z dH dH P ( H ) P ( H ) f ( x, y, H , H ) (6)where f ( x, y, H , H ) = sinh ( βX )cosh ( βX ) + exp ( − βJ ) cosh ( βY ) (7)and X = ( x + y + H + H ) , Y = h (2 J ) + ( x − y + H − H ) i / , (8)with β = 1 / ( k B T ), k B is the Boltzmann constant and T is the temperature. With the help of theBinomial expansion, Eq. (3) can be written as m = z X p =0 z X q =0 z X r =0 C ′ pqr m p + q + r (9)where the coefficients are C ′ pqr = (cid:18) z p (cid:19) (cid:18) z q (cid:19) (cid:18) z r (cid:19) A z − px A z − qy A z − rxy B px B qy B rxy F ( x, y ) | x =0 ,y =0 (10)and these coefficients can be calculated by using the definitions given in Eqs. (4) and (5). Let uswrite Eq. (9) in more familiar form as m = z X k =0 C k m k (11)and C k = z X p =0 z X q =0 z X r =0 δ p + q + r,k C ′ pqr (12)where δ i,j is the Kronecker delta. It can be shown from the symmetry properties of the functiondefined in Eq. (6) and operators defined by Eq. (4) that for even k , the coefficient C k is equal tozero. This property is derived in Sec. A.For a given set of Hamiltonian parameters ( J ), temperature ( k B T /J ) and field distributionparameter ( σ ), we can determine the coefficients from Eq. (12) and we can obtain a non linearequation from Eq. (11). By solving this equation, we can get the magnetization ( m ) for a givenset of parameters and temperature. Since the magnetization is close to zero in the vicinity of thecritical point, we can obtain a linear equation by linearizing the equation given in Eq. (11) whichallows us to determine the critical temperature. Since we have not calculated the free energy inthis approximation, we can locate only second order transitions from the condition C = 1 , C < . (13)The tricritical point at which the second and first order transition lines meet can be determinedfrom the condition C = 1 , C = 0 . (14)3 Results and Discussion
The effect of the zero centered Gaussian magnetic field distribution on the isotropic quantumHeisenberg model problem has one parameter as a measure of randomness, namely the width ofthe distribution σ . As σ increases then randomly distributed magnetic fields with greater absolutestrengths start to act on the lattice sites.In order to see the effect of the random field distribution width ( σ ) on the critical temperatureof the model, we depict the variation of the critical temperature with σ in ( k B T c /J, σ ) plane inFig. 1 for some selected lattices, namely honeycomb ( z = 2 , z = 0), square ( z = 3 , z = 0) andsimple cubic ( z = 5 , z = 0) lattices. As we can see from Fig. (1) that increasing σ values decreasethe critical temperature of the whole lattices continuously. This is due to increasing randomnessin the system: Increasing the randomness via raising the width of the magnetic field distributiondrags the system to the disordered phase while the spin-spin interaction tends to keep the systemin an ordered phase. After a critical value of the width, randomness prevails and for the valuesthat provide σ > σ c , the spin-spin interaction can not maintain an ordered phase even at the zerotemperature. At a certain value of σ , the critical temperature of the simple cubic lattice is greaterthan that of the square lattice, due to the excess number of nearest neighbors of square lattice. Thesame relation is also valid between square and honeycomb lattices. Also due to the same reason,simple cubic lattice has a wider ferromagnetic region in a ( k B T c /J, σ ) plane than the square latticeand the square lattice has a wider one in comparison with the honeycomb lattice. Besides, as wecan see from Fig. (1) that the whole phase transitions between ordered and disordered phases areof the second order and the system does not exhibit a reentrant behavior.In Fig. (1) (b), we depict the low temperature magnetization versus σ curves for the latticesmentioned above. Here, we fixed the temperature as k B T /J = 0 .
001 and this may be consideredas the ground state due to the fact that thermal energy supplied by the temperature on thesystem may be neglected in comparison with the spin-spin interaction energy. The behavior atthis temperature is found to be different from the case in the vicinity of the critical temperatures,since increasing σ first can not change the ground state magnetization value while it changes thecritical temperature continuously. After than, the ground state magnetization starts to reduce, andvanishes with increasing σ . Due to the significant strength of the spin-spin interaction originatingfrom the large number of nearest neighbors concerning the simple cubic lattice, the ground statemagnetization of the simple cubic lattice can become saturated at 1 . σ values, incontrast to the square and honeycomb lattices. The σ c values at which the critical temperaturesreduce to zero corresponding to different lattices can be seen in Table (1). As seen in the Table(1), the EFT-2 formulation is able to distinguish between the lattices with the same coordinationnumber, but different geometry, i.e. the critical width values of the Kagome and square lattices,or triangular and simple cubic lattices are different. For instance, σ c = 2 .
