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Exactly solvable model with a multicritical point
605 EXACTLY SOLVABLE MODEL W I T H A M U L T I C R I T I C A L POINT M. L U B A N * Department of Physics, Bar-Ran University, Ramat-Gan, Israel
R.M. H O R N R E I C H and SHTRIKMAN:~ Department of Electonics, The Weizmann Institute of Science, Rehovot, Israel A hypercubic d-dimensional lattice of spins with multineighbor ferro and antiferromagnetic coupling is studied in the spherical model limit (n ~ ) and is found to exhibit a multicritical point of the uniaxial Lifshitz type. Analytic expressions are given for the shift exponent dJ(d) and its amplitudes A.(d).
We deal here with an n-component, ddimensional lattice spin model that can be solved exactly in the spherical model limit (n ~) and which exhibits a multicritical point on the X line, o f the uniaxial Lifshitz type [1], i.e. a triple point for paramagnetic, ferromagnetic, and helicoidal ferromagnetic phases. Order parameter fluctuations play a primary role for the present model as evidenced by the fact that the A line exhibits a singularity of the type T~(p, d ) - T c ( d ) + A , ( d ) [ p l t/~a) as one varies a parameter p. Here TL(d) = Tc(0, d) is the Lifshitz temperature, ~ is the shift exponent and the amplitudes A÷, A apply for p > 0 , p < 0, respectively. We present explicit analytic expressions for @ and A+ as functions of dimensionality d in the range 2 < d---3.5. For d > 3 . 5 , ~ "sticks" at the value tp= 1. An unexpected feature is the change of sign of A_(d) at d = 3. As discussed below, there are several rare earth alloy systems that appear to display a multicritical point of the uniaxial Lifshitz type, and the existing data for the shape of the A line is suggestive of that calculated here for d = 3. We consider a system of classical, n ( ~ ) c o m p o n e n t spins occupying the sites of a d-dimensional hypercubic lattice with unit lattice spacing. A given spin at lattice site R, is coupled to each of its 2d nearest neighbors via a ferromagnetic interaction J1 (>0), and also, b y an interaction J 2 = ~ ( p - 1)J~, to the spins at the pair of sites R~ - 2~, where ~ is a unit vector in the xl direction. * Also at Department of Electronics, The Weizmann Institute of Science, Rehovot, Israel. ~tWork supported in part by the Commission for Basic Research of the Israel Academy of Sciences and Humanities.
Physica 86-88B (1977) 605-606 O North-Holland
For p > 0 the ordered state is a spatially homogeneous ferromagnetic phase, characterized by the wave vectors ___k~(=k ~ ) , where cos ks = ( 1 - p ) - ~ . In the immediate vicinity of the Lifshitz point at p = 0 we have k ~ - ( - p ) ~ as p ~ 0 - , with/3k = 1/2 for all d. We have obtained an analytic expression for the leading singular term of Tc for small values of Ipl and 2 < d <-3.5. The strong dependence on dimensionality can be summarized as follows: For 2 < d < 2.5, TL(d) is zero and we have = 12.5 - dl-' with A+/A= k/2 sin ½~r(3- d). For d = 2.5, TL(2.5) is still zero and Tc(p~0, 2.5) decreases to zero as (In [Pl) ' with A÷ = A_. For d > 2 . 5 , the Lifshitz temperature TL(d) increases from zero monotonically with increasing d, and, in the interval 2.5 < d < 3.5, ~ - - 1 2 . 5 - d l -~ as before, but now A_/A+= X/2 sin ½~r(3- d). There is a dramatic change in the shape of the A line as one passes through d = 3 due to the change of sign of A_ at this dimensionality, and Tc increases linearly with Ipl for d = 3 and p < 0. We remark that we have obtained the result A_ = 0 for d = 3 also for a continuous spin model in the n ~ limit. This reflects the more general fact that the ratio A÷/A_ is universal for a system with a uniaxial Lifshitz point, at least in the n ~ limit. At d = 3.5, Tc displays a very weak singularity of the form [ p l n l P l l with A = - A ÷ . Finally, for d > 3 . 5 , Tc(p~O,d) is dominated by a term linear in Ipl with A = - A ÷ , so that the shift exponent "sticks" at @ = 1 for all d > 3.5. By contrast, for a uniaxial Lifshitz point, the thermodynamic critical exponents adopt classical values only when d > 4.5 [1]. A more complete description of our results for this model are given elsewhere [2]. As is well known, scaling arguments predict
