The Thermodynamic Transitions of Antiferromagnetic Ising Model on the Fractional Multi-branched Husimi Recursive Lattice
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] J u l The Thermodynamic Transitions of Antiferromagnetic IsingModel on the Fractional Multi-branched Husimi Recursive Lattice
Ran Huang ∗ and Chong Chen † School of Chemistry and Chemical Engineering,Shanghai Jiao Tong University, Shanghai 200240, China RushandRan Research Co., Shanghai 200433, China
Abstract
The multi-branched Husimi recursive lattice has been extended to a virtual structure with frac-tional numbers of branches joined on one site. Although the lattice is undrawable in real space, theconcept is consistent with regular Husimi lattice. The Ising spins of antiferromagnetic interactionon such a sets of lattices were calculated to check the critical temperatures ( T c ) and ideal glasstransition temperatures ( T k ) variation with fractional branch numbers. Besides the similar resultsof two solutions representing the stable state (crystal) and metastable state (supercooled liquid)and indicating the phase transition temperatures, the phase transitions show a well-defined shiftwith branch number variation. Therefore the fractional branch number as a parameter can be usedas an adjusting tool in constructing a recursive lattice model to describe real systems. Keywords: critical temperature, ideal glass transition, Ising model, fractional multi-branched, Husimi lattice ∗ Correspondence to: [email protected] † Co-correspondence to: [email protected] . INTRODUCTION Husimi lattice as a classical recursive lattice has been intensively studied and employedin many statistical modelings for decades . The recursive lattice is believed to be a reliableapproximation of regular lattices because of the same coordination numbers. In a recursivelattice the particles fixed on sites have the same interaction environment as in the regularlattices, while the fractal structure avoids the sharing of interactions on neighbor unitsand consequently provides the advantage of the exact calculation
Among the works onHusimi lattice, the thermodynamics of Ising model on Husimi is a vigorously interest-drawingsubject . Previous works on exact calculation of Ising model on Husimi successfullypresented comparable results with other techniques, e.g. mean-field approximation, MonteCarlo or series expansion .With the nature of the approximation to regular lattices, the setup of the coordinationnumber is critical for the Husimi lattice modeling. There are two ways to settle the coordina-tion number: the basic unit selection and the number of units joined on site, i.e. the numberof branches. The original structure of Husimi lattice is featured as recursively constructedby square units (Fig.1a) and it has the coordination number of 4, and the extensions ofthe same principle with units such as hexagon, tetrahedron and cube, correspondingly withcoordination number of 4, 6 and 6 have also been studied and applied in different physicalmodeling .In this work, we follow the second way and the Husimi square lattice is extended to bemulti-branched structure. Different than others’ works on branch number setup, besidesthe 2-Dimensional case of 2 branches on one site with coordination number of 4 and 3-Dimensional case of 3 branches with coordination number of 6, the branch number is treatedas a variable parameter, which is unnecessary an integer, in the model formulation andcalculation. Although the fractional number of branches makes the structure virtual, or tosay, cannot be realized in drawing, the calculation shows consistent and realistic results. Insection II we will review the lattice geometry of original 2D, and multi-branched 3D case ofHusimi lattice. In section III the Ising spins modeled on Husimi lattice will be calculatedby recursive calculation technique. In section IV the parameter representing branch numberwill be set as fractional values and we are going to check the thermodynamics under thiscircumstance. And Section V is the summary and conclusion.2 a) (b)
FIG. 1: The structure of Husimi square lattice with branches 2 (a) and 3 (b). Note that allsquares have the same size and all the bonds represent the same distance or interactionsregardless of the length drawn on the page.
