Introduction to Statistical Mechanics and a Two-Dimensional Ising Model J.F. Nystrom University of Idaho Moscow, Idaho

Transcription

1 Introduction to Statistical Mechanics and a Two-Dimensional Ising Model J.F. Nystrom University of Idaho Moscow, Idaho nystrom@ee.uidaho.edu Keywords: Statistical mechanics, Quantum statistics, Ising model, Critical exponents, Scaling hypothesis. PACS numbers:.3.rr, 5.2.-y, 5.5.+q, 75.4.Cx

2 Abstract In order to analyze macroscopic systems while taking into account knowledge of quantum behaviors, it is required to use such techniques as are available under the label of statistical mechanics. Herein we present an introduction to statistical mechanics and provide a suite of models to demonstrate the eectiveness of the methodology. 2

3 Introduction Quantum mechanics (QM) uses the idea of quantization to postulate the existence of discrete states of a system. = H The time-dependent Schrodinger gives information about the time evolution of the system, while the timeindependent Schrodinger equation H = E () gives the energy values of a system. Using the time-independent equation, it is easy to show that the quantum oscillator in one{dimension has energy levels of E n = n + h! 2 while the oscillator in three{dimensions has energy levels of E n = n + 3 h! : 2 For macroscopic systems though, it is unreasonable to expect that we could write an exact Hamiltonian to describe the systems time-evolution, or even nd the exact energy levels available to said macroscopic system. Therefore, quantum mechanics is only appropriate for isolated few-body systems. To attempt to describe actual macroscopic systems, one utilizes many of the ideas from quantum mechanics to perform calculations which fall under the label of statistical mechanics. The paper starts by giving an overview of the fundamentals of statistical mechanics. This introduction, combined with the thermodynamic review 3

4 in the Appendix, gives a foundation for the statistical mechanical models discussed in the remainder of the paper. In x the entropy representation is utilized to show the eectiveness of the statistical mechanics methodology. Callen's excellent treatise [] on thermodynamics and statistical mechanics is utilized extensively throughout the paper. The models in x are rederived in x2 using the canonical ensemble before attention is turned to the analysis of quantum spin systems. Included in x2 is an introduction to the Ising model and a review of a full two{dimensional renormalization group calculation. The discussion summarizes the theoretical eectiveness of statistical mechanical techniques, including comments on the universality hypothesis and cooperative phenomena. The way of statistical mechanics For macroscopic systems, with given values of extensive parameters U, V, and N (i.e., the energy, volume and particle number), we expect to have many discrete states available that are consistent with the specic values of U, V, and N. That is, many dierent congurations are available to the system, such that, the systems extensive parameters are always equal to the given values of U, V, and N. If we could write a quantum mechanical Hamiltonian for a given macroscopic system, then using this Hamiltonian, we theoretically could describe the time evolution of the system, and provide probability estimates on future states. For an isolated atomic system, it is then the act of measurement that tells us what actual state the system has evolved into. But, no physical system can be isolated totally. Furthermore, for transitions to occur between the quantum states of a macroscopic system, 4

5 it is enough that the system interacts with the so-called \vacuum" []. This is a fact, because, for macroscopic systems, the energy dierence between diering quantum states that are consistent with the imposed constraints is unperceptible. Thus [] A realistic view of a macroscopic system is one in which thesys- tem makes enormously rapid random uctuations among quantum states. A macroscopic measurement senses only an average of the properties of myriads of quantum states. Statistical mechanics further assumes that \a macroscopic system samples every permissible quantum state with equal probability" []. This assumption of equal probability of occupation is the fundamental postulate of statistical mechanics. Now, since we have made such a big deal about the number of permissible quantum states, we naturally ask if we can somehow measure the \number" of these states. According to Callen [] The number of microstates among which the system undergoes transitions, and which thereby share uniform probability of occupation, increases to the maximum permitted by the imposed constraints. We now conjecture that the thermodynamic entropy should be associated with the number of available states (consistent with the imposed macroscopic constraints). But entropy, being an extensive parameter, has an additive character when two systems are brought together, and the number of microstates available to two systems is a product of the number of states 5

6 available to each, as shown in Figure. A unique answer to the question of how to formulate the entropy is to identify the entropy with the logarithm of the number of available states. Let be the number of microstates consistent with xed values of U, V and N, then S = k B ln (2) is the entropy, where k B =:387 ;23 J=K (Boltzman's constant) is the prefactor chosen so that the tempature, given as V N (3) agrees with the Kelvin tempature scale. This denition nows givesaphysical foundation for the entropy associated with a macroscopic system in equilibrium. In this representation, we are basically doing statistical mechanics in the microcanonical formalism. That is, we set a xed energy, U, and nd the number of available states of the system consistent with this constraint. Later we will nd it advantageous to work in the canonical formalism, wherein the temperature T is xed, and the fundamental relation of choice is the Helmholtz free energy. Statistical mechanical modeling In this section we utilize the entropy representation to calculate the specic heat for two classical models. Both models deal with systems of identical particles. In the rst model, the \two{state" model, the occupation number follows the Fermi-Dirac statistics, while the second model, the Einstein 6