262 for Kagome latticeis slightly lower than the same value for the square lattice σ c = 2 . z = 2 , z = 1) lattice and the square lattice ( z = 3 , z = 0) .The critical value of σ for the 3D lattices can be compared with the same values for the Isingmodel which are σ c = 3 .
850 for the simple cubic, σ c = 5 .
450 for the body centered cubic and σ c = 8 .
601 for the face centered cubic lattices[21]. For the isotropic Heisenberg model thesecritical values are slightly greater than the Ising counterparts.Now let us investigate the evolution of the variation of the magnetization curves with tempera-ture with increasing randomness in the system. In order to achieve this, we depict the variation ofthe magnetization with temperature curves for some selected values of σ in Fig. (2) for honeycomband square lattices. As seen in Fig.(1), since the difference between the 2D and 3D lattices regard-ing to the behavior of the critical temperature and ground state magnetization is only quantitative,we do not expect a qualitative difference between the magnetization profiles of 2D and 3D latticeswith zero centered Gaussian random field distribution. As seen in Fig. (2), magnetization versustemperature curves related to the honeycomb and square lattices are qualitatively similar. As σ increases then both critical temperature and the ground state magnetization become reduced.However, σ = 1 . σ < . k B T c / J σ A:z =2, z =0 B:z =3, z =0C:z =5, z =0(a) A B C 0 1 2 3 4 5 0 1 2 3 4 5 k B T c / J σ A:z =2, z =0 B:z =3, z =0C:z =5, z =0(a) A B C 0 1 2 3 4 5 0 1 2 3 4 5 k B T c / J σ A:z =2, z =0 B:z =3, z =0C:z =5, z =0(a) A B C 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 m σ A:z =2, z =0 B:z =3, z =0C:z =5, z =0(b) A B C 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 m σ A:z =2, z =0 B:z =3, z =0C:z =5, z =0(b) A B C 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 m σ A:z =2, z =0 B:z =3, z =0C:z =5, z =0(b) A B C Figure 1: (a) Phase diagrams of the isotropic quantum Heisenberg model with Gaussian magneticfield distribution in ( k B T c /J, σ ) plane for selected lattices. (b) Variation of the ground statemagnetization of the system with ( σ ) for selected lattices. In (b), the temperature has been fixedas k B T c /J = 0 . σ c ) of the isotropic quantum Heisen-berg model with zero centered Gaussian mag-netic field distribution.Lattice z z σ c ( H )Honeycomb 2 0 1.466Kagome 2 1 2.262Square 3 0 2.320Triangular 3 2 3.875Simple cubic 5 0 3.985Body centered cubic 7 0 5.563Face centered cubic 7 4 8.844transition in the curves with increasing σ . This also indicates that increasing σ can not induce afirst order transition in the isotropic Heisenberg model. In conclusion, the effect of the zero centered Gaussian random magnetic field distribution on thephase diagrams of the isotropic quantum Heisenberg model has been investigated in detail. In thisregard, the effects of the random magnetic fields have been discussed for 2D and 3D lattices. Thephase diagrams, which are the variation of the critical temperature with the width of the Gaussiandistribution have been depicted for the honeycomb, square and simple cubic lattices. It has beenfound that there is not any reentrant behavior or first order transition in the system. The criticalGaussian distribution width at which the critical temperature of the system vanishes has beenobtained for several 2D and 3D lattices within the isotropic model.Besides, the effects of the increasing randomness of the magnetic field distribution on thebehavior of the magnetization versus temperature curves have been investigated. We have notobserved any qualitative difference