Liang-Sheng Li(李糧生)

Science and Technology on Electromagnetic Scattering Laboratory,Beijing 100854,China

1 Introduction

Phase transitions in society,[1]polymer,[2?3]network,[4?6]and quantum systems[7]have attracted much attention due to their scientific interest and technological significance.In the physics of critical phenomena,the universality class of systems with short-range interactions is determined by the system dimensionality(d)and the order-parameter symmetry number.[8?9]In 1994,Kim,et al.investigated the critical behaviors of the diluted Ising model with Monte Carlo simulation and estimated the critical exponents which differ from the Ising model.[10]Later Nijmeijer,et al.simulated the Heisenberg fluid and found that the values of critical exponents depart from the lattice case.[11?12]The order parameter for the Heisenberg fluid is mixture of density and magnetization by using the density functional theory.[13]Recently,the critical exponent of correlation length is found to dependent on the density in the two-dimensional magnetic lattice gas model.[14]These results challenge traditional viewpoint of critical phenomena and are not well-understood.

On the other hand,critical exponents for systems with long-range interactions,decaying as 1/r(d+σ),are dependent on σ,but when σ >2 the exponents take their short-range values.[15]As σ is equal to ?d,models tend to the mean- field case and might become simpler and easily solvable.For mean- field models,the canonical critical exponents of specific heat,order parameter,and susceptibility,defined by C ～ t?α,M ～ tβ,and ξ～ t?γ,take the values α =0, β =1/2,and γ =1 at a critical point or α =1/2, β =1/4,and γ =1 at a tricritical point,where these exponents satisfy the scaling law α +2β + γ =0.[8?9,16]Whereas,the situation is less simple for tricritical points in long-range models.Nagle found that C ～ |t|?1/2for temperature(T)below tricritical temperature(Tc),in one-dimensional long-range Ising model,but when T>TcC～t0,where two exponents(0= α 6= α′=1/2)violate the Widom homogeneity equality(α = α′)and the scaling law for T>Tc.[17]Microcanonical and canonical ensembles could be inequivalent in long-range interactions systems which obey Boltzmann statistics.Indeed,it is found that the microcanonical and canonical tricritical points,although close to each other,are not identical.[18?19]Ellis,et al.have presented a general mathematical theory of inequivalence between canonical and microcanonical ensembles,and shown that the local entropy,when a single canonical state contains many microcanonical states,is not one-to-one correspondence to the canonical temperature.[20?22]Therefore,systems within long-range forces display many interest phenomena observed in microcanonical ensembles,such as negative specific heat,temperature jumps,the violation of the zeroth law of thermodynamics,etc.This phenomenon are not observed in the equilibrium canonical ensemble.[23?27]

In this paper,we investigate,in the microcanonical ensemble,the tricritical and critical behaviors of both the Blume–Emery–Griffiths(BEG)and the Ising model with long-range interactions.The tricritical and critical exponents in the microcanonical ensemble are estimated by using scaling analysis to test the Rushbrooke inequality and the Widom homogeneity equality.It will be shown that the well-known relations could be violated in systems with long-range interactions in the microcanonical ensemble.

2 BEG Model and Phase Diagram

In the BEG model,[18,28?29]spins site on a lattice and have in finite range interactions.The Hamiltonian is given by

where the coupling constant J>0 and the single spin anisotropy parameter?>0 are chosen to be larger than zero.Then,we define a new parameter K=J/2?>0,so that there is a ferromagnetic phase transition.The transition line separates a paramagnetic phase from a ferromagnetic one in the canonical ensemble,and the transition is second order for large K and becomes first order below a canonical tricritical point located at a coupling KCTP=3/ln16.[18]

To analyze the model within the microcanonical ensemble,the entropy per site normalized by kBin the large N limit is given by[19]

where

are the quadrupole moment and magnetization per site,respectively.Let ?=H/(N?)=q?Km2be the energy per site.In order to locate the second order transition line between the paramagnetic and ferromagnetic phases we expand the entropy in powers of m,and the expansion takes the form

where

In the paramagnetic phase both A and B are negative,and the entropy is maximized by m=0.In order to obtain the critical line in the(K,?)plane,the continuous transition to the ferromagnetic phase takes place at A=0 for B<0 and the transition line(black solid line)is shown in Fig.1.The microcanonical tricritical point is obtained at A=B=0,and the authors of Ref.[18]found that the coupling KMTP?1.08129645(black star),which does not coincide with the canonical tricritical point(red square).In the region between the two tricritical points,the microcanonical ensemble yields a continous transition at a smaller coupling parameter,while in the canonical ensemble the transition isfirst order.In this region,negative specific heat and temperature jumps may be observed at transition energies.[18]

Fig.1 (Color online)The microcanonical(K,?)phase diagram.The large K transition is second order(black solid line)down to the microcanonical tricritical point(black star),where it becomes first order(red line).Due to the ensemble inequivalence,the coupling parameter KCTPcorresponding to the canonical tricritical point(red solid square)is larger than KMTP.In the region between KMTPand KCTP,the microcanonical ensemble still yields a continuous transition.