7 model of a crystalline solid, has an occupation number that follows the Bose- Einstein statistics.. Two{state model For the two{state model, a total of N atoms comprise the system, and it is assumed that each atom can have an energy value of or. The total energy of the system is xed at U. Given a total energy of U, along with the fact that each atom can be in only one of the two states, it is required that U= atoms be in the excited state, thus (N ; U=) are in the ground state. To calculate the entropy, using Eqn. (2), we are required to determine the number of possible states consistent with the current constraints. Recall from combinatorics, that the number of ways to choose k items among n things is given by n k C A = n! k!(n ; k)! i.e., \n choose k". Thus, the number of ways to distribute the U= packets (or quanta) of energy among the N atoms is given as = N!! (4) U! N ; U!! which is exactly equal to the number of states accessible to the macroscopic system. Using Eqn. (4) in Eqn. (2) the entropy for the two{state system is "! # S = k B ln N! ; ln U! ; ln N ; U!! : (5) 7

8 Stirling's approximation, ln N! ' N ln N ; N, can be applied to Eqn. (5) to obtain the fundamental relation: S(U N) = Nk B N ln N ; U U ln ; N ; U ln N ; N ; U U U = ;Nk B ln ; N N N Now calculate the temperature for this system: ln ; U N ; U N ln ; U N : (6) = k B N ln U ; (7) and invert Eqn. (7) to obtain an equation of state for the energy: U = N +e =k BT = N e;=kbt +e ;=k BT (8) where e ;=k BT is the Boltzmann factor, which is proportional to the probability that the state will occur for a given T. Let u = U=N be the energy per atom, then the specic heat is given as C = du dt = 2 k B T 2 e =kbt ( + e =k BT ) 2 : (9) A plot of C = C=k B as a function of x = k B T= is given in Figure 2. This gure exhibits a so{called Schottky hump, indicating the presence of a pair of low energy states lying below some considerably higher energy states. The two{state models assumption of only two energy levels is equivalent to assuming that the next energy state of each atom is so high that it can not be reached. We nd that this model, just as with the examples in the Appendix, show that simple thermal calculations can give information on the atomic or molecular structure of material substances. 8

9 .2 Einstein model for the heat capacity of crystals The specic heat for a crystalline solid at high-temperature can be calculated 2 from the equipartition theorem to be C = 3R. But at low temperatures it was found experimentally that the heat capacity goes to zero. This behavior of the specic heat as a function of tempature is shown qualitatively in Figure 3. Classically, there is no way to explain this behavior for the specic heat. We nowmodelthecrystalasmadeupofn atoms, with each atom bound harmonically to six nearest neighbors. Each atom is free to vibrate around an equilibrium position in any of the three Cartesian directions with a natural frequency!. This model is shown in Figure 4. As the lattice is built up, we assign three oscillators to each atom, and neglect edge eects. Assume now that each of the 3N oscillators can only take on discrete values of energy, such that the energy of each oscillator can be written as E n = nh! n= 2 ::: () which are the well known energy levels for the one{dimensional oscillator (neglecting the /2h! zero point energy). Therefore, each oscillator can be occupied by an integral number of energy quanta. In the microcanonical formalism, as usual, we x the energy at U, giving U=h! quanta to distribute throughout the system. Unlike thetwo{state model presented earlier, in this model we do allow theplacement of all the quanta into one oscillator. To calculate the entropy, we require the number of possible states that are consistent with the imposed constraint (i.e., a total energy of U). Let represent the number of ways to distribute U=h! quanta among the 3N modes. To count the number of states, consider a diagrammatic approach 9

10 which uses 's to represent each of the U=h! quanta, and to represent the end of a mode (which means each list has U=h! 's and 3N 's). If all the quanta are in the rst oscillator, this conguration is written as ::: :::, i.e., U=h! 's followed by 3N 's. If the quanta are distributed such that each of the rst U=h! modes has only one quanta, the list would look like ::: :::. Using this diagrammatic technique, we basically want to know how many ways there are to place U=h! among (U=h! +3N) locations, that is, = U + 3N h!! U! h! 3N!!!! Applying Stirling's approximation, and letting u=u write the molar entropy, s = S=n, as s(u v) = 3R + u u = 3R ln + u u The tempature is calculated as markers : () ln + u ; u ln u u u u + u ln u + u u = k B ln + 3Nh!! h! U which is inverted to give the energy per mode: U 3N = u u h! = = U=3Nh!, we can : (2) h! (e h! =k BT ; ) : (3) After rewriting s in terms of T, the specic heat for this crystalline model is given as C v =3R h!! 2 e h! =k B T k B T (e h! =k BT ; ) : (4) 2

11 A graph of Eqn. (4) is shown in Figure 3. Taking the limit of Eqn. (4) as T!, we nd C v! 3R (the expected result), and as T!, C v! 3R(h! =k B T ) 2 e ;h! =k BT, approaching zero as T!. Even though the Einstein model is very simple, it gives a good qualitative agreement with the specic heat at low temperatures. While Eqn. (4) approaches zero with an exponential decay, which is dierent from the experimental T 3 dependence [, 5], the model deserves attention because of its elegance, and the fact that this was the rst quantum theory of the heat capacity of solids..3 Quantum statistics Using Eqn. (3), we nd that the average number of quanta per mode, (U=3N)=(h! ), i.e., the occupation number [2], is given by n boson = e h! =k BT ; (5) which is just the Bose-Einstein distribution. This should be expected since we do allow more than one quanta to occupy any given mode (to wit: these quanta act like bosons). Compare Eqn. (5) to the occupation number, n = U=N, that is obtained from Eqn. (8): n fermion = e h! =k BT + (6) which is just the Fermi-Dirac distribution. Again appropriate in light of the fact that at most one quanta can occupy each mode (to wit: these quanta act like fermions). In the limit as T!, both Eqn. (5) and Eqn. (6) give a behavior similar in functional form to the Maxwell-Boltzmann distribution, going e ;h! =k BT.

