Critical properties of the XY model

556 CRITICAL PROPERTIES OF THE X Y MODEL D.D. B E T T S Theoretical Physics Institute, University of Alberta, Edmonton, Canada (Invited paper) The spin one half X Y model is of interest as the probably simplest q u a n t u m mechanical m a n y body s y s t e m , as a model for liquid "He near the lambda transition, as a model for a class of antiferromagnetic insulators near TN and as a model for granular superconductors. The latest estimates of the critical properties of the three dimensional spin one half X Y model are presented. The evidence for and nature of a phase transition in the two dimensional X Y model are discussed. Recent experimental m e a s u r e m e n t s on the c o m p o u n d s CoCI.,.6H.,O and CoBr2.6H~,O and the c o m p o u n d s CO(C,H~NO)~ (CIO~)2 and Co(C~HsNO)6 (BF4)2 are compared with theoretical predictions for the spin I/2 X Y model on the square and simple cubic lattices, respectively. Renormalization group calculations are touched on.

1. Introduction

A rather general model for the interaction of magnetic ions localized on the sites of a lattice is specified by the interaction Hamiltonian of the anisotropic Heisenberg model expressed in terms of pseudospin variables S ~,

~0-

± x x+ S~'S~) +JijSiSt]. 2 S ( S 3 + 1) ~~,j [J~j(S~Sj II z z

(1) The indices i and j label the sites of the lattice while Ji~ and Jll are " e x c h a n g e " energy parameters. In theoretical studies calculational simplicity is achieved by studying the special cases of the Ising model, J± = 0, the X Y model, jII = 0, or the isotropic Heisenberg model, J ± = J". The above restrictions are not so severe as they might seem because of the principle of universality [1]. According to this principle all systems which undergo second order phase transitions fall into a small number of universality classes distinguished by a very few parameters notably the dimensionality, d, of the system and the dimensionality, n, of the order parameter. In the critical region the thermodynamic properties of all systems of the same universality class are identical apart from typically two scale factors. The universality principle has been well verified so far [2], so we concentrate on the nearest neighbour, spin 1/2 X Y model as a simple example of an n = 2 system. As the properties of the X Y model have been reviewed [3] recently, we emphasize subsequent Physica 86-88B (1977) 556-561 (~) North-Holland

developments. In section 2 we outline the relation of the spin 1 / 2 X Y model to a class of magnetic insulators, to liquid helium and to granular superconductors. The dimensionality of the lattice is of supreme importance with regard to the existence and nature of a phase transition for the X Y model. The one dimensional model is paramagnetic at all temperatures, while there seems little doubt that the three dimensional model undergoes a second order phase transition at a finite temperature to a phase with conventional long range order. The existence of a phase transition, the nature of a low temperature phase and the type of transition are all controversial questions for the two dimensional model, which we consider last. A thorough review has been published [4] on magnetic systems as realizations of the d -- I, 2 and 3 Ising, X Y and Heisenberg models. A very recent exciting experimental development, discussed below, has been the investigation of the properties of good examples of systems approximating both the two and the three dimensional spin one half X Y models. 2. Experimental realizations of the X Y model

A magnetic system of pseudo spins governed by the Hamiltonian (1) will belong to the n = 2 universality class of the X Y model provided j l > jll. The examples recently studied in detail include [5] the antiferromagnetic layer (d = 2) compounds, COC12.6H20 and CoBr26H20, and [6] the antiferromagnetic pyradine compounds C o ( C s H s N O ) 6 ( C l O 4 ) 2 and Co(CsHsNO)6(BF4)2 in which the Co 2+ ions form a nearly simple cubic magnetic lattice.