606 that tO is equal to the crossover exponent. In the present context 4~ is the crossover exponent in the scaling form for the free energy F ( T , p ) t2-~f(p/t~), where t = ( T - To(p, d))/Tc(p, d). Using renormalization group methods we have found [1] (b = v14(2-*/t4), and, for the uniaxial Lifshitz point considered in the spherical model limit, we have 4~ = (d - 2.5) -1 for 2.5 < d < 4.5. Thus we confirm the equality tO -- (b for 2 . 5 < d < 3 . 5 , although for d = 3 one has the additional feature that the amplitude A vanishes. There are several magnetic systems which appear to display a multicritical point of the type described here. Of particular interest are the alloys G d - Y and G d - S c where neutron diffraction data [3] indicate that the helicoidal phase is characterized by a wavevector which decreases continuously to zero, as required at a Lifshitz point, when the parameter p (here alloy composition) is varied. As regards the dependence of Tc on p in the vicinity of the multicritical point, only a limited amount of data is
available. H o w e v e r , for both G d - Y [4] and G d Sc [5], the existing data is suggestive of that derived here for the case d = 3. The same is true for the alloys G d - L a [4] and G d - D y [6], although for these compounds no data exists for the composition dependence of the wavevector characterizing the helicoidal phase. Precise measurements of both the A line and the critical exponents in the vicinity of the Lifshitz point would be of great value. References [l] R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett. 35 (1975) 1678; Phys. Lett. 55A (1975) 269. [2] R.M. Hornreich, M. Luban and S. Shtrikman, Physica A (in press). [3] H.R. Child and J.W. Cable, J. Appl. Phys. 40 (1969) 1003. [4] W.C. Thoburn, S. Legvold and F.H. Spedding, Phys. Rev. l0 (1958) 1298. [5] H.E. Nigh, S. Legvold, F.H. Spedding and B.J. Beaudry, J. Chem. Phys. 41 (1964) 3799. [6] F. Milstein and L.B. Robinson, Phys. Rev. 159 (1967) 466. D.M. Sweger, R. Segnan and J.J. Rhyne, Phys. Rev. B9 (1974) 3864.
R.M. H O R N R E I C H and SHTRIKMAN:~ Department of Electonics, The Weizmann Institute of Science, Rehovot, Israel A hypercubic d-dimensional lattice of spins with multineighbor ferro and antiferromagnetic coupling is studied in the spherical model limit (n ~ ) and is found to exhibit a multicritical point of the uniaxial Lifshitz type. Analytic expressions are given for the shift exponent dJ(d) and its amplitudes A.(d).
We deal here with an n-component, ddimensional lattice spin model that can be solved exactly in the spherical model limit (n ~) and which exhibits a multicritical point on the X line, o f the uniaxial Lifshitz type [1], i.e. a triple point for paramagnetic, ferromagnetic, and helicoidal ferromagnetic phases. Order parameter fluctuations play a primary role for the present model as evidenced by the fact that the A line exhibits a singularity of the type T~(p, d ) - T c ( d ) + A , ( d ) [ p l t/~a) as one varies a parameter p. Here TL(d) = Tc(0, d) is the Lifshitz temperature, ~ is the shift exponent and the amplitudes A÷, A apply for p > 0 , p < 0, respectively. We present explicit analytic expressions for @ and A+ as functions of dimensionality d in the range 2 < d---3.5. For d > 3 . 5 , ~ "sticks" at the value tp= 1. An unexpected feature is the change of sign of A_(d) at d = 3. As discussed below, there are several rare earth alloy systems that appear to display a multicritical point of the uniaxial Lifshitz type, and the existing data for the shape of the A line is suggestive of that calculated here for d = 3. We consider a system of classical, n ( ~ ) c o m p o n e n t spins occupying the sites of a d-dimensional hypercubic lattice with unit lattice spacing. A given spin at lattice site R, is coupled to each of its 2d nearest neighbors via a ferromagnetic interaction J1 (>0), and also, b y an interaction J 2 = ~ ( p - 1)J~, to the spins at the pair of sites R~ - 2~, where ~ is a unit vector in the xl direction. * Also at Department of Electronics, The Weizmann Institute of Science, Rehovot, Israel. ~tWork supported in part by the Commission for Basic Research of the Israel Academy of Sciences and Humanities.
