arXiv:cond-mat/0507633v2 [cond-mat.stat-mech] 19 Jun 2006 iglrpr ftefe nrydensity energy free the of part singular sculdt h re aaee.Tesaigfunctions scaling The parameter. order the to coupled is ecie yol w needn exponents. independent two only by described where rmvrosdrvtvso h reeeg nE.() hr r o are there (1), Eq. contributions in singular energy free the the Because of xx] derivatives notation various of from change note [xx elkonepnn hti enda h rtclpiti h anoma the is infinite. point critical are derivatives the at the defined of is most that [xx exponent known values. well infinite on take usually ea fteodrprmtrcreainfunction: correlation parameter order the of decay specify thus and system the of on behavior critical the characterize where osat qain()yed h elkonFse cln identity: scaling Fisher well-known the yields (5) Equation constant. aaee est iha xoetdfie by: defined exponent an with density parameter ytm hthv otnospaetasto,sc sflisat fluids as such th transition, systems, phase magnetic continuous to a appropriate have notation that a systems use will we Although H ···i · h· where susceptibility n h nryvrino q 5 hc eae h hra susce thermal the relates which (5) Eq. of version energy the and h orlto length correlation The ( tteciia on h hroyai eiaie r o useful not are derivatives thermodynamic the point critical the At ti elkonta h rtclbhvo fasse eriscontin its near system a of behavior critical the that known well is It h erhfrascn needn xoetlast h fourt the to leads exponent independent second a for search The h exponents The r t ( ) nE.()last oe-a iglrt o h pcfi heat, specific the for singularity power-law a to leads (1) Eq. in eoea nebeaeae and average, ensemble an denote t δm H ∼ ≡ ( m r ( ( r stelclHmloino h neato nrydniy oetha Note density. energy interaction the or Hamiltonian local the is ) ( T = ) r utainrsos qainfrteseicha.W als We heat. specific the for equation fluctuation-response a eae oteepnnsdtriigtebhvo fthermo of behavior the determining exponents the to related rtclpit oee,a h rtclpitol h expo the only point critical the at However, point. critical h orlto ucin suulydsusd eemphasi We discussed. usually is function, correlation the exponent − ) 2 hr r w needn rtclepnnsta describe that exponents critical independent two are There T sqartci h re aaee density parameter order the in quadratic is ) m c η ) ( /T ′ r eateto hsc,Cleeo cecs hrzUnivers Shiraz Sciences, of College Physics, of Department and ) c h − η sterdcdtemperature, reduced the is ′ ξ η htdsrbstedcyo h orhodrcreainfu correlation fourth-order the of decay the describes that m r eae oteuulepnnsvaaflcuto-epneeq fluctuation-response a via exponents usual the to related are hrceie h xoeta ea ftecorrelations, the of decay exponential the characterizes i steodrprmtrdniyflcuto rmtema tpoint at mean the from fluctuation density order-parameter the is G m ( r C ) sing h ≡ C/k cln asa h rtclpoint critical the at laws Scaling G χ d E = δm f stesaildmnin oethat Note dimension. spatial the is = ( sing = r | t ( ) β | β r − h ≡ Z = ) 2 α δm Z c d | H ± t 2 d 3 d | ( ( rG 2 ( 0 d h/ hsdsrpincnb xrse ytesaigbhvo fthe of behavior scaling the by expressed be can description This − r rG .Davatolhagh S. ) T α ) i γ m c | H f t = E ( | steciia on eprtr,and temperature, point critical the is ± (2 = ∆ r ( ( ( 0 h r ) = ) h/ m f ) ) ∼ h − i ± ( ∼ | − ( Z t r c x | Z ) ∆ ± η m eedon depend ) ξ m H ) ) (0) ξ ν, ( r , ( r i d 0 r d 2 − | ). t nvraiycas o xml,tedependence the example, For class. universality its d tblt rteseicheat: specific the or ptibility ) l w needn xoet,say exponents, independent two nly t d − d h − i hi rtclpoint. critical their 2+ | ∼ r − -re orlto ucini em fteorder the of terms in function correlation h-order d 2+ ics cln a for law scaling a discuss o eta hr sascn independent second a is there that ze α r ( η 4 ovrostemdnmcqatte r found are quantities thermodynamic various to olwn icsinas sapial oother to applicable is also discussion following e C 1 nent yai ucin erciiaiyvia criticality near functions dynamic t osdmninexponent dimension lous , o hrceiigteciia eaira they as behavior critical the characterizing for η /r sing ∼ m → ′ h eairo ytm ertheir near systems of behavior the o’ nesadti etnex]Teonly The xx] sentence this understand don’t d ∼ i ξ − t 2 0 η 2 ospaetransition phase uous − ∼ 2+ and ± ξ − G hc ecie h ea of decay the describes which , ( ∼ t,Sia 15,Iran 71454, Shiraz ity, 2 h (1) ) η m η − , 1 ′ 0) = η ( ∂ , cin h exponent The nction. /r h r ′ h oa neato nrydensity energy interaction local the t 2 . satopitcreainfunction. correlation two-point a is ) f/∂t nyi h combination the in only d − . β 2+ 1 = 2 η ainfrteodrparameter order the for uation , when /kT h η steetra edthat field external the is and , h ′ . r η 0: = h nua braces angular The . hc ecie the describes which , 1 a ecompletely be can k η ′ sBoltzmann’s is α is n ,that ∆, and x = h/ | t (2) (3) (7) (4) (5) (6) | ∆ . 2 where γ is the linear response susceptibility exponent defined by χ ∼ |t|−γ, and ν is the correlation length exponent, ξ ∼ |t|−ν . Equation (6) yields an energy version of Fisher scaling:3,5,6
α = (2 − η′)ν. (8)