557 In terms of the true spin, S = 3/2, of the Co 2+, we may assume [6] a Hamiltonian containing the basic isotropic Heisenberg exchange term and a single ion anisotropy due to a local axial crystalline field,

=-2J~

S,. S ~ - / 5 ~ [ ~ ) z _ ~(~ + 1)/3)]. (i,i~

i

(2) The effect of the single ion term is to split in energy the Kramers doublet, only the lowest doublet being appreciably populated, so that there is an effective spin, S--1/2, in the interesting temperature range. The components of the effective spin are related to those of the true spin by S ~= 2S~/g ~. On comparing (1) and (2) we see that J~ = (g~/2)2j and fir= (glr/2)2j. In the above compounds jll~ 0.3 J±. In the halogen compounds there is a slight easy plane anisotropy, JY/JX ~0.96, important only as T ~ T¢. The S = 1/2 anisotropic Heisenberg model was derived [7] as a model of liquid 4He in the vicinity of the lambda transition. It is necessary that J ~ > jH for the order parameter to have the appropriate dimensionality, n = 2. The superfluid order parameter, tO, of liquid aHe is the analogue of the perpendicular magnetization, M ~, of an X Y like ferromagnet. The conjugate field, r t, which is physically unrealizable, is the analogue of H 1. The axial field, H", of the ferromagnet is analogous to the pressure or chemical potential in 4He while the axial magnetization, M ~, is linearly related to the density of 4He [3]. For a quite different realization of the S = 1/2 X Y model [8] consider a regular lattice of spherical grains of superconducting material embedded in a nonconducting matrix. As the temperature is lowered each grain will undergo, at T¢, a transition to a superconducting state which should become fully coherent for grain size a -> ~(T), the coherence length. The phases of the superconducting order parameter in different grains should remain at first completely random. All electrons are coupled into spin zero Cooper pairs which can tunnel from one grain to another governed by a tunneling Hamiltonian,

= - ~ i.j

~,.j(s+,s~, + s ,s+j),

(3)

where S~ =EkS~, S~ being the Cooper pair creation and annihilation operators for momentum k in grain i. This is precisely the S = I/2 X Y model, and thus a further phase transition is expected at a temperature To(< Tc) below which there will be long range order in the sense that phases of the superconducting order parameters in the individual grains achieve a long range correlation. Finally certain ferrodistortive lattice dynamical models with quartic anisotropy [9], though not quite X Y models, nevertheless seem to belong to the same universality class. 3. Critical properties of the three dimensional

XY model The order parameter of the X Y model is the perpendicular or transverse magnetization M ~ = (MX, M ~) for ferromagnets or the staggered transverse magnetization, N l for antiferromagnets where M ~=/zag~EST. For an X Y ferromagnet quantities of particular interest and their expected critical behaviour include: (i) the reduced specific heat, Cu/NkB ~ ~'A(I - T J T ) -~ T > Tc I. A'(1 - T[Tc) -~' T < T~,

(4a)

(ii) the fluctuation in the long range order, Y

(( M X)2)/ N (txng ±) 2

~ C ( 1 - T d T ) ~ T > T¢ (C'(I - TITc) -v T < Tc

(4b) The perpendicular susceptibility X1= OMX/OH x will behave in the critical region like Y. The values of the amplitudes and critical temperatures depend on the specific properties of the model such as the underlying lattice and the spin value and likewise for experimental systems vary from substance to substance, but the critical exponents should have the values of the simple X Y model for all models and systems of the same universality class. The methods of deriving high temperature series expansions and the actual coefficients as far as they were then known have already been reviewed [3]. Dekeyser and Rogiers [10] have recently extended the high temperature series for the free energy and the transverse magnetization fluctuation, Y, to include arbitrary

558 longitudinal field, H ~. Their analysis of the fluctuation series in zero field, which contains one additional coefficient, confirming the previous estimate, yields y = 1.333 _+0.01. It is attractive to assume that 3' = 4/3 exactly. The high t e m p e r a t u r e specific heat series is an even function of T for loose packed lattices, so the only existing series in three dimensions which can be analyzed directly is that for the f.c.c, lattice [3], for which a = 0.02-+ 0.05. It is attractive to assume a logarithmic singularity (a = 0). Scaling laws then give for the other exponents /3 = 1/3, 8 = 5, A = 5/3, v = 2/3 and r / = 0. The amplitude, A F, and additive constant, B0F have also been estimated [3] in

It is now of experimental interest to have the specific heat curve for the simple cubic (S) lattice. Assuming the same form, (5), for the s.c. lattice and adopting the a b o v e values for the critical exponents and estimates for C F, A F and C s one finds [1 l] with the aid of one scale factor universality [12] that As = 0.319_+ 0.005. The additive constant, B0s, can next be found by evaluating at Kc Pad6 approximants to the series for the specific heat with the logarithmic singularity subtracted off. Unfortunately there seem to be errors in the last two coefficients of the original series [13] so we reproduce the revised series in its entirity.

