THERMAL PHASE TRANSITIONS AND GROUND STATE PHASE TRANSITIONS: THE COMMON FEATURES AND SOME INTERESTING MODELS

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1 THERMAL PHASE TRANSITIONS AND GROUND STATE PHASE TRANSITIONS: THE COMMON FEATURES AND SOME INTERESTING MODELS Ján Greguš Department of Physcs, Faculty of the Natural Scences, Unversty of Constantne the Phlosopher, Treda A. Hlnku, SK Ntra, Slovaka Emal: Abstract We pont out the common features of the thermal phase transtons and the ground state quantum phase transtons and descrbe the coherent anomaly method (CAM) as a sutable tool for the theoretcal treatng of these collectve phenomena. We use CAM for the studyng of the unversalty of the crtcal exponents n the ground state phase transtons n the quantum spn Hesenberg XYZ model on the one-, two- and three dmensonal lattces and present here some prelmnary results. Keywords: phase transton, ground state, mean-feld, unversalty, XYZ model. Introducton Many nterestng (and presently wdely studed) physcal phenomena lke ferromagnetsm, antferromagnetsm, superconductvty, superfludty, etc., can be well understood n the framework of the theory of phase transtons and crtcal phenomena whch provdes a sutable qualtatve and quanttatve descrpton of them. Thus, a further development and mprovement of the sutable models and methods n ths area s desrable. Generally, a phase transton n a physcal system s characterzed by the breakng of certan symmetry of the system and the appearng of a nonzero value of some order parameter together wth a sngular behavor of the characterstc quanttes near the crtcal value of the system s parameters [], []. Thermal Phase Transtons and Ground State Phase Transtons The physcal phenomena mentoned above (ferromagnetsm, superconductvty, etc.) are the examples of thermal phase transtons. The characterstc feature of these s that thermal fluctuatons about an equlbrum state wth thermodynamc temperature near some crtcal value T c (and wth crtcal values of external felds) nduce the long-range correlatons n the system together wth appearng of nonzero value of some order parameter and sngular behavor of the thermodynamc quanttes of the system. We focus on the magnetc cooperatve phenomena caused prncpally by the exchange nteracton of the spns lke ferromagnetsm and antferromagnetsm, because here one can fnd most smple models, gvng possblty to understand the phase transtons and crtcal phenomena more deeply. Here, the parameters characterzng an equlbrum state of the system are the temperature T of the system and an external magnetc feld h. 3

2 If the temperature value T s near some T c, and h=0, the thermal fluctuatons nduce the long-range correlatons between the spns and for T<T c and h=0 there are two equvalent equlbrum states (nstead of only one for T>T c ) dfferng only by the sgn of the mean value of total spn magnetc moment,.e. the symmetry wth respect to spn flps s broken and the spontaneous magnetzaton (or sublattce magnetzaton, n case of antferromagnetsm) appears, together wth the sngulartes of the free energy and ts dervatves (magnetzaton, susceptblty, specfc heat, etc.) as functons of the temperature and magnetc feld. The typcal spontaneous magnetzaton [3], and susceptblty [4], curves are n Fg. a, b, respectvely. Fg. (a) (b) One can observe analogcal phase transtons n quantum systems n the ground state. Here, the symmetry breakng, the long-range correlatons and the nonzero value of an order parameter are nduced by the quantum fluctuatons of the quantty representng the order parameter about ts ground state mean value. Thus, whle n the descrpton of thermal phase transtons the mportant parameters were the ones characterzng an equlbrum state (.e. temperature and an external parameter), here these are the ones characterzng the ground state,.e. the parameters characterzng the nteracton tself (together wth an external parameters). At the transton pont, agan one can observe the sngular behavor of the ground state energy (as a functon of these parameters) and ts dervatves. In quantum spn systems, these represent e.g. the ground state mean total spn,.e. the ground state magnetzaton and correspondng magnetc susceptblty. The nteracton parameter (whch depends on the composton of the materal) dependence of these s smlar as that n the Fg. a, b. (see e.g. [5], [6]). Let us state several good reasons to study the models of the ground state phase transtons: t turns out that a smple model of superconductvty, the Hubbard model, n certan values of ts parameters, s the same as that of antferromagnetc quantum Hesenberg model [7] and studyng ts ground state propertes (ncludng the phase transtons) helps to clarfy the structure of ths ground state (e.g. the symmetry, the order); another reason s a rather theoretcal one: the ground state crtcal behavor of the d-dmensonal quantum spn system should be the same as that of (d+)-dmensonal system of classcal spns [8], whch means that studyng the ground state phase 33