12 2 Ising model In x we utilized the microcanonical formalism to analyze two systems: one with quanta that behave like fermions, the other with quanta that behave like bosons. In this section weintroduce the basics of the canonical formalism and then reformulate the two{state model and Einstein's model in this formalism. Following the introduction of the canonical formalism, the classical magnetic interaction is discussed, and the Heisenberg Hamiltonian for quantum spin systems is derived. A spin Hamiltonian is then applied to a linear chain of spin variables, the so-called Ising model in one{dimension. This section ends with a description of a fully two-dimensional exact renormalization group calculation involving an interesting geometrical conguration of spin variables. 2. The canonical formalism The convenient fundamental relation in the canonical formalism is the the Helmholtz free energy, F (T V N). To obtain the free energy, we can convert fundamental relations from the microcanonical formalism using a Legendre transformation, or develop the fundamental relation using the partition function. If the energy levels of a system are the sum over the energies of the individual elements of the system, then the partition sum factors, such that the partition function is given as Z = z z 2 z n 2

13 where z i are the partition functions for the individual elements. Thus, for N identical elements, Z = z N, where z = X j e ; j with ==k B T and j is the energy of the j th mode of one of the elements. That is, z is just the sum of the Boltzmann factors associated with each energy state. The Helmholtz free energy is then given by log of the partition function: F = ; ln Z: (7) For the two{state model of x., there are two separate energy states, thus the partition function for each element is z = e + e ; =+e ; : Accordingly, forn atoms: ln zn F = ; = ;Nk B T ln( + e ; ) : (8) The specic heat is derivable from the free energy: C = = = ; T 2 @ ln Z! (9) which gives the same result as shown in Eqn. (9). For the Einstein model, each modecanhave zero, one, two, or basically an innite number of quanta. Thus, for each mode: z =+e ;h! + e ;2h! + = 3 X n= e ;nh! : (2)

14 Recognizing a binomial expansion, Eqn. (2) is rewritten as which gives a free energy of z = ( ; e ;h! ) F = ; ln z3n =3Nk B T ln ; e ;h! : (2) Calculating the specic heat for the Einstein model as is done in Eqn. (9) gives the same result as shown in Eqn. (4). Using the canonical formalism for the two{state model and the Einstein model eliminated the need to count states. Recall, in the microcanonical ensemble we x the energy and then calculate the entropy based on the number of states consistent with the set value of energy. For the canonical ensemble, we only need know the energy levels available to each element in the system, and with this knowledge we construct the partition function, wherefrom the free energy is obtained. 2.2 Heisenberg Hamiltonian In classical electromagnetics, the interaction energy for a magnetic dipole, m, inanappliedmagnetic eld, B, is given as [4] U int = ;m B : (22) Using a standard expression for the eld due to a dipole gives the dipole{ dipole interaction energy: U int = r 3 [m m 2 ; 3(m ^r)(m 2 ^r)] (23) 4

15 where r is the relative position vector from m to m 2. If we limit the type of relative orientations that the dipoles m and m 2 can have, we gain insight into the preferred orientations. When m km 2 kr, suchthatm m 2 >, the energy, using Eqn. (23), is given as U = ;2m m 2 =r 3, which is the minimum energy conguration, i.e., the preferred orientation. If we specify that m?r?m 2, then we force the dipoles into the type of congurations shown in Figure 5. These strict relationships between dipoles not only restricts the relative location of the dipoles, but also only allows for two projections of each dipole, on, say, the z-axis (e.g., m z ). The energy of the orientation in part (a) of Figure 5 is U a = m m 2 =r 3, while for the orientation in part (b), the energy is U b = ;m m 2 =r 3. Evidently, the orientation in part (b) is preferred over that in part (a). Unfortunately, this classical analysis of the interaction between two magnetic dipole moments is insucient to explain some types of magnetic ordering, as these energies are on the order of the thermal energy (i.e., k B T ), when T is near.3 K [3]. This means that above this temperature, thermal energy (stored in the hidden atomic modes of the spin system) will dominate the magnetic interaction in determining the thermodynamics of the system. In order to model the spontaneous magnetism of some metals, we must therefore consider quantum mechanical eects. To begin, we consider two interacting electrons, and assume that the spin interactions of this system will dominate in determining thermal properties. Here the spin represents the intrinsic spin of the electron, h=2. Restricting the dipoles to only two orientations as done in Figure 5 will thus represent a spin up, or spin down electron. We will nd that the interactions between electrons that 5