The correlation length exponent ν appearing in Eqs. (7) and (8) must be the same because of the hyperscaling hypothesis, which states that close to criticality the correlation length ξ is the only relevant length scale, and thus ξ is solely responsible for the singular contributions to the thermodynamic quantities. Another important outcome of the hyperscaling hypothesis is the Josephson scaling identity, 2 − α = dν, which provides another relation between α and ν and thus gives a constraint on Eq. (8). If we substitute α =2 − dν into Eq. (8), we obtain another scaling law involving η′: 2 η′ = d +2 − . (9) ν We see that the critical exponent η′ is determined by the correlation length exponent ν and the spatial dimension d. To verify the scaling equation (9) and to test its predictions, we first consider the d = 2 ferromagnetic Ising model in the absence of an external field for which exact analytic solutions are available.7,8 In particular, we are interested in the four-point correlation function of the form
w4(r)= hS1S2SrSr+1i − hS1S2ihSrSr+1i (10) which also can be interpreted as a two-point energy correlation, with S1S2 characterizing the energy of a reference bond and SrSr+1 the energy of a bond at a distance r from the reference bond, as can be seen in Fig. 1. The Ising spins take on the values, Sr = ±1. The fourth-order correlation function w4(r), when summed over r for all distinct 2 pairs of bonds is related to the specific heat in agreement with Eq. (6): r w4(r)= ∂ǫ/∂β − 1+ ǫ , where ǫ = hS1S2i characterizes the mean energy per bond.9 P From Eq. (9) we see that η′ = 2 for the d = 2 Ising model, where we have used the exactly known correlation length exponent ν = 1. We note that when η′ = 2 is substituted into the Eq. (6), we find a logarithmic divergence for the specific heat as we should. The asymptotic behavior of w4(r) has been calculated rigorously for the triangular and the square Ising lattice and has been found to be identical for the two lattices (apart from an amplitude factor of 1/2) and has the usual Ornstein-Zernick forms10 e−2r/ξ w (r) ∼ , (r → ∞). (11) 4 r2 We see that the predicted value η′ = 2 agrees with results from exact analytic calculations. For comparison the critical exponents relevant to our discussion for the d = 2 and d = 3 Ising models are summarized in Table I. For d = 3 Eq. (9) predicts η′ = 1.822 (3), where we have used the numerical estimate, ν = 0.629 (1).11 A direct measurement of the ratio α/ν = 2 − η′ can be obtained by a finite size scaling analysis12 of the specific heat obtained by Monte Carlo simulations.13 The slope of a log-log plot of C versus L provides an estimate for the ratio α/ν: C ∼ t−α ∼ Lα/ν . A reliable estimate of this ratio for d = 3 is α/ν = 0.178 (5).11 Thus we have η′ =2 − α/ν =1.822 (5), which is in good agreement with the value predicted by Eq. (9). In summary, the exponent η′ of the four-point correlation function provides a second independent exponent at the critical point. Thus {η′, η} form an independent exponent pair. The discussion presents a convenient framework for introducing the concept of fourth-order correlation functions, which are important in the study of viscous liquids5 and glasses.14
1 See, J. Tobochnik, “Resource letter CPPPT-1: Critical point phenomena and phase transitions,” Am. J. Phys. 69, 255–263 (2001). 2 K. Huang, Statistical Mechanics (Wiley, New York, 1987), 2nd ed., pp. 399–406. 3 M. Kardar, “Statistical mechanics of fields” (lecture notes),
7 L. Onsager, “ Crystal statistics. I. A two-dimensional model with an order-disorder transition,” Phys. Rev. 65, 117–149 (1944). 8 B. M. McCoy and T. T. Wu, The Two-Dimensional Ising Model (Harvard University Press, Cambridge, MA, 1973). 9 M. F. Thorpe and D. Beeman, “Themodynamics of an Ising model with random exchange interactions,” Phys. Rev. B 14, 188–199 (1976). 10 J. Stephenson, “Ising model spin correlations on the triangular lattice. II. Fourth-order correlations,” J. Math. Phys. 7, 1123–1132 (1966). 11 H. G. Ballesteros, L. A. Fernandez, V. Martin-Mayor, A. M. Sudupe, G. Parisi, and J. J. Ruiz-Lorenzo, “Scaling corrections: Site percolation and Ising model in three dimensions,” J. Phys. A: Math. Gen. 32, 1–13 (1999). 12 Finite-Size Scaling and Numerical Simulations of Statistical Systems, edited by V. Privman (World Scientific, Singapore, 1990). 13 D. P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, 2nd ed., (Cambridge University Press, Cambridge, 2005). 14 C. Donati, S. Franz, S. C. Glotzer, and G. Parisi, “Theory of non-linear susceptibility and correlation length in glasses and liquids,” J. Non-Cryst. Sol. 307, 215–224 (2002) and references therein.
TABLE I: Comparison of various critical point exponents for the d = 2 and d = 3 Ising models.
′a ′b System α ν η η η 2D Ising model (exact) 0 1 1/4 2 2 3D Ising model (approx) 0.112(3) 0.629 (1) 0.037 (1) 1.822 (3) 1.822 (5)
aFrom Eq. (9) bFrom analytical10/numerical11 calculations
FIG. 1: Schematic of four-point spin correlations in the d = 2 Ising model. The line segments represent bonds whose energies are determined by the product of nearest-neighbor spins at each end.