Physica 86-88B (1977) 605-606 O North-Holland
For p > 0 the ordered state is a spatially homogeneous ferromagnetic phase, characterized by the wave vectors ___k~(=k ~ ) , where cos ks = ( 1 - p ) - ~ . In the immediate vicinity of the Lifshitz point at p = 0 we have k ~ - ( - p ) ~ as p ~ 0 - , with/3k = 1/2 for all d. We have obtained an analytic expression for the leading singular term of Tc for small values of Ipl and 2 < d <-3.5. The strong dependence on dimensionality can be summarized as follows: For 2 < d < 2.5, TL(d) is zero and we have = 12.5 - dl-' with A+/A= k/2 sin ½~r(3- d). For d = 2.5, TL(2.5) is still zero and Tc(p~0, 2.5) decreases to zero as (In [Pl) ' with A÷ = A_. For d > 2 . 5 , the Lifshitz temperature TL(d) increases from zero monotonically with increasing d, and, in the interval 2.5 < d < 3.5, ~ - - 1 2 . 5 - d l -~ as before, but now A_/A+= X/2 sin ½~r(3- d). There is a dramatic change in the shape of the A line as one passes through d = 3 due to the change of sign of A_ at this dimensionality, and Tc increases linearly with Ipl for d = 3 and p < 0. We remark that we have obtained the result A_ = 0 for d = 3 also for a continuous spin model in the n ~ limit. This reflects the more general fact that the ratio A÷/A_ is universal for a system with a uniaxial Lifshitz point, at least in the n ~ limit. At d = 3.5, Tc displays a very weak singularity of the form [ p l n l P l l with A = - A ÷ . Finally, for d > 3 . 5 , Tc(p~O,d) is dominated by a term linear in Ipl with A = - A ÷ , so that the shift exponent "sticks" at @ = 1 for all d > 3.5. By contrast, for a uniaxial Lifshitz point, the thermodynamic critical exponents adopt classical values only when d > 4.5 [1]. A more complete description of our results for this model are given elsewhere [2]. As is well known, scaling arguments predict
606 that tO is equal to the crossover exponent. In the present context 4~ is the crossover exponent in the scaling form for the free energy F ( T , p ) t2-~f(p/t~), where t = ( T - To(p, d))/Tc(p, d). Using renormalization group methods we have found [1] (b = v14(2-*/t4), and, for the uniaxial Lifshitz point considered in the spherical model limit, we have 4~ = (d - 2.5) -1 for 2.5 < d < 4.5. Thus we confirm the equality tO -- (b for 2 . 5 < d < 3 . 5 , although for d = 3 one has the additional feature that the amplitude A vanishes. There are several magnetic systems which appear to display a multicritical point of the type described here. Of particular interest are the alloys G d - Y and G d - S c where neutron diffraction data [3] indicate that the helicoidal phase is characterized by a wavevector which decreases continuously to zero, as required at a Lifshitz point, when the parameter p (here alloy composition) is varied. As regards the dependence of Tc on p in the vicinity of the multicritical point, only a limited amount of data is
available. H o w e v e r , for both G d - Y [4] and G d Sc [5], the existing data is suggestive of that derived here for the case d = 3. The same is true for the alloys G d - L a [4] and G d - D y [6], although for these compounds no data exists for the composition dependence of the wavevector characterizing the helicoidal phase. Precise measurements of both the A line and the critical exponents in the vicinity of the Lifshitz point would be of great value. References [l] R.M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett. 35 (1975) 1678; Phys. Lett. 55A (1975) 269. [2] R.M. Hornreich, M. Luban and S. Shtrikman, Physica A (in press). [3] H.R. Child and J.W. Cable, J. Appl. Phys. 40 (1969) 1003. [4] W.C. Thoburn, S. Legvold and F.H. Spedding, Phys. Rev. l0 (1958) 1298. [5] H.E. Nigh, S. Legvold, F.H. Spedding and B.J. Beaudry, J. Chem. Phys. 41 (1964) 3799. [6] F. Milstein and L.B. Robinson, Phys. Rev. 159 (1967) 466. D.M. Sweger, R. Segnan and J.J. Rhyne, Phys. Rev. B9 (1974) 3864.