16 give rise to ferromagnetism are based on the Pauli exclusion principle and the Coulomb interaction, and thus are electrostatic, not magnetic in nature. We now commence to build the Hamiltonian for a pair of interacting, not necessarily bound, electrons. For this two electron system, there are four possible congurations of spins, which in the braket notation can be written as: j"#i, j""i, j#"i, and j##i. These four possible congurations partition into two distinct energy states, the singlet state, given [5, 6], for example, as = p 2(j"#i ; j#"i), and a triplet energy state. To satisfy the time-independent Schrodinger equation, Eqn. (), we require a Hamiltonian such that H s = E s s and H t = E t t where E s and E t are the eigenvalues, i.e., the energies, of the singlet and triplet state, and s, for example, is the wavefunction for the singlet state. Just such a Hamiltonian is given in [5] as H spin = 4 (E s +3E t ) ; (E s ; E t ) S S 2 : (24) For the Hamiltonian in Eqn. (24) to give the correct eigenvalues, note that in the singlet state, S S 2 < (i.e., the spins are antiparallel), and in the triplet state, S S 2 > (i.e., the spins are parallel). We will show now that the energy levels of the two states are such that E s > E t. The Pauli exclusion principle requires that two electrons with the same spin can not occupy the same location. To ensure this, the wave functions for a system of two electrons should be antisymmetric under interchange of coordinates. While the triplet states do have antisymmetric wave 6

17 functions, the singlet state does not, which requires the singlet state to have a larger electrostatic repulsion between electrons [7] than the triplet state (to ensure that the two electrons do not ever occupy the same location). We now view the quantity (E s ; E t ) as an exchange energy splitting between the two states, and accordingly, weletj = E s ; E t, where J is the so-called exchange coupling constant. Using J in Eqn. (24) and setting a new zero for the energy, the spin Hamiltonian for this two electron system becomes: H = ;JS S 2 : Note that if J >, parallel spins are preferred, and if J <, antiparallel spins give the lower energy. Over a large system of spins, we then write the Heisenberg Hamiltonian as X H = ; J ij S i S j : (25) Technically, all the electrons in the system interact with each other, so that the coupling constant J ij will vary, say, as the separation between spins become larger. When an external magnetic eld is present, the Hamiltonian will also include terms which represent the interaction of each spin with the applied magnetic eld. 2.3 Ising model in one{dimension The one-dimensional Ising model consists of a linear sequence of spin variables, s i, where each spin variable can take on only two possible values (s i = ). Each spin variable is connected to its neighbor's with a coupling constant k, and potentially interacts with an externally imposed magnetic 7

18 eld. The Hamiltonian for the one-dimensional Ising model is a simplication to the Heisenberg Hamiltonian, Eqn. (25), where if we let J be the exchange coupling and B the external magnetic eld strength, then the Hamiltonian for a linear chain of spins is: H(s i )=;J X < i j > s i s j ; B X i s i (26) where < i j > means to sum only over nearest neighbors, and we have assumed all nearest neighbors interact with the same eective strength (i.e., J ij = J jk = ). Note that in formulating Eqn. (26) we have simplied Eqn. (25) by looking at nearest neighbors only, and have also added a term to describe the interaction of the spins with an external eld. Let ( k B T ), and the dimensionless Hamiltonian, H, is given by: where H = ;H = k X < i j > s i s j + h X i s i (27) k = J=k B T = J (28) h = B=k B T = B : The dimensionless couplings k and h dened in Eqn. (28) and used in Eqn. (27) are referred to as a nearest-neighbor spin-spin coupling and as a singlespin (eld) coupling, respectively. The partition function in the canonical ensemble is given by: Z = X i e ;E i : (29) 8

19 Using Eqn. (27) in Eqn. (29), we can write the partition function for the one-dimensional Ising model in an equivalent form: Z = Tr e ;H = Tr exp(k s i s j ) exp(h P i s i ), fs i g fs i g <ij> P (3) where Tr means to trace over all possible congurations of spins, and <ij> says to operate on nearest neighbors only. The (Helmholtz) free energy is given by: F = ;k B T ln Z = ;= ln Z (3) and the reduced free energy per site is given by: f = ;(=N) F =(=N) ln Z : (32) The use of Eqn. (3) on a system of two spins, s ands2, with couplings k and h as shown in Eqn. (27) provides an instructive example: Tr e ;H = Tr fs s2g fs s2g = = X+ e k(ss2)+h(s+s2) X+ s=;(by2) s2=;(by2) X+ s=;(by2) e ;ks e hs e ;h {z } s2=; e k(ss2) e hs e hs2 + e ks e hs e h {z } s2=+ =(e k e ;h e ;h + e ;k e h e ;h )+(e ;k e ;h e h + e k e h e h ) {z } s=+ {z } s=+ =2e k cosh (2h)+2e ;k : (33) If k = (no exchange coupling), using Eqn. (33) in Eqn. (3): Z = Tr e ;H = cosh(2h) + 2 = 4 cosh 2 h, fs s2g (34) 9

20 and using Eqn. (34) in Eqn. (32) the free energy for this system of two spins simplies to: f =(=N) ln Z = 2 (ln cosh2 h + ln 4) = ln cosh h +ln2: (35) Evidently the ln 2 term in Eqn. (35) is the entropy contribution (F = U ; TS), associated with the two possible states per site. To calculate the magnetization, M, using Eqn. (35), we nd M related to the reduced free energy via a derivative: = sinh h =tanhh (36) cosh h which is the required result for the paramagnetism of a free electron (or ion) in an imposed eld. 2.4 The two{dimensional tetrahedral gasket Here we rst introduce the geometry of the tetrahedral gasket, shown in Figure 6, and then the spin Hamiltonian for the gasket is developed and the real-space renormalization group (RSRG) procedure is formulated. Using the RSRG we can identify a critical point, produce a phase diagram, and calculate critical exponents which describe how many thermodynamic response functions vary near the critical point. The complete calculation for this model is contained in [9]. To nd the dimension of the tetrahedral gasket, we rst consider the idea of dimension in general. To calculate the dimension of an object, one can consider [8] how the area or volume changes with the length of the object's edges. Let b be the scaling factor and d the dimensionality. When we expand 2