CS/NkB = (3/4)K 2 + (1/32)K 4 + (51/180)K 6 + 8.60052077 K s + 65.7598958 K t ° + • • •

(6)

F r o m the revised series B0s = - 0 . 3 9 - + 0.05, only slightly larger than the previous estimate [l 1]. Although the critical properties of the X Y model for T < T~ cannot yet be predicted, the simple spin wave theory result that Cu oc T 3 at low t e m p e r a t u r e is useful. XY

H-

)--

• T

7

/

/

/

?

/

/

/

/

/

8 )__2_

(5)

C~/NkB ~ - A F In (l - T,F/T) + BFo.

4. N a t u r e of t h e t w o d i m e n s i o n a l

illustrated in fig. 1. Fig. l(a) illustrates the N6el state of longitudinal staggered magnetization, N", with spins pointing alternatively up and down as for an Ising antiferromagnet. Fig. l(b) illustrates a state of nonzero transverse magnetization, M ~, while fig. l(c) shows a state of non-zero transverse staggered magnetization, N ±. Finally fig. l(d) illustrates a vorticity state in which clockwise and counter clockwise vertices f o r m a lattice of spacing 28.

model

A conventional type of phase transition would involve a transition at a t e m p e r a t u r e Tc b e t w e e n a low t e m p e r a t u r e phase having a nonzero thermal expectation value of some long range order parameter, O, and a high t e m p e r a t u r e phase for which ( 0 ) = O. For a square lattice some possible types of long range order are

T

~"

Y

~

/

,t

/_

7

K

_K

/

/_

x~

,~ N

/

/ ff

_N c

d

Fig. 1. Possible ordered states of the two dimensional X Y model. (a) staggered longitudinal magnetization, N ~, (N6el state), (b) transverse magnetization, M x, (c) staggered transverse magnetization, N X, (d) longitudinal vorticity, W.

For a class of two dimensional spin models including the X Y and isotropic Heisenberg models no phase of nonzero magnetization or staggered magnetization exists for any zero t e m p e r a t u r e [14]. Meanwhile Stanley and Kaplan [15] conjectured, on the basis of 6 term high t e m p e r a t u r e susceptibility series [16], that some sort of phase transition takes place at a finite Tc for the isotropic Heisenberg model for S > 1/2. The low t e m p e r a t u r e phase would be characterized b y correlations which decay with distance sufficiently slowly to yield an infinite susceptibility without a finite spontaneous magnetization. On the basis of longer series [17] a S t a n l e y - K a p l a n transition in the two dimensional isotropic Heisenberg model (n = 3) now seems unlikely for any S. Models with n = 2 which have been in-

559 vestigated by high temperature series expansion include the plane rotator model [18], the classical X Y model [19] and the S = 1/2 X Y model [20]. All of these models show stronger evidence of a divergent susceptibility at a finite Tc than does the isotropic Heisenberg model. One puzzling feature however is that the same methods of analysis yield 3' = 3.0 for the S - models and 3' = 1.5 for the S = 1/2 model. For the d -- 2 classical X Y model Berezinskii [21] has argued that for spins separated sufficiently far apart the relative phase may greatly exceed 2~-. Kosterlitz and Thouless [22] introduced the concept of a low temperature phase of "topological long range order" in which point dislocations or vortices of opposite Burgers vector are bound together. Above a transition temperature Tc ~ ~rJ~/kB the dislocations b e c o m e unbound and isolated vortices will be found. Fig. l(d) illustrates a simple form of topological long range order. Kosterlitz [23] then predicted that the susceptibility would have the unusual critical behaviour, g ~ X0 exp A(I - T/To) ~