between the results for 2D and 3D lattices. We have found thatas σ increases then the ferromagnetic region in ( m − k B T c /J ) plane becomes narrower and finallyferromagnetic region disappears right after the critical value of the Gaussian distribution width.5 m k B T/J A: σ =0.0 B: σ =0.5 C: σ =1.0 D: σ =1.2 (a) ABCD 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 m k B T/J A: σ =0.0 B: σ =0.5 C: σ =1.0 D: σ =1.2 (a) ABCD 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 m k B T/J A: σ =0.0 B: σ =0.5 C: σ =1.0 D: σ =1.2 (a) ABCD 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 m k B T/J A: σ =0.0 B: σ =0.5 C: σ =1.0 D: σ =1.2 (a) ABCD 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 m k B T/J A: σ =0.0 B: σ =0.5 C: σ =1.0 D: σ =1.5 E: σ =2.0 (b) ABCDE 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 m k B T/J A: σ =0.0 B: σ =0.5 C: σ =1.0 D: σ =1.5 E: σ =2.0 (b) ABCDE 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 m k B T/J A: σ =0.0 B: σ =0.5 C: σ =1.0 D: σ =1.5 E: σ =2.0 (b) ABCDE 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 m k B T/J A: σ =0.0 B: σ =0.5 C: σ =1.0 D: σ =1.5 E: σ =2.0 (b) ABCDE 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 3.5 m k B T/J A: σ =0.0 B: σ =0.5 C: σ =1.0 D: σ =1.5 E: σ =2.0 (b) ABCDE Figure 2: (a) Phase diagrams of the isotropic quantum Heisenberg model in ( k B T c /J, σ ) planewith Gaussian magnetic field distribution for selected lattices. (b) Variation of the ground statemagnetization of the system with ( σ ) for selected lattices. In (b), the temperature has been fixedas k B T c /J = 0 . A Symmetry properties of the coefficients
In order to see the symmetry properties coefficients given in Eq. (12), let us start with writingcoefficients given Eq. (10) as C ′ pqr = (cid:18) z p (cid:19) (cid:18) z q (cid:19) (cid:18) z r (cid:19) Θ pqr (15)where Θ pqr = Z dH dH P ( H ) P ( H )Θ ′ pqr ( H , H ) (16)and Θ ′ pqr ( H , H ) = A z − px A z − qy A z − rxy B px B qy B rxy f ( x, y, H , H ) | x =0 ,y =0 . (17)Here the function defined by Eq. (7) and the distribution function for the magnetic field on thesites labeled by 1 and 2 is given by Eq. (2). From the definitions given in Eqs. (4), (5) and withusing Binomial expansion, (17) can be written in the formΘ ′ pqr ( H , H ) = z − p X t =0 p X v =0 z − q X t =0 q X v =0 z − r X t =0 r X v =0 K t , v f ( a , a , H , H ) (18)where t , v stands for the ( t , t , t , v , v , v ) and a = ( z − t − v + z − t − v ) J z a = ( z − t − v + z − t − v ) J z K t , v = (cid:18) z − pt (cid:19) (cid:18) pv (cid:19) (cid:18) z − qt (cid:19) (cid:18) qv (cid:19) (cid:18) z − rt (cid:19) (cid:18) rv (cid:19) × ( − v + v + v − (2 z + z ) . (19)We can see from Eq. (18) that, expanded form of the expression have both of the terms whichhas f ( a , a , H , H ) and f ( − a , − a , H , H ) for all possible values of a and a . In other words,every term which include f ( a , a , H , H ) has a corresponding term f ( − a , − a , H , H ). Let usfocus on these two term. We can see from the definitions given in Eq. (19) that the transformation t → z − p − t v → p − v t → z − q − t v → q − v t → z − r − t v → r − v (20)6ransforms the terms in Eq. (19) as a → − a a → − a K t , v → ( − p + q + r − v + v + v ) K t , v (21)i.e. the terms which have f ( a , a , H , H ) and f ( − a , − a , H , H ) in the expanded form of theEq. (18) have same coefficients if p + q + r is even and same but opposite signed coefficients if p + q + r is odd. Thus we can write (18) asΘ ′ pqr ( H , H ) = X t , v K t , v [ f ( a , a , H , H ) + f ( − a , − a , H , H )] , p+q+r is even X t , v K t , v [ f ( a , a , H , H ) − f ( − a , − a , H , H )] , p+q+r is odd . (22)Here, the limits of the sums are identical to those in Eq. (18) and instead of using six sum as inEq. (18), only one sum is used in short notation.From Eqs. (7) and (8) we can see that the function satisfies f ( a , a , H , H ) = − f ( − a , − a , − H , − H ) (23)then we can write (22) asΘ ′ pqr ( H , H ) = X t , v K t , v [ f ( a , a , H , H ) − f ( a , a , − H , − H )] , p+q+r is even X t , v K t , v [ f ( a , a , H , H ) + f ( a , a , − H , − H )] , p+q+r is odd . (24)Thus, we can conclude from Eq. (24) that,Θ ′ pqr ( H , H ) = (cid:26) − Θ ′ pqr ( − H , − H ) , p+q+r is evenΘ ′ pqr ( − H , − H ) , p+q+r is odd . (25)Now, if we look at the integrant of Eq. (16), with the help of the Eqs. (2) and (25), we cansee that it is symmetric about the origin for the odd valued p + q + r and antisymmetric aboutthe origin for the even valued p + q + r in the ( H , H ) plane. Thus, with using this result in Eqs.(16) and (15) then in Eq. (12) we arrive the property C k (cid:26) = 0 , p+q+r is even = 0 , p+q+r is odd (26)and this completes of our derivation. 7 eferences [1] A. I. Larkin, Sov. Phys. JETP , 784 (1970).[2] Y. Imry and S. K. Ma, Phys. Rev. Lett. , 1399 (1975).[3] S. Fishman and A. Aharony, J. Phys. C , L729 (1979).[4] J. L. Cardy, Phys. Rev. B , 505 (1984).[5] Daniel S. Fisher, Geoffrey M. Grinsrein, Anil Khurana, Physics Today , December (1988).[6] M.E. Fisher, J. Chem. Soc. Faraday Trans. , 1569 (1986).[7] G. Forgacs, R. Lipowsky, T.M. Nieuwenhuizen C. Domb, J. Lebowitz (Eds.), Phase Transitionsand Critical Phenomena, vol. 14 Academic Press, London (1991), p. 136[8] R. Blossey, T. Kinoshita, J. Dupont-Roc, Physica A , 247 (1998).[9] Y. El Amraoui, A. Khmou J. Magn. Magn. Mater. , 182 (2000).[10] Y. El Amraoui, A. Hamid, S. Sayouri J. Magn. Magn. Mater. , 89 (2000).[11] A. Aharony, Phys. Rev. B , 3318 (1978).[12] D. C. Mattis, Phys. Rev. Lett. , 3009 (1985).[13] E. F. Sarmento and T. Kaneyoshi, Phys. Rev. B , 9555 (1989).[14] N. G. Fytas, A. Malakis, and K. Eftaxias, J. Stat. Mech. Theory Exp. (2008), 03015.[15] I. A. Hadjiagapiou, Physica A , 3945 (2010).[16] T. Schneider and E. Pytte, Phys. Rev. B , 1519 (1977).[17] T. Kaneyoshi, Physica A 139, 455 (1985).[18] Y. Q. Liang, G. Z. Wei, Q. Zhang, Z. H. Xin, and G. L. Song, J. Magn. Magn. Mater. 284,47 (2004).[19] N. Crokidakis and F. D. Nobre, J. Phys. Condens. Matter 20, 145211 (2008).[20] O. R. Salmon, N. Crokidakis, and F. D. Nobre, J. Phys. Condens. Matter 21, 056005 (2009).[21] ¨U. Akıncı, Y. Y¨uksel, H. Polat, Phys. Rev. E , 061103 (2011).[22] Douglas F. de Albuquerque, A.S. de Arruda Physica A , 13 (2002).[23] A. Oubelkacem, K. Htoutou, A. Ainane, M. Saber Chin. J. Phys. , 717 (2004).[24] Douglas F. de Albuquerque, Sandro L. Alves, A.S. de Arruda Phys. Lett. A , 128 (2005).[25] J. Ricardo de Sousa, Douglas F. de Albuquerque, Alberto S. de Arruda, Physica A , 3361(2012).[26] R. Honmura, T. Kaneyoshi J. Phys. C , 3979 (1979).[27] H.B. Callen Phys. Lett. , 161 (1963).[28] H. Suzuki Phys. Lett , 267 (1965).[29] T. Kaneyoshi, Acta Phys. Pol. A , 703 (1993).[30] A. Bob´ak, M. Jaˇsˇcur Phys. Status Solidi B , K9 (1986).[31] T. Idogaki, N. Uryˆu Physica A , 173 (1992).832] J. Mielnicki, G. Wiatrowski, T. Balcerzak, J. Magn. Magn. Mater , 186 (1988).[33] Ijan´ılio G. Ara´ujo, J. Cabral Neto, J. Ricardo de Sousa Physica A , 150 (1998).[34] J. Ricardo de Sousa, Douglas F. de Albuquerque Physica A , 419 (1997).[35] J. Ricardo de Sousa, Physica A , 383 (1998).[36] Y. Miyoshi, A. Tamaka, J.W. Tucker, T. Idogaki J. Magn. Magn. Mater. , 110 (1999).[37] T. Idogaki, A. Tanaka, J.W. Tucker J. Magn. Magn. Mater. −181