21 the length of the side of an object by a factor of b, the relationship between the new volume V and the original volume V is given by V = b d V : (37) When the length of all sides of a regular structure is doubled (b = 2), the relationship between the new volume and original volume is then V =2 d V : (38) For the tetrahedral gasket, when the octahedral segment is removed from the tetrahedron, this can be viewed as adding three more tetrahedrons to our base one, in a sense doubling the length of the side of the original tetrahedron. Thus, with V gasket =4V gasket and b = 2, the dimensionality of the tetrahedral gasket is calculated as d gasket =2. Therefore the tetrahedral gasket is a full two-dimensional object. To utilize the tetrahedral gasket for this statistical mechanical calculation, we place binary spin variables at the vertices of all the tetrahedrons. Consider the arrangement in Figure 7, which can be viewed as the state after the rst octahedral segment is removed from a base tetrahedron, where the four tetrahedrons are enclosed by the following four sets of spins: f s s5 s6 s7 g f s2 s6 s9 s g f s3 s5 s8 s9 g f s4 s7 s8 s g and the octahedral void is enclosed by the six spins labeled with dots: fs5 s6 s7 s8 s9 sg : The decimation procedure of the RSRG technique consists of summing over all congurations of the six interior spins (s i =+ ; 8i 5 i ), and 2

22 allowing the exterior spins, fs s2 s3 s4g, to take on dierent congurations. For each conguration of spins fs s2 s3 s4g, we equate the decimated system to the same conguration on a base tetrahedral system consisting only of four tetrahedral oriented spins. This decimation gives a set renormalized couplings in terms of the original (or rather the previous) couplings. The functions that relate the renormalized couplings to the previous set are the recurrence relations. The details of the model are now presented by specifying the couplings used in the Hamiltonian and by describing the development of the recurrence relations. For the system displayed in Figure 7, we have the following type of (even) spin interactions: k : a zero-spin coupling, k(ss5) : a nearest-neighbor spin-spin interaction, l(ss8) : a second nearest-neighbor spin-spin interaction, and m(ss5s6s7) : a four-spin interaction for each tetrahedron. These even spin couplings are functions of an even number of spin variables. Later we add the odd couplings, h and p. When the interactions k, l, andm are used, the spin Hamiltonian for the system in Figure 7 is: ;H = k (ss5+ss6+ss7+s5s6+s5s7+s6s7+s2s6+s2s9 +s2s + s6s9 + s6s + s9s + s3s5 + s3s8 + s3s9 + s5s8 +s5s9 + s8s9 + s4s7 + s4s8 + s4s + s7s8 + s7s + s8s) + l (ss8 + ss9+ ss + s2s5+ s2s7 + s2s8 + s4s6 22

23 +s4s5+ s4s9 + s3s7+ s3s + s3s6) + m (ss5s6s7 + s2s6s9s + s3s5s8s9 + s4s7s8s) : (39) Compare Eqn. (39) to the Hamiltonian for the base tetrahedron below (with spins fs s2 s3 s4g only): ;H = 2 k + k (ss2+ss3+ss4+s2s3+s2s4+s3s4) +m (ss2s3s4) : (4) The zero spin coupling, k, contributes to the reduced free energy. For the tetrahedral gasket, the outside spins in Figure 7 are shared between two separate decimations, therefore the four exterior spins each contribute a =2 k term to the Hamiltonian for the base tetrahedron, as shown in Eqn. (4). The partition function for the system in Figure 7 involves both the interior set of spins, and exterior set of spins, and is given as Z N = Tr e ;H. fs s2 s3 s4g fs5 s6 s7 s8 s9 sg (4) For the base tetrahedron, only the exterior spins are used. The partition function for the base tetrahedron is Z N = Tr e ;H. fs s2 s3 s4g (42) The renormalization-group calculation requires that the partition function be preserved (Z N = Z N ). Thus we equate Eqn. (4) and Eqn. (42), which relates the primed and original couplings: Tr Tr e ;H = Tr e. ;H fs s2 s3 s4g fs5 s6 s7 s8 s9 sg fs s2 s3 s4g (43) 23

24 Now, we can calculate the recurrence relations between the interaction couplings given in Eqn. (39) and Eqn. (4), which give functions for the renormalized couplings in terms of the previous couplings: k(k l m) k (k l m) and m (k l m) : (44) The details associated with obtaining the recurrence relations is contained in [9]. With the recurrence relations, the reduced free energy per spin, Eqn. (32), can be calculated for the tetrahedral gasket. Note that the decimation procedure (from the spin arrangement in Figure 7 to the base tetrahedron) reduces the number of spins by a fourth. This factor is not two-fths since the corners of the tetrahedron are counted as one{half of a spin. Using Eqn. (43), we expand out Eqn. (32), where N is the number of spins on the decimated lattice, with each spin contributing k to the free energy. Note that k is a prefactor in Eqn. (4), and Z the partition function, Eqn. (4), with the primed couplings used in the corresponding Hamiltonian. This last step corresponds to setting up for the next decimation by grouping four tetrahedrons (which resulted from the previous decimation) into a conguration as shown in Figure 7. This, in essence, is the RSRG procedure, which yields a derivation of the free energy: f s = (=N)lnZ N =(=N)lnZ N =(=N)lne [(N=4) k ] Z (k l m ) = (=4) k (k l m)+(=n)(n =N ) ln Z (k l m ) = (=4) k (k l m)+(=4) (=N ) ln e [(N =4) k ] Z (k l m ) = (=4) k +(=4) 2 k +(=4) 3 k (k (3;) l m (3;) )+ 24