3 Exponents in BEG Model

By using the second derivatives of the entropy with respect to the energy and magnetization,one can obtain the specific heat

and susceptibility

where s?= ?s/?? and sm= ?s/?m.When K ≥ KMTP,in the vicinity of a critical energy ?c,the magnetization(m),the susceptibility(χ),and the specific heat(C)can be written into a scaling form as

These critical exponents,therefore,can be estimated by the scaling relations.We numerically obtain the magnetization,the specific heat,and the susceptibility as a function of the energy from Eq.(2).The plots of log(m)and log(|χ|)versus log(?c? ?)and log(|?c? ?|)become straight lines with slopes β and γ±,respectively.The results for K=KMTP,K=1.08142,and K=KCTPare shown in Figs.2 and 3.

For the tricritical point in the microcanonical phase diagram,we obtain the tricritical exponents β=1/4,?1/2= α+6= α?=0,and 1/2= γ+6= γ?=1.The exponents β and γ?are in agreement with the prediction of classical theory(mean field theory),but the values of exponents γ+, α+,and α?deviate from the classical expectations. Recently,Deng,et al.generalized the Fisher renormalization mechanism to describe that tricritical exponents are renormalized under the constraint when the system has a divergent specific heat at tricritically.[34]However,because of the finite specific heat in the BEG model at the microcanonical tricritical point,the change of tricritical exponents,here,could not be explained by the Fisher renormalization.Additionally,the exponents of susceptibility(γ+6= γ?)and specific heat(α+6= α?)violate the homogeneity equality.[30?31]This violation was also observed for the long-range Ising model in the canonical ensemble.[17]Additionally,we find that α++2β + γ+=1/2<2 break down the famous Rushbrooke inequality.[32?33]This inequality between critical exponents assumes that specific heat must be positive,and consequently a system with negative specific heat in microcanonical ensemble could display the violation.

Fig.2(Color online)(a)Log-log plot of the magnetization versus(?c??)for different parameters KMTP,KCTP and K=1.08142.(b)Plot of the estimated critical exponent(β)against the coupling parameter K.

When K≥KMTP,the transition turns into second order and the exponents take classical values,where the homogeneity equality and scaling law α±+2β + γ±=2 are recovered.For K=KCTP,the transition is still second order.However we find that 1= γ?6= γ+=2 and 0= α?6= α+=1,where α+is estimated from the plot of log(C)versus log(?c? ?)as shown in Fig.4.The homogeneity equality is violated by exponents of susceptibility and specific heat,where the violation results from the inequivalence between the microcanonical and the canonical ensemble in a model within long-range interactions.Although,this set of critical exponents obeys the thermodynamics inequality α++2β + γ+=4>2,the scaling law is still broken.

Fig.3 (Color online)(a)Log-log plot of the susceptibility as a function of(|?c ? ?|)for different parameters KMTP,KCTPand K=1.08142.(b)Plot of the estimated critical exponent(γ+and γ? )against the coupling parameter K.

Fig.4 (Color online)Log-log plot of the heat capacity as a function of(?c ? ?)for KCTPand KMTP.

4 Ising Model Within Long-Range Interactions

The Hamiltonian of Ising model combining long-with short-range interactions is given by

where si±1.One has obtained the canonical tricritical point KCTP=≈ ?0.317142 and the microcanonical tricritical point KMTP=?0.359 45674,respectively.[17,19]Furthermore,the entropy per site can be written as[19]

Fig.5 (Color online)Log-log plot of the magnetization(a),the heat capacity(b),and the susceptibility(c)versus(?c ? ?)for KMTPand KCTP,in the Ising model within long-range interactions,respectively.

Table 1 Summary of results.K is the coupling parameter;β,γ+,γ?,α+and α? are exponents.Our results are shown in the rows denoted as microcanonical ensemble.The third and last rows show the tricritical and critical exponents for the classical mean- field theory from Refs.[8,15–16].

5 Conclusion

In summary,we have studied the critical behavior of both BEG and Ising model within long-range interactions in the microcanonical ensembles.By numerically calculating the magnetization,the susceptibility,and the specific heat of two solvable models in the vicinity of critical and tricritical points,we have shown that,for K=KMTPand K=KCTP,the values of critical and tricritical exponents deviate from the prediction of the classical theory as presented in Table 1.Additionally,we found that the ho-mogeneity equality and scaling law of exponents are also broken for systems with long-range interactions at the microcanonical and canonical tricritical points.It is noted that the violation of Rushbrooke inequality is only observed at the microcanonical tricritical point in our paper,where a vanishing specific heat is yielded.This violation should not be limited to the microcanonical ensemble,but may be found in the canonical ensemble if the system is out of equilibrium.

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