(7)

rather than the conventional power law divergence. The series for the classical, d = 2, n = 2 models favours (7) over a power law [24], but for S = 1/2 we favour a power law divergence. Contrary to general belief the Bethe-Peierls approximation to the solution of the S = 1/2 Ising model has been found not to become exact on the Cayley tree [25] but rather the free energy is of the form [26]. F ( T , H ) = Freg(T, H 2) + G(T)IH[ -~T'

(8)

to lowest order in the field H. The exponent K varies from 1 at T = 0 to ~ at T = T ~ . For T > T' where T' >i T~ F ( T , 0) is analytic. Thus there is a line of phase transitions varying from first order at T = 0 to infinite order at T = T~. Zittartz [27,28] has shown that the d = 2 plane rotator model undergoes a continuous phase transition of the above type. As a consequence there are the following properties all of which are in accord with previous information about the two dimensional X Y model: (i) the spontaneous magnetization vanishes identically [14]; (ii) the specific heat is analytic at all temperatures [29]; (iii) at low temperatures

the singular exponents should depend on temperature [21]; (iv) the initial susceptibility should diverge at a temperature T: such that K(T2) = 2 [18-20]. We have made a detailed study of the ground state properties of clusters of 4, 8 and 16 spins coordinated as in a square lattice with periodic boundary conditions [30]. By extrapolation we estimate the ground state energy per bond for the infinite square lattice, - E o / 2 N J l ~ - 0 . 5 4 compared with the mean field value for the completely alligned state of 0.50. For the 16 spin "lattice" we have computed exactly all inequivalent two spin correlations. The correlations (o~tr~) for five inequivalent values of r are all of the order of +0.5 and fall very slowly with distance. The nearest neighbour correlation, ( o ' ~ ) ~ - 0 . 1 8 and others are negligible. The ground state is more complex than any of the states of fig. 1 but closest to fig. la. The renormalization group method [31,32] provides an alternative to series expansion methods for investigating phase transitions and critical phenomena. The approach of Niemeijer and van L e e u w e n [33] in real space is most convenient for d = 2, finite S, and can be adapted to n > l systems such as the S = I / 2 X Y model. For the plane rotator model Lublin [34] finds a Tc 30% lower than the series estimate [18] and critical exponents 3' = 2.4 _+0.5 and a = - 0 . 7 + 0.3. The estimate of 3' is not inconsistent with the series estimate, a = - 1 would indicate an analytic specific heat in agreement with Zittartz [28]. In the renormalization group calculations for the S = I / 2 X Y model [35, 36, 37] the chosen lattice is divided into identical cells each containing a small number, I d, of sites and in such a way that the cells themselves form the same lattice (with lattice spacing 16). All properties of the system within the cells can be calculated exactly while interactions between spins in different cells are treated as a perturbation. In order to make the system of cells look like the original system some of the degrees of freedom are summed over in a partial trace operation. The partial trace must be performed in such a way as to preserve the free energy,

exp (--/3~¢e,,) = tr' exp ( - f l ~ i , ~ ) -

(9)

560 By equating corresponding matrix elements on both sides of (9) a set of nonlinear recurrence relations can be obtained for the various interaction parameters in the Hamiltonian. These relations can then be solved numerically to obtain fixed points, critical temperatures, exponents etc. in the usual way. Calculations have been carried out for symmetrical three [35], five [36] and seven [37] spin cells on the triangular, square and triangular lattices to second order in the cumulant expansion. Starting with nearest neighbour X Y interactions the renormalization operation generates second and third neighbour X Y interactions and first neighbour Ising interactions. Solution of the recurrence relations for the four interaction strengths gave a finite critical temperature for the triangular lattice about 30% lower than the series estimate and a negative value for 3! H o w e v e r the zero field free energy curves for both lattices are in excellent agreement with the curves from high temperature series expansions. Extensions of these calculations either by increasing the cell size or by going to third order in the cumulant expansion would be very laborious and are not being contemplated at present. H o w e v e r it would be desirable to have longer high temperature series and very shortly two additional coefficients will be available* [38].