25 which is summarized as: f s = X i= (=4) i k i (k (i;) l m (i;) ) (45) where k i and the couplings k i and m i are calculated using the recurrence relations. Note that, k in Eqn. (45) is k (k l m), the rst renormalized coupling k 2 k (k l m ), is the second renormalized coupling, and k () is just k. The reduced free energy per spin, Eqn. (45), can be used to calculate the free energy surface for a segment of the km{plane around the origin. A plot of this surface, using l = :, is shown in Figure 8. The free energy surface eectively shows the value of the free energy, written as f s (k l m), at each point on the km{plane. This gure has many features in common with the phase diagram of the model (see [9]). Now, the Hamiltonian for the base tetrahedron, Eqn. (4), is expanded to include the odd interaction couplings, h and p. Therefore, the interactions for the base tetrahedron now include: k : a zero-spin coupling, k(ss2+ss3+ss4+s2s3+s2s4+s3s4): a nearest-neighbor coupling, m(ss2s3s4) : a four-spin (tetrahedral) coupling, h(s + s2 + s3 + s4) : a single-spin (eld) coupling, and p(ss2s3 + ss2s4 + ss3s4 + s2s3s4) : a triangular (face) interaction. The Hamiltonian for the system in Figure 7 is derived in the same way as in Eqn. (39), and is shown in [9] along with the Hamiltonian for the base 25

26 tetrahedron. According the RSRG program, we again desire the recurrence relations (cf. Eqn. (44)): k(k m l h p) k (k m l h p) m (k m l h p) h (k m l h p) and p (k m l h p) : (46) The novelty associated with the tetrahedral gasket spin model is that it is a two-dimensional model for which exact recurrence relations can be found, even in the presence of a non-zero applied eld (i.e., h 6= ). 2.5 Critical exponents for the tetrahedral gasket According to the scaling hypothesis, near a critical point, some thermodynamic properties diverge according to strict power laws. For example, the critical exponents,,, and can be dened [, 5] in terms of how various thermodynamic quantities behave near a critical point. Let T c be the critical temperature, and dene t such that it gives a measure of how far one is from the critical temperature: Then, the equations, t (T c ; T ) T c : (47) M t t ; C t ; =a t ; I B = (48) 26

27 describe the temperature dependence near the critical point of the thermodynamic quantities: C, the specic heat =a, the scaled correlation length M, the magnetization, thesusceptibility andi, themagnetic moment (with B the external magnetic eld strength, I is related to M (in Gaussian units) as R I = 2c MdV). If all the exponents are positive, Eqn. (48) says, for instance, that the specic heat has a singularity at the critical point and diverges as t ;. Typically, the correlation length and susceptibility also have singular behavior at the critical point. One of the great theoretical achievements of this century, shows that, evidently, the critical exponents, dened in Eqn. (48), are not necessarily independent of each other. This is manifested in the so-called scaling laws: where Eqn =2 = ( ; ) (49) (49a) is the Essam-Fisher scaling law[] (also referred to as Rushbrooke's scaling law[]) and Eqn. (49b) is Widom's scaling law. In light of Eqn. (49), it is natural to seek exponents which are independent[]. Using scaling arguments, one such pair of exponents are found to be the thermal exponent, y T, and the magnetic exponent, y H. The critical exponents are then given as: = 2 ; d y T = y T = d ; y H y T = 2 y H ; d y T (5) 27

28 where d is the dimensionality of the model. Therefore, to determine the critical behavior of a system, it suces to nd y T and y H. A critical point for the tetrahedral gasket in the h = p = phase space (i.e., the km-plane) is an unstable equilibrium point of the renormalization, e.g., if (k m ) is the critical point, then k (k m ) = k and m (k m ) = m. For the tetrahedral gasket the critical point, (k m ), is found numerically to be given approximately as: k ' :525 m ' ;:28 : (5) Th critical point shown in Eqn. (5) compares favorably to those found for other two-dimensional models (and the three{dimensional face centered cubic), as seen by comparing Eqn. (5) with the values given in Table. Also note that it is standard practice to relate the temperature dependence of an Ising model to the nearest neighbor coupling, k. Using Eqn. (28), we can x a relationship between k and T by scaling the exchange coupling J such that T = k : (52) Therefore, in Eqn. (47), we uset c ==k for the critical tempature at which the model undergoes, in this case, a nontrivial phase transition. To obtain values for the thermal exponent, y T, and the magnetic exponent, y H,we consider a linearization of the renormalization about the critical point. From nonlinear system theory[2], we know we can linearize around a critical point, and if both eigenvalues of the Jacobian are positive we have an unstable equilibrium point. This unstable equilibrium point is in fact a 28