II. LATTICE GEOMETRY AND ISING MODEL ON HUSIMI LATTICE
Original Husimi square lattice is featured as a uniformly fractal structure in which onesquare unit linking other four identical squares on its four sites, it is an approximation toa regular square lattice (the 2-Dimensional case) because the coordination numbers are asthe same as four. Extending this recursive construction method can provide a structurewith more units linked on one site, for example taking the number of branches B = 3with the coordination number of six may simulate the 3-Dimensional Case (a regular cubiclattice). Previous works have been examined this 3D representation to be a reliable modeland compared it with the cubic recursive representation . Fig.1 shows the structure ofHusimi square lattice with branches 2 and 3.Now we apply Ising spins S i = ± J , andenergy of the entire system is the sum of the interactions: E = X − J ij S i S j . (1)In this work we do not consider the magnetic field, i.e. the energy of particle itself,and taking a uniform interaction parameter J to be negative to make the spins preferringalternative arrangement for the lowest energy state. In this way the S = +1, −
1, +1... alter-3ating state corresponds to the stable (crystal) state, other arrangements with an obtainablestationary solution represent the metastable states. With four binary spins on the sites onesquare unit has four interactions and 2 = 16 possible configurations, the Boltzmann weightof one configuration γ is w ( γ ) = exp ( − β X − J S i S j ) , (2)where β = 1 /k B T , in the following discussion the Boltzmann weight k B is set to be 1 andwe set J = − Z = X Γ Y α w ( γ α ) , (3)the Γ = N α γ α denotes the state of the lattice as an ensemble of unit α . Taking the original2-branched Husimi lattice as an example, for any site in the lattice there are two independentidentical sub-trees contributing onto it. Define the partial partition function (PPF) Z i ( S i )as the contribution of one sub-tree onto the site S i , e.g. the Z i (+1) is the sum of weightsof one sub-tree with S i = +1. Then if we name a random site as the origin point S ofthe whole lattice, the partition function can be expressed by the sum of two terms of PPFscontributing to S = ± Z = Z (+1) + Z ( − . (4)The exponent of 2 is because of two sub-trees (branches) contributing onto S , thereforefor a generalized B -branched lattice we have the total partition function: Z = Z B (+1) + Z B ( − , (5)and for the lattice shown in Fig.1b we have B = 3.4 II. CALCULATIONS OF ISING MODEL ON HUSIMI LATTICE
Starting from Eq.5, like the total partition function as a function of PPFs, the PPFon level 0 (level n refers to the sub-trees contributing to S n ) can similarly be expressedas function of lower levels’ PPFs plus the local weight. Taking the n th level PPF as anexample: Z n (+) = X γ =1 Z B ′ n +1 ( S n +1 ) Z B ′ n +1 ( S ′ n +1 ) Z B ′ n +2 ( S n +2 ) w ( γ ) , (6) Z n ( − ) = X γ =9 Z B ′ n +1 ( S n +1 ) Z B ′ n +1 ( S ′ n +1 ) Z B ′ n +2 ( S n +2 ) w ( γ ) . (7)By defining B ′ = B −
1, in Eq.6 and 7 on the two sites S n +1 and S ′ n +1 neighboring to S n there are B ′ sub-trees contributing onto level n + 1, and similar to the S n +2 diagonal to S n .The term w ( γ ) is the local weight of the square confined by the four sites S n , S n +1 , S ′ n +1 ,and S n +2 .With the PPFs we can introduce a ratio x ( S n ) (shorten as x n ) on one site x n = Z n (+) Z n (+) + Z n ( − ) , (8)Since x n is a function of PPFs and vice versa, and from Eq.6 and 7 PPFs are also functionsof the PPFs on lower two levels, then x n must also be a function of x s on lower levels, wethen can derive x n by the recursive relation: x n = f ( x n +1 , x n +2 ) . (9)With the solution x and partition functions Z we can exactly calculate the thermodynamicsof the system, i.e. the free energy and entropy per spin, and the energy density. The detailedderivation of x and thermodynamics are presented in appendix.The x is so-called “solution” of the model and it determines the probability that one siteis occupied by the spin S = +1, and subsequently determines the entire configuration andthermodynamics of the system. Two stable solutions can be obtained from the calculation:One is a uniform solution regardless of temperature, and without magnetic field this solutionis 0.5 everywhere, which is named as one cycle solution; The other solution presents an5lternating form of x , x on two successive levels at low temperature, which we call a 2-cycle solution: x = f ( x , x ) and x = f ( x , x ) . (10)The solutions of the 2 and 3-branched Husimi lattice ( B = 2 and 3) with J = − x = 0 . S = +1 is fifty percent regardless of temperature,while the 2-cycle solution is also 0.5 at high temperature but below some point it differs tobe two solutions, one approaches to 1 and the other branch approaches to 0 as T →
0, i.e. wehave more probability that S = +1 and S = − B = 3 (the 1-cycle is not shownbecause it is also 0.5 and therefore not interesting) has a similar behavior with howeverhigher transition temperatures, which can be expected as the more coordination sites bringslarger interactions.Two transitions can be indicated by the thermodynamics of 1 and 2-cycle solutions (Fig.3:free energy and Fig.4: entropy). With the cooling process at the critical temperature T c ,where the 2-cycle solution occurs, the system may undergo an ordering transition (crys-tallization featured as entropy drop in the 2-cycle curves), or stay in the metastable state(supercooled liquid featured as continuous entropy in the 1-cycle curves). Thus obviously itis the melting transition at T c . With the continuous temperature decrease, the entropy ofthe metastable state will extrapolate to zero at another featured temperature T k and thengo to negative (corresponding to the unphysical free energy bending down), which is theKauzmann paradox and ideal glass transition . Similar to the solution implication, thethermodynamic results indicate that both transition temperatures are higher in the B = 3system. 6 c, B=2 T c, B=3 FIG. 2: The 1 and 2-cycle solutions and thermodynamic behaviors of antiferromagneticIsing model on Husimi square lattice with B = 2. IV. FRACTIONAL NUMBER OF BRANCHES
In the calculation process introduced above, the branch number B is generalized as aparameter in Eq.5, 6 and 7. Without concerning the physical meaning in real space, in thecalculation program we are freely to setup B to be fractional values. This setup though doesnot give any results abnormal, as an example the 2-cycle solution and thermal behaviorsof the stable and metastable states with B = 2 . B = 2 .