good agreement with liquid 4He near the lambda transition [3], but until recently good magnetic examples of the X Y model have not been studied. Algra et al. [6] have measured the specific heats of Co(C~HsNO)6(CIO4)~ and Co(CsHsNO)6(BF4)2 below I K in an adiabatic demagnetization apparatus using the heat pulse technique with a cerium magnesium nitrate thermometer. Phase transitions, marked by lambda anomalies in the specific heat, were observed at T~ = 0.428 K and T~ = 0.357 K, respectively. The two specific heat curves can be brought into very near coincidence by plotting against T/Tc. To compare theory with experiment, for T > T~, Algra et al. constructed a theoretical curve by smoothly joining the critical part, (6), to a high temperature part obtained more directly from the series. (Since K s ~ 0 . 5 0 the errors noted in the coefficients of K s and K~° make no detectable contribution to the curve.) In fig. 2 we have plotted the dimensionless specific heat,

O8

06

c_ R o4

5. Comparison with experiment The S - - I / 2 X Y model in three dimensions behaves in the critical region in remarkably Table 1 Critical properties of the S = 1/2 Ising, X Y and Heisenberg models on the s.c. lattice compared with experiment [6] System

kTc/J

(S~- Sc)/NkB

Ising

2.255 2.02 1.68 1.98 2.02

0.135 0.20 0.26 0.25 0.24

XY Heisenberg Co(PyNO)dBF, h Co(PyNO)dCIO,)2

-E,:/NkBTc 0.220 0.41 0.63 0.47 0.43

* In the course of these investigations it has been discovered that the original series for the zero field partition function for the square and triangular lattices [29] contain errors in coefficients beyond the sixth degree. The errors do not effect the previous conclusion [29] that the free energy is analytic.

02

01

02

03

04

Kc

K Fig. 2. Smoothed experimental specific heat data for C o ( P y N O M B F 4 h (upper curve) compared with the specific heat of S = 1/2 X Y model on the s.c. lattice (lower curve) versus reduced inverse temperature, K = Jl/kBT.

C]Nkn, for the S = 1/2 X Y model on the simple cubic lattice using the corrected coefficients in (5) and (6) and for comparison the experimental curve [6] for Co(PyNO)6(BF4)2. Possible reasons for the small discrepancy between theory and experiment are: (i) in the experimental system jII ~ 0.3 j l (ii) some anisotropy in the easy plane is to be expected (iii) in the calculation there are significant uncertainties in estimate of K s, A s and particularly B0s.

561

De Jongh et al. [39] have compared the experimental specific heat and the initial susceptibility of COC12.6H20 and C o B r 2 . 6 H 2 0 with theoretical predictions for the S = 1/2 X Y antiferromagnet on the square lattice. The theoretical specific heat curve is obtained from the high temperature series for T ~> 1.5 Jl/kB (the errors in the last two coefficients as originally reported have no effect on this portion of the curve). For T ~< 0.3 Jl/kB the simple spin wave result [40] C/Nk~ = 0.28 (kBT/Jl) 2 is used, and in the intermediate region a smooth curve is drawn so that the area under the curve yields the correct entropy [5, 40]. The exchange constants for the chlorine and bromine salts, J ± / k B = - 2 . 0 5 K and Ji/kB = - 2 . 4 5 K respectively, are obtained by fitting experimental data to the high temperature end of the theoretical curve. U p o n fixing j l the specific heat curve for COC12.6H20 is in excellent agreement with the theoretical curve for all T > 1.5{JlllkB. There is however a sharp spike in the experimental curve just below this temperature, which could be due either to easy plane anisotropy causing a d = 2 Ising logarithmic singularity or to the effect of interplanar interactions causing a d = 3 X Y nearly logarithmic singularity. The initial susceptibility of both salts is also in excellent agreement with the theoretical for the X Y model for T > Tc [39]. References [1] L.P. Kadanoff in Critical Phenomena, M.S. Green, ed. (Academic Press, N e w York, 1971). [2] See e.g.D.D. Betts, A.J. Guttmann and G.S. Joyce, J. Phys. C4 (1971) 1994. M. Ferer and M. Wortis, Phys. Rev. B6 (1972) 3426. P.C. Hohenberg, A. Aharony, B.I. Halperin and E.D. Siggia, Phys. Rev. BI3 (1976) 2986, where other references may be found. [3] D.D. Bens in Phase Transitions and Critical Phenomena, Vol. 3, C. Domb and M.S. Green, eds. (Academic Press, London, 1974). [4] L.J. de Jongh and A.R. Miedema, Advances in Phys. 23 (1974) I. [5] J.W. Metselaar, L.J. de Jongh and D. de Klerk, Physica 79B (1975) 53.