29 critical point of the RSRG calculation. For the full tetrahedral gasket model with odd interactions, the linearization is k m h p =@p C A k m h p k m h p : (53) To calculate the magnetic exponent, consider the eigenvectors of the Jacobian in Eqn. (53), evaluated at the critical point (k m h = p = ). If one of these eigenvectors is associated with the direction of the renormalization ows away from the critical pointtowards larger h, thenwe use the associated eigenvalue to nd the magnetic exponent. This same technique is used when only even spin couplings are present to calculate the thermal exponent, y H. The use of y T and y H in Eqn. (5) then shows how the thermodynamics of the system behave near the critical point. 3 Discussion The statistical mechanical methodology has been employed to study simple models that have energy quanta which behave according to statistics which are associated with fermions and bosons. The methodology was also employed to study the phase transition associated with the spontaneous magnetization of a two-dimensional fractal lattice. In all the models presented, basic quantum mechanical ideas are extended from the microscopic realm for use in a macroscopic calculation. It should not be surprising that once we leave the microscopic realm of analysis, we no longer speak of the 29

30 time-development of a system, but rather talk in terms of thermodynamic measurements, thus the statistical mechanical methods discussed herein are appropriate to equilibrium situations only. The Ising model is concerned with the dynamics of spin-variable systems, and as such lends itself directly to the modeling of ferromagnetic and antiferromagnetic systems. Evidently the ferromagnetic phase transition of a magnetic system has much in common with the more familiar liquid-gas phase transition. Experiment and theory have shown that the relationship between the ferromagnetic phase transition and the liquid-gas phase transition is such that the two systems in fact share common values of their critical exponents [, 3]. These critical exponents describe how the systems order parameter varies near the critical temperature. An order parameter is a variable that is non-zero only in the ordered phase of the system. While one systems order parameter might be dierent from another systems order parameter (for instance, let l and g represent the liquid and gas densities, then ( l ; g ) is the order parameter for the liquid-gas phase transition, while the magnetization, M, is the order parameter for ferromagnetic systems), the apparent similarities in behavior near the critical point oftwo such disparate systems eventually has led to the so-called universality hypothesis. The universality hypothesis states that if two systems have the same (spatial) dimensionality and have order parameters with the same dimensionality, then these two systems will have similar values for their critical exponents [, 3]. This quite remarkable hypothesis has been applied across a wide range of physical systems, including binary alloys, binary mixtures, superuids, absorbed lms and of course uids and ferromagnets [3]. The study of phase transitions 3

31 in general is now commonly grouped into a subject refereed to as critical phenomena, and the statistical mechanical methodology is currently the best theoretical tool available to study such critical phenomena. Appendix Some basic denitions and techniques from thermodynamics are reviewed. Specic details are included on how thermal measurements can be used to uncover the molecular topology of simple gases. An extensive parameter is a parameter whose value in a composite system is equal to the sum of the values over all subsystems. Examples of extensive parameters include S, U, V and N (i.e., the entropy, energy, volume, and particle number). Thus, it is easily seen that S(U V N) =S(U V N). An intensive parameter is a value that is constant throughout a macroscopic system, e.g., the temperature T, the pressure P and the chemical potential. Fundamental relations are functions that contain all possible thermodynamic information about a system. In the microcanonical formalism, S(U V N) is the fundamental relation in the entropy representation, while U(S V N) is the fundamental relation in the energy representation. Taking a full dierential in the energy representation we obtain: V N = TdS; PdV + dn: From Eqn. (54), we can identify, e.g., V N 3! S N dv + = T (S V N) dn V S

32 Such a function for the intensive parameter T is referred to as an equation of state. If all the equations of state are known for a system, the fundamental relation can be constructed. In the microcanonical formalism, we can use the entropy (or energy) representation, which assumes that the energy (or entropy) of the system is xed. There are two principles in the microcanonical formalism that guide the system to various equilibrium conditions. The entropy maximum principle states that the equilibrium value of any unconstrained internal parameter is such as to maximize the entropy foragiven value of the total internal energy. The energy minimum principle states that the equilibrium value of any unconstrained internal parameter is such as to minimize the energy for agiven value of the total entropy. In the canonical formalism we assume the tempature is xed and the fundamental relation is the Helmholtz Free Energy, F = F (T V N). move between dierent formalisms, the Legendre transformation can be employed. Recall that in classical mechanics, the Lagrangian is a function of coordinates and velocities: L = L(x i v i ). To obtain the Hamiltonian from the Lagrangian, we employ a Legendre transformation: ;H = L ; k v k = ;H(x i P i ) where P k k are the generalized momenta. To move from the microcanonical to the canonical formalism, note that F = U ; TS, where T =(@U=@S) V N, therefore the transformation F = S To 32

33 eliminates the variable S to give F = F (T V N). dierential of F : df dv = ;P dv; SdT + dn: Now consider the We can readily identify, for instance, that S = ;(@F=@T ), which gives a Legendre transformation from the canonical to microcanonical formalism: U = F + TS = F The molar heat capacity, at constant volume, C v, in the microcanonical formalism is given as C v V = V : (55) For a xed particle number, du = TdS;PdV, such that at constant volume, from Eqn. (55), C v : (56) V The heat capacity is the quasi-static heat ux per mole required to produce unit increase in the tempature of a system maintained at a constant volume. Heat is evidently a form of energy transfer. Thermodynamics describes the macroscopic properties apparently a result of interactions involving the atomic coordinates of a system. The \hidden" atomic modes of a system thus act as repositories for energy. An energy transfer via the hidden atomic modes is called heat []. Thermodynamics shows that if we can measure how a system stores heat, we can obtain information about the structure of these hidden modes. An 33