5: 1) the transitions are in between of these are in B = 2 and B = 3; 2) the free energy at near-zero temperature (the ground state energy) isjust 2.5, i.e. the coordination number. A series of fractional value of 2 < B < T c shows linearincrease with B , while the T k increase is better to be fitted as a slight logarithmic behavior(very close to linear increase). So far it is not clear about the reason behind the logarithmicfitting of T k . However with the observations of as expected, we may conclude that it is safeto take the parameter B as an option control on the transition temperatures and ground7 c, B=2 T k, B=2 T c, B=3 T k, B=3 FIG. 3: The 2-cycle solutions and thermodynamic behaviors of antiferromagnetic Isingmodel on Husimi square lattice with B = 3. T k, B=3 T c, B=3 T k, B=2 T c, B=2 FIG. 4: The 2-cycle solutions and thermodynamic behaviors of antiferromagnetic Isingmodel on Husimi square lattice with B = 3.state energy, and it provides an tool in fitting the lattice model to describe the real system.With the non-integer value of B , the lattice model is generalized to be an abstract concept,however enables it to be more flexible and practical.8 X s Temperature (a) F r ee E ne r g y Temperature (b) E n t r op y Temperature (c)
FIG. 5: The properties of antiferromagnetic Ising model on Husimi square lattice with B = 2 .
5: (a)the 2-cycle solution (b)the free energy (c)the entropy.9 .0 2.5 3.0345 y = 2.1183x - 1.4497R = 0.9999 T c Branch Number B (a)(b)
FIG. 6: The critical (a) and ideal glass transition temperatures (b) variation with differentfractional branch numbers. The T c s show a linear behavior and T k s has a logarithmicfitting.10 . CONCLUSION To summarize, the parameter representing branch numbers in Husimi lattice calculationwas set to be non-integer values. This setup virtualized the lattice structure and made itundrawable in real space, however in the program calculation it is merely a simple settingof one parameter. We have checked and confirmed that the imaginary structure can becalculated as a realistic model. The fractional number of branches show nothing abnormalin calculation results for Ising spins modeled on the lattice. In this way, the number ofinteractions acting on one site directly determined by the coordination number, i.e. thenumber of branches, which could be any real number in our practice, corresponds to thesame value of ground-state energy at near-zero temperature. With subsequently the shift oftransition temperatures T c and T k , the general thermodynamic behaviors of Ising systemson recursive lattice are consistent, and that implies the branch number may serve as anadjustable option to make the model more flexible in specific fittings to describe real system. K. Husimi, J. Chem. Phys. , 682 (1950) F. Harary, G.E. Uhlenbeck, Proc. Nat. Acad. Sci. , 315 (1953) C. Thompson, J. Phys. Lett. A , 23 (1974) J.R. Heringa, Phys. Rev. Lett. , 1546 (1989) J.L. Monroe, J. Stat. Phys. , 255 (1991) J.L. Monroe, J. Stat. Phys. , 1185 (1992) M.F. Thorpe, D. Weaire, R. Alben, Phys. Rev. B , 3777 (1973) P. Chandra, B. Doucot, J. Phys. A , 1541 (1994) H. Rieger, T.R. Kirkpatrick, Phys. Rev. B , 9772 (1992) T. Yokota, Physica A , 534 (2007) A. Lage-Castellanos, R. Mulet, Eur. Phys. J. B , 117 (2008) T. Morita, J. Phys. A , 169 (1976) R.A. Zara, M. Pretti, J. Chem. Phys. , 184902 (2007) E. Jurˇciˇsinov, M. Jurˇciˇsin, J. of Stat. Phys. , 1077 (2012) F. Semerianov and P. D. Gujrati, Phys. Rev E , 011102 (2005) P.D. Gujrati, Phys. Rev. Lett. , 809 (1995) P.D. Gujrati, J. Chem. Phys. , 5089 (1998) R. Huang and P.D. Gujrati, arXiv:1209.2090 [cond-mat.stat-mech] (2012) W. Geertsma and J. Dijkstra, J. Phys. C: Solid State Phys. , 5987 (1985) J.F. Stilck and M.J. de Oliveira, Phys. Rev. A , 5955 (1990) J.A. Verges and F. Yndurain, J. Phys. F: Met. Phys. , 873 (1978) W. Kauzmann, Chemical Reviews , 2 (1948) B T c T k T c /T k
1. The calculation of x By starting from x n = Z n (+) Z n (+) + Z n ( − ) ,y n = Z n ( − ) Z n (+) + Z n ( − ) , we define a compact note z n ( S n ) = x n if S n = +1 y n if S n = − A B ′ n = Z n (+) + Z n ( − ) , we have A B ′ n z n ( ± ) = X A B ′ n +1 z B ′ n +1 ( S n +1 ) A B ′ n +1 z B ′ m +1 ( S ′ n +1 ) A B ′ n +2 z B ′ n +2 ( S n +2 ) w ( γ ) ,z n ( ± ) = X z B ′ n +1 ( S n +1 ) z B ′ n +1 ( S ′ n +1 ) z B ′ n +2 ( S n +2 ) w ( γ ) /Q ( x n +1 , x n +2 ) , where the sum is over γ = 1 , , , . . . , S n = +1, and over γ = 9 , , , . . . ,
16 for S n = −
1, and where Q ( x n +1 , x n +2 ) ≡ (cid:2) A n /A n +1 A n +2 (cid:3) B ′ ;it is related to the polynomials Q + ( x n +1 , x n +2 ) = X γ =1 z B ′ n +1 ( S n +1 ) z B ′ n +1 ( S ′ n +1 ) z B ′ n +2 ( S n +2 ) w ( γ ) ,Q − ( x n +1 , x n +2 ) = X γ =9 z B ′ n +1 ( S n +1 ) z B ′ n +1 ( S ′ n +1 ) z B ′ n +2 ( S n +2 ) w ( γ ) , Q ( x n +1 , x n +2 ) = Q + ( x n +1 , x n +2 ) + Q − ( y n +1 , y n +2 ) . In terms of the above polynomials, we can express the recursive relation for the ratio x n interms of x n +1 and x n +2 : x n = Q + ( x n +1 , x n +2 ) Q ( x n +1 , x n +2 ) . Then the 2-cycle solution can be calculated as: x = Q + ( x , x ) Q ( x , x ) , x = Q + ( x , x ) Q ( x , x ) . and the 1-cycle solution is just a special case when x = x .
2. Free Energy
Since the lattice is infinite, we are only interested in the free energy per site. Thecalculation technique is following the Gujrati trick in reference . Here we give a briefintroduction: Consider the B squares meeting at the origin site S . Surrounding S we have S and S on next levels. If we cut off the BB ′ branches on level 1 and hook up themtogether to form B ′ smaller lattices, the partition function of each of these smaller latticesis: Z = Z B (+) + Z B ( − ) . Similarly for the cutting off and hook-up of branches on level 2: Z = Z B (+) + Z B ( − ) . The free energy of the left out squares is F local = − T log (cid:20) Z ( Z Z ) B ′ (cid:21) .
15e have 4 /B cites in a square and B squares in the local origin region. The free energy persite is: F = − F local . By substituting Z n (+) = A n x n and Z n ( − ) = A n y n , we have F = − T log( Q B ′ { [ x B + (1 − x ) B ] [ x B + (1 − x ) B ] } B ′ ) ..