[6] H.A. Algra, L.J. de Jongh, W.J. Huiskamp and R.L. Carlin, Physica 83B (1976) 71. [7] T. Matsubara and H. Matsuda, Prog. Theor. Phys. 16 (1956) 416. [8] J. Rosenblatt, A. Raboutou and R. Pellan in Proc. LT 14, Vol. 2, M. Krusius and M. Vuorio, eds. (NorthHolland, Amsterdam, 1975). [9] T. Schneider and E. Stoll, Phys. Rev. Lett. 36 (1976) 1501. [10] R. Dekeyser and J. Rogiers, Physica 81A (1975) 72. [11] D.J. Austen and D.D. Betts, Phys. Lett. 53A (1975) 313. [12] D.D. Betts and D.S. Ritchie, Phys. Rev. Lett. 34 (1975) 788. [13] J.T. Tsai and C.J. Elliott, Phys. Lett. 45A (1973) 295. [14] N.D. Mermin and H. Wagner, Phys. Rev. Lett. 17 (1966) 1133. [15] H.E. Stanley and T.A. Kaplan, Phys. Rev. Lett. 17 (1966) 913. [16] G.S. Rushbrooke and P.J. Wood, Mol. Phys. 1 (1958) 257. [17] K. Yamaji and J. Kondo, J. Phys. Soc. Japan 35 (1973) 25. [18] M.A. Moore, Phys. Rev. Lett. 23 (1969) 861. [19] M. Ferer, M.A. Moore and M. Wortis, Phys. Rev. B8 (1973) 5205. [20] D.D. Betts, C.J. Elliott and R.V. Ditzian, Can. J. Phys. 49 (1971) 1327. [21] V.L. Berezinskii, Soviet Phys. JETP 32 (1971) 493. [22] J.M. Kosterlitz and D.J. Thouless, J. Phys. C6 (1973) 1181. [23] J.M. Kosterlitz, J. Phys. C7 (1974) 1046. [24] W.J. Camp and J.P. Van Dyke, J. Phys. C8 (1975) 336. [25] T.P. Eggarter, Phys. Rev. B9 (1974) 2989. [26] E. Miiller-Hartmann and J. Zittartz, Phys. Rev. Lett. 33 (1974) 893. [27] J. Zittartz, Z. Physik B23 (1976) 55. [281 J. Zittartz, Z. Physik B23 (1976) 63. [29] D.D. Betts, J.T. Tsai and C.J. Elliott, in Proc. Int. Conf. on Magnetism, Moscow, 1973, Vol. IV (Nauka, Moscow, 1974). [30] J. Oitmaa and D.D. Betts, to be published. [31] K.G. Wilson, Phys. Rev. B4 (1971) 3174. [32] K.G. Wilson and J. Kogut, Phys. Reports 12C (1974) 75. M.E. Fisher, Rev. Mod. Phys. 46 (1974) 597. [33] Th. Niemeijer and J.M.J. van Leeuwen, Phys. Rev. Lett. 31 (1973) 1411. [34] D.M. Lublin, Phys. Rev. Lett. 34 (1975) 568. [35] J. Rogiers and R. Dekeyser, Phys. Rev. B13 (1976) 4886. [36] D.D. Betts and M. Plischke, Can. J. Phys. 54 (1976) 1553. [37] J. Rogiers and D.D. Betts, Physica, in press. [38] J. Rogiers and D.D. Betts, to be published. [39] L.J. de Jongh, D.D. Betts and D.J. Austen, Solid State Commun. 15 (1974) 1711. [40] P. Bloembergen, Physica 79B (1975) 467.