34 equation of state for the energy of an ideal gas is given as U = cnrt (57) which describes an ideal gas fairly well at high tempature. In Eqn. (57), R = N A k B = 8:344 J=mole K is the universal gas constant, n is the number of moles, and c = 3=2 for monatomic ideal gases. Using Eqn. (57) in Eqn. (56), we nd that C v = 3R=2 for a monatomic gas. This value of C v is due to a R=2 contribution from each quadratic term in the classical energy. For a monatomic gas, the energy is due to the kinetic energy of the molecules, thus the Hamiltonian is U p2 x + p 2 y + p 2 z 2m Classically, for conservative systems, we can write the energy (or Hamiltonian) as a sum of quadratic terms (of the momenta and coordinates): U p 2 + q 2 where p is the generalized momenta and q a generalized coordinate. The equipartition theorem of classical statistical mechanics (with ~ N = nn A ) then says []: At suciently high tempature every quadratic term in the energy contributes a term =2 ~Nk B to the heat capacity. We can now use the equipartition theorem to investigate the molecular topological structure of some simple gases. Consider the molecular shapes shown in Figure 9. If we quantify the type of modes each of these molecules possess, as in done in Table 2, we nd that the specic heat can tell us something 34 :

35 about the topology (or shape) of the molecule. In this way, we are basically measuring how many ways the system can store energy. For a diatomic gas, the molecule still has three momentum terms in the Hamiltonian for the molecule as a whole, but for this bonded molecule, we also attribute a vibrational mode which has a potential and momentum term in the Hamiltonian. Lastly, since we require two angles to specify the orientation, we need two more terms in the Hamiltonian. When all the modes are totaled up, the specic heat for a diatomic gas is given at high tempature as C v =7R=2 per mole. 35

36 That is, the average number of quanta occupying each state. 2 Table 2 shows that this is due to the vibrational modes of 3N oscillators. 36

37 References [] H.B. Callen, Thermodynamics and an Introduction to Thermostatistics (John Wiley &Sons,985). [2] K. Stowe, Introduction to Statistical Mechanics and Thermodynamics (John Wiley &Sons,984). [3] J.R. Hook, H.E. Hall, Solid State Physics (John Wiley & Sons, 99). [4] J.D. Jackson, Classical Electrodynamics (John Wiley & Sons, 975). [5] N.W. Ashcroft, N.D. Mermin, Solid State Physics (W.B. Saunders, 976). [6] A. Goswami, Quantum Mechanics (Wm. C. Brown Publishers, 992). [7] C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics: Volumes I and II (John Wiley & Sons, 977). [8] J.T. Sandefur, \Using Self-Similarity to Find Length, Area, and Dimension," The Am. Mathematical Monthly, (2), 7-2 (996). [9] J.F. Nystrom, \An Exact Finite Field Renormalization Group Calculation on a Two-Dimensional Fractal Lattice," Int. J. Mod. Phys. C, (2), (2). [] G.M. Bell, D.A. Davis, Statistical Mechanics of Lattice Models Volume : closed form and exact theories of cooperative phenomena (Ellis Horwood, 989). 37

38 [] G.L. Sewell, Quantum Theory of Collective Phenomena (Oxford University Press, 986). [2] M. Vidyasagar, Nonlinear System Analysis (Prentice Hall, 993). [3] G. Careri, Order and Disorder in Matter (Benjamin/Cummings, 984). 38

39 Figure : Entropy is additive. Figure 2: Specic heat for a two{state system. Figure 3: Specic heat for a crystalline solid. Figure 4: Bound oscillator model for a crystalline solid. Figure 5: Sample spin congurations. Figure 6: Three levels of the tetrahedral gasket. Figure 7: Tetrahedral decimation. Figure 8: Free energy surface using l =:. Figure 9: Simple molecular gases. 39

40 Lattice Critical Value k honeycomb :6585 k square :447 k triangular :2747 k fcc ' : Table : Critical value of the spin-spin coupling for common lattices. 4

41 Type of molecule Dierent modes p i q i (mode) Total Monatomic 3 translational =2k B =3=2k B! 3=2k B Diatomic 3 translational =2k B =3=2k B vibrational =2k B =2k B = k B 2 rotational =2k B = k B! 7=2k B Triatomic 3 translational =2k B =3=2k B 3 vibrational =2k B =2k B =3k B 3 rotational =2k B =3=2k B! 6k B Linear Triatomic 3 translational =2k B =3=2k B 4 vibrational =2k B =2k B =4k B 2 rotational =2k B = k B! 3=2k B Table 2: Hidden atomic modes of ideal gases.

42 Entropy W W states S W 2 W 2 states S 2 W W 2 W W 2 states S + S 2

43 .4.3 C x

44 C R T

45

46 (b) (a)

47

48 S S6 S7 S5 S2 S S9 S4 S8 S3

49 -2 - m 2 6 f(k,.,m) k 2

50 Shape triatomic Molecule Diatomic Triatomic Linear