3 transtons one can get an nformaton about the crtcal behavor of the correspondng classcal system (and vce versa). 3. Some Interestng Models Collectve behavor of many magnetc systems wth domnant spn exchange nteracton can be successfully descrbed by the lattce models of nteractng spn systems wth exchange nteracton between the spns occupyng the stes of the lattce. Probably the frst (but very mportant) such model s the mean feld (MF) model [9]. The energy of a confguraton of the spns n ths model s qj E( σ, h) = hmf σ, hmf = h+ σ j () N Here σ s a classcal spn n the -th ste of the lattce (ts value s ether or ), q s a number of nearest neghbors of the ste (t depends on the dmenson of the lattce), J s a couplng parameter characterzng the strength of exchange nteracton, N s the total number of stes on the lattce and h s the external magnetc feld. Thus, n ths model every spn nteracts wth the mean feld of all other spns on the lattce (unlke other models, n most of whch the nteracton s only between the nearest neghborng spns). Gven ths mcroscopc descrpton of the system, one can employ statstcal physcs to calculate thermodynamc quanttes n thermal equlbrum wth the temperature T: f ( T, h) = lm ( kt ln Z), Z = exp ( E( σ, h) / kt ) () N N σ ( j ) mt (, h) = exp( E(, h) / kt), m( T) = lm mt (, h) Z mt (, h) f( T, h) χ ( Th, ) = = h h σ 0 N σ (3) h 0 σ where f(t,h) s the free energy per one ste of the lattce, Z s the partton functon, m(t,h) s the magnetzaton, m 0 (T) s the spontaneous magnetzaton (becomng nonzero for temperature values under the crtcal one) and χ(t,h) s the magnetc susceptblty. Sums over σ are over all possble spn confguratons on the lattce. Ths way, for m(t,h) n the MF model one can derve the self-consstency relaton m tanh qjm + h = (5) kt Detaled analyss of (5) and of the free energy f(t,h) shows that n the neghborhood of the pont T=T c =qj/k, h=0 the free energy, the magnetzaton and the susceptblty exhbt sngular behavor of the form: f( T,0) ( T T c ) ( ) ( ) β m0 T Tc T for T < T c γ χ ( T,0) ( T T ) for T > α (6a) c (,0) ( γ c ) for T c χ T T T T < (6d) where α, β, γ, γ are called the crtcal exponents. For MF model, there s T c 34 (4) (6b) (6c)

4 α = 0, β = /, γ = γ = (7) the same crtcal temperature and exponents one can obtan also from the Landau theory by expandng the free energy n the neghborhood of the crtcal pont nto a power seres n order parameter (n ths case, the magnetzaton) f ( T, h, m) = a ( T) + a ( T) m + a ( T) m +... mh (8) and by examnng the (global) mnma of ths functon []. When h=0, the crtcal pont s ndcated by the appearance of two symmetrc global mnma and a nonzero value of the magnetzaton (compared to one global mnmum above the crtcal temperature). At ths pont, the sngle global mnmum becomes a local maxmum and the second dervatve of f wth respect to m becomes zero, whch yelds the crtcal temperature. The MF crtcal exponent values are nterestng, because, accordng to the renormalzaton group calculatons [], for the lattces wth the dmenson d$4 these values correctly descrbe crtcal behavor of classcal spn system wth short range nteracton. The crtcal exponents (.e. the character of crtcal sngulartes) are unversal for a broad range of collectve phenomena. For spn systems wth short range nteracton the unversalty means that the crtcal exponents depend only on the spatal dmensonalty of the system and on the symmetry of the system (they are ndependent e.g. of the detals of nteracton). However, there are several models, for whch the unversalty hypothess doesn t hold [9]. We focus on the ground state of the quantum Hesenberg XYZ model, characterzed by the energy operator (9) H = ( J σ σ + J σ σ + J σ σ ) ( hσ + hσ + h σ ) (, j) j j 3 j 3 where σ,,3 are the spn operators on the -th ste of the lattce, and h,,3 are the magnetc feld components. The frst sum s over all pars of the nearest neghborng spns. In one dmenson wth zero external feld, the ground state energy of ths model s known as an explct functon of the couplng parameters J, J, J 3. The ponts of phase transton are gven by the sngulartes of ths functon at the border J 3 =J of the basc parameter regon [5] J 3 J J. Its sngular part and the order parameter (we take ( h, h, h ) = ( h,0,0) ) n the neghborhood of the crtcal pont are [9], [0] E 3 ( 4π µ ) J snµ cot( π µ ) / sn g = p π µ (0a) N ( π µ )/4µ σ ( π µ ) p (0b) J3 J J p cos µ =, 0 < µ < π (0c) 6( J J ) J where. s the ground state mean value. Clearly, the crtcal exponents of the sngular part of the ground state energy and of the magnetzaton (n the x drecton) depend on the couplng parameters J, J, J 3.e. they are not unversal. The am of our work s to check f unversalty n the XYZ model s restored for the two- and three-dmensonal lattce. To reach the am, we use the coherent anomaly method (CAM) developed by Suzuk []. CAM s based on the constructon of the sequence of sutable MF-type approxmatons from whch t s possble to obtan the true crtcal value of a parameter 35

5 lke temperature or couplng parameter, and the correctons to the MF crtcal exponents. For the quantum XYZ model, we construct ths sequence as a sequence of varatonal approxmatons []. Each of these s characterzed by the tral functon ψ L (x) contanng free parameter(s) x (L s the degree of approxmaton). The ground state energy approxmaton s then a global mnmum of the functon E ( J, h, x) ψ ( x) H ψ ( x) ψ ( x) ψ ( x) L = L L L L (for gven J, J, J 3, h) wth respect to x. The MF crtcal behavor s then obtaned smlarly to the Landau theory-by expandng E L (J, h, x) n J, h, x, (two of the couplng parameters are fxed here, the true crtcal exponents wll turn to depend on them) near the crtcal pont (J (L) c,0,0). and examnng the global mnma. Accordng to CAM, the ( L) MF approxmaton sequence should be such that, J where Jc s the true crtcal temperature and the prefactors of the MF sngulartes (.e. the proportonalty coeffcents n expressons smlar to (6)) exhbt coherent anomaly, for example, the spontaneous ground state magnetzaton (e.g. for J, J 3 fxed) s: m J L ψ x σ ψ x ψ x ψl x (a) 0(, ) L( ) L( ) L( ) ( ) N m ( J, L) m ( L, J, J )( J J ) (b) ( L) / c m ( L, J, J ) = a( J, J, J )( J J ) ( L) ( L) -ϕ c c c L x( e e L ) e L L + e (3) + = c L J c () (c) By determnng parameters J c and ϕ n (c) one can get the true crtcal value of J and the true crtcal exponent β=/-ϕ. 4. Prelmnary results So far we have been dong calculatons for the one dmensonal case (when they are complete they wll be compared to the exact results): we have constructed the sutable tral functon havng symmetry propertes necessary for descrpton of the phase transton and parameterzed such that t allows for the dsordered as well as for the ordered (.e. wth a nonzero magnetzaton) ground state. On the L-th degree of approxmaton t s an N/L-folded tensor product of vectors of the form where { } L e = s the bass of L -dmensonal Eucldean space at each ste of the chan. For L=,, 3 we have calculated the approxmate crtcal lnes (Fg. 3b). The calculatons were done wth J =, J 3 fxed and we were searchng for the global mnma ( L) for varous values of J, lookng for J. It can be found as n Landau theory, but for c L> t had to be found numercally, because the equatons are too complcated to be solved analytcally. The calculatons were programmed n FORTRAN, usng the MINUIT routne of the CERN program lbrary to fnd the mnma. The results agree wth those obtaned wth the help of Mathematca package [3], the calculatons programmed n FORTRAN were much qucker. The approxmate crtcal lnes obvously approach the true crtcal lne shown n Fg. 3a, whch s a promsng result 36

6 from the CAM pont of vew. At the pont J =, J =, J 3 =- they touch the true crtcal lne, whch means that our tral functon s sutable n the neghborhood of ths pont. Fg. 4a, b, c show the ground state spontaneous magnetzaton for L=,, 3, as ( L) functon of J, J < Jc, for J3=-, -0.9, -0.8 respectvely. It exhbts the MF behavor, the next step wll be to calculate the prefactors (b), check the coherent anomaly property of our sequence of approxmatons, fnd the correctons to the MF crtcal exponents and compare them wth the exact soluton. Fg. 3 (a) (b) Fg. 4 (a) (b) (c) 5. References [] Ma, S. K., 976, Modern Theory of Crtcal Phenomena, Addson-Wesley, Wokngham [] Yeomans, J., 99 Statstcal Mechancs of Phase Transtons, Clarendon Press [3] Lynn, J. W. et al. 000, Phys. Rev. B 6, pp. R4964-R4967 [4] Chnchure, A. D. et al. 00, Physca B 99, pp [5] Ztzler, R. et al. 00, Eur. Phys. Journal B 7, [6] Dawson, A. LeR. et al. 996, Phys. Rev. B 54, p. 38 [7] Anderson, P. W., 959, Phys. Rev. 5, pp. -3 [8] Suzuk, M., 976, Progr. Theor. Phys. 56, 454 [9] Baxter, R. J., 98, Exactly Solved Models n Statstcal Mechancs, Academc Press [0] Lukyanov, S., Zamolodchkov, A., 997, Nucl. Phys. B 493 [FS}, p.57 [] Suzuk, M., et al., 995, Coherent Anomaly Method: Mean Feld, Fluctuatons and Systematcs, World Scentfc, Sngapore [] Šamaj, L., et al., 997, J. Phys. A: Math. Gen. 30, p.47 [3] Greguš, J., 00, n Physcal and Thermal Engneerng, (Eds. Novák, J., Mkš, A